GNU MCSim User's Manual

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# GNU MCSim User’s Manual

GNU MCSim manual, version 5.6.4

GNU MCSim is a general purpose modeling and simulation program which can performs "standard" or "Markov chain" Monte Carlo simulations. It allows you to specify a set of linear or nonlinear algebraic equations or ordinary differential equations. They are solved numerically using parameter values you choose or parameter values sampled from statistical distributions. Simulation outputs can be compared to experimental data for Bayesian parameter estimation (model calibration).

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# 1. Software and Documentation Licenses

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## 1.1 Software license

GNU MCSim is free software; you can redistribute it and/or modify it under the terms of the GNU General Public License as published by the Free Software Foundation; either version 3 of the License, or (at your option) any later version. This program is distributed in the hope that it will be useful, but WITHOUT ANY WARRANTY; without even the implied warranty of MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU General Public License for more details.

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## 1.2 Documentation license

The GNU Free Documentation License

Version 1.2, November 2002

    Copyright © 2000,2001,2002 Free Software Foundation, Inc.
    51 Franklin St, Fifth Floor, Boston, MA  02110-1301, USA

    Everyone is permitted to copy and distribute verbatim copies
    of this license document, but changing it is not allowed.

1. PREAMBLE

   The purpose of this License is to make a manual, textbook, or other functional and useful document free in the sense of freedom: to assure everyone the effective freedom to copy and redistribute it, with or without modifying it, either commercially or noncommercially. Secondarily, this License preserves for the author and publisher a way to get credit for their work, while not being considered responsible for modifications made by others.

   This License is a kind of “copyleft”, which means that derivative works of the document must themselves be free in the same sense. It complements the GNU General Public License, which is a copyleft license designed for free software.

   We have designed this License in order to use it for manuals for free software, because free software needs free documentation: a free program should come with manuals providing the same freedoms that the software does. But this License is not limited to software manuals; it can be used for any textual work, regardless of subject matter or whether it is published as a printed book. We recommend this License principally for works whose purpose is instruction or reference.

2. APPLICABILITY AND DEFINITIONS

   This License applies to any manual or other work, in any medium, that contains a notice placed by the copyright holder saying it can be distributed under the terms of this License. Such a notice grants a world-wide, royalty-free license, unlimited in duration, to use that work under the conditions stated herein. The “Document”, below, refers to any such manual or work. Any member of the public is a licensee, and is addressed as “you”. You accept the license if you copy, modify or distribute the work in a way requiring permission under copyright law.

   A “Modified Version” of the Document means any work containing the Document or a portion of it, either copied verbatim, or with modifications and/or translated into another language.

   A “Secondary Section” is a named appendix or a front-matter section of the Document that deals exclusively with the relationship of the publishers or authors of the Document to the Document’s overall subject (or to related matters) and contains nothing that could fall directly within that overall subject. (Thus, if the Document is in part a textbook of mathematics, a Secondary Section may not explain any mathematics.) The relationship could be a matter of historical connection with the subject or with related matters, or of legal, commercial, philosophical, ethical or political position regarding them.

   The “Invariant Sections” are certain Secondary Sections whose titles are designated, as being those of Invariant Sections, in the notice that says that the Document is released under this License. If a section does not fit the above definition of Secondary then it is not allowed to be designated as Invariant. The Document may contain zero Invariant Sections. If the Document does not identify any Invariant Sections then there are none.

   The “Cover Texts” are certain short passages of text that are listed, as Front-Cover Texts or Back-Cover Texts, in the notice that says that the Document is released under this License. A Front-Cover Text may be at most 5 words, and a Back-Cover Text may be at most 25 words.

   A “Transparent” copy of the Document means a machine-readable copy, represented in a format whose specification is available to the general public, that is suitable for revising the document straightforwardly with generic text editors or (for images composed of pixels) generic paint programs or (for drawings) some widely available drawing editor, and that is suitable for input to text formatters or for automatic translation to a variety of formats suitable for input to text formatters. A copy made in an otherwise Transparent file format whose markup, or absence of markup, has been arranged to thwart or discourage subsequent modification by readers is not Transparent. An image format is not Transparent if used for any substantial amount of text. A copy that is not “Transparent” is called “Opaque”.

   Examples of suitable formats for Transparent copies include plain ASCII without markup, Texinfo input format, LaTeX input format, SGML or XML using a publicly available DTD, and standard-conforming simple HTML, PostScript or PDF designed for human modification. Examples of transparent image formats include PNG, XCF and JPG. Opaque formats include proprietary formats that can be read and edited only by proprietary word processors, SGML or XML for which the DTD and/or processing tools are not generally available, and the machine-generated HTML, PostScript or PDF produced by some word processors for output purposes only.

   The “Title Page” means, for a printed book, the title page itself, plus such following pages as are needed to hold, legibly, the material this License requires to appear in the title page. For works in formats which do not have any title page as such, “Title Page” means the text near the most prominent appearance of the work’s title, preceding the beginning of the body of the text.

   A section “Entitled XYZ” means a named subunit of the Document whose title either is precisely XYZ or contains XYZ in parentheses following text that translates XYZ in another language. (Here XYZ stands for a specific section name mentioned below, such as “Acknowledgements”, “Dedications”, “Endorsements”, or “History”.) To “Preserve the Title” of such a section when you modify the Document means that it remains a section “Entitled XYZ” according to this definition.

   The Document may include Warranty Disclaimers next to the notice which states that this License applies to the Document. These Warranty Disclaimers are considered to be included by reference in this License, but only as regards disclaiming warranties: any other implication that these Warranty Disclaimers may have is void and has no effect on the meaning of this License.

3. VERBATIM COPYING

   You may copy and distribute the Document in any medium, either commercially or noncommercially, provided that this License, the copyright notices, and the license notice saying this License applies to the Document are reproduced in all copies, and that you add no other conditions whatsoever to those of this License. You may not use technical measures to obstruct or control the reading or further copying of the copies you make or distribute. However, you may accept compensation in exchange for copies. If you distribute a large enough number of copies you must also follow the conditions in section 3.

   You may also lend copies, under the same conditions stated above, and you may publicly display copies.

4. COPYING IN QUANTITY

   If you publish printed copies (or copies in media that commonly have printed covers) of the Document, numbering more than 100, and the Document’s license notice requires Cover Texts, you must enclose the copies in covers that carry, clearly and legibly, all these Cover Texts: Front-Cover Texts on the front cover, and Back-Cover Texts on the back cover. Both covers must also clearly and legibly identify you as the publisher of these copies. The front cover must present the full title with all words of the title equally prominent and visible. You may add other material on the covers in addition. Copying with changes limited to the covers, as long as they preserve the title of the Document and satisfy these conditions, can be treated as verbatim copying in other respects.

   If the required texts for either cover are too voluminous to fit legibly, you should put the first ones listed (as many as fit reasonably) on the actual cover, and continue the rest onto adjacent pages.

   If you publish or distribute Opaque copies of the Document numbering more than 100, you must either include a machine-readable Transparent copy along with each Opaque copy, or state in or with each Opaque copy a computer-network location from which the general network-using public has access to download using public-standard network protocols a complete Transparent copy of the Document, free of added material. If you use the latter option, you must take reasonably prudent steps, when you begin distribution of Opaque copies in quantity, to ensure that this Transparent copy will remain thus accessible at the stated location until at least one year after the last time you distribute an Opaque copy (directly or through your agents or retailers) of that edition to the public.

   It is requested, but not required, that you contact the authors of the Document well before redistributing any large number of copies, to give them a chance to provide you with an updated version of the Document.

5. MODIFICATIONS

   You may copy and distribute a Modified Version of the Document under the conditions of sections 2 and 3 above, provided that you release the Modified Version under precisely this License, with the Modified Version filling the role of the Document, thus licensing distribution and modification of the Modified Version to whoever possesses a copy of it. In addition, you must do these things in the Modified Version:

   1. Use in the Title Page (and on the covers, if any) a title distinct from that of the Document, and from those of previous versions (which should, if there were any, be listed in the History section of the Document). You may use the same title as a previous version if the original publisher of that version gives permission.
   2. List on the Title Page, as authors, one or more persons or entities responsible for authorship of the modifications in the Modified Version, together with at least five of the principal authors of the Document (all of its principal authors, if it has fewer than five), unless they release you from this requirement.
   3. State on the Title page the name of the publisher of the Modified Version, as the publisher.
   4. Preserve all the copyright notices of the Document.
   5. Add an appropriate copyright notice for your modifications adjacent to the other copyright notices.
   6. Include, immediately after the copyright notices, a license notice giving the public permission to use the Modified Version under the terms of this License, in the form shown in the Addendum below.
   7. Preserve in that license notice the full lists of Invariant Sections and required Cover Texts given in the Document’s license notice.
   8. Include an unaltered copy of this License.
   9. Preserve the section Entitled “History”, Preserve its Title, and add to it an item stating at least the title, year, new authors, and publisher of the Modified Version as given on the Title Page. If there is no section Entitled “History” in the Document, create one stating the title, year, authors, and publisher of the Document as given on its Title Page, then add an item describing the Modified Version as stated in the previous sentence.
   10. Preserve the network location, if any, given in the Document for public access to a Transparent copy of the Document, and likewise the network locations given in the Document for previous versions it was based on. These may be placed in the “History” section. You may omit a network location for a work that was published at least four years before the Document itself, or if the original publisher of the version it refers to gives permission.
   11. For any section Entitled “Acknowledgements” or “Dedications”, Preserve the Title of the section, and preserve in the section all the substance and tone of each of the contributor acknowledgements and/or dedications given therein.
   12. Preserve all the Invariant Sections of the Document, unaltered in their text and in their titles. Section numbers or the equivalent are not considered part of the section titles.
   13. Delete any section Entitled “Endorsements”. Such a section may not be included in the Modified Version.
   14. Do not retitle any existing section to be Entitled “Endorsements” or to conflict in title with any Invariant Section.
   15. Preserve any Warranty Disclaimers.

   If the Modified Version includes new front-matter sections or appendices that qualify as Secondary Sections and contain no material copied from the Document, you may at your option designate some or all of these sections as invariant. To do this, add their titles to the list of Invariant Sections in the Modified Version’s license notice. These titles must be distinct from any other section titles.

   You may add a section Entitled “Endorsements”, provided it contains nothing but endorsements of your Modified Version by various parties—for example, statements of peer review or that the text has been approved by an organization as the authoritative definition of a standard.

   You may add a passage of up to five words as a Front-Cover Text, and a passage of up to 25 words as a Back-Cover Text, to the end of the list of Cover Texts in the Modified Version. Only one passage of Front-Cover Text and one of Back-Cover Text may be added by (or through arrangements made by) any one entity. If the Document already includes a cover text for the same cover, previously added by you or by arrangement made by the same entity you are acting on behalf of, you may not add another; but you may replace the old one, on explicit permission from the previous publisher that added the old one.

   The author(s) and publisher(s) of the Document do not by this License give permission to use their names for publicity for or to assert or imply endorsement of any Modified Version.

6. COMBINING DOCUMENTS

   You may combine the Document with other documents released under this License, under the terms defined in section 4 above for modified versions, provided that you include in the combination all of the Invariant Sections of all of the original documents, unmodified, and list them all as Invariant Sections of your combined work in its license notice, and that you preserve all their Warranty Disclaimers.

   The combined work need only contain one copy of this License, and multiple identical Invariant Sections may be replaced with a single copy. If there are multiple Invariant Sections with the same name but different contents, make the title of each such section unique by adding at the end of it, in parentheses, the name of the original author or publisher of that section if known, or else a unique number. Make the same adjustment to the section titles in the list of Invariant Sections in the license notice of the combined work.

   In the combination, you must combine any sections Entitled “History” in the various original documents, forming one section Entitled “History”; likewise combine any sections Entitled “Acknowledgements”, and any sections Entitled “Dedications”. You must delete all sections Entitled “Endorsements.”

7. COLLECTIONS OF DOCUMENTS

   You may make a collection consisting of the Document and other documents released under this License, and replace the individual copies of this License in the various documents with a single copy that is included in the collection, provided that you follow the rules of this License for verbatim copying of each of the documents in all other respects.

   You may extract a single document from such a collection, and distribute it individually under this License, provided you insert a copy of this License into the extracted document, and follow this License in all other respects regarding verbatim copying of that document.

8. AGGREGATION WITH INDEPENDENT WORKS

   A compilation of the Document or its derivatives with other separate and independent documents or works, in or on a volume of a storage or distribution medium, is called an “aggregate” if the copyright resulting from the compilation is not used to limit the legal rights of the compilation’s users beyond what the individual works permit. When the Document is included in an aggregate, this License does not apply to the other works in the aggregate which are not themselves derivative works of the Document.

   If the Cover Text requirement of section 3 is applicable to these copies of the Document, then if the Document is less than one half of the entire aggregate, the Document’s Cover Texts may be placed on covers that bracket the Document within the aggregate, or the electronic equivalent of covers if the Document is in electronic form. Otherwise they must appear on printed covers that bracket the whole aggregate.

9. TRANSLATION

   Translation is considered a kind of modification, so you may distribute translations of the Document under the terms of section 4. Replacing Invariant Sections with translations requires special permission from their copyright holders, but you may include translations of some or all Invariant Sections in addition to the original versions of these Invariant Sections. You may include a translation of this License, and all the license notices in the Document, and any Warranty Disclaimers, provided that you also include the original English version of this License and the original versions of those notices and disclaimers. In case of a disagreement between the translation and the original version of this License or a notice or disclaimer, the original version will prevail.

   If a section in the Document is Entitled “Acknowledgements”, “Dedications”, or “History”, the requirement (section 4) to Preserve its Title (section 1) will typically require changing the actual title.

10. TERMINATION

    You may not copy, modify, sublicense, or distribute the Document except as expressly provided for under this License. Any other attempt to copy, modify, sublicense or distribute the Document is void, and will automatically terminate your rights under this License. However, parties who have received copies, or rights, from you under this License will not have their licenses terminated so long as such parties remain in full compliance.

11. FUTURE REVISIONS OF THIS LICENSE

    The Free Software Foundation may publish new, revised versions of the GNU Free Documentation License from time to time. Such new versions will be similar in spirit to the present version, but may differ in detail to address new problems or concerns. See http://www.gnu.org/copyleft/.

    Each version of the License is given a distinguishing version number. If the Document specifies that a particular numbered version of this License “or any later version” applies to it, you have the option of following the terms and conditions either of that specified version or of any later version that has been published (not as a draft) by the Free Software Foundation. If the Document does not specify a version number of this License, you may choose any version ever published (not as a draft) by the Free Software Foundation.

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# 2. Overview

GNU MCSim is a simulation and statistical inference tool for algebraic or differential equation systems. Other programs have been created to the same end, the Matlab family of graphical interactive programs being some of the more general and easy to use. Still, many available tools are not optimal for performing computer intensive and sophisticated Monte Carlo analyses. GNU MCSim was created specifically to this end: to perform Monte Carlo analyses in an optimized, and easy to maintain environment. The software consists in two pieces, a model generator and a simulation engine:

- The model generator, "mod", was created to facilitate structural model definition and maintenance, while keeping execution time short. You code your model using a simplified syntax and `mod` translates it in C.

- The simulation engine is a set of routines are linked to your model to produce executable code. After linking, you will be able to run simulations of your structural model under a variety of conditions, specify an associated statistical model, and perform Monte Carlo simulations.

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## 2.1 General procedure

Model building and simulation proceeds in four stages:

1. You create with any text editor (e.g., `emacs`) a model description file. The reference section on `mod`, later in this manual gives you the syntax to use (see section Setting-up Structural Models). This syntax allows you to describe the model variables, parameters, equations, inputs and outputs in a C-like fashion without having you to actually know how to write a C program.
2. You instruct the model generator, `mod`, to preprocess your structural model description file. `Mod` creates a C file, called ‘`model.c`’.
3. You compile and link the newly created ‘`model.c`’ file together with a library containing the other C routines (or with the other C files of the ‘`mcsim/sim`’ directory). GNU MCSim C code is standard, so you should be able to compile it with any standard C compiler, for example GNU `gcc`. After compiling and linking, an executable simulation program is created, specific of your particular model. These preprocessing and compilation steps can be performed in Unix with a single shell command `makemcsim` (in which case, the ‘`model.c`’ is created only temporarily and erased afterward). This produces the most efficient code for your particular machine.
4. You then write any number of simulation specification files and run them with the compiled `mcsim` program. These simulation files describe the kind of simulation to run (simple simulations, Monte Carlo etc.), various settings for the integration algorithm if needed, and a description of one or several simulation conditions (eventually with a statistical model and data to fit) (see section Running Simulations). The simulation output is written to standard ASCII files.

Little or no knowledge of computer programming is required, unless you want to tailor the program to special needs, beyond what is described in this manual (in which case you may want to contact us).

Under Unix, a graphical user interface written in Tcl/Tk, XMCSim (called by the command `xmcsim`), is also provided. This menu-driven interface automatizes the compilation and running tasks. It also offers a convenient interface to 2-D and 3-D plotting of the simulation results.

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## 2.2 Types of simulations

Five types of simulations are available:

- A simple simulation will solve (eventually integrate) the equations you specified, using the default parameter values eventually overridden in the simulation specification file. User-requested outputs are sent to an output file of your choice.
- "Monte Carlo" simulations will perform repeated (stochastic) simulations across a randomly sampled region of the model parameter space (see section `MonteCarlo()` specification).
- A Markov-chain Monte Carlo (MCMC) simulation performs a series of simulations along a Markov chain in the model parameter space (see section `MCMC()` specification). In MCMC simulations the random choice of a new parameter value is influenced by the current value. They can be used to obtain the Bayesian posterior distribution of the model parameters, given a statistical model, prior parameter distributions (that you need to specify) and data for which a likelihood function can be computed. The program handles hierarchical (e.g., random effects and mixed effects) statistical models (see section Setting-up statistical models).
- A "SetPoints" simulation solves the model for a series of specified parameter sets, listed in a separate ASCII file (see section `SetPoints()` specification). You can create these parameter sets yourself (on a regular grid, for example) or use the output of a previous Monte Carlo or MCMC simulation.
- An "OptimalDesign" procedure optimizes the number and location of observation times for experimental conditions, in order to minimize the variance of a parameter or an output you specify, given a structural model, a statistical model, and prior distributions for their parameters (see section `OptimalDesign()` specification).

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## 2.3 Major changes introduced with version 5.4.0

- GNU MCSim is now distributed under version 3 of the GNU General Public License.
- The installation scripts have been rewritten using GNU `autoconf`, `automake` and `libtool`. This should make GNU MCSim easier to install and more portable.
- Systems Biology Markup Language (SBML) models are processed by libSBML if it is installed.
- Tempered MCMC (for hard, multimodal posterior densities) and stochastic optimizations are offered as options of the `MCMCx()` specification.

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## 2.4 Major changes introduced with version 5.5.0

- The installation scripts have been regenerated using GNU `autoconf` version 2.69 which fixes a potential security problem in the installation. That should be transparent to the user.
- The `mod` utility can now generate C model files suitable for use with the R package `deSolve`. Use `mod -R` for that.

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## 2.5 Major changes introduced with version 5.6.0

- The keyword `End` is now mandatory at the end of every model. This is not backward compatible (you will need to modify your older models accordingly) but we had to implement this for technical reasons.
- The `StartTime()` specification, setting the initial time for integration can now accept a symbolic parameter. That allows you to treat the initial time as a random variable in error-in-variable problems (when the initial time is an unknown).
- The `PrintStep()` specification can read a list of variables to print (as `Print()` does).
- In MCMC simulation, the jump kernel is now output in a file with the .kernel extension. If the simulations are restarted in a continuation mode and if the kernel file is present the jump kernel will restored to its saved value.
- Delay differential equations can now be coded and solved using the `CalDelay` function.
- Discontinuities in state variables can now be forced to happen at predefined times through the `Events()` specification.

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# 3. Installation

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## 3.1 System requirements

GNU MCSim is written in ANSI-standard C language. We are distributing the source code and you should be able to compile it for any system, provided you have an ANSI C compliant compiler.

Starting with version 5.0.0 GNU MCSim is using a few routines from the GNU Scientific Library (`libgsl`). We recommend that you install version 1.5 (or higher) of the shared GSL library, gslcblas library, and GSL include files on your system. Otherwise a few obscure features will not be available (you’ll get a error message if you are trying to use them.)

Version 5.4.0 and higher of GNU MCSim can take advantage of (`libSBML`) to read SBML models. If you choose to install libSBML on your system, we recommend that you use version 3.3.2 (or higher) of libSBML. LibSBML needs an XML parser library (either Expat, Xerces, or libxml2). The Expat library has worked well for us under Linux.

On any system we recommend the GNU `gcc` compiler (freeware). The automated installation script checks for the availability on your system of the tools needed for compilation and proper running of the software. It should warn you of missing component and eventually adapt the installation to your needs (for example by installing the package locally if you do not have superuser’s priviledges).

To run the graphical user interface XMCsim, you need a GNU/Linux or Unix system with "XWindows", "Tcl/Tk" and "wish" installed.

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## 3.2 Distribution

GNU MCSim source code is available on Internet through:

- http://savannah.gnu.org/projects/mcsim.

Packaged distributions are available at:

- http://ftp.gnu.org/gnu/mcsim,

- http://www.gnu.org/software/mcsim,

and mirror sites of the GNU project.

Three mailing lists are available for GNU MCSim users:

General info on GNU MCSim is broadcasted through:

- http://lists.gnu.org/archive/html/info-mcsim

You can subscribe to the info list by going to:

- http://lists.gnu.org/mailman/listinfo/info-mcsim.

You can request help from us, and from other GNU MCSim users, by sending email to:

- help-mcsim@gnu.org

(see http://lists.gnu.org/mailman/listinfo/help-mcsim for subscribing).

Help archives are found at:

- http://news.gmane.org/gmane.comp.gnu.mcsim,

- http://lists.gnu.org/archive/html/help-mcsim.

You can report bugs to us, by sending email to:

- bug-mcsim@gnu.org

(http://lists.gnu.org/mailman/listinfo/bug-mcsim for subscribing).

Bugs archives are located at:

- http://news.gmane.org/gmane.comp.gnu.mcsim.bugs,

- http://lists.gnu.org/archive/html/bug-mcsim.

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## 3.3 Machine-specific installation

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### 3.3.1 Unix and GNU/Linux operating systems

To install on a Unix or GNU/Linux machine, download (in binary mode) the distributed archive file to your machine. Place it in a directory where there is no existing ‘`mcsim`’ subdirectory that could be erased (make sure you check that). Decompress the archive with GNU gunzip (`gunzip <archive-name>.tar.gz`). Untar the decompressed archive with tar (`tar xf <archive-name>.tar`) (do `man tar` for further help). Move to the ‘`mcsim`’ directory just created and issue the following commands:

    ./configure
    make
    make check

The first command above checks for the availability of the tools needed for installation and proper running of the software. The second compiles the `mod` program and the dynamic `libmcsim.so` library and eventually compiles this manual in various formats. The third checks whether the software is running and producing meaningful results in test cases. In case of error messages, don’t panic: check the actual differences between the culprit output file and the file ‘`sim.out`’ produced by the checking. Small differences may occur from different machine precision. This can happen for random numbers, in which case the Markov chain simulations (MCMC) can diverge greatly after a while.

If you are logged in as "root" or have sufficient access rights, you can then install the software in common directories in ‘`/usr`’ by typing at the shell prompt:

    make install

If this system-wide installation is successful the executable files `mod`, `makemcsim`, `xmcsim` are installed in ‘`/usr/local/bin`’. The library `libmcsim` is placed in ‘`/usr/local/lib`’. A copy of the ‘`mcsim`’ source directory (with the ‘`mod`’, ‘`sim`’, ‘`doc`’, ‘`samples`’, and ‘`xmcsim`’ subdirectories) is placed in ‘`/usr/local/share`’. If you have the GNU `info` system available, an `mcsim` node is added to the main `info` menu, so that `info mcsim` will show you this manual. Finally, a symbolic link to ‘`/usr/local/share/mcsim/doc`’, which contains the documentation files and this manual (if it was generated), is created as ‘`/usr/share/doc/mcsim`’.

If you do not have the necessary access rights and want to install GNU MCSim in a directory such as ‘`/home/me`’, type:

    ./configure prefix=/home/me

This will copy or move ‘`mod`’, ‘`makemcsim`’, and ‘`xmcsim`’ in a ‘`/bin`’ directory in the ‘`/home/me`’ directory, creating it if necessary. The library `libmcsim.so` will be moved to the ‘`/home/me/lib`’ directory, etc.

On certain platforms (Linux...), you will also need to do one of the following:
1) run ’ldconfig’ (see the man page if this is unfamiliar)
2) set the LD_LIBRARY_PATH (or equivalent) environment variable to contain the path "/usr/local/lib" or whatever you set so that programs can find the libSBML library at run-time.

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### 3.3.2 Other operating systems

Under other operating systems (Windows, etc.) or if everything else fails you should be able to both uncompress and untar the archive with widely distributed archiving tools. Refer to the documentation of your C compiler to create an executable ‘`mod`’ file from the source code files (getopt.c, lex.c, lexerr.c, lexfn.c, mod.c, modd.c, modi.c, modiSBML.c, modiSBML2.c, modo.c, strutil.c) provided in the ‘`mod`’ directory. If you want to process SBML models it is best to install the libSBML library first. You would then compile mod with the HAVE_LIBSBML flag defined (option `-DHAVE_LIBSBML`) and link with the library (using the `-lsbml` directive). Place then the executable ‘`mod`’ on your command path.

The ‘`sim`’ directory contains all the source files (delays.c, getopt.c, lex.c, lexerr.c, lexfn.c, list.c, lsodes1.c, lsodes2.c, matutil.c, matutilo.c, mh.c, modelu.c, optdsign.c, random.c, sim.c, simi.c, siminit.c, simmonte.c, simo.c, strutil.c, yourcode.c) to create a dynamic library or a set of objects to link with the ‘`model.c`’ files generated by `mod` after processing your models. Compilation also requires reference to the ‘`config.h`’ file sitting in the main folder (one level above the ‘`sim`’ directory). The -I.. option should make the compiler aware of the correct location of ‘`config.h`’. Alternatively, ‘`config.h`’ can be copied into the ‘`sim`’ directory to make the package complete (apart of model.c).

The final product should be an executable able to run your model. Linking with the GNU Scientific Library (`gsl`) is recommended (but not mandatory. In that case, define the `HAVE_LIBGSL` flag and link with the `-lgsl` and `-lgslcblas` (in that order!).

You are now ready to use GNU MCSim. We recommend that you go through the next section of this manual, which walks you through an example of model building and running.

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# 4. Working Through an Example

Several models and simulation specification files are provided with the package as examples (they are in the ‘`samples`’ directory. You can try any of them. The linear regression model is particularly simple, but to be more complete we will try here a nonlinear implicit model, specified through differential equations.

Pharmacokinetics models describe the transport and transformation of chemical compounds in the body. These models often include nonlinear first-order differential equations. The following example is taken from our own work on the kinetics of tetrachloroethylene (a solvent) in the human body (Bois et al., 1996; Bois et al., 1990) (see section Bibliographic References).

Go to the ‘`mcsim/samples/perc`’ directory (installed either locally or by default in ‘`usr/share`’ under Unix or GNU/Linux). Open the file ‘`perc.model`’ with any text editor (e.g., `emacs` or `vi` under Unix). This file is an example of a model definition file. It is also printed at in Appendix the end of this manual (see section ‘`perc.model`’: A sample model description file). You can use it as a template for your own model, but you should leave it unchanged for now. In that file, the pound signs (`#`) indicate the start of comments. Notice that the file defines:

- state variables for the model (for which differentials are defined), for example:

      States = {Q_fat,   # Quantity of PERC in the fat (mg)
                Q_wp,    #   ...   in the well-perfused compartment (mg)
                Q_pp,    #   ...   in the poorly-perfused compartment (mg)
                Q_liv,   #   ...   in the liver (mg)
                Q_exh,   #   ...   exhaled (mg)
                Q_met}   # Quantity of metabolite formed (mg)

- output variables (obtainable at any time as analytical functions of the states, inputs and parameters), for example:

      Outputs = {C_liv,           # mg/l in the liver
                 C_alv,           # ... in the alveolar air
                 C_exh,           # ... in the exhaled air
                 C_ven,           # ... in the venous blood
                 Pct_metabolized, # % of the dose metabolized
                 C_exh_ug}        # ug/l in the exhaled air

- input variables (independent of the others variables, and eventually varying with time), for example:

      Inputs = {C_inh,   # Concentration inhaled (ppm)
                Q_ing};  # Quantity ingested (mg)

- model parameters (independent of time), such as:

      LeanBodyWt = 55; # lean body weight (kg)

- model initialization and parameters’ scaling (the parameters used in the dynamic equations can be made functions of other parameters: for example volumes can be computed from masses and densities, etc.),
- system’s dynamics (differential or algebraic equations defining the model per se),
- equations to compute the output variables.

This model definition file as a simple syntax, easy to master. It needs to be turned into a C program file before compilation and linking to the other routines (integration, file management etc.) of GNU MCSim. You will use `mod` for that. First, quit the editor and return to the operating system.

To start `mod` under Unix just type `mod perc.model`. After a few seconds, with no error messages if the model definition is syntactically correct, `mod` announces that the ‘`model.c`’ file has been created. It should operate similarly under other operating systems.

The next step is to compile and link together the various C files that will constitute the simulation program for your particular model. Note that each time you want to change an equation in your model you will have to change the model definition file and repeat the steps above. However, changing just parameter values or state initial values does not require recompilation since that can be done through simulation specification files.

- Under Unix, the simplest is to use the `makemcsim` script. Just type `makemcsim` and compilation will be done automatically (see section Using `makemcsim` to fully process model files). An executable ‘`mcsim.perc`’ is created. You can rename it if you wish.
- Under other operating systems, you should use the command `make` or its equivalent to compile and link together the ‘`model.c`’ file created by `mod` and the other C files of the ‘`sim`’ directory (see section Installation). That should create an application (you should give it a name specific to the model you are developing, e.g., ‘`mcsim.perc`’). Refer to your compiler manual for details on how to use your programming environment. Your executable ‘`mcsim.perc`’ program is now ready to perform simulations.

To start your GNU MCSim program just type `mcsim.perc` (if you gave it that name) under Unix. After an introductory banner (telling in particular which model file the program has been compiled with), you are prompted for an input file name: type in `perc.lsodes.in` (see section ‘`perc.lsodes.in`’, to see this file in Appendix), then a space, and then type in the output file name: `perc.lsodes.out`. After a few seconds or less (depending on your machine) the program announces that it has finished and that the output file is ‘`perc.lsodes.out`’. You can open the output file with any text editor or word processor, you can edit it for input in graphic programs etc.

You can try the various demonstration models provided in the ‘`samples`’ directory and observe the output you obtain. You can then start programming you own models and doing simulations. The next sections of this manual reference the syntax for model definition and simulation specifications.

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# 5. Setting-up Structural Models

The model generator, "mod", was created to facilitate structural model definition and maintenance, while keeping short execution time through compilation. This chapter explains how to use `mod`, and how to code your models using a simplified syntax that `mod` can translate in C (creating thereby a ‘`model.c`’ file).

After compiling and linking of the newly created ‘`model.c`’ file together with the other C files of the ‘`mcsim/sim`’ directory (or after linking with a dynamic library ‘`libmcsim.so`’), an executable simulation program is created, specific of your particular model. These preprocessing and compilation steps can be performed in Unix with a single shell command `makemcsim` (in which case, the ‘`model.c`’ is created only temporarily and erased after that).

Several examples of model simulation files are included in the ‘`mcsim/samples`’ directory. Some of them are reproduced in Appendix (see section Examples).

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## 5.1 Using `mod` to preprocess model description files

The `mod` program is a stand-alone facility. It takes a model description file in the "user-friendly" format described below (see section Syntax of the model description file) and creates a C language file ‘`model.c`’ which you will compile and link to produce the simulation program. `Mod` allows the user to define equations for the model, assign default values to parameters or default initial values to model variables, and to initialize them using additional algebraic equations. `Mod` lets the user create and modify models without having to maintain C code. Under Unix or GNU/Linux, the command line syntax for the mod program is:

    mod [input-file [output-file]]

where the brackets indicate that the input and output filenames are optional. If the input filename is not specified, the program will prompt for both. If only the input filename is specified, the output is written by default to the file ‘`model.c`’. Unless you feel like doing some makefile programming, we recommend using this default since the makefile for GNU MCSim assumes the C language model file to have this name. You have to have prepared a text file containing a description of the model following the syntax described in the following (see section Syntax of the model description file).

The following options are available:

-h, -H gives a short online help.

-R generate a C file of the format requested for use by the `deSolve` package of the `R` software for statistical analysis; `deSolve` implements differential equations solvers with interesting capabilities.

Most error messages given by `mod` are self-explanatory. Where appropriate, they also give the line number in the model file where the error occurred. Beware, however, of cascades of errors generated as a consequence of a first one; so don’t panic: start by fixing the first one and rerun `mod`. Note that when using the -R option, care has to be taken to adopt the `deSolve` code conventions (see the `deSolve` manual on R CRAN). If you get really stuck you can send a message to the help mailing list (see section Installation) or to the authors of this manual.

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## 5.2 Using `makemcsim` to fully process model files

`makemcsim` is a Unix `sh` shell script that further facilitates preprocessing and compilation. You run `makemcsim` by entering it at the command prompt:

    makemcsim [model-file]

where the brackets indicate that the model filename is optional. If a model filename is not specified, the first file having extension ‘`.model`’ (by alphabetical order) is used. Makemcsim calls `mod` if the model file has changed since last compilation, compiles the ‘`model.c`’ generated, links it to the shared ‘`libmcsim.so`’ library to create an executable ‘`mcsim.<root-model-name>`’. The extension ‘`root-model-name`’ corresponds to your model filename (without its last extension if it has one; i.e.,typically, without the ‘`.model`’ extension). The ‘`model.c`’ file is deleted afterward; if you want to inspect it (for example, if you got error messages from `mod`), run `mod` on your model definition file.

Two variants of `makemcsim` are also available: `makemcsims`, which creates a standalone version (no dynamic libraries needed), and `makemcsimd`, which creates a standalone version with debugging symbols included (so that you can use gdb, for example, to check what the code does).

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## 5.3 Syntax of the model description file

The model description file is a text (ASCII) file that consists of several sections, including global declarations, dynamics specifications (with derivative calculations), model initialization ("scaling"), and output computations. Here is a template for such a file (for further examples see section Examples):

    # Model description file (this is a comment)
    <Global variable specifications>
    Initialize {
      <Equations for initializing or scaling model parameters>
    }
    Dynamics {
      <Equations for computing derivatives of the state variables>
    }
    Jacobian {
      <Equation for the Jacobian of the state derivatives>
    }
    CalcOutputs {
      <Equations for computing output variables>
    }
    End. # mandatory ending keyword

`Initialize`, `Dynamics`, `Jacobian` and `CalcOutputs` are reserved keywords and, if used, must appear as shown, followed by the curly braces which delimit each section (see section Model initialization; Dynamics section; Output calculations). Please note that at least one of the sections `Dynamics` or `CalcOutputs` should be defined, and that `Dynamics` must be used if the model includes differential equations. Finally the model definition file must have the `End` keyword at the beggining of a line, eventually preceeded by white spaces or tabs. Text after the `End` keyword is ignored.

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### 5.3.1 General syntax

The general syntax of the model description file is as follows:

- Comments begin with a pound sign (`#`) and continue to the end of the line.
- Blank lines are allowed and ignored.
- All commands can span several lines and are terminated by a semi-colon (`;`).
- Four types of variables are used: state variables, output variables, input variables, and parameters (see section Global variable declarations). The name of a variable should be a valid C identifier, starting with a letter or underscore (`_`) and followed by any number of alpha-numeric characters or underscores, up to a maximum of 80. Variable names are case sensitive. Note that the name `IFN`, in capital letters, is reserved by the program and should not be used as parameter or variable name.
- Variable assignments have the following syntax:

      <variable-name> '=' <constant-value-or-expression> ';'

  The equal sign is needed. The right-hand side expression can be a valid C mathematical expression including numerical constants, already defined variables, standard ANSI C mathematical functions, and GNU MCSim’s "special functions" (see section Special functions) or "input functions" (see section Input functions). Special functions can take already defined variables, constant values or expressions as parameters. Input functions can only be used on the right hand side of assignments to input variables.

  Colon conditional assignments have the following syntax:

      <variable-name> = (<test> ? <value-if-true> : <value-if-false>);

  For example:

      Adjusted_Param = (Input_Var > 0.0 ? Param * 1.1 : Param);

  In this example, if ‘`Input_Var`’ is greater than 0, the parameter ‘`Adjusted_Param`’ is computed as the product of ‘`Param`’ by ‘`1.1`’; otherwise ‘`Adjusted_Param`’ is equal to ‘`Param`’. Note that conditional assignments can be nested (i.e., <value-if-true> or <value-if-false> can themselves be a conditional expression). The comparison operators allowed are the equality operator `==`, and non-equality operators `!=`, `<`, `>`, `<>`, `<=` and `>=`.

  More complex conditions can also be specified, but the Boolean AND, OR and NOT operations have not yet been implemented. You can use:

  `(('A'*'B')>0)` for AND `(('A'+'B')>0)` for OR `('A'==0)` for NOT

- Vectors: (Warning: for now, the square bracket notation can be only used in the model definition file. It is not recognized in simulation specification files; for those, the underscore ((`_`) unrolling syntax can be used to address vector components. This limits the usefulness of this feature).

  Vectors’ declaration: To declare a variable as a vector use the one of the two following syntaxes when you first define it:

      <variable-name> '[' <integer> ']'
      <variable-name> '[' <integer> '-' <integer> ']'

  The variable name is immediately followed by an opening square bracket (’[’). The array index bounds (which define the valid indices) can be given as (long) positive or null integers separated by an hyphen (’-’) (spaces are allowed). In this case the second integer must be higher the first. They are followed by a closing bracket (’]’). The hyphen and second integer are optional. If only one bound (integer) is given, only the component with corresponding index is declared. Both syntaxes can be mixed. For example:

      States = {y[0-9]};
      alpha[0-2] = 1;
      beta[0] = 1;
      beta[1] = 2;
      beta[2-4];

  The previous lines define a state variable ‘`y`’ as a vector of length 10, with valid indices ranging between 0 and 9, included. The parameter vector ‘`alpha`’ is defined with range 0 to 2, each component being initialized to value 1. For parameter ‘`beta`’, components 0, 1 and 2 to 4 are initialized separately (components 2 to 4 are initialized with default value 0).

  Accessing vectors’ components: After declaration, vector’s components can be accessed individually using the square bracket syntax:

      <variable-name> '[' <integer> ']'

  For example:

      Outputs = {x[0-1]};
      beta[0] = 0;
      beta[1] = beta[0] + 1;
      CalcOutputs {
        x[0] = beta[0] * t;
        x[1] = beta[1] * t;
      }

  In the above example, ‘`beta[0]`’, ‘`beta[1]`’, ‘`x[0]`’, and ‘`x[1]`’ are accessed individually. The variable ‘`t`’ refers to the implicit variable ’time’.

  Vectorization of equations: The equations specifying the model, which consist in assignments, can be vectorized in the `Initialize`, `Dynamics` and `CalcOutputs` sections (but not in the global section) (see section Global variable declarations). Vectorization allows you to specify an operation for an entire vector or parts of it. The following syntax should be used:

      <var-name>'['<integer>'-'<integer>']' = <vectorized-expression>;

  On the right-hand side, the vectorized expression should be a valid C mathematical expression including numerical constants, already defined state, input, output, other (parameter) variables or vectors, and standard ANSI C mathematical functions or special functions (see section Special functions). Here also, input functions (see section Input functions) can only be used on the right hand side of assignments to input variables. Vector indices on the right-hand side can take the special form of "bracketed expressions". Bracketed expressions can be composed of integers, the 4 basic arithmetic operators (’+’, ’-’, ’*’, ’/’), parentheses and the index letter ’i’. The running index ’i’ points in turn to each component in the range specified on the left-hand side (imagine that the range given on the left-hand side corresponds to a ’for’ loop with index ’i’ running from the lower bound to the upper bound). This is best understood by looking at some code. In the previous example, the assignments to x[0] and x[1] obviously deserve vectorization. This is achieved by the following statements:

      CalcOutputs {
        x[0-1] = beta[i] * t;
      }

  Here, the index ’i’ refers to the values 0 and 1. Here is another example:

      Outputs{x[1-10]};
      CalcOutputs {
        x[1] = 0;
        x[2-10] = x[i-1] + 1;
      }

  This is equivalent to:

      Outputs{x[1-10]};
      CalcOutputs {
        x[1] = 0;
        x[2] = x[1] + 1;
        ...
        x[10] = x[9] + 1;
      }

  and will assign value 1 to ‘`x[2]`’, 2 to ‘`x[3]`’, etc. On the right-hand side, more complicated bracketed expressions like ‘`[(2*i-1)/(i+3)]`’ can be used. Another, working, example of vector use is given in the ‘`mcsim/samples/pde2`’ directory.

  Alternative ’underscore’ (’_’) syntax: Individual vector components can be declared and used (everywhere in the model file) with the following syntax:

      <variable-name>'_'<integer>

  The integer indicates which component of the vector is referred to. For example ‘`x_1`’ is strictly equivalent to ‘`x[1]`’. Note!: No space are allowed between the variable name, the underscore and the integer. Note also!: This syntax is the only one that can be used in simulation specification files. If you declare a parameter ‘`beta[1-10]`’ in your model definition file, the only way to refer to it in the simulation input file will be through its individual components ‘`beta_1`’, ‘`beta_2`’, ... etc. This limitation will be removed in a future release of the software.

- The `End` keyword must used to indicate model termination.

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### 5.3.2 Global variable declarations

Commands not specified within the delimiting braces of another section are considered to be global declarations. In the global section, you first declare the state, input, and output variables. There should be at least one state or output variable in your model.

- States are variables for which a first-order differential equation is defined in the `Dynamics` section (see section Dynamics section) (higher orders or partial differential equations are not allowed).
- Inputs are variables independent of the others variables, but eventually varying with time (for example an exposure concentration to a chemical).
- Outputs are dependent model variables (obtainable at any time as analytical functions of the states, inputs or parameters) that do not have dynamics. They must receive assignments in either the `Dynamics` or `CalcOutputs` sections.

The format for declaring each of these variables is the same, and consists of the keyword `States`, `Inputs` or `Outputs` followed by an equal sign and a list of the variable names enclosed in curly braces as shown here:

    States =   {Qb_fat,  # Benzene in the fat
                Qb_bm,   # ...     in the bone marrow
                Qb_liv}; # ...     in the liver and others

    Inputs =   {Q_gav,   # Gavage dose
                C_inh};  # Inhalation concentration

    Outputs =  {Cb_exp,  # Concentration in expired air
                Cb_ven}; # ...           in venous blood

After being defined, states, inputs and outputs can then be given initial values (constants or expressions). Inputs can also be assigned input functions, described below (see section Input functions). Some examples of initialization are shown here:

    Qb = 0.1; # Default initial value for state variable Qb

    # Input variable assigned a periodic exponential input function
    Q = PerExp(1, 60, 0, 1); # Magnitude of 1.0,
                             # period of 60 time units,
                             # T0 in period is 0,
                             # Rate constant is 1.0

If a state, input, or output variable is not explicitly given an initial value, that value will be set to zero by default. Initial values are reset to their specified value by the simulation program at the start of each simulation of an `Experiment` (see section `Simulation` sections).

All the other variables are "parameters". Model parameters you want to be able to change in simulation input files should be declared in the global section. For example:

    Wind_speed; # (m/s) Local wind speed

Parameters are by default assigned a value of zero. To assign a different nominal values, use the assignment rules given above. For example:

    BodyWt = 65.0 + sqrt(15.0); # Weight of the subject (in kg)

All parameters and variables are computed in double precision floating-point format. Initial values should not be such as to cause computation errors in the model equations; this is likely to lead to crashing of the program (so, for example, do not assign a default value of zero to a parameter appearing alone in a denominator). Note that the order of global declarations matters within the global section itself (i.e., parameters and variables should be defined and initialized before being used in assignments of others), but not with respect to other blocks. A parameter defined at the end of the description file can be used in the `Dynamics` section which may appear at the beginning of the file. Still, such an inverse order should be avoided. For this reason, the format above, where global declarations come first, is strongly suggested to avoid confusing results. Note again that the name `IFN`, in capital letters, is reserved by the program and should not be used as parameter or variable name. Finally, if a parameter is defined several times, only the first definition is taken into account (a warning is issued, but beware of it).

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### 5.3.3 Model types

This section deals with structural models. Statistical models that you setup for model calibration and data analysis are defined in the simulation input files, through statistical distribution functions. They are dealt with later in this manual (see section Setting-up statistical models).

GNU MCSim can easily deal with purely algebraic structural models. You do not need to define state variables or a `Dynamics` section for such models. Simply use input and output variables and parameters and specify the model in the `CalcOutputs` section. You can use the time variable `t` if that is natural for your model. If your model does not use `t`, you will still need to specify "output times" in `Print()` or `PrintStep()` statements to obtain outputs: you can use arbitrary times. If you do not use `t` as "independent" model variable, you will also need do define a `Simulation` section (see section `Simulation` sections) for each combination of values for the independent variables of your model. This may be clumsy if many values are to be used. In that case, you may want to use the variable `t` to represent something else than time.

Ordinary differential models, with algebraic components, can be easily setup with GNU MCSim. Use state variables and specify a `Dynamics` section. Time, `t` is the integration variable, but here again, `t` can be used to represent anything you want. For partial differential equations some problems might be solved by implementing line methods (see examples in ‘`mcsim/samples/pde1`’ and ‘`mcsim/samples/pde2`’)...

You can use GNU MCSim for discrete-time dynamic models (or difference models). That is a bit tricky. Assignments in the `CalcOutputs` section are volatile (not memorized), so the model equations have to be given in a `Dynamics` section. But the model variables should still be declared as outputs, because they should not be updated by integration. However, you need at least one true differential equation in the `Dynamics` section, so you should declare a dummy state variable (and assign to its derivative a constant value of zero). You also want the calls to `Dynamics` to be precisely scheduled, so it is best to use the `Euler` integration routine (see section `Integrate()` specification) which uses a constant step. Since `Euler` may call repeatedly `Dynamics` at any given time, you want to guard against untimely updating... Altogether, we recommend that you examine the sample files in the ‘`mcsim/samples/discrete`’ directory provided with the source code for GNU MCSim.

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### 5.3.4 Special functions

The following special functions (whose name is case-sensitive) are available to the user for assignment of parameters and variables in the model definition file:

- `BetaRandom(alpha, beta, a, b)`: returns a Beta distributed variate on the interval [a,b] with shape parameters alpha and beta;
- `BinomialBetaRandom(E, alpha, beta)`: return random variate, of mathematical expectation E, drawn from a binomial distribution with probability p, p being Beta distributed with parameters alpha and beta;
- `BinomialRandom(p, N)`: returns a binomially distributed random variate;
- `CauchyRandom(s)`: returns a Cauchy distributed random variate with scale s;
- `CDFNormal(x)`: the normal cumulative density function;
- `Chi2Random(dof)`: returns a Chi-squared random variate with dof degrees of freedom;
- `erfc(x)`: the complementary error function;
- `ExpRandom(beta)`: returns an exponential variate with inverse scale beta;
- `GammaRandom(alpha)`: returns a gamma distributed random variate with shape parameter alpha and inverse scale equal to 1;
- `GetSeed()`: returns the current value of the random generator seed;
- `GGammaRandom(alpha, beta)`: returns a gamma distributed random variate with shape parameter alpha and inverse scale beta;
- `InvGGammaRandom(alpha, beta)`: returns an inverse gamma distributed random variate with shape parameter alpha and scale parameter beta;
- `lnDFNormal(x, mean, sd)`: the natural logarithm of the normal density function;
- `lnGamma(x)`: the natural logarithm of the gamma function;
- `LogNormalRandom(mean, sd)`: returns a lognormally distributed variate with geometric mean mean and geometric standard deviation sd (i.e., the log of the returned variate is normally distributed with mean log(mean) and standard deviation log(sd));
- `LogUniformRandom(a, b)`: returns variate log-uniformly distributed on the interval [a,b];
- `NormalRandom(mean, sd)`: returns a normally distributed random variable with prescribed mean and standard deviation;
- `PiecewiseRandom(min, a, b, max)`: the distribution of the returned variate has the form of a truncated triangle, with base from min to max and a plateau between a and b. If a = b, the distribution is the triangular distribution;
- `PoissonRandom(mu)`: returns a Poisson-distributed random variate, of rate mu;
- `SetSeed(seed)`: sets the current value of the pseudo-random generator seed to the specified seed. That seed can be any positive real number. Seeds between 1.0 and 2147483646.0 are used as is, the others are rescaled within those bounds (and a warning is issued);
- `StudentTRandom(dof, mean, sd)`: returns a Student t distributed random variate with dof degrees of freedom and given mean and standard deviation;
- `TruncInvGGammaRandom(alpha, beta, a, b)`: returns a truncated inverse gamma distributed random variate with shape parameter alpha and scale beta, in the range [a,b]. Explicit specification of a,b is required;
- `TruncLogNormalRandom(mean, sd, a, b)`: returns a truncated lognormal variate with geometric mean mean and geometric standard deviation sd, in the range [a,b]. Explicit specification of a,b is required;
- `TruncNormalRandom(mean, sd, a, b)`: returns a truncated normal variate with prescribed mean and standard deviation, in the range [a,b]. Explicit specification of a,b is required;
- `UniformRandom(min, max)`: returns a uniformly distributed random variable, sampled between min and max. The algorithm used is that of Park and Miller (Barry, 1996; Park and Miller, 1988; Vattulainen et al., 1994) (see section Bibliographic References). A default random generator seed (314159265.3589793) is used.

Note: for all the above random number generating functions, a default random generator seed is used. It can be changed with the function `SetSeed`. Note also that assignment of a random number generating function to a state variable derivative will define a form of stochastic differential equation. GNU MCSim’s integration routines are not particularly suited to the resolution of such equations. If you wish to try it anyway, you may want to consider using the "robust" Euler method (see section `Integrate()` specification).

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### 5.3.5 Input functions

These functions can be used in special assignments, valid only for input variables. Inputs can be initialized to a constant or to a standard mathematical expression, or assigned one of the following input functions:

- `PerDose()` specifies a periodic input of constant <magnitude>. The input begins at <initial-time> in the <period> and lasts for <exposure-time> time units. Syntax:

      <input variable> = PerDose(<magnitude>, <period>, <initial-time>, <exposure-time>);

- `PerExp()` specifies a periodic exponential input. At time <initial-time> in the <period> the input rises instantaneously to <magnitude> and begins to decay exponentially with the constant <decay-constant>. The input is turned off once the magnitude reaches a negligible fraction of its original value. Note that the input does not accumulate across periods, it resets at each period start. Syntax:

      <input variable> = PerExp(<magnitude>, <period>, <initial-time>, <decay-constant>);

- `NDoses()` specifies a number of stepwise inputs of variable magnitude and their starting times. The first argument, <n>, is the number of input steps and start times. Next come a list of magnitudes and a list of corresponding initial times. Each list is comma-separated. The duration of each input step is computed automatically by difference between the listed times. Currently this function can only be used in the simulation description file, and not in the model description file (which simply implies that you cannot use it as a default). Syntax:

      <input variable> = NDoses(<n>, <list-of-magnitudes>, <list-of-initial-times>);

  Note that the list of times must begin at the starting time of the simulation (typically time zero), even if the magnitude at that first time is zero.

- `Spikes()` specifies a number of instantaneous inputs of variable magnitude and their exact times of occurrence. The first argument, <n>, is the number of inputs and input times. Next come a list of magnitudes and a list of times. Each list is comma-separated. Currently this function can only be used in the simulation description file, and not in the model description file (which simply implies that you cannot use it as a default). Syntax:

      <input variable> = Spikes(<n>, <list-of-magnitudes>, <list-of-times>);

The arguments of input functions can either be constants or variables. For example, if ‘`Mag`’ and ‘`RateConst`’ are defined model parameters, then the input variable ‘`Q_in`’ can be defined as:

    Q_in = PerExp(Mag, 60, 0, RateConst);

In this way the parameters of input functions can, for example, be assigned statistical distributions in Monte Carlo simulations (see section `Distrib()` specification). Variable dependencies are resolved before each simulation specified by an `Experiment` (see section `Simulation` sections).

For each of the periodic functions, a single exposure beginning at time initial-time can be specified by giving an effectively infinite period, e.g. 1e10. The first period starts at the initial time of the simulation. Magnitudes change exactly at the times given.

Input variables assigned input functions can be combined to give a lot of flexibility (e.g., an input variable can be declared as the sum of others). Separate inputs can also be declared in the global section of the model definition file and combined in the `Dynamics` (see section Dynamics section) and `CalcOutputs` (see section Output calculations) sections.

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### 5.3.6 In line functions

`Inline()` functions can be placed in the various sections of a model file to introduce standard C code (or whatever) in your models. Text placed between the parentheses of an `Inline` function will be passed as is to the C compiler. That text can span several lines but its size should not exceed MAX_EQN (defined in ‘`lex.h`’); In case it does, you can increase MAX_EQN (and recompile ‘`mod`’...) or you can split you text between any number of `Inline()` in a row. It is your responsibility to make sure that the code passed can be compiled without errors!

Example:

    Inline( printf("hello/n"); );

Note also that the inlined code is likely to be dependent on whether or not you are using the `-R` option of `mod`.

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### 5.3.7 Model initialization

The model initialization section begins with the keyword `Initialize` (the keyword `Scale` is obsolete but is still understood) and is enclosed in curly braces. The equations given in this section will define a function (subroutine) that will be called by GNU MCSim after the assignments specified in each `Experiment` are done (see section `Simulation` sections). They are the last initializations performed. The model file in ‘`mcsim/samples/perc`’ gives an example of the use of `Initialize` (see section ‘`perc.model`’: A sample model description file, in Appendix).

All model variables and parameters, except inputs, can be changed in this section. Modifications to state variables affect initial values only. In this section, state variables, outputs and parameters (but not input variables) can also appear at the the right-hand side of equations. The integration variable can be accessed if referred to as `t`

Warning: Assignments to state variables in the `Initialize` section override the same assignments made in input files.

Additional parameters (to those declared in the global section) may be used within the section. They will be declared as local temporary variables and their scope will be limited to the `Initialize` section (i.e., their value and existence will be forgotten outside the section).

The `dt()` operator (see section Dynamics section) cannot be used in this section, since derivatives have not yet been computed when the scaling function is called.

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### 5.3.8 Dynamics section

The dynamics specification section begins with the keyword `Dynamics` and is enclosed in curly braces. The equations given in this section will be called by the integration routines at each integration step. `Dynamics` must be used if the model includes differential equations.

Additional parameters (to those declared in the global section) may be used for any calculations within the section. They will be declared as local temporary variables. (Note, for example, the use of ‘`Cout_fat`’ and ‘`Cout_wp`’ in the ‘`perc.model`’ sample file). Local variables are not accessible from the simulation program, or from other sections of the model definition file, so don’t try to output them.

Each state variable declared in the global section must have one corresponding differential equation in the `Dynamics` section. If a differential equation is missing, `mod` issues an error message such as:

    Error: State variable 'Q_foo' has no dynamics.

and no ‘`model.c`’ file or executable program will be created.

The derivative of a state variable is defined using the `dt()` operator, as shown here:

    dt(state-variable) '=' constant-value-or-expression ';'

The right-hand side can be any valid C expression, including standard math library calls and the special functions mentioned above (see section Special functions). Note that no syntactic check is performed on the library function calls. Their correctness is your responsibility.

The `dt()` operator can also be used in the right-hand side of equations in the dynamics section to refer to the value of a derivative at that point in the calculations. For example:

    dt(Qm_in) = Qmetabolized - dt(Qm_out);

The integration variable (e.g., time) can be accessed if referred to as `t`, as in:

    dt(Qm_in) = Qmetabolized - t;

Output variables can also be made a function of `t` in the `Dynamics` section.

Note that while state variables, input variables and model parameters can be used on the right-hand side of equations, they cannot be assigned values in the `Dynamics` section. If you need a parameter to change with time, you can declare it as an output variable in the global section. Assignments to states, inputs or parameters in this section causes an error message like the following to be issued:

    Error: line 48: 'YourParm' used in invalid context.
    Parameters cannot be defined in Dynamics{} section.

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### 5.3.9 Delay differential equations

GNU MCSim can solve delay differential equations.

Delay differential equations are equations that depend on past values of the state variables, say at time t-tau instead of at time t.

This is done very easily in GNU MCSim models with the `CalcDelay` funtion. Its syntax is:

      CalcDelay (<variable>, <delay>);

The `variable` must be a declared state or output variable. `CalcDelay()` will return its past value (at time delay). The delay specified must be either a declared parameter or a constant floating point or integer value. For example:

      tau = 100;
      dt (Q1) = k * CalcDelay(Q3, tau);
      dt (Q2) = k * CalcDelay(Q3, 10);

An example of a model using `CalcDelay()` and input file is given in the ‘`example/delay_diff_eqns`’ folder. Note that currently, the `CalcDelay()` function cannot be used with the `-R` option for `mod` (i.e., for use with the deSolve R package).

GNU MCSIm stores required variables past values in a arrays of size MAX_DELAY (equal by default to 1000). If the needed past recall exceeds that capacity you will need to increase the value of MAX_DELAY in the C file ‘`delays.c`’ and recompile the library and your model.

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### 5.3.10 Output calculations

The output calculation section begins with the keyword `CalcOutputs` and is enclosed in curly braces. The equations given in this section will be called by the simulation program at each output time specified by a `Print()` or `PrintStep()` statement (see section `Print()` specification, and see section `PrintStep()` specification). In this way, output computations are done efficiently, only when values are to be saved.

Only variables that have been declared with the keyword `Outputs`, or local temporary variables, can receive assignments in this section. Assignments to other types of variables cause an error message like the following to be issued:

    Error: line 56: 'Qb_fat' used in invalid context.
    Only output and local variables can be defined in CalcOutputs section.

Any reference to an input or state variable will use the current value (at the time of output). The `dt()` operator can appear in the right-hand side of equations, and refers to current values of the derivatives (see section Dynamics section). Like in the `Dynamics` section, the integration variable can be accessed if referred to as `t`, as in:

    Qx_out = DQx * t;

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### 5.3.11 Comments on style

For your model file to be readable and understandable, it is useful to use a consistent notation style. The example file ‘`perc.model`’ tries to follow such a style (see section ‘`perc.model`’: A sample model description file). For example we suggest that:

- All variable names begin with a capital letter followed by meaningful lower case subscripts.
- Where two subscripts are necessary, they can be separated by an underscore, such as in ‘`Qb_fat`’.
- Where there is only one subscript an underscore can still be used to increase readability as in ‘`Q_fat`’.
- Where two words are used in combination to name one item, they can be separated visually by capitalizing each word, as in ‘`BodyWt`’.

These conventions are suggestions only. The key to have a consistent notation that makes sense to you. Consistency is one of the best ways to:

1. Increase readability, both for others and for yourself. If you have to suspend work for a month or two and then come back to it, the last thing you want is to have to decipher your own file.
2. Decrease the likelihood of mistakes. If all of the equations are coded with a consistent, logical convention, mistakes stand out more readily.

Last, but not least, do use comments to annotate your code! Also: make sure your comments are accurate and update them when you change your code. In our experience, an enormous number of hours has been wasted in trying to figure out inconsistencies that existed only because of inaccurate comments (e.g., erroneous comments about the reasons for choice of default parameter values). That does not decrease the value of good comments, however...

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## 5.4 Reading SBML models and applying a template

To read models written in SBML, you simply need to create a text file of the same format as an MCsim model definition file (comments starting with `#`, file ending with `End`, etc.) In that file, you specify a list of one or more SBML model files with the keyword `SBMLModels` followed by an equal sign and a list of SBML model file names (of maximum size MAX_FILENAMESIZE, 80 characters, as defined in mod.h), enclosed in curly braces, as shown here:

    #---------------------------------------------------------
    # SBML_List.in
    # Use it as "mod SBML_List.in"
    #---------------------------------------------------------

    SBMLModels = {useID, "default", "C_central.xml", "C_periph.xml"};

    End.

The first two tokens, `useID` and `"default"`, at the very beginning of the list, are optional (but if the appear they must both appear). The first indicates that species are recognized by "IDs" (at the moment this the only possibility, hardly an option indeed, but in the future we would like to be able also to use Names"). The second one gives the name of the default (external) compartment in your SBML files. Here the default is named "default", which works for CellDesigner, but JDesigner, for example, uses "compartment"... The list of files comes after, with the filenames being enclosed with double quotes. There is no restriction about their extension (".xml" is just an example). They just have to be valid filenames.

Give that file as input to `mod`, by typing in `mod SBML_List.in` on your shell command line. A ‘`model.c`’ file is produced, suitable for further compilation by `makemcsim`. SBML model files are typically ASCII text files with a xml extension (for examples see ‘`mcsim/sim/samples/SBML`’). SBML Level 1 and Level 2 are recognized (but some features of Level 2 are not yet understood by `mod`, such as `functionDefinition`, `unit`, `rateRule` and a few others.) Omitted reaction stoichiometries are set to value 1 by default. Compartments are ignored unless a model template for circulating species is given (see below).

If two or more of the SBML files define a same chemical species, `mod` merges the models: namely, the rate equation for that species will be the sum of the rate equations implied by each model. In fact, `mod` constructs the rate equations from the reaction descriptions given by the SBML format. State variables and parameters keep the same name after merging, so they should be unique from the start to each SBML model, to avoid confusion.

To automatically extend SBML (level 2) models (e.g., with transport component terms, an example which will be used throughout this part of the manual), a template model can be applied to all the species defined in SBML which are placed in the default compartment (conventionally called `compartment`), and outside of other compartments whose names are specified by the template itself. Specifically: the template model should define compartments and differential equations terms for the transport of an unspecified species between those compartments (the syntax for that will be described below). For each species placed in SBML outside one of the defined compartments, a differential equations is created for each compartment using the template transport terms. The differentials for the compartments found in both the template and the SBML models will contain both transport and kinetic terms. With the exception of the generic `compartment`, the SBML models merged can only use the compartments defined by the template (compartment name recognition is case-sensitive). Species placed in SBML inside one of the defined compartments are not transported and stay local to that compartment. Their differentials will contain only the reaction kinetics terms defined by the SBML models. For simplicity, the default initial values of the model state variables (species) specified by the user are ignored and set to zero.

Warning: At the moment, libSBML does not guarantee the type of operands or their format for a division. While `5.4/2.3` will appear correctly as a float division, `5.0/2.0` is likely to be translated in `5/2` which is an integer division in C! It is probably best to use the multiplication by the inverse in that case, but that forces you to check SBML models manually for all divisions...

There might be many applications of the template mechanism, and they are left to the reader to imagine.

The template model to use is specified with `PKTemplate` keyword, followed by an equal sign and the file name of a template model file (of maximum size MAX_FILENAMESIZE, 80 characters, as defined in mod.h), enclosed in curly braces, like in:

    #---------------------------------------------------------
    # SBML_List.in
    # Use it as "mod SBML_List.in"
    #---------------------------------------------------------

    PKTemplate = {"input_f/2cpt_PBPK.model"};

    SBMLModels = {"C_central.xml", "C_periph.xml"};

    End.

The template structure is similar to that of other GNU MCSim models, with two exceptions: First, the variables and parameters which are supposed to apply to several chemical species defined in SBML are preceeded with an underscore (`_`). Second, the compartments allowed in SBML and for which the template is defined are declared using the keyword `Compartment`, followed by an equal sign and a list of the compartment names enclosed in curly braces. Here is an example of template file:

    #---------------------------------------------------------
    # Template for pharmacokinetic modeling
    #---------------------------------------------------------

    States = {_central, _periph};
    Inputs = {_Dose_rate};
    Outputs = {_Q_total};
    Compartments = {central, peripheral};

    # Species-dependent parameters
    _PC = 2;
    _K_urine;

    # Species-independent parameters
    V_central;
    V_periph;

    Initialize { _K_urine = _K_urine * 2; }

    Dynamics {
      _Q_rate_c_p = 5 * (_central - _periph / _PC);

      # Central compartment quantity
      _pk_central  = (_Dose_rate - _Q_rate_c_p -
                      _K_urine * _central) / V_central;
      dt(_central) = _pk_central;

      # Peripheral compartment quantity
      _pk_periph  = _Q_rate_c_p / V_periph;
      dt(_periph) = _pk_periph;
     }

    CalcOutputs { _Q_total = _central / V_central + _periph / V_periph; }

    End.

With that template, each species, say S1, defined in SBML to be outside of the `central` or `peripheral` compartments will be cloned to form the state variables S1_central and S1_peripheral. Those will have associated differential equations using specific parameters S1_PC or unspecific ones like V_central. The chemical reactions defined in SBML to take place inside the central or peripheral compartments will be translated into specific terms added to the `dt(_central)` and `dt(_periph)` equations.

Warning: the automatic creation of a model by merging SBML files with a template may shuffle the order of the differential equation declarations. Therefore you should not use an already defined `dt()` term in a subsequent differential equation in the template model.

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## 5.5 Working with the R package deSolve

GNU MCSim ‘`mod`’ model generator can be passed the `-R` option. For example:

      mod -R perc.model

In that case, the C code produced can be used by the `deSolve` package of the R statistical software (see http://www.r-project.org/) to perform simulations of your models. The numerical integators provided by deSolve are improved implementations of the lsode family of integrators used by GNU MCSim), and `deSolve` provides a few more options than GNU MCSim (see the `deSolve` user manuals). However, if you need raw speed (say, for Markov chain Monte Carlo simulations) GNU MCSim is probably the fastest option.

In addition to producing a ‘`model.c`’ file in C language, ‘`mod`’ called with the `-R` option also generates ‘`model_inits.R`’ file. That file can be loaded in R and provides the R two functions `initParms`, `initStates` and the variable Outputs, which can be handy in R scripts:

- `initParms()` without parameters reset the model parameters to their default values. The newparms parameter takes a vector of named parameters and values, assign the given values to the corresponding model parameters and reset the others to their default values.
- `initStates()` with just the parms parameter reset the model initial states to their default values. If the newparms parameter is used it takes a vector of named states and values, assign the given values as initial values to the corresponding model state variables and reset the others to their default values.
- Outputs is simply an array of output variable names.

An example of R script using `GNU MCsim` to generate and run a model is given in the ‘`example/R`’ folder.

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# 6. Running Simulations

After having your model processed by `mod` or `makemcsim`, and obtained an executable ‘`mcsim_...`’ file, you are ready to run simulations. For this you need to write simulation files. This chapter explains how to write such files with the proper syntax and how to run the executable program.

You may want to first give a look at the examples given in the ‘`mcsim/samples`’ directory. An sample file ‘`perc.lsodes.in`’, which works with the perchloroethylene model ‘`perc.model`’, is also given in an Appendix to this manual (see section ‘`perc.lsodes.in`’).

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## 6.1 Using the compiled program

GNU MCSim provides several types of simulations for the models you create. Simulations are specified in a text file of format similar to that of the model description file.

Assume that your model ‘`a.model`’ has been preprocessed and compiled by `makemcsim` (see section Using `makemcsim` to fully process model files) to generate an executable ‘`mcsim_my`’. If you have renamed the executable file, substitute ‘`mcsim_my`’ by the name of your executable in the following. In Unix the command-line syntax to run that executable is simply:

    mcsim_a [input-file [output-file]]

where the brackets indicate optional arguments. If no input and output file names are specified, the program will prompt you for them. You must provide an input file name. That file should describe the simulations to perform and specify which outputs should be printed out (see section Syntax of the simulation definition file). If you just hit the return key when prompted for the output name, the program will use the name you have specified in the input file, if any, or a default name (see section `OutputFile()` specification). If just one file name is given on the command-line, the program will assume that it specifies the input file. For the output filename, the program will then use the name you have specified in the input file, if any, or a default name.

When the program starts up, it announces which model description file was used to create it. While the input file is read or while simulations are running, some informations will be printed on your computer screen. They can help you check that the input file is correctly interpreted and that the program runs as it should. GNU MCSim can also post error messages, which should be self-explanatory. Where appropriate, they show the line number in the input file where the error occurred. Beware, however, of cascades of errors generated as a consequence of a first one; also errors may be detected after the line in which they really occur and the line number shown will be unhelpful; don’t panic: start by fixing the first error in the input file and rerun your executable. You should not need to recompile your executable, unless you have changed the model itself. If you get really stuck you can send a message to the mailing list "help-mcsim@prep.ai.mit.edu" (see section Installation) or to the authors of this manual.

The program ends (if everything is fine) by giving you the name of the output file generated. If you want to run the program in batch mode (in the background), you may want to redirect the screen output and error messages; refer for this to the `man` pages for your shell.

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## 6.2 Syntax of the simulation definition file

A simulation specification file is a text (ASCII) file that consists of several sections, starting with global specifications and assignments (valid throughout the file), followed by a number of `Simulation` sections (see section `Simulation` sections), eventually enclosed in `Level` sections. (The keyword `Experiment` is now obsolete but can still be used as a synonym for `Simulation`.)

Each `Simulation` section defines simulation conditions, from an initial time (or whatever the dependent variable represents, see section Model types) to a final time. Initial values of the model state variables, parameter values, input variables time-course, and which outputs are to be printed at which times, can all be changed in a given `Simulation` section.

In simple cases, the general layout of the file is therefore (see also the sample file in ‘`perc.lsodes.in`’):

    # Input file (text after # are comments)
    <Global assignments and specifications>
    Simulation {
      <Specifications for first simulation>
    }
    Simulation {
      <Specifications for second simulation>
    }
    # Unlimited number of simulation specifications
    End. # Mandatory End keyword. Everything after this line is ignored

For Markov chain Monte Carlo simulations (see section `MCMC()` specification), the general layout of the file must include `Level` sections. `Level` sections are used to define a hierarchy of statistical dependencies (see section Setting-up statistical models). In that case, the general layout of the file is:

    # Input file
    <Global assignments and specifications>
    Level {
      # Up to 10 levels of hierarchy
      Simulation {
        <Specifications and data for first simulation>
      }
      Simulation {
        <Specifications and data for second simulation>
      }
      # Unlimited number of simulation specifications
    } # end Level
    End. # Mandatory keyword.

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### 6.2.1 General input file syntax

The general syntax of the file is the same as that of structural model definition files (see section General syntax) except that:

- No new variable can be created (all variables must have been defined in the model definition file). Assignments can only modify the value of already defined model variables. This implies that parameters needed to set up a statistical model must be declared in the model definition file, even if the structural model does not use them.
- Assignments to state variables or parameters can only use constant values; mathematical expressions are not allowed.
- Input variables’ assignments can use any input function (including the `NDoses()` and `Spikes()` functions), an `Events()` specification or a constant values.
- Output variables cannot receive assignments.

At the program start, all model parameters are initialized to the nominal values specified in the model description file. Next, after the input file is read, modifications given in its global section (including random sampling) are applied, then those specified at each `Level`, and finally any modifications specified by the `Simulation` sections. Computations specified in the `Initialize` section of the model definition file are the last initialization statements executed.

Structural changes to the model (e.g., addition of a state, input, output or parameter) cannot be done here and must be done in the model description file. The simulation specification file is read until a mandatory `End` keyword is reached.

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### 6.2.2 Input functions (revisited)

Input variables can be assigned all the input functions defined previously (see section Input functions). Briefly, these are:

- `PerDose()`:

      PerDose(<magnitude>, <period>, <initial-time>, <exposure-time>);

- `PerExp()`:

      PerExp(<magnitude>, <period>, <initial-time>, <decay-constant>);

- `NDoses()`:

      NDoses(<n>, <list-of-magnitudes>, <list-of-initial-times>);

- `Spikes()`:

      Spikes(<n>, <list-of-magnitudes>, <list-of-times>);

In addition, they can be assigned a `Events()` specification (see section `Events()` specification for state discontinuities).

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### 6.2.3 Global specifications

In the global section you can modify, by assignment, the value of already defined state or input model variables or parameters (you cannot assign a value to an output variable). These assignments will be in effect throughout the input file, unless they are overridden later in the file. Here is an exemple of assignment (assuming that `x` and `Pi` have been properly defined in the model definition file):

    x = 10; # set the initial value if x is a state variable
    Pi = 3; # to stop worrying about little decimals...

In the global section, you can also give specifications relevant to all `Simulation` or `Level` sections. These specifications are not needed if you just want to perform simple simulations. They should also not appear inside `Simulation` or `Level` sections (with the notable exception of `Distrib()` specifications which can appear inside `Level` sections). They are used to call for and define the parameters of special computations (e.g., the number of Monte Carlo simulations to run, which sampling distributions to use for a given parameter, the data likelihood, etc.) These specifications are the following:

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#### `OutputFile()` specification

The `OutputFile()` specification allows you to specify a name for the output file of basic simulations. If this specification is not given the name ‘`sim.out`’ is used if none has been supplied on the command-line or during the initial dialog. The corresponding syntax is:

    OutputFile("<OutputFilename>");

where the character string <OutputFilename>, enclosed in double quotes, should be a valid file name for your operating system.

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#### `Integrate()` specification

The integrator settings can be changed with the `Integrate` specification. Two integration routines are provided: `Lsodes` (which originates from the SLAC Fortran library and is originally based on Gear’s routine) (Gear, 1971b; Gear, 1971a; Press et al., 1989) (see section Bibliographic References) and `Euler` (Press et al., 1989).

The syntax for `Lsodes` is:

    Integrate(Lsodes, <rtol>, <atol>, <method>);

where <rtol> is a scalar specifying the relative error tolerance for each integration step. The scalar <atol> specifies the absolute error tolerance parameter. Those tolerances are used for all state variables. The estimated local error for a state variable y is controlled so as to be roughly less (in magnitude) than rtol*|y| + atol. Thus the local error test passes if, for each state variable, either the absolute error is less than <atol>, or the relative error is less than <rtol>. Set <rtol> to zero for pure absolute error control, and set <atol> to zero for pure relative error control. Caution: actual (global) errors may exceed these local tolerances, so choose them conservatively.

The <method> flag should be 0 (zero) for non-stiff differential systems and 1 or 2 for stiff systems. If you specify <method> 2 you should provide the Jacobian of your differential system (see the “Perc” model in the example folder); otherwise the Jacobian will be computed by numerical differentiation (which is about as fast if the Jacobian is dense). You should try flag 0 or 1 and select the fastest for equal accuracy of output, unless insight from your system leads you to choose one of them a priori. In our experience, a good starting point for <atol> and <rtol> is about 1e-6.

The syntax for `Euler` is:

    Integrate(Euler, <time-step>, 0, 0);

where <time-step> is a scalar specifying the constant time increment for each integration step. The next two scalars are reserved for future use and should be set to zero.

Note: if the `Integrate()` specification is not used, the default integration method is `Lsodes` with parameters 1e-5, 1e-7 and 1. We recommend using `Lsodes`, since is it highly accurate and efficient. `Euler` can be used for special applications (e.g., in system dynamics) where a constant time step and a simple algorithm are needed.

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#### `MonteCarlo()` specification

Monte Carlo simulations (Hammersley and Handscomb, 1964; Manteufel, 1996) (see section Bibliographic References) randomly sample parameter values and run the model for each parameter set so generated. The statistical distribution of the model outputs can be studied for uncertainty analysis, sensitivity analysis etc. Such simulations require the use of two specifications, `MonteCarlo()` and `Distrib()`, which must appear in the global section of the file, before the `Simulation` sections. They are ignored if they appear inside a `Simulation` section.

The `MonteCarlo` specification gives general information required for the runs: the output file name, the number of runs to perform, and a starting seed for the random number generator. Its syntax is:

    MonteCarlo("<OutputFilename>", <nRuns>, <RandomSeed>);

The character string <OutputFilename>, enclosed in double quotes, should be a valid filename for your operating system. If a null-string "" is given, the default name ‘`simmc.out`’ will be used. The number of runs <nRuns> should be an integer, and is only limited by either your storage space for the output file or the largest (long) integer available on your machine. The seed <RandomSeed> of the pseudo-random number generator can be any positive real number. Seeds between 1.0 and 2147483646.0 are used as is, the others are rescaled within those bounds (and a warning is issued). Here is an example of use:

    MonteCarlo("percsimmc.out", 50000, 9386.630);

The parameters’ sampling distributions are specified by a list of `Distrib()` specifications, as explained in the following (see section `Distrib()` specification). The format of the output file of Monte Carlo simulations is discussed later (see section Analyzing simulation output).

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#### `MCMC()` specification

Markov chain Monte Carlo (MCMC) can be defined as stochastic simulations following a Markov chain in a given parameter space. In MCMC simulations, the random choice of a new parameter value is influenced by the current value. They can be used to obtain a sample of parameter values from complex distribution functions, eventually intractable analytically. Such complex distribution functions are typically encountered during Bayesian data analysis, under the guise of posterior distributions of a model’s parameters. The reader wishing to use the MCMC capabilities of GNU MCSim is referred to the published literature (for example, Bernardo and Smith, 1994; Gelman, 1992; Gelman et al., 1995; Gelman et al., 1996; Gilks et al., 1996; Smith, 1991; Smith and Roberts, 1993) (see section Bibliographic References).

MCMC simulation chains (which in GNU MCSim start from a sample from the specified prior) need to reach "equilibrium". Checking that equilibrium is obtained is best achieved, in our opinion, by running multiple independent chains (cf. Gelman and Rubin, 1992, and other relevant statistical literature). GNU MCSim does not deal (yet) with convergence issues.

The Bayesian analysis of data with GNU MCSim requires you to setup:

Setting-up a statistical model requires `Level` sections and `Data()` specifications. Assigning priors and likelihoods is achieved through the `Distrib()` statements (or its equivalents `Density()` and `Likelihood()`). Please refer to the corresponding sections of this manual, if you are not familiar with them. The `MCMC()` statement, gives general directives for MCMC simulations and has the following syntax:

    MCMC("<OutputFilename>", "<RestartFilename>", "<DataFilename>",
         <nRuns>, <simTypeFlag>, <printFrequency>, <itersToPrint>,
         <RandomSeed>);

The character strings <OutputFilename>, <RestartFilename>, and <DataFilename>, enclosed in double quotes, should be valid file names for your operating system. If a null-string "" is given instead of the output file name, the default name ‘`MCMC.default.out`’ will be used.

If a restart file name is given, the first simulations will be read from that file (which must be a text file). This allows you to continue a simulated Markov chain where you left it, since an MCMC output file can be used as a restart file with no change. Note that the first line of the file (which typically contains column headers) is skipped. Also, the number of lines in the file must be less than or equal to <nRuns>. The first column of the file should be integers, and the following columns (tab- or space-separated) should give the various parameters, in the same order as specified in the list of `Distrib()` specifications in the input file.

If a data file name is given, the observed (data) values for the simulated outputs will be read from that file (in ASCII format); otherwise, `Data()` specifications (see section `Data()` specification) should be provided. We recommend that you use `Data()` specifications rather that the data file, which is much more error prone. The first line of the data file is skipped and can be used for comments. The total number of data points should equal the total number of outputs requested. The data values should be given on the second and following lines, separated by white spaces or tabs. A data value of "-1" will be treated as "missing data" and ignored in likelihood calculations. The convention "-1" can be changed by changing INPUT_MISSING_VALUE in the header file ‘`mc.h`’ and recompiling the entire library.

The integer <nRuns> gives the total number of runs to be performed, including the runs eventually read in the restart file.

The next field, <simTypeFlag> should be between 0 and 4 (included):

- It should be set at zero to start a chain of MCMC simulations. In that case, parameters are updated by Metropolis steps, one at a time.
- If <simTypeFlag> = 1, a restart file must also be specified. The output file will contain codes for the level sequence, simulation numbers, printing times, data values and the corresponding model predictions, computed using the last parameter vector of the restart file. This is useful to quickly check the model fit to the data.
- If <simTypeFlag> is equal to 2, a restart file must also be specified and that entire file is used to compute the parameters’ covariance matrix. All parameters are then updated at once using a multivariate normal kernel as proposal distribution of the Metropolis steps. This may result in large improvement in speed. However, we recommend that this option be used only when convergence is approximately obtained (therefore, you should run MCMC simulations with <simTypeFlag> set to 0 first, up to approximate convergence, and then restart the chain with the flag at 2).
- With <simTypeFlag> equal to 3 ’tempered’ component by component MCMC is performed MCMC (Geyer and Thompson, 1995) (see section Bibliographic References). In that case a grid of inverse temperatures is used (along an annealing scheme). By default 5 inverse temperatures are used (0.4, 0.512, 0.64, 0.8, 1), but you can define your own scale with the `InvTemperature()` specification, with the following syntax:

      InvTemperature(<nValues>, <value 1>, <...>, <value n>);

  The first number of the specification gives the number of inverse temperatures to use and is followed by a list of them. Those should be numbers in the interval ]0,1]. You usually would want to end with a value of 1 (since temperature = 1 is the cold end of the temperature scale, corresponding to your target distribution.)

- Finally with <simTypeFlag> set to 4, GNU MCSim does stochastic optimization (an MCMC in which only, but all, jumps leading posterior density improvement are accepted).

The integer <printFrequency> should be set to 1 if you want an output at each iteration, to 2 if you want an output at every other iteration etc. The parameter <itersToPrint> is the number of final iterations for which output is required (e.g., 1000 will request output for the last 1000 iterations; to print all iterations just set this parameter to the value of <nRuns>). Note that if no restart file is used, the first iteration is always printed, regardless of the value of <itersToPrint>. Finally, the seed <RandomSeed> of the pseudo-random number generator can be any positive real number. Seeds between 1.0 and 2147483646.0 are used as is, others are rescaled silently within those bounds.

In the case of component by component jumps (<simTypeFlag> set to 0), tempered or stochastic optimization (<simTypeFlag> set to 3 or higher), the jump kernel is saved with the same name as the output file, with a .kernel extension. If the simulations are restarted in a continuation mode and if the kernel file is present that file will be read so that the jump kernel is restored.

Finally, the format of the output file of MCMC simulations is quite similar to that of straight Monte Carlo simulations and will discussed in a later section (see section Analyzing simulation output).

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#### `SetPoints()` specification

To impose a series of set points (i.e., already tabulated values for the parameters), the global section can include a `SetPoints()` specification. It allows you to perform additional simulations with previously Monte Carlo sampled parameter values, eventually filtered. You can also generate parameters values in a systematic fashion, over a grid for example, with another program, and use them as input to GNU MCSim. Importance sampling, Latin hypercube sampling, grid sampling, can be accommodated in this way.

This command specifies an output filename, the name of a text file containing the chosen parameter values, the number of simulations to perform and a list of model parameters to read in. It has the following syntax:

    SetPoints("<OutputFilename>", "<SetPointsFilename>", <nRuns>,
              <parameter identifier>, <parameter identifier>, ...);

If a null string is given for the output filename, the set points output will be written to the same default output file used for Monte Carlo analyses, ‘`simmc.out`’.

The SetPointsFilename is required and must refer to an existing file containing the parameter values to use. The first line of the set points file is skipped and can contain column headers, for example. Each of the other lines should contain an integer (e.g., the line number) followed by values of the various parameters in the order indicated in the `SetPoints()` specification. If extra fields are at the end of each line they are skipped. The first integer field is needed but not used (this allows you to directly use Monte Carlo output files for additional `SetPoints` simulations).

The variable <nRuns> should be less or equal to the number of lines (minus one) in the set points file. If a zero is given, all lines of the file are read. Finally, a comma-separated list of the parameters to be read in the SetPointsFilename is given. The format of the output file of set points simulations is discussed below (see section Analyzing simulation output).

Following the `SetPoints()` specification, `Distrib()` statements can be given for parameters not already in the list (see section `Distrib()` specification). These parameters will be sampled accordingly before to performing each simulation. The shape parameters of the distribution specifications can reference other parameters, including those of the list.

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#### `OptimalDesign()` specification

The "OptimalDesign" procedure optimizes the number and location of observation times for experimental conditions you specify, in order to minimize the variance of a parameter or an output you designate. It requires a structural model (see section Setting-up Structural Models), a statistical model in the form of a `likelihood()` function (see section Setting-up statistical models), and a random set of parameter vectors sampled from a prior distribution (using Monte Carlo or MCMC simulations) (for example and details, see Bois et al., 1999) (see section Bibliographic References). The statistical model used should be quite simple and cannot not use `Level` sections (and hence cannot be hierarchical).

The OptimalDesign command has the following syntax:

    OptimalDesign("<OutputFilename>", "<ParameterSampleFilename>",
                  <nSamples>, <RandomSeed>, <Style>,
                  <parameter identifier>, <parameter identifier>, ...);

The character strings <OutputFilename>, and <ParameterSampleFilename>, enclosed in double quotes, should be valid file names for your operating system. If a null-string "" is given instead of the output file name, the default name ‘`simopt.default.out`’ will be used.

A parameter sample file name must be given (that file must be a text file). The first line of the file (which typically contains column headers) is skipped. The number of lines in the file must be less than or equal to <nSamples>. The first column of the file should be integers (typically row numbers), and the following columns (tab- or space-separated) should be values of the various parameters in the order indicated in the list at the end of the `OptimalDesign()` specification. If extra fields are at the end of each line they are skipped. The first integer field is needed but not used (this allows you to directly use Monte Carlo output files for `OptimalDesign` simulations).

The integer <nSamples> indicates the number of lines to read from the <ParameterSampleFilename> file. The seed <RandomSeed> of the pseudo-random number generator can be any positive real number. Seeds between 1.0 and 2147483646.0 are used as is, others are rescaled silently within those bounds. The directive Style should be either the keyword `Forward` or the keyword `Backward`. Forward optimization will start from no new data and will add, sequentially, optimal observation times. Backward optimization starts with the full set of observation times you propose and delete the least informative ones, sequentially. We recommand that you try both options. Finally, a comma-separated list of the parameters to be read in the ParameterSamplFilename should be given.

The input file must then contain two sets of `Simulation` definitions. You should look at the sample optimal design files provided in ‘`mcsim/samples`’.

The first set specifies all experimental conditions and the set of observation times to optimize, for one or several output variables given in `Print` statements. The output times you specify for each output variable define an array of observation time values that the optimization algorithm will rank by order of the estimated variance reduction they permit for variables or parameters you will specify in the second set of `Simulation` definitions. Data will be simulated for each of the required output. There must be one Data statement per output specified (the data values are arbitrary). An error model must be specified for those data, using a `Likelihood` statement (see section `Distrib()` specification).

The second set of `Simulation` specifies optimization target parameters or outputs. The algorithm will select time-points (in the first section’s `Simulation` specifications) that minimize the estimation variance of those parameters or outputs. When a parameter is targeted no inputs are needed. If you optimize for an output variable variance (i.e., for the variance of a model prediction), the experimental conditions can be very different from those of the experiment whose conditions you optimize. The link is afforded solely by the parameters (in the first set you are trying to determine the conditions that will optimally identify the parameter values conditioning the predictions – or trivially, the parameters – of the second set)

The format of the output file of design optimization simulations is quite specific. The first column is an iteration number. At each iteration one observation point is added (`Forward` mode) or removed (`Backward` mode). Each step is therefore conditioned by the selection of an observation time-point made by the previous step. The following columns give, for each observation time point you specify, the average variance of the target outputs or parameters achieved if this point is added (`Forward` mode) or removed (`Backward` mode). Next the chosen time point at this step is given (the one minimizing average variance), followed by the variance it leads to (in expectation) and the corresponding standard deviation. The last column "Utility" is zero, unless you uncomment the function `Compute_utility` and modify its code in ‘`optdesign.c`’ to compute a utility of your own.

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#### `Distrib()` specification

The specification of distributions for simple Monte Carlo simulations is quite straighforward. MCMC simulations require the definition of a full statistical model and the use of distributions is somewhat more complex in that case, but the use of `Distrib()` is basically the same.

In the context of `MonteCarlo()` or `SetPoints()` simulations (see section `MonteCarlo()` specification, and `SetPoints()` specification), one (and only one) `Distrib()` specification must be included for each model parameter to randomly sample. State, input or output variables cannot be randomly sampled by `Distrib()` in this context. A simulation specification file can include any number of `Distrib()` commands at the global level.

`Distrib()` specifies the name of the parameter to sample, and its sampling distribution. Its syntax is:

    Distrib(<parameter identifier>, <distribution-name>, [<shape parameters>]);

The <parameter identifier> gives the name of the parameter to sample. The <distribution-name> and the corresponding <shape parameters> indicate the sampling distribution to use (Bernardo and Smith, 1994; Gelman et al., 1995) (see section Bibliographic References). They are specified as follow:

- `Beta`, takes at least two strictly positive real shape parameters: A and B. By default the Beta distribution is defined over the interval [0;1]. If a range is given for the beta distribution, the [0;1] interval is mapped onto the specified range.
- `Binomial`, needs two strictly positive numbers: the probability p (a real in the interval [0;1]), and the sample size N, an integer. If N is not given as an integer it will be rounded down during the computations.
- `Cauchy`, takes one strictly positive real number as parameter: its scale s.
- `Chi2`, takes one strictly positive real number as parameter: n. This distribution is the same as Gamma(n/2, 1/2).
- `Exponential`, uses one strictly positive real number: the inverse-scale b.
- `Gamma`, uses two strictly positive real parameter: the shape and the inverse scale.
- `HalfCauchy`, takes one strictly positive real number as parameter: the scale s. The mode is at zero, on the lower boundary. The random variates returned are strictly positive.
- `HalfNormal`, takes one reals number as parameter: the standard deviation, strictly positive. The mode is at zero, on the lower boundary. The random variates returned are strictly positive.
- `InvGamma` (inverse gamma distribution), needs two strictly positive real parameters: the shape and the scale.
- `LogNormal`, takes two reals numbers as parameters: the geometric mean (exponential of the mean in log-space) and the geometric standard deviation (exponential, strictly superior to 1, of the standard deviation in log-space).
- `LogNormal_v`, is the lognormal distribution with the variance (in log space!) instead of the standard deviation as second parameter. You can use it to specify a hierarchical model with a conjugate prior on the variance (see section Setting-up statistical models).
- `LogUniform`, with two shape parameters: the minimum and the maximum of the sampling range (real numbers) in natural space.
- `Normal`, takes two reals numbers as parameters: the mean and the standard deviation, the latter being strictly positive.
- `Normal_v`, is also the normal distribution with the variance instead of the standard deviation as second parameter. You can use it to specify a hierarchical model with a conjugate prior on the variance (see section Setting-up statistical models).
- `Piecewise`, uses four reals as parameters: the minimum, A, B, and the maximum. The distribution has the form of a truncated triangle, with a plateau between A and B. If A = B, the distribution is the triangular distribution.
- `Poisson`, needs a strictly positive real: the rate A.
- `StudentT`, requires three parameters: its number of degrees of freedom (an integer), its mean, and its standard deviation.
- `TruncInvGamma` (truncated inverse gamma distribution), needs four strictly positive real parameters: the shape, the scale, the minimum and the maximum.
- `TruncLogNormal` (truncated lognormal distribution), uses four real numbers: the geometric mean and geometric standard deviation (strictly superior to 1), the minimum and the maximum in natural space. For example:

      Distrib(Var, TruncLogNormal, 1, 2.718, 0.01, 10)

  samples ‘`Var`’ such that log(‘`Var`’) is a standardized normal variate of mean log(1) and standard deviation log(2.718) - while ‘`Var`’ is truncated to fall between 0.01 to 10.

- `TruncLogNormal_v`, is like the truncated lognormal, except that it takes the variance (in log space!) instead of the standard deviation as second parameter. You can use it to specify a hierarchical model with a conjugate prior on the variance (see section Setting-up statistical models).
- `TruncNormal` (truncated normal distribution), takes four real parameters: the mean, the standard deviation (strictly positive), the minimum and the maximum.
- `TruncNormal_v`, is like the truncated normal distribution with the variance instead of the standard deviation as second parameter.
- `Uniform`, with two shape parameters: the minimum and the maximum of the sampling range (real numbers).

The shape parameters of the above distributions can symbolically reference other model parameters, even if distributions for these have already been defined. For example:

    Distrib(A, Normal, 0, 1);
    Distrib(B, Normal, A, 2);

In the context of MCMC sampling, GNU MCSim provides extensions of the above `Distrib()` specification syntax.

First, when `Distrib()` is used to specify the distribution of a model parameter, that parameter can also appear as a shape parameter, if a distribution has already been specified for the parameter at an upper `Level` of the file. For example:

    Level {                      # upper level
      Distrib(A, Normal,   0, 1);
      Distrib(B, InvGamma, 2, 2);
      Level {                    # sub-level
        Distrib(A, Normal_v, A, B);
        ...
      }                          # end sub-level
    }                            # end upper level

In that case, the parameter A, used for shape specification (as the mean of a Normal distribution) in the sub-level, refers to the "parent" A parameter, for which a standard Normal distribution is defined at the upper `Level`. The sampled values of the parent parameters A and B will be used as mean and variance for their "child" parameter, A, when it will be its turn to be randomly sampled. This forms the basis of the specification of multilevel (hierarchical) models (see section Setting-up statistical models).

Next, in MCMC simulations, you usually assign a probability distribution (or a likelihood) to the data you are trying to analyze. Typically, your model’s state and/or output variables will attempt to predict some aspect of the observed data distributions (mean, variance, etc.). GNU MCSim gives you the possibility to specify a distribution for your data, using model parameters, input, state, or output model variables, or even other data, to define the distribution shape. This is achieved through the use of the `Data()` and `Prediction()` "qualifiers".

`Data()` can be used at the first position of a `Distrib()` statement, or as a distribution shape parameter. It uses the following syntax:

    Data(<identifier>)

where <identifier> corresponds to a valid input, state or output model variable for which data are available. Model parameters cannot be used (but you can assign a simple parameter value to an output variable in your model definition file and use that output here). The actual data values need to be given later in the simulation input file through `Data()` specifications (which, in addition to a variable identifier, give a list of numerical data values, see `Data()` specification) or in a separate datafile (see section `MCMC()` specification).

Working hand in hand with `Data()`, and using the same syntax, the `Prediction()` qualifier can be used to designate actual model inputs, states and outputs for any shape parameter of a specified distribution (therefore `Prediction()` must appear after the distribution name). The actual predicted values, matching exactly the corresponding data, need to be given later in the simulation input file through `Print()` or `PrintStep()` specifications [see section `Print()` specification and `PrintStep()` specification).

Here are some example of use of `Data()` and `Prediction()` in the extended syntax of a `Distrib()` specification:

    Distrib (Data(y), Normal,   Prediction(y), 0.01);
    ...
    Data (y, 0.1, 2,  5,  3,  9.2);
    Print(y, 10,  20, 40, 60, 100);

    Distrib (Data(y), Normal,   Prediction(y), Prediction(sigma));
    ...
    Data (y, 1.01, 1.20, 0.97, 0.80, 1.02);
    PrintStep(y,     10, 50, 10);
    PrintStep(sigma, 10, 50, 10);

    Distrib (Data(R), Binomial, Prediction(P), Data(N));
    ...
    Data (R, 0,  2,  5, 5,  8, 9, 10, 10);
    Data (N, 10, 10, 9, 10, 9, 9, 11, 10);
    Print(P, 10, 20, 30,40, 50,60,70, 80);

(these could not appear all as such in an input file, they would need to be embedded in `Level` and `Simulation` sections.)

Last, for more readable input files, two keywords, `Density()` and `Likelihood()`, can be used instead of `Distrib()`. They are equivalent to `Distrib()` and have the same syntax.

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#### `SimType()` specification

This specification is now obsolete and should not be used. It is left for compatibility with old input files. It specifies the type of analysis to perform. Syntax:

    SimType(<keyword>);

The following keywords can be used: `DefaultSim` (the list of specified simulations is simulated), `MonteCarlo`, `MCMC` (previously `Gibbs`), `SetPoints`. If `MonteCarlo`, `MCMC`, or `SetPoints` analyses are requested, additional specifications are needed (see below).

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### 6.2.4 Specifying basic conditions to simulate

Any simulation file must define at least one `Simulation` section. `Simulation` sections include particular specifications, which are presented in the following.

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#### `Simulation` sections

After global specifications, if any, `Simulation` sections must be included in the input file. Expectedly, these sections start with the keyword `Simulation` and are enclosed in curly braces.

A `Simulation` section can make assignments to any state variable, input variable or parameter defined in the global section of the model description file. Output variables cannot receive assignments in simulation input files.

State variables and parameters can only take constant values (see section General input file syntax). For state variables, this sets the initial value only. So, for example, in a `Simulation` section the parameter `Speed`, if properly defined, can be set using:

    Speed = 83.2;

This overrides any previously assigned values, even if randomly sampled, for the specified parameter.

Inputs can be redefined with input functions (see section Input functions (revisited)) or constant values. Input functions can reference other variables (eventually randomly sampled), as in:

    Q_in = PerExp(InMag, 60, 0, RateConst);

The maximum number of `Simulation` sections allowed in an input file is 200. This can be changed by changing MAX_INSTANCES and MAX_EXPERIMENTS in the header file ‘`sim.h`’ and recompiling the program (this requires re-installation).

Within a `Simulation` section, several additional specifications can be used:

- `StartTime()`,
- `Print()`,
- `PrintStep()`,
- `Data()`.

The `Data()` specification is used only when a statistical model is set up and will be covered in the corresponding section of this manual (Setting-up statistical models).

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#### `Events()` specification for state discontinuities

You can impose state variables discontinuities at a given set of times with the `Events` specification. `Events` has a syntax similar to the `NDoses` input function (see section Input functions). It is in fact a special type of input function that resets a state variable, hence the need to assign it to a (dummy) input variable. Its syntax is:

    <input variable> = Events(<state variable>, <n>, <list-of-times>, <list-of-operation>, <list-of-scalars>);

The first argument, <state variable>, is the state whose value you want to reset at given times. The integer <n> is the number of reset times you want to specify; <list-of-times> is a comma separated list of those reset times; <list-of-operations> is a comma separated list of operations that will affect the given state variable at the specified times (see next paragraph), and <list-of-scalar> is the list of floating point values used for by the operations specified, at the corresponding times.

The three keywords operations are `Add`, `Multiply` and `Replace`. `Add` adds the corresponding scalar to the target state variable at the specified time; `Multiply` multiplies the state variable by the specified scalar; `Replace` simply replaces the value of the state variable by the given scalar.

The assigned input value takes the value 1 at the specified times and is zero otherwise. If you don’t have a use for such an input, simply define a dummy input variable. For example:

In the model definition file define:

    Inputs = {events_v1, events_v2};

and in the simulation specification file you can request, within a `Simulation` specification:

    events_v1 = Events (v1, 2, 1,   9,
                               Add, Add,
                               1,   4);

    events_v2 = Events (v2, 2, 1,        5,
                               Multiply, Replace,
                               2,        6);

At time 1, a value of 1 will be added to the state variable v1 and v2 will be multiplied by 2; At time 5 v2 will be reset to 5, and at time 9 the value 4 will be added to v1.

See (and run) the example model and input file provided in the example/events folder.

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#### `StartTime()` specification

The origin of time for a simulation, if it needs to be defined, can be set with the `StartTime()` specification, whose syntax is:

    StartTime(<initial-time>);

It just shifts the time scale. If this specification is not given, a value of zero is used by default. A parameter can be used as initital time value, so that initial time can be sampled in error-in-variable models, for example:

    T0 = 20;
    StartTime(T0);

The final time is automatically computed to match the largest output time specified in the `Print()` or `PrintStep()` statements. Output times cannot be inferior to the initial time.

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#### `Print()` specification

The value of any model variable or parameter can be requested for output with `Print()` specifications. Their arguments are a comma-separated list of variable names (at least one and up to MAX_PRINT_VARS, which is set to 10 by default), and a comma-separated list of increasing times at which to output their values:

    Print(<identifier1>, <identifier2>, ..., <time1>, <time2>, ...);

where <identifier1>, <identifier2> etc. correspond to valid input, state or output model variables. Model parameters cannot be used (but you can assign a parameter value to an output variable in your model definition file and use that output here).

The same output times are used for all the variables specified. The size of the time list is only limited by the available memory at run time. The limit of 10 variables names can be increased by changing MAX_PRINT_VARS in the header file ‘`sim.h`’ and re-installing the whole software. The number of `Print()` statements you can used in a given `Simulation` section is only limited by the available memory at run time. The same variable or parameter can appear in more than one `PrintStep()` in a given `Simulation` section.

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#### `PrintStep()` specification

The value of any model variable can be also output with `PrintStep()` specifications. They allow dense printing, suitable for smooth plots, for example. Their arguments are: a comma-separated list of variable names (at least one and up to MAX_PRINT_VARS, which is set to 10 by default), the first output time, the last one, and a time increment:

    PrintStep(<identifier1>, <identifier2>, ...,, <start-time>, <end-time>, <time-step>);

The final time has to be superior to the initial time and the time step has to be less than the time span between end and start. If the time step is not an exact divider of the time span the last printing step is shorter and the last output time is still the end-time specified. The number of outputs produced is only limited by the memory available at run time. You can use several `PrintStep()`, and the same variable or parameter can appear in more than one `PrintStep()`, in a given `Simulation` section.

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### 6.2.5 Setting-up statistical models

With GNU MCSim, you must define a statistical model to use the `MCMC()` specification. `MCMC` simulations will give you a sample from the joint posterior distribution of the parameters that you designate as randomly sampled through `Distrib()` specifications. You do not need to specify explicitly that joint posterior distribution (in fact, in most case, this is impossible). The posterior distribution is implicitly defined by a statistical model, that is simply a set of conditional relationship between the parameters and some data.

GNU MCSim handles multilevel (hierarchical) random effects and mixed effects statistical models in a Bayesian framework. These models need to be defined in the simulation specification file, rather than in the structural model definition file. Yet, due to compilation constraints, if you need special parameters for your statistical model (e.g., variances) you have to declare them in the structural model file, even if they are not used by the structural model itself.

So, how do we go about specifying a statistical model with GNU MCSim? Take for example the following simple linear regression model:

y_i = N(Mu_i, Sigma^2), Mu_i = Alpha + Beta * (x_i - x_bar).

where the observed (x,y) pairs are (1,1), (2,3), (3,3), (4,3) and (5,5). Assume that the parameters Alpha and Beta are given N(0,10000) priors, and that 1/Sigma^2 is given a Gamma(1e-2,1e-2) prior. x_bar is the average of the above values for x. We want the posterior distributions of Alpha, Beta, and Sigma^2.

The first thing to do is to define a structural (or link) model to compute y as a function of x. Here is such a model (quite similar to the one distributed with GNU MCSim source code (see section ‘`linear.model`’):

    # ---------------------------------------------
    # Model definition file for a linear model
    # ---------------------------------------------
    Outputs = {y};

    # Structural model parameters
    Alpha = 0;
    Beta = 0;
    x_bar = 0;

    # Statistical parameter
    Sigma2 = 1;

    CalcOutputs { y = Alpha + Beta * (t - x_bar); }
    # ---------------------------------------------

The parameters’ default values are arbitrary, and could be anything reasonable. They will be changed or sampled through the input file. Note that Sigma2 is not used in the model equations, but still needs to be defined here in order to be part of the statistical model. On the other hand, Mu is not defined, since we do not really need it. Finally x is replaced by the time, `t`, for convenience. An alternative would be to define an input ‘`x`’ and use it instead of `t`.

We now need to write an input file specifying the distribution of y (i.e., the likelihood), and the prior distributions of the various parameters. Technically, GNU MCSim uses Metropolis sampling and you do not need to worry about issues of conjugacy or log-concavity of your prior or posterior distributions. Here is what a simulation file with a statistical model looks like:

    # ---------------------------------------------------------------
    # Simulation input file for a linear regression
    # ---------------------------------------------------------------
    MCMC ("linear.MCMC.out", "", "", 50000, 0, 5, 40000, 63453.1961);
    Level {
      Distrib(Alpha,  Normal_v, 0, 10000);
      Distrib(Beta,   Normal_v, 0, 10000);
      Distrib(Sigma2, InvGamma, 0.01, 0.01);
      Likelihood(Data(y), Normal_v, Prediction(y), Sigma2);
      Simulation {
        x_bar = 3.0;
        PrintStep (y, 1, 5, 1);
        Data (y, 1, 3, 3, 3, 5);
      }
    } # end Level
    End.
    # ---------------------------------------------------------------

The file begins with `MCMC()` (see section `MCMC()` specification). The keyword `Level` comes next. `Level` is used to specify hierarchical dependences between model parameters. There should be at least one `Level` in every MCMC input file, even for a non-hierarchical model like the one above. See below for further discussion of the `Level` keyword. You can also look at the MCMC input files provided as examples with GNU MCSim source code. The `Distrib()` statements define the parameter priors. `Normal_v` specifications are used since we use variances instead of standard deviations. The inverse-Gamma distribution is used for the variance component, since the precision is supposed to be Gamma-distributed. The likelihood is the distribution of the data, given the model: it is specified by a `Likelihood()` specification, valid for every y data point. Again, note that the Mu variable is not used. Instead, the `Prediction(y)` specification designates the linear model output. The distributions and likelihoods specified are in effect for every sub-level or every `Simulation` section included in the current `Level`.

The "simulations" to perform, and the corresponding data values, are specified by a `Simulation` section. Only one `Simulation` section is needed here, but several could be specified. In this section, the value of x_bar is provided. The different values of x (time in our formulation of the model) can be specified via `PrintStep()` (see section `PrintStep()` specification), since they are equally spaced. More generally, `Print()` can also be used (see section `Print()` specification). The data values are given in a `Data()` statement (see below).

The following paragraphs deal with `Level` sections and `Data()` specifications.

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#### `Level` sections

Markov chain Monte Carlo simulations require the definition of a statistical model structured with "levels". Think for example of the definition of a prior distribution as a top level in a hierarchy, with the data likelihood being at the lowest level. The hierarchy levels are defined in GNU MCSim with the help of `Level` sections. At least one `Level` section must be defined in the simulation input file (you cannot use `Level` in a structural model definition file). A `Level` section starts with the corresponding keyword and is enclosed in curly braces (’{}’). It can include any number of sub-levels or `Simulations` sections. `Simulations` (where the data are specified) form the lowest level of the hierarchy (see section `Simulation` sections). In terms of structure, `Simulation` sections behave like `Level` sections (in particular with regard to "cloning" of random variables, see below) except that they cannot include further levels. There must be one and only one top `Level` and at most 10 nested sub-levels in the hierarchy. This limit of 10 can be increased (up to 255) by changing MAX_LEVELS in the header file ‘`sim.h`’ and re-installing GNU MCSim.

A `Level` can specify or change the sampling distribution of any model parameter properly defined in the global section of the structural model description file. These distribution specifications apply to all sub-levels of the `Level` where they take place. For example:

    MCMC("samp.out", "", "", 1, 1, 1, 1, 1); # we are in an MCMC context
    Level { # this is the top level
      Distrib(A, Uniform, 0, 1);
      Likelihood(Data(y), Normal, Prediction(y), 1);
      Level { # sub-level 1
        Distrib(A, Normal, A, 1);
        Simulation { ... } # simulation 1
        Simulation { ... } # simulation 2
      } # End sub-level 1
    } # End top, end file
    End.

A `Level` can also make simple assignments to any model parameter (see section General input file syntax). So, for example, in an simulation, the parameter A could be modified with:

    A = 2.0;

This overrides any previously assigned values for the specified parameter, even if randomly sampled, and applies to the sub-levels of the `Level` where it take place.

An important concept to grasp here is that of parameter "cloning". Cloning automatically creates, using templates, as many new parameters as you need in your multilevel model. One of the characteristic feature of multilevel models is the same parameters appear at several levels. For example, in a random effect model, a parameter (e.g., size) will be assumed to be randomly distributed in a population of individuals. If you have 100 individuals in your database, your model will have to deal with 100 individual size parameters and an average size. To spare you the tedium of defining the same distribution for many parameters, GNU MCSim creates an appropriate number of parameters for your model on the basis of its level structure. Assume that you have specified a distribution for a parameter A at a given level (that we label L1 for clarity). GNU MCSim will automatically create new parameters ("clones") with the same distribution as A to match the number of immediate sub-levels in L1. For example, if there are three sub-levels included in L1, GNU MCSim creates two clones to form a total of three instances of A (the original and its two clones). This convention saves a lot writing and effort in the long run.

In the sample of code given above, the parameter A, defined at the top level, will be simply moved to sub-level 1 (cloning is not necessary since there is only on sub-level directly included in the top level). Within sub-level 1, the normally-distributed A will be cloned once in order to create another normal variate with the same distribution. Each one of those two will be moved to a lower `Simulation`, where they will be conditioned by the data of that simulation only. A total of three variables of "type" A will be sampled and will be printed in the output file (coded so that the position in the hierarchy is apparent): the "parent" A(1), a priori uniformly distributed, and two "dependents" A(1.1) and A(1.2), a priori normally distributed around A(1).

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#### `Data()` specification

Experimental observations of model variables, inputs, outputs, or parameters, can be specified with the `Data()` command. Markov chain Monte Carlo sampling requires that you specify `Data()` statements (see section `MCMC()` specification; see section Setting-up statistical models). The data are then used internally to evaluate the likelihood function for the model. The arguments are the name of the variable for which observations exist, and a comma-separated list of data values:

    Data(<variable identifier>, <value1>, <value2>, ...);

This specification can only be used with a matching `Print()` or `PrintStep()` for the same variable (see section `Print()` specification; see section `PrintStep()` specification). You must make sure that there are as many data values in the `Data()` specification as output time requested in the corresponding `Print()` or `PrintStep()`. A data value of "-1" is treated as "missing data" and ignored in likelihood calculations. The convention "-1" can be changed by changing INPUT_MISSING_VALUE in the header file ‘`mc.h`’ and recompiling.

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## 6.3 Analyzing simulation output

The output from Monte Carlo or `SetPoints` simulations is a tab-delimited text file with one row for each run (i.e., parameter set) and one column for each parameter and output in the order specified. Thus each line of the output file is in the following order:

    <# of run> <parameters> <outputs for Exp 1> <outputs for Exp2> ...

The parameters are printed in the order they were sampled or set.

The first line gives the column headers. A variable called name requested for output in an simulation i at a time j is labeled name_i.j.

The output of Markov chain Monte Carlo simulations is also a text file with one row for each run. It displays a column of iteration labels, and one column for each parameter sampled. The last three columns contain respectively, the sum of the logarithms of each parameter’s density given its parents’ values (‘`LnPrior`’), the logarithm of the data likelihood (‘`LnData`’), and the sum of the previous two values (‘`LnPosterior`’). The first line gives the column headers. On this line, parameters names are tagged with a code identifying their position in the hierarchy defined by the Level sections. For example, the second instance of a parameter called name placed at the fist level of the hierarchy is labeled name(2); the first instance of the same parameter placed at the second instance of the second level of the hierarchy is labeled name(2.1), etc.

The tab-delimited file can easily be imported into your favorite spreadsheet, graphic or statistical package for further analysis.

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## 6.4 Error handling

If integration fails for a `imulation` in `DefaultSim` simulations no output is generated for that simulation, and the user is warned by an error message on the screen. In `MonteCarlo` or `SetPoints` simulations, the corresponding simulation line is not printed, but the iteration number is incremented. Finally, in MCMC simulations, the parameter for which the data likelihood was computed is simply not updated (which implicitly forbids the uncomputable region of the parameter space). In all cases an error message is given on the screen, or wherever the screen output has been redirected.

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# 7. Common Pitfalls

The following mistakes are particularly easy to make, and sometimes hard to notice, or understand at first.

- Forgetting about type-related arithmetics in C: ‘`1000/882`’ gives ‘`1`’ since it is interpreted as an integer division by the compiler. To get a floating-point (usual) division use ‘`1000./882.`’.
- Forgetting a semi-colon (’;’) at the end of statements: the error is usually detected at the following line(s) where in fact nothing may be wrong.

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# 8. XMCSim

XMCSim is a menu-driven interface which automatizes the compilation and running tasks of GNU MCSim. It also offers a convenient interface to 2-D and 3-D plotting of the simulation results. Note that you need `XWindows`, `Tcl/Tk` and `wish` installed to run XMCSim. `xemacs` is also recommended.

Just type `xmcsim` at the command promt. A windows appear, with a menu bar. Menu items are:

- `File`, which allows you to choose an existing model file or to exit the program. Once you have chosen a model file, its file name appears as a reminder at the bottom of the window.
- `Edit`, which calls `xemacs` for you to create a new model file or edit any file of your choice (for example an input or output file). Note: if you do not have `xemacs` installed you can change the file ‘`xmcsim`’ to replace the call to `xemacs` by a call to your editor.
- `Compile` has two items: `Compile model` will compile the current model file or prompt you for one and will call `mod` to generate a ‘`model.c`’ file from it; `Compile mcsim` will first call `mod` and will then go on to create an executable mcsim filevia a call to `makemcsim` create an executable program.
- `Run` with three items: `Run` which will prompt you for an executable mcsim file, an input file and an output file (the latter is optional) and will then launch the executable; `Stop` will just stop a running executable; `Debug` will produce a standalone executable with a name starting with ‘`debugmcsim`’ and will launch `xemacs` for you (you will then need to call `gdb` or another debugger by yourself; if you find a way to start gdb on an executable via xemacs on the command line please tell me...).
- `Plot` will start an Xgnuplot-based interface to `gnuplot` An `Help` menu available there to guide you further in the arcanes of `gnuplot`, but we recommend that you also browse `gnuplot` documentation.

At some point GNU MCSim may do symbolic computations, wash dishes, clothes and cars, and write poems, but for now, that’s all, folks!

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# Bibliographic References

Barry T.M. (1996). Recommendations on the testing and use of pseudo-random number generators used in Monte Carlo analysis for risk assessment. Risk Analysis 16:93-105.

Bernardo J.M. and Smith A.F.M. (1994). Bayesian Theory. Wiley, New York.

Bois F.Y., Gelman A., Jiang J., Maszle D., Zeise L. and Alexeef G. (1996). Population toxicokinetics of tetrachloroethylene. Archives of Toxicology 70:347-355.

Bois F.Y., Smith T.J., Gelman, A., Chang H.Y., Smith A.E. (1999). Optimal design for a study of butadiene toxicokinetics in humans. Toxicological Sciences 49:213-224.

Bois F.Y., Zeise L. and Tozer T.N. (1990). Precision and sensitivity analysis of pharmacokinetic models for cancer risk assessment: tetrachloroethylene in mice, rats and humans. Toxicology and Applied Pharmacology 102:300-315.

Gear C.W. (1971a). Algorithm 407 - DIFSUB for solution of ordinary differential equations [D2]. Communications of the ACM 14:185-190.

Gear C.W. (1971b). The automatic integration of ordinary differential equations. Communications of the ACM 14:176-179.

Gelman A. (1992). Iterative and non-iterative simulation algorithms. Computing Science and Statistics 24:433-438.

Gelman A., Bois F.Y. and Jiang J. (1996). Physiological pharmacokinetic analysis using population modeling and informative prior distributions. Journal of the American Statistical Association 91:1400-1412.

Gelman A., Carlin J.B., Stern H.S. and Rubin D.B. (1995). Bayesian Data Analysis. Chapman & Hall, London.

Gelman A. and Rubin D.B. (1992). Inference from iterative simulation using multiple sequences (with discussion). Statistical Science 7:457-511.

Geyer C.J. and Thompson E.A. (1995). Annealing Markov chain Monte Carlo with applications to ancestral inference. Journal of the American Statistical Association90: 909-920.

Gilks W.R., Richardson S. and Spiegelhalter D.J. (1996). Markov Chain Monte Carlo In Practice. Chapman & Hall, London.

Hammersley J.M. and Handscomb D.C. (1964). Monte Carlo Methods. Chapman and Hall, London.

Manteufel R.D. (1996). Variance-based importance analysis applied to a complex probabilistic performance assessment. Risk Analysis 16:587-598.

Park S.K. and Miller K.W. (1988). Random number generators: good ones are hard to find. Communications of the ACM 31:1192-1201.

Press W.H., Flannery B.P., Teukolsky S.A. and Vetterling W.T. (1989). Numerical Recipes (2nd ed.). Cambridge University Press, Cambridge.

Smith A.F.M. (1991). Bayesian computational methods. Philosophical Transactions of the Royal Society of London, Series A 337:369-386.

Smith A.F.M. and Roberts G.O. (1993). Bayesian computation via the Gibbs sampler and related Markov chain Monte Carlo methods. Journal of the Royal Statistical Society Series B 55:3-23.

Vattulainen I., Ala-Nissila T. and Kankaala K. (1994). Physical tests for random numbers in simulations. Physical Review Letters 73:2513-2516.

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# A. Keywords List

You should use the following reserved keywords as prescribed when building your models and input files:

- Add
- Beta
- BetaRandom
- Binomial
- BinomialBetaRandom
- BinomialRandom
- CDFNormal
- CalcOutputs
- CauchyRandom
- Chi2
- Chi2Random
- Data
- DefaultSim
- Density
- Distrib
- dt
- Dynamics
- End
- Euler
- ExpRandom
- Experiment
- Exponential
- GGammaRandom
- Gamma
- GammaRandom
- GetSeed
- Gibbs
- IFN
- Initialize
- Inputs
- Integrate
- InvGGammaRandom
- InvGamma
- Level
- Likelihood
- lnDFNormal
- lnGamma
- LogNormal
- LogNormalRandom
- LogNormal_v
- LogUniform
- LogUniformRandom
- Lsodes
- MCMC
- MonteCarlo
- Multiply
- NDoses
- Normal
- NormalRandom
- Normal_v
- OptimalDesign
- OutputFile
- Outputs
- PerDose
- PerExp
- Piecewise
- PiecewiseRandom
- Poisson
- PoissonRandom
- Prediction
- Print
- PrintStep
- Replace
- Scale
- SetPoints
- SetSeed
- SimType
- Simulation
- Spikes
- StartTime
- States
- StudentTRandom
- t
- TruncLogNormal
- TruncLogNormalRandom
- TruncLogNormal_v
- TruncNormal
- TruncNormalRandom
- TruncNormal_v
- Uniform
- UniformRandom

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# B. Examples

You will find here some examples of model description files and simulation input files.

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## B.1 ‘`linear.model`’

    # Linear Model with a random component
    # y = A + B * time + N(0,SD_true)
    # Setting SD_true to zero gives the deterministic version
    #---------------------------------------------------------

    # Outputs
    Outputs = {y};

    # Model Parameters
    A = 0;
    B = 1;
    SD_true = 0;
    SD_esti = 0;

    CalcOutputs { y = A + B * t + NormalRandom(0,SD_true); }

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## B.2 ‘`1cpt.model`’: A sample model description file

    # One Compartment Model
    # First order input and output
    #---------------------------------------------------------

    # Inputs
    Inputs = {Dose};

    # Outputs
    Outputs = {C_central, AUC, ln_C_central, ln_AUC,
               SD_C_computed, SD_A_computed};

    # Model Parameters
    ka = 1;
    ke = 0.5;
    F  = 1;
    V  = 2;

    # Statistical Parameters
    SDb_ka = 0;
    SDw_ka = 0;
    SDb_ke = 0;
    SDw_ke = 0;
    SDb_V  = 0;
    min_F  = 0;
    max_F  = 0;
    SD_C_central = 0;
    SD_AUC       = 0;
    CV_C_cen     = 0;
    CV_AUC       = 0;
    CV_C_cen_true = 0;
    CV_AUC_true   = 0;

    # Calculate Outputs
    CalcOutputs {

      # algebraic equation for C_central
      C_central = (ka != ke ?
                   (exp(-ke * t) - exp(-ka * t)) *
                   F * ka * Dose / (V * (ka - ke))):
                   exp(-ka * t) * ka * t * F * Dose / V);

      # algebraic equation for AUC
      AUC = (ka != ke ?
             ((1 - exp(-ke * t)) / ke -
              (1 - exp(-ka * t)) / ka) * F * ka * Dose / (V * (ka - ke))) :
             F * Dose * (1 - (1 + ka * t) * exp(-ka * t)) / (V * ke));

      C_central = C_central + NormalRandom(0, C_central * CV_C_cen_true);
      AUC       = AUC + NormalRandom(0, AUC * CV_AUC_true);

      ln_C_central = (C_central > 0 ? log (C_central) : -100);
      ln_AUC = (AUC > 0 ? log (AUC) : -100);

      SD_C_computed = (C_central > 0 ? C_central * CV_C_cen : 1e-10);
      SD_A_computed = (AUC > 0 ? AUC * CV_AUC : 1e-10);

    } # End of output calculations

    End.

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## B.3 ‘`perc.model`’: A sample model description file

    #---------------------------------------------------------
    # perc.model
    # A four compartment model of Tetrachloroethylene (PERC)
    # and total metabolites.
    #---------------------------------------------------------
    # States are quantities of PERC and metabolite formed, they can be
    # output

    States = {Q_fat,        # Quantity of PERC in the fat
              Q_wp,         #   ...   in the well-perfused compartment
              Q_pp,         #   ...   in the poorly-perfused compartment
              Q_liv,        #   ...   in the liver
              Q_exh,        #   ...   exhaled
              Qmet};        # Quantity of metabolite formed

    # Extra outputs are concentrations at various points

    Outputs = {C_liv,               # mg/l in the liver
               C_alv,               # ... in the alveolar air
               C_exh,               # ... in the exhaled air
               C_ven,               # ... in the venous blood
               Pct_metabolized,     # % of the dose metabolized
               C_exh_ug};           # ug/l in the exhaled air

    Inputs = {C_inh}                # Concentration inhaled

    # Constants
    # Conversions from/to ppm: 72 ppm = .488 mg/l

    PPM_per_mg_per_l = 72.0 / 0.488;
    mg_per_l_per_PPM = 1/PPM_per_mg_per_l;

    #---------------------------------------------------------
    # Nominal values for parameters
    # Units:
    # Volumes: liter
    # Vmax:    mg / minute
    # Weights: kg
    # Km:      mg / minute
    # Time:    minute
    # Flows:   liter / minute
    #---------------------------------------------------------

    InhMag = 0.0;
    Period = 0.0;
    Exposure = 0.0;

    C_inh = PerDose (InhMag, Period, 0.0, Exposure);

    LeanBodyWt = 55;    # lean body weight

    # Percent mass of tissues with ranges shown

    Pct_M_fat  = .16;   # % total body mass
    Pct_LM_liv = .03;   # liver, % of lean mass
    Pct_LM_wp  = .17;   # well perfused tissue, % of lean mass
    Pct_LM_pp  = .70;   # poorly perfused tissue, recomputed in scale

    # Percent blood flows to tissues

    Pct_Flow_fat = .09;
    Pct_Flow_liv = .34;
    Pct_Flow_wp  = .50; # will be recomputed in scale
    Pct_Flow_pp  = .07;

    # Tissue/blood partition coeficients

    PC_fat = 144;
    PC_liv = 4.6;
    PC_wp  = 8.7;
    PC_pp  = 1.4;
    PC_art = 12.0;

    Flow_pul   = 8.0;    # Pulmonary ventilation rate (minute volume)
    Vent_Perf = 1.14;    # ventilation over perfusion ratio

    sc_Vmax = .0026;     # scaling coeficient of body weight for Vmax

    Km = 1.0;

    # The following parameters are calculated from the above values in
    # the Scale section before the start of each simulation.
    # They are left uninitialized here.

    BodyWt = 0;

    V_fat = 0;           # Actual volume of tissues
    V_liv = 0;
    V_wp  = 0;
    V_pp  = 0;

    Flow_fat = 0;        # Actual blood flows through tissues
    Flow_liv = 0;
    Flow_wp  = 0;
    Flow_pp  = 0;

    Flow_tot = 0;        # Total blood flow
    Flow_alv = 0;        # Alveolar ventilation rate

    Vmax = 0;            # kg/minute

    #---------------------------------------------------------
    # Dynamics
    # Define the dynamics of the simulation. This section is
    # calculated with each integration step. It includes
    # specification of differential equations.
    #---------------------------------------------------------

    Dynamics {

    # Venous blood concentrations at the organ exit

    Cout_fat = Q_fat / (V_fat * PC_fat);
    Cout_wp  = Q_wp  / (V_wp  * PC_wp);
    Cout_pp  = Q_pp  / (V_pp  * PC_pp);
    Cout_liv = Q_liv / (V_liv * PC_liv);

    # Sum of Flow * Concentration for all compartments

    dQ_ven = Flow_fat * Cout_fat + Flow_wp * Cout_wp
             + Flow_pp * Cout_pp + Flow_liv * Cout_liv;

    # Venous blood concentration

    C_ven =  dQ_ven / Flow_tot;

    # Arterial blood concentration
    # Convert input given in ppm to mg/l to match other units

    C_art = (Flow_alv * C_inh / PPM_per_mg_per_l +  dQ_ven) /
            (Flow_tot + Flow_alv / PC_art);

    # Alveolar air concentration

    C_alv = C_art / PC_art;

    # Exhaled air concentration

    C_exh = 0.7 * C_alv + 0.3 * C_inh / PPM_per_mg_per_l;

    # Differentials

    dt (Q_exh) = Flow_alv * C_alv;
    dt (Q_fat) = Flow_fat * (C_art - Cout_fat);
    dt (Q_wp)  = Flow_wp  * (C_art - Cout_wp);
    dt (Q_pp)  = Flow_pp  * (C_art - Cout_pp);

    # Quantity metabolized in liver

    dQmet_liv = Vmax * Q_liv / (Km + Q_liv);
    dt (Q_liv) = Flow_liv * (C_art - Cout_liv) - dQmet_liv;

    # Metabolite formation

    dt (Qmet)  = dQmet_liv;

    } # End of Dynamics

    #---------------------------------------------------------
    # Scale
    # Scale certain model parameters and resolve dependencies
    # between parameters. Generally the scaling involves a
    # change of units, or conversion from percentage to actual
    # units.
    #---------------------------------------------------------

    Scale {

    # Volumes scaled to actual volumes

    BodyWt = LeanBodyWt/(1 - Pct_M_fat);

    V_fat = Pct_M_fat  * BodyWt/0.92;        # density of fat = 0.92 g/ml
    V_liv = Pct_LM_liv * LeanBodyWt;
    V_wp  = Pct_LM_wp  * LeanBodyWt;
    V_pp  = 0.9 * LeanBodyWt - V_liv - V_wp; # 10% bones

    # Calculate Flow_alv from total pulmonary flow

    Flow_alv = Flow_pul * 0.7;

    # Calculate total blood flow from the alveolar ventilation rate and
    # the V/P ratio.

    Flow_tot = Flow_alv / Vent_Perf;

    # Calculate actual blood flows from total flow and percent flows

    Flow_fat = Pct_Flow_fat * Flow_tot;
    Flow_liv = Pct_Flow_liv * Flow_tot;
    Flow_pp  = Pct_Flow_pp  * Flow_tot;
    Flow_wp  = Flow_tot - Flow_fat - Flow_liv - Flow_pp;

    # Vmax (mass/time) for Michaelis-Menten metabolism is scaled
    # by multiplication of bdw^0.7

    Vmax = sc_Vmax * exp (0.7 * log (LeanBodyWt));

    } # End of model scaling

    #---------------------------------------------------------
    # CalcOutputs
    # The following outputs are only calculated just before values
    # are saved.  They are not calculated with each integration step.
    #---------------------------------------------------------

    CalcOutputs {

    # Fraction of TCE metabolized per day

    Pct_metabolized = (InhMag ?
                       Qmet / (1440 * Flow_alv * InhMag * mg_per_l_per_PPM):
                       0);

    C_exh_ug  = C_exh * 1000; # milli to micrograms

    } # End of output calculation

    End.

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## B.4 ‘`perc.lsodes.in`’

    #---------------------------------------------------------
    # perc.lsodes.in
    #
    #---------------------------------------------------------

    Integrate (Lsodes, 1e-4, 1e-6, 1);

    #---------------------------------------------------------
    # The following is a simulation of one of Dr. Monster's
    # exposure experiments described in "Kinetics of Tetracholoroethylene
    # in Volunteers; Influence of Exposure Concentration and Work Load,"
    # A.C. Monster, G. Boersma, and H. Steenweg,
    # Int. Arch. Occup. Environ. Health, v42, 1989, pp303-309
    #
    # The paper documents measurements of levels of TCE in blood and
    # exhaled air for a group of 6 subjects exposed to
    # different concentrations of PERC in air.
    #
    # Inhalation is specified as a dose of magnitude InhMag for the
    # given Exposure time.
    #
    # Inhalation is given in ppm
    #---------------------------------------------------------

    Simulation {

    InhMag = 72;            # ppm
    Period = 1e10;          # Only one dose
    Exposure = 240;         # 4 hour exposure

    # measurements before end of exposure
    # and at [5' 30'] 2hr 18 42 67 91 139 163

    Print (C_exh_ug, 239.9 245 270 360 1320 2760 4260 5700 8580 10020 );
    Print (C_ven, 239.9 360 1320 2760 4260 5700 8580 10020 );

    }

    END.

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# Concept Index

Jump to: | ' A B C D E F G H I K L M N O P Q R S T U V W X

| Index Entry | Section
' |  |
| ’!=’ operator | 5.3.1 General syntax
| ’#’ sign | 5.3.1 General syntax
| ’-’ (hyphen) sign | 5.3.1 General syntax
| ’:’ sign | 5.3.1 General syntax
| ’;’ sign | 5.3.1 General syntax
| ’<’ operator | 5.3.1 General syntax
| ’<=’ operator | 5.3.1 General syntax
| ’<>’ operator | 5.3.1 General syntax
| ’=’ sign | 5.3.1 General syntax
| ’==’ operator | 5.3.1 General syntax
| ’>’ operator | 5.3.1 General syntax
| ’>=’ operator | 5.3.1 General syntax
| ’?’ sign | 5.3.1 General syntax
| ’[]’ (square brackets) notation | 5.3.1 General syntax
| ’_’ (underscore) in templates | 5.4 Reading SBML models and applying a template
| ’_’ (underscore) vector notation | 5.3.1 General syntax
A |  |
| Algebraic models | 5.3.3 Model types
| Analyzing simulation output | 6.3 Analyzing simulation output
| Arrays | 5.3.1 General syntax
| Assignment | 5.3.1 General syntax
B |  |
| `Beta` distribution | `Distrib()` specification
| `BetaRandom()` function | 5.3.4 Special functions
| Bibliographic references | Bibliographic References
| `Binomial` distribution | `Distrib()` specification
| `BinomialBetaRandom()` function | 5.3.4 Special functions
| `BinomialRandom()` function | 5.3.4 Special functions
| Blank lines | 5.3.1 General syntax
C |  |
| `CalcOutputs`, output section | 5.3.10 Output calculations
| `Cauchy` distribution | `Distrib()` specification
| Cauchy distribution | `Distrib()` specification
| `CauchyRandom()` function | 5.3.4 Special functions
| `CDFNormal()` function | 5.3.4 Special functions
| Chi-square distribution | `Distrib()` specification
| `Chi2` distribution | `Distrib()` specification
| `Chi2Random()` function | 5.3.4 Special functions
| Colon conditional assignment | 5.3.1 General syntax
| Comments | 5.3.1 General syntax
| Common pitfalls | 7. Common Pitfalls
| Comparison operators | 5.3.1 General syntax
| `Compartment` keyword | 5.4 Reading SBML models and applying a template
| Complementary error function | 5.3.4 Special functions
| Conditional assignment | 5.3.1 General syntax
| Cumulative density function, Normal | 5.3.4 Special functions
D |  |
| `Data()` qualifier, for use in `Distrib()` | `Distrib()` specification
| `Data()` specification | `Data()` specification
| Default | 2.2 Types of simulations
| Defining models | 5. Setting-up Structural Models
| Delay differential equations | 5.3.9 Delay differential equations
| Density function, Normal | 5.3.4 Special functions
| `Density()` specification | `Distrib()` specification
| Derivative specification | 5.3.8 Dynamics section
| Differential models | 5.3.3 Model types
| Discrete-time models | 5.3.3 Model types
| `Distrib()` specification | `Distrib()` specification
| Distribution, `Beta` | `Distrib()` specification
| Distribution, `Binomial` | `Distrib()` specification
| Distribution, `Cauchy` | `Distrib()` specification
| Distribution, Cauchy | `Distrib()` specification
| Distribution, Chi-square | `Distrib()` specification
| Distribution, `Chi2` | `Distrib()` specification
| Distribution, `Exponential` | `Distrib()` specification
| Distribution, `Gamma` | `Distrib()` specification
| Distribution, `HalfCauchy` | `Distrib()` specification
| Distribution, `HalfNormal` | `Distrib()` specification
| Distribution, inverse-gamma | `Distrib()` specification
| Distribution, `InvGamma` | `Distrib()` specification
| Distribution, `Lognormal` | `Distrib()` specification
| Distribution, `Lognormal_v` | `Distrib()` specification
| Distribution, `Loguniform` | `Distrib()` specification
| Distribution, `Normal` | `Distrib()` specification
| Distribution, `Normal_v` | `Distrib()` specification
| Distribution, `Piecewise` | `Distrib()` specification
| Distribution, `Poisson` | `Distrib()` specification
| Distribution, `StudentT` | `Distrib()` specification
| Distribution, triangular | `Distrib()` specification
| Distribution, truncated inverse-gamma | `Distrib()` specification
| Distribution, truncated lognormal | `Distrib()` specification
| Distribution, truncated normal | `Distrib()` specification
| Distribution, `TruncInvGamma` | `Distrib()` specification
| Distribution, `TruncLogNormal` | `Distrib()` specification
| Distribution, `Trunclognormal_v` | `Distrib()` specification
| Distribution, `TruncNormal` | `Distrib()` specification
| Distribution, `Truncnormal_v` | `Distrib()` specification
| Distribution, `Uniform` | `Distrib()` specification
| Documentation license | 1. Software and Documentation Licenses
| `Dt()` operator | 5.3.8 Dynamics section
| Dynamics section | 5.3.8 Dynamics section
E |  |
| End keyword | 5.3.1 General syntax
| `erfc()` function | 5.3.4 Special functions
| Error function, complementary | 5.3.4 Special functions
| Error handling | 6.4 Error handling
| `Euler` integrator | `Integrate()` specification
| `Events` specification | `Events()` specification for state discontinuities
| Examples | 5. Setting-up Structural Models
| Examples | B. Examples
| `Experiment` sections | `Simulation` sections
| Experimental design optimization | 2.2 Types of simulations
| `Exponential` distribution | `Distrib()` specification
| `ExpRandom()` function | 5.3.4 Special functions
F |  |
| Function, `BetaRandom()` | 5.3.4 Special functions
| Function, `BinomialBetaRandom()` | 5.3.4 Special functions
| Function, `BinomialRandom()` | 5.3.4 Special functions
| Function, `CauchyRandom()` | 5.3.4 Special functions
| Function, `CDFNormal()` | 5.3.4 Special functions
| Function, `Chi2Random()` | 5.3.4 Special functions
| Function, `erfc()` | 5.3.4 Special functions
| Function, `ExpRandom()` | 5.3.4 Special functions
| Function, `GammaRandom()` | 5.3.4 Special functions
| Function, `GetSeed()` | 5.3.4 Special functions
| Function, `GGammaRandom()` | 5.3.4 Special functions
| Function, Inline, in line | 5.3.6 In line functions
| Function, input | 5.3.1 General syntax
| Function, `InvGGammaRandom()` | 5.3.4 Special functions
| Function, `lnDFNormal()` | 5.3.4 Special functions
| Function, `lnGamma()` | 5.3.4 Special functions
| Function, `LogNormalRandom()` | 5.3.4 Special functions
| Function, `LogUniformRandom()` | 5.3.4 Special functions
| Function, `NDoses()` | 5.3.5 Input functions
| Function, `NDoses()` | 6.2.2 Input functions (revisited)
| Function, `NormalRandom()` | 5.3.4 Special functions
| Function, `PerDose()` | 5.3.5 Input functions
| Function, `PerDose()` | 6.2.2 Input functions (revisited)
| Function, `PerExp()` | 5.3.5 Input functions
| Function, `PerExp()` | 6.2.2 Input functions (revisited)
| Function, `PiecewiseRandom()` | 5.3.4 Special functions
| Function, `PoissonRandom()` | 5.3.4 Special functions
| Function, `SetSeed()` | 5.3.4 Special functions
| Function, special | 5.3.1 General syntax
| Function, `Spikes()` | 5.3.5 Input functions
| Function, `Spikes()` | 6.2.2 Input functions (revisited)
| Function, `StudentTRandom()` | 5.3.4 Special functions
| Function, `TruncInvGGammaRandom()` | 5.3.4 Special functions
| Function, `TruncLogNormalRandom()` | 5.3.4 Special functions
| Function, `TruncNormalRandom()` | 5.3.4 Special functions
| Function, `UniformRandom()` | 5.3.4 Special functions
| Functions, input | 5.3.5 Input functions
| Functions, input | 6.2.2 Input functions (revisited)
| Functions, special | 5.3.4 Special functions
G |  |
| `Gamma` distribution | `Distrib()` specification
| Gamma function | 5.3.4 Special functions
| `GammaRandom()` function | 5.3.4 Special functions
| General input file syntax | 6.2.1 General input file syntax
| `GetSeed()` function | 5.3.4 Special functions
| `GGammaRandom()` function | 5.3.4 Special functions
| Global specifications | 6.2.3 Global specifications
H |  |
| `HalfCauchy` distribution | `Distrib()` specification
| `HalfNormal` distribution | `Distrib()` specification
I |  |
| Initialize, initialization section, | 5.3.7 Model initialization
| Inline, in line functions | 5.3.6 In line functions
| Input functions | 5.3.1 General syntax
| Input functions | 5.3.5 Input functions
| Input functions | 6.2.2 Input functions (revisited)
| Input variables | 5.3.2 Global variable declarations
| Installation | 3. Installation
| `Integrate()` specification | `Integrate()` specification
| Integration routine, `Euler` | `Integrate()` specification
| Integration routine, `Lsodes` | `Integrate()` specification
| Integration variable | 5.3.8 Dynamics section
| Inverse-gamma distribution | `Distrib()` specification
| `InvGamma` distribution | `Distrib()` specification
| `InvGGammaRandom()` function | 5.3.4 Special functions
| `InvTemperature()` specification | `MCMC()` specification
K |  |
| Keyword, `Compartment` | 5.4 Reading SBML models and applying a template
| Keywords list | A. Keywords List
L |  |
| `Level` sections | `Level` sections
| Licenses | 1. Software and Documentation Licenses
| `Likelihood()` specification | `Distrib()` specification
| `lnDFNormal()` function | 5.3.4 Special functions
| `lnGamma()` function | 5.3.4 Special functions
| Logical tests | 5.3.1 General syntax
| `Lognormal` distribution | `Distrib()` specification
| `LogNormalRandom()` function | 5.3.4 Special functions
| `LogNormal_v` distribution | `Distrib()` specification
| `LogUniform` distribution | `Distrib()` specification
| `LogUniformRandom()` function | 5.3.4 Special functions
| `Lsodes` integrator | `Integrate()` specification
M |  |
| Major changes in version 5.4.0 | 2.3 Major changes introduced with version 5.4.0
| Major changes in version 5.5.0 | 2.4 Major changes introduced with version 5.5.0
| Major changes in version 5.6.0 | 2.5 Major changes introduced with version 5.6.0
| `makemcsim` script | 5.2 Using `makemcsim` to fully process model files
| Markov-chain Monte Carlo simulations | 2.2 Types of simulations
| Markov-chain Monte Carlo simulations | `MCMC()` specification
| `MCMC` simulations | 2.2 Types of simulations
| `MCMC` simulations | `MCMC()` specification
| `MCMC()` specification | `MCMC()` specification
| `mod` syntax | 5.3 Syntax of the model description file
| `mod` usage | 5.1 Using `mod` to preprocess model description files
| Model definition files | 5. Setting-up Structural Models
| Model types | 5.3.3 Model types
| Models, algebraic | 5.3.3 Model types
| Models, differential | 5.3.3 Model types
| Models, discrete-time | 5.3.3 Model types
| Models, statistical | 6.2.5 Setting-up statistical models
| Monte Carlo simulations | 2.2 Types of simulations
| Monte Carlo simulations | `MonteCarlo()` specification
| `MonteCarlo()` specification | `MonteCarlo()` specification
N |  |
| `NDoses()` function | 5.3.5 Input functions
| `NDoses()` function | 6.2.2 Input functions (revisited)
| Normal cumulative density function | 5.3.4 Special functions
| Normal density function | 5.3.4 Special functions
| `Normal` distribution | `Distrib()` specification
| `NormalRandom()` function | 5.3.4 Special functions
| `Normal_v` distribution | `Distrib()` specification
O |  |
| `OptimalDesign()` specification | `OptimalDesign()` specification
| Output specification | 5.3.10 Output calculations
| Output variables | 5.3.2 Global variable declarations
| `OutputFile()` specification | `OutputFile()` specification
| Overview | 2. Overview
P |  |
| Parameter declaration | 5.3.2 Global variable declarations
| Parameter scaling | 5.3.7 Model initialization
| `PerDose()` function | 5.3.5 Input functions
| `PerDose()` function | 6.2.2 Input functions (revisited)
| `PerExp()` function | 5.3.5 Input functions
| `PerExp()` function | 6.2.2 Input functions (revisited)
| `Piecewise` distribution | `Distrib()` specification
| `PiecewiseRandom()` function | 5.3.4 Special functions
| Pitfalls | 7. Common Pitfalls
| `Poisson` distribution | `Distrib()` specification
| `PoissonRandom()` function | 5.3.4 Special functions
| `Prediction()` qualifier, for use in `Distrib()` | `Distrib()` specification
| `Print()` specification | `Print()` specification
| `PrintStep()` specification | `PrintStep()` specification
Q |  |
| Qualifier, `Data()` | `Distrib()` specification
| Qualifier, `Prediction()` | `Distrib()` specification
R |  |
| R deSolve | 5.5 Working with the R package deSolve
| Random number, beta | 5.3.4 Special functions
| Random number, binomial | 5.3.4 Special functions
| Random number, binomial-beta | 5.3.4 Special functions
| Random number, Cauchy | 5.3.4 Special functions
| Random number, Chi-squared | 5.3.4 Special functions
| Random number, exponential | 5.3.4 Special functions
| Random number, gamma | 5.3.4 Special functions
| Random number, general-gamma | 5.3.4 Special functions
| Random number, inverse-gamma | 5.3.4 Special functions
| Random number, lognormal | 5.3.4 Special functions
| Random number, loguniform | 5.3.4 Special functions
| Random number, normal | 5.3.4 Special functions
| Random number, piecewise | 5.3.4 Special functions
| Random number, Poisson | 5.3.4 Special functions
| Random number, Student t | 5.3.4 Special functions
| Random number, truncated inverse-gamma | 5.3.4 Special functions
| Random number, truncated lognormal | 5.3.4 Special functions
| Random number, truncated normal | 5.3.4 Special functions
| Random number, uniform | 5.3.4 Special functions
| Random seed, reading its value | 5.3.4 Special functions
| Random seed, setting its value | 5.3.4 Special functions
| Reserved keywords | A. Keywords List
| Running simulations | 6. Running Simulations
S |  |
| SBML models | 5.4 Reading SBML models and applying a template
| Scale, scaling section | 5.3.7 Model initialization
| Semi-colon | 5.3.1 General syntax
| `SetPoints` simulations | 2.2 Types of simulations
| `SetPoints()` specification | `SetPoints()` specification
| `SetSeed()` function | 5.3.4 Special functions
| Setting-up statistical models | 6.2.5 Setting-up statistical models
| Setting-up structural models | 5. Setting-up Structural Models
| `SimType()` specification | `SimType()` specification
| Simulation definition files | 6. Running Simulations
| Simulation file, syntax | 6.2 Syntax of the simulation definition file
| `Simulation` sections | `Simulation` sections
| Software license | 1. Software and Documentation Licenses
| Special functions | 5.3.1 General syntax
| Special functions | 5.3.4 Special functions
| Specification, `Data()` | `Data()` specification
| Specification, `Density()` | `Distrib()` specification
| Specification, `Distrib()` | `Distrib()` specification
| Specification, `Integrate()` | `Integrate()` specification
| Specification, `InvTemperature()` | `MCMC()` specification
| Specification, `Likelihood()` | `Distrib()` specification
| Specification, `MCMC()` | `MCMC()` specification
| Specification, `MonteCarlo()` | `MonteCarlo()` specification
| Specification, `OptimalDesign()` | `OptimalDesign()` specification
| Specification, `OutputFile()` | `OutputFile()` specification
| Specification, `Print()` | `Print()` specification
| Specification, `PrintStep()` | `PrintStep()` specification
| Specification, `SetPoints()` | `SetPoints()` specification
| Specification, `SimType()` | `SimType()` specification
| Specification, `StartTime()` | `StartTime()` specification
| Specifications, global | 6.2.3 Global specifications
| Specifying simulations | 6. Running Simulations
| `Spikes()` function | 5.3.5 Input functions
| `Spikes()` function | 6.2.2 Input functions (revisited)
| Square brackets | 5.3.1 General syntax
| `StartTime()` specification | `StartTime()` specification
| State discontinuities | `Events()` specification for state discontinuities
| State variables | 5.3.2 Global variable declarations
| Statistical models | 6.2.5 Setting-up statistical models
| Structural models | 5. Setting-up Structural Models
| Structural models | 5.3.3 Model types
| `StudentT` distribution | `Distrib()` specification
| `StudentTRandom()` function | 5.3.4 Special functions
| Style | 5.3.11 Comments on style
| Syntax for `mod` | 5.3 Syntax of the model description file
| Syntax of simulation files | 6.2 Syntax of the simulation definition file
T |  |
| Template models | 5.4 Reading SBML models and applying a template
| Tests, logical | 5.3.1 General syntax
| Triangular distribution | `Distrib()` specification
| Truncated Inverse-gamma distribution | `Distrib()` specification
| Truncated lognormal distribution | `Distrib()` specification
| Truncated normal distribution | `Distrib()` specification
| `TruncInvGamma` distribution | `Distrib()` specification
| `TruncInvGGammaRandom()` function | 5.3.4 Special functions
| `TruncLogNormal` distribution | `Distrib()` specification
| `TruncLogNormalRandom()` function | 5.3.4 Special functions
| `TruncLogNormal_v` distribution | `Distrib()` specification
| `TruncNormal` distribution | `Distrib()` specification
| `TruncNormalRandom()` function | 5.3.4 Special functions
| `TruncNormal_v` distribution | `Distrib()` specification
U |  |
| Underscore syntax for vectors | 5.3.1 General syntax
| Underscore use for template models | 5.4 Reading SBML models and applying a template
| `Uniform` distribution | `Distrib()` specification
| `UniformRandom()` function | 5.3.4 Special functions
| Using deSolve | 5.5 Working with the R package deSolve
V |  |
| Variable names | 5.3.1 General syntax
| Vectorization | 5.3.1 General syntax
| Vectors | 5.3.1 General syntax
W |  |
| Working Through an Example | 4. Working Through an Example
X |  |
| `xmcsim` | 8. XMCSim

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