J Pharmacokinet Pharmacodyn (2010) 37:435–474
DOI 10.1007/s10928-010-9167-z

Tikhonov adaptively regularized gamma variate ﬁtting
to assess plasma clearance of inert renal markers
Carl A. Wesolowski • Richard C. Puetter • Lin Ling
Paul S. Babyn

•

Received: 11 February 2010 / Accepted: 2 September 2010 / Published online: 24 September 2010
Ó The Author(s) 2010. This article is published with open access at Springerlink.com

Abstract The Tk-GV model ﬁts Gamma Variates (GV) to data by Tikhonov
regularization (Tk) with shrinkage constant, k, chosen to minimize the relative error
in plasma clearance, CL (ml/min). Using 169Yb-DTPA and 99mTc-DTPA (n = 46,
8–9 samples, 5–240 min) bolus-dilution curves, results were obtained for ﬁt
methods: (1) Ordinary Least Squares (OLS) one and two exponential term (E1 and
E2), (2) OLS-GV and (3) Tk-GV. Four tests examined the ﬁt results for: (1)
physicality of ranges of model parameters, (2) effects on parameter values when
different data subsets are ﬁt, (3) characterization of residuals, and (4) extrapolative
error and agreement with published correction factors. Test 1 showed physical
Tk-GV results, where OLS-GV ﬁts sometimes-produced nonphysical CL. Test 2
showed the Tk-GV model produced good results with 4 or more samples drawn
between 10 and 240 min. Test 3 showed that E1 and E2 failed goodness-of-ﬁt testing
whereas GV ﬁts for t [ 20 min were acceptably good. Test 4 showed CLTk-GV
clearance values agreed with published CL corrections with the general result that
CLE1 [ CLE2 [ CLTk-GV and ﬁnally that CLTk-GV were considerably more robust,
precise and accurate than CLE2, and should replace the use of CLE2 for these renal
markers.

C. A. Wesolowski (&) Á L. Ling
Nuclear Medicine, The General Hospital, HSC, 300 Prince Philip Drive,
St. John’s, NF A1B 3V6, Canada
e-mail: carl.wesolowski@gmail.com
R. C. Puetter
PixonImaging LLC, 4930 Longford Street, San Diego, CA 92117-2156, USA
P. S. Babyn
Department of Diagnostic Imaging, The Hospital for Sick Children, 555 University Avenue,
Toronto, ON M5G 1X8, Canada

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Keywords Clearance curves Á Renal function Á Tikhonov regularization Á
Minimization of errors Á Gamma variate

Introduction
Plasma clearance (CL, ml/min) measurement has multiple clinical uses. Inert
markers are routinely employed to calculate CL for triage of chronic renal disease
[1], transplant kidney and donor evaluation, and so forth. As one example, the inert
markers 51Cr-EDTA (ethylenediamine tetra-acetic acid) and 99mTc-DTPA (diethylenetriamine penta-acetic acid) have been used to estimate the Area Under the
Curve (AUC) of the plasma concentration of carboplatin and other chemotherapeutic agents to reduce toxicity [2]. The total economic impact of adverse drug
events from cancer chemotherapy alone is a 76 billion dollar annual expenditure [3].
Thus, the potential impact of ﬁnding efﬁcient, accurate and precise estimates for
inert marker CL to avoid toxicity is substantial.
This paper presents detailed results of four tests of three models for estimating
CL of intravenously bolus-injected inert markers, which have only been outlined
prior as abstracts and patent applications [4–7]. Statistical hypothesis testing of
curve ﬁt models (goodness-of-ﬁt testing, lag plotting, etc.) is common practice, e.g.,
see Trutna et al. Chapter 5 [8]. For hypothesis testing, we examine curve ﬁt
suitability for temporal concentration–curves by inspecting (i) ﬁt parameter ranges
to see if these values agree with proper physical models, (ii) ﬁts to subsets of the
data that include the broadest possible range of mean sample times, (iii) model
goodness of ﬁt to see if the model credibly matches the data, and (iv) extrapolation,
because only proper extrapolation is likely to conserve mass (Eq. 6).
The historical ﬁrst model type used for estimating inert marker clearance is the
Sums of Exponential Terms model, SETs, which is often ﬁt to concentration curves
using Ordinary Least Squares regression, OLS. For most markers, whether
metabolized or not, current practice is dominated by ﬁtting the observed
concentrations, Cobs , with SET models using an arbitrary choice of from one to
four exponential terms [9]. We call a single exponential term an E1 SET model and
SET models using 2, 3, 4,. . ., n exponential terms E2 ; E3 ; E4 ; . . .; En models. Current
recommendations for assessing renal function or drug elimination for nuclear
medicine and clinical pharmacology after venous bolus injection are to use En [ 1
for ﬁtting marker concentration curves with 8–13 blood samples [9–11]. The use of
SET models inspired a physical model of linearly coupled, fast-mixing compartments. The ﬁrst semiquantitative, graphic methods for E2 SET ﬁtting were too
inexact for meaningful statistical testing of the quality of the ﬁt, with the ﬁrst E2
SET graphic solution being published in 1944 and predating the general availability
of digital computation [12–14]. Some test tools, such as Monte Carlo simulation
[15] and bootstrap [16], are more recent and are primarily digital computer
techniques. Most criticisms of the unrealistic nature of E2 SET modeling [17–20]
have fallen on deaf ears [21]. Goodness-of-ﬁt testing was published in 1922 [22],
but has seldom been applied to the regression results of SETs, perhaps because of a
lack of alternative ﬁt models.

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A variation of the SET method of estimating CL uses numerical integration to
ﬁnd AUC of the concentrations of multiple samples at early time and extrapolation
of the unmeasured late time behavior using mono-exponential ﬁts to the last 2 h of
data [23, 24]. SETs and ‘‘AUC plus terminal mono-exponentials’’ are currently the
only bolus models in use for estimating radiometric CL. Using mono-exponential
extrapolations, Moore et al. [24] estimated a 10% difference between the 4- and
24-h AUC. Thus, the longer one waits to perform the terminal ﬁt extrapolation with
a mono-exponential the less the mono-exponential extrapolation underestimates the
actual concentrations.
OLS regression of a gamma variate function is less often used than SETs, and
shown here to have a more accurate terminal, or limiting, ﬁt to the data at late times
than SET functions. The GV function has been used in two physiologic models: (i)
The bolus ﬁrst-pass blood-ﬂow model, e.g. [25–30] and (ii) The bolus-dose total
plasma-clearance model, e.g. [19, 20, 31, 32]. These two physiologic models
approximate distinct physical phenomena. Concerning (i), bolus transit models
apply to the rapid early-time behavior exhibited by an organ’s vasculature as a
source loaded with all the tracer that will empty, and for which there is initially no
emptying and a minimum vascular ﬁrst transit time of several seconds applies [33].
In this case, the system is not in equilibrium, and the GV approximates the temporal
behavior of the concentration in one location (e.g., a vein draining the organ).
Concerning (ii), a GV renal model uses the dynamic-equilibrium behavior of
elimination over several hours, which exhibits gradual changes of concentration in a
systemically distributed volume. Here the body acts as a sink for marker and renal
GV model shape-parameter values do not overlap with those of the organ-source
(i.e., not a sink) ﬁrst-pass GV model—see the Theory section, below.
Unlike SET models, for which sporadic testing has been performed, to our
knowledge, no critical appraisal of GV plasma-clearance models has been
presented. Olkkonen et al. [19], using 131I-hippuran, compared GV ﬁts to E2 SET
ﬁtting, but did not detect different clearances between models. Wise [20] suggested
that the GV be used to model concentration curves and calculate CL of most renal
eliminated substances, but did no hypothesis testing. Macheras [31] extended
Wise’s [20] work without critical testing. Perkinson et al. found the OLS GV model
to ﬁt better than other models, but did not test the GV model itself [32]. We shall
ﬁnd, however, that OLS GV ﬁtting should not be used to ﬁnd CL and this is at odds
with prior work. The gamma variate, GV, model, as presented here, sometimes
produces unphysical CL-values when the OLS (ordinary least squares) method is
used to ﬁt the data. As we shall see, the problem is that OLS-GV ﬁtting of the
concentration versus time is an ill-posed problem. The solution is to introduce
regularization. We name this method the Tk-GV method, where ‘‘Tk’’ stands for our
implementation of Tikhonov regularization for optimizing CL-values from GV ﬁts
to the dilutions curves. The Tk-GV method further selects the shrinkage constant, k,
multiplying the regularization term to minimize the coefﬁcient of variation (CV) of
CL. This makes the technique adaptive to the particular data set used and ensures the
most reliable result in all cases. This adaptation is unusual in that CL and its errors
are not estimated from the ﬁt-data range of times, but from t = 0 to ?. By ﬁnding
that GV function that minimizes the CV of CL, as we shall see, the Tk-GV model

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overcomes the instabilities observed in OLS ﬁtting of the GV function to marker
concentrations, and thus provides good CL values.
The Theory section below presents the general relationships governing the
dynamic plasma clearance of inert markers. The focus here is on inert Glomerular
Filtration Rate, GFR, markers as these produce numerically similar (to within 2 or
3% of 99mTc-DTPA) CL-values [34, 35]. However, inert markers have different
rates of body tissue extraction from plasma for [36], which are likely to be related to
the different molecular sizes [17, 37]. Markers for GFR are (i) generally exogenous
and not naturally occurring, (ii) metabolically inert, (iii) extracellular markers, (iv)
only eliminated by ﬁltration in the renal glomeruli and thus largely lack active
transport by the renal tubules, and (v) have poor plasma protein binding. Examples
of GFR markers include the DTPA metallic chelates used here, 51Cr-EDTA,
125
I-iothalamate, and inulin (an indigestible oligosaccharide). There are occasional
quality assurance problems with manufacture of 99mTc-DTPA, but not 169Yb-DTPA
[38]. As we shall see, GFR is CLurine or renal clearance of plasma measured using
urinary collection of a GFR marker and is less than CLtotal , the total plasmaclearance of the GFR marker dosage. For any non-metabolized drug, CLurine
corresponds to the rate of drug elimination. However, CLtotal relates to the systemic
effect of (pharmacodynamics), or body exposure to, that drug.

Theory
Measurement systems
Intravenously administered exogenous-marker clearance determinations can be
obtained by bolus injection (dynamic) or constant infusion (steady state) methods.
An independent classiﬁcation for clearance determinations arises from the use of
either CLtotal , total clearance (a.k.a. input, dose or plasma clearance) or CLurine ,
urinary-clearance (a.k.a. output, renal clearance or glomerular ﬁltration rate; GFR)
sampling techniques. For the CLurine technique, two sets of samples, urine and
plasma, are assayed. Since the dosage is easy to assay, the majority of papers in the
literature, as here, use the total (plasma) clearance method and do not obtain the
inconvenient urinary catheterizations for timed urinary collections.
To illustrate the difference between CLtotal and CLurine , let us ﬁrst consider the
steady state (equilibrium) method performed with constant infusion. After several
hours of constant infusing, CLtotal (ml/min) is mg/min of substance infused divided
by the constant concentration in plasma in mg/ml, whereas CLurine (ml/min) is the
timed urinary output of substance in mg/min divided by the plasma concentration
in mg/ml. That the CLtotal and CLurine sampling techniques yield different estimates
was shown by Florijn et al. and others [39]. Florijn et al. constant-infused inulin
for 4 h after a loading dose and calculated an 8.3 ml/min/1.73 m2 greater CLtotal
than CLurine in humans. This suggests that inulin sequestered itself somewhere in
the body at residence times that are substantial in comparison to the duration of
their 4-h infusion experiment. Indeed, inulin storage ‘‘in a slow compartment’’

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439

(sic, within the body; the compartmental argument is superﬂuous) has been
invoked to explain same-day repeat-measurement results [40].
For dynamic (bolus) methods there is also an inert marker difference between
CLtotal and CLurine [41]. Radiolabeled DTPA, a smaller molecule than inulin,
redistributes more quickly [37]. For an even smaller molecule than labeled DTPA,
51
Cr-EDTA, Moore et al. [24] used the AUC plus terminal mono-exponentials
method and found a 7.6% higher 24-h CLtotal than CLurine , and Brochner-Mortensen
et al. found 4.5% at 72 h using whole body counting [41].
Thus, both constant infusion experiments, and bolus experiments have shown
that CLtotal [ CLurine over long time scales. These results strongly suggest that the
difference between CLtotal and CLurine is a physiologic redistribution (i.e., for inert
markers, a clearance) within the body (CLbody ). Thus, one can deﬁne
CLtotal ¼ CLbody þ CLurine ;
ð1Þ
À
Á
such that CLtotal contains both marker redistribution in the body CLbody and
CLurine over long time scales. CLbody is likely a function of the time elapsed during
an experiment. For example, during ﬁrst pass of a 99mTc-DTPA bolus, the mass
extraction of that marker from plasma by the body is approximately 50% and at a
maximum rate [36]. Therefore from Eq. 1, CLtotal is also likely a function of the
time elapsed during an experiment.
Theory common to all bolus models of total (plasma) clearance
It is not generally appreciated that plasma clearance (CL) and volume of marker
distribution (V) are actually concentration weighted values, i.e., hCLi and hV i. The
common practice is to assume that CLtotal is a constant. However, this is not
reasonable. For one thing, CL varies physiologically in time [42]. Moreover, as per
the previous section, CLtotal likely varies during a bolus experiment. In any case, it is
more general to assume that CLtotal is a function of time, CLtotal ðtÞ, than a constant,
and one can write
Z1
D¼
CLtotal ðtÞCobs ðtÞdt;
ð2Þ
0

where D is the injected dose (in percent, mg, or Bq) and Cobs are the observed
concentrations as D (dose units) per ml, as a function of time. With reference to
Eq. 2, taken outside of the integral, CLtotal ðtÞ becomes a concentration-weighted,
time average value, i.e., hCLtotal i, allowing Eq. 2 to be rewritten as
Z1
D ¼ hCLtotal i Cobs ðtÞdt;
0

or,
hCLtotal i ¼ R 1
0

D
;
Cobs ðtÞdt

ð3Þ

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where hCLtotal i is concentration Rweighted for the entire time interval from zero
1
to inﬁnity, and where the term 0 Cobs ðtÞdt is referred to as the Physical AUC,
Phy-AUC (in percent dosage-min, mg-min, or Bq-min). Better documented than
Eq. 3 is the similarly derived concentration-weighted, average time named the
Mean Residence Time, MRT, by Hamilton et al. [43],
R1
t Á Cobs ðtÞdt
R
MRT ¼ hti ¼ 0 1
:
0 Cobs ðtÞdt

ð4Þ

Finally hVtotal i, the concentration-weighted average volume of distribution of the
marker, can be deﬁned as the product of MRT and hCLtotal i from Eqs. 3 and 4, i.e.,
hVtotal i ¼ MRT hCLtotal i:

ð5Þ

Equations 3 through 5 are common to all bolus models of total plasma clearance.
For any speciﬁc function used to ﬁt Cobs , the formulas for that model can be
obtained by substituting the ﬁt function into these general equations.
In the following development, when we refer to CL and V values, these
should be understood as concentration-weighted, time average values. In
practice, using Eq. 3 to evaluate CL is awkward. In particular, one cannot wait
inﬁnite time to construct Cobs over the required interval from 0 to ?. Moreover,
for discrete blood sampling, one does not have the necessary continuous
recording of Cobs to apply Eq. 3. One practical solution is to use a continuous
function, C(t), to ﬁt the available data and thereby estimate Phy-AUC, i.e., make
an Estimated AUC, Est-AUC. The consequences of doing this can be examined
by breaking Est-AUC into three pieces: (i) before the earliest plasma
concentration measurement time, t1 (in min), (ii) during the times the plasma
concentration is measured (from t1 to tm , where t1 ; t2 ; t3 ; ; ; tm are the sequential
sampling times) and (iii) beyond the last measurement time of plasma
concentration, tm . This yields
CL ¼

D
D
D
%
¼ R t1
Rt
R1
;
PhyÀAUC EstÀAUC
CðtÞdt þ t m C ðtÞdt þ t CðtÞdt
0
1

ð6Þ

m

where C(t) approximates Cobs , and where typically both the ﬁrst and third integrals
in the summed denominator are extrapolations, and only the second integral is from
interpolation (as in integration of an OLS regression ﬁt function, or occasionally,
with numerical integration followed by third integral mono-exponential extrapolation [23, 24]). In practice, one only has a C(t) model that accurately ﬁts the data,
Cobs , over the times for which samples have been collected. The average hCLtotal i
derived is consequently in error. To conserve the dose, D, the C(t) ﬁt from t1 to tm
must also extrapolate correctly over the entire interval from time zero to inﬁnity.
Therefore, in order to estimate CL correctly, especially for low renal function when
the third integral is much larger (e.g., 50 times larger) than the second, one needs to
show that the AUC beyond tm is correctly estimated. Test 4 below examines this
extrapolation process.

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Theory speciﬁc to SET models
SET models are deﬁned as follows
C ðt Þ ¼ E n ¼

n
X

Ci eÀki t ; ki [ 0;

ð7Þ

i¼1

where Ci is in 100 ml-1 (Note, this unit is percent of D per ml, which standardizes
D between experiments), MBq/ml or mg/ml, and ki is in min-1. Note that Eq. 7 is
used to ﬁt the concentrations from t1 to tm , and concentration approaches zero
rapidly after tm , i.e., Eq. 7 may underestimate the third integral of Eq. 6.
Improvements to the E2 SET model for better testing. The common expression
[44] for an E2 SET has the form
CðtÞ ¼ C1 eÀk1 t þ C2 eÀk2 t :

ð8Þ

This equation has a poorly diagonalized covariance matrix and degraded regression
performance. The use of more statistically independent parameters improves convergence. Thus, Eq. 8 was reparameterized (e.g., see [45]) using the substitutions
C1 ¼ ka, and C2 ¼ k to yield
À
Á
Cobs % E2 ¼ k aeÀk1 t þ eÀk2 t ;
À
Á
ln Cobs % ln E2 ¼ ln k þ ln aeÀk1 t þ eÀk2 t :

ð9Þ

Regression of Eq. 9 yields a more normally distributed ln k (i.e., the
concentration measurement errors are proportional to the concentrations) than C1
or C2 of Eq. 8. While Eqs. 8 and 9 ﬁnd the same regression solutions, regression of
Eq. 9 is more numerically stable with a markedly reduced required number of
iterations for convergence.
Once convergence has been achieved, CL is then calculated from this ﬁt using the
substitution of Eq. 9 into Eq. 3
CL ¼ R 1
0

D
D=k
¼
;
E2 dt a=k1 þ 1=k2

ð10Þ

where D is the injected dose. Subsequently, Mean Residence Time, MRT, is calculated from substitution of Eq. 9 for Cobs in Eq. 4 yielding
MRT ¼

a=k2 þ 1=k2
1
2
:
a=k1 þ 1=k2

ð11Þ

Then V ¼ MRT Á CL gives
V¼

D a=k2 þ 1=k2
1
2
;
k ða=k1 þ 1=k2 Þ2

ð12Þ

from the product of Eqs. 10 and 11.

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While reparametrization considerably simpliﬁed ﬁnding ﬁts to the data, this does
not insure convergence, and, for Test 2, constrained global search techniques were
applied to E2 ﬁtting to guarantee convergence [46–48].
Theory speciﬁc to gamma variate models
One might think that a ﬁrst principles approach to the plasma-clearance marker
problem would use a slow-mixing model. One such approach might adapt the Local
Density Random Walk (LDRW) function [26, 29]. Speciﬁcally, LDRW, which
models blood ﬂow with Brownian motion, might be extended to include
recirculation. However, and although Brownian motion occurs, turbulent vascular
ﬂow [49], core and plug ﬂow of blood [50], delay channels [33] and molecular
sieving at capillary walls [14, 51], also occur. Given this complex physiology, (i)
use of the term dilution in time is preferred to mixing, and (ii) a dilution in time
model derived from ﬁrst principles may be intractably complicated and position
dependent. Fortunately, there is a simple model that approximates the observed
marker concentrations, Cobs , (in Bq per ml or percent dose per ml or mg/ml), in
some circumstances, i.e., the GV function,
Cobs % GVðt; a; b; K Þ ¼ C ðtÞ ¼ KtaÀ1 eÀbt :

ð13Þ

Note that the GV function is described by only three parameters: a, b and K. The
parameter a is dimensionless and is called the shape parameter. The rate constant b
is used in Eq. 13 rather than its more common reciprocal, h = 1/b, since b has no
discontinuity in the derivative at b = 0. This is important since regression analyses
often use gradient techniques (e.g. Levenberg–Marquardt and sequential quadratic
programming). And so here, as for the statistical software package SPSS, we adopt
the convention of using the rate constant b to describe GV.
The power function multiplier of Eq. 13, taÀ1 can assume non-integer powers of
time, to have proper physical units, the quantity raised to the a - 1 power must be
dimensionless. In other words, Eq. 13 should be understood as meaning, for
example, that
CðtÞ ¼ jðbtÞaÀ1 eÀbt ;

ð14Þ

where j is a constant with units of concentration and the product bt is dimensionless. For equivalence for regression when b C 0 in Eqs. 13 and 14, let
K  j b aÀ1 . Equation 13 is used to ﬁt the data herein, and is the historic form for
C(t) (e.g., see [20]). In that form, KtaÀ1 has the units of concentration. With these
caveats aside, one can derive the formula for CL, the weighted-average rate of
total plasma-clearance of dose for the GV model, Eq. 15. After substituting the
GV Eq. 13 for Cobs in the integral of Eq. 3 and performing the integration, one
ﬁnds


Dba
Db
for b ! 0 ;
ð15Þ
CL ¼

j C ð aÞ
KCðaÞ

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R1
where CðaÞ ¼ 0 taÀ1 eÀt dt is the gamma function and is widely available either as
the function itself (e.g., Mathematica) or as its logarithm (Excel and SPSS).
Substituting GV from Eq. 13 into Eq. 4 yields
R1
t Á taÀ1 eÀbt dt a
R
ð16Þ
¼ :
MRT ¼ 0 1 aÀ1 Àbt
b
e dt
0 t
Indeed, this mean residence time formula has been published for a GV ﬁt to ﬁrst
transit of a bolus through the brain [27], and the form of the equation is
identical to that of the CL model, even though the models are physically different. MRT has certain restrictions regarding unbounded integrals [52] that
sometimes apply here (see below). To calculate V, the physical volume of the
system, one recalls Eq. 5, i.e., V ¼ MRT Á CL. Thus, Eq. 16 substituted into
Eq. 5 yields
a
V ¼ CL:
ð17Þ
b
Note that this new expression for V is a times the virtual volume, CL/b, the
latter (misleadingly) given by Wise [20] as V. For incrementally small renal
function, both CL and b can become small, yet maintain a stable ratio, if the
method for extracting the CL/b ratio information is well behaved. So V (Eq. 17)
may remain constant even when MRT (Eq. 16) becomes inﬁnite time and CL
becomes vanishingly small. As per the Results below, the use of Tk-GV ﬁtting
can allow CL/b to be preserved for vanishingly small CL. However, the volume
information is indeterminate when CL is exactly zero (and b = 0) as sometimes
occurs in b C 0 constrained OLS GV ﬁtting, which latter does not hold CL/b
constant.
When Cobs(t) follows a GV law, the rate of change of concentration with
time is
1 dC ðtÞ a À 1
¼
À b;
CðtÞ dt
t

ð18Þ

which can be derived from differentiating the GV function with respect to time. The
(a - 1)/t term is presumed to be loss of marker to the interstitium and related to
CLbody . The second term, -b, is the more familiar ﬁrst order kinetic rate constant,
herein renal loss only, and as the markers are inert, associated with CLurine . So when
one uses the AUC method for determining CL with a GV for the concentration, both
clearances add up to CLtotal . Indeed, ða À 1Þ=t, hence CLbody ðtÞ are time dependent.
Hence, CLtotal ðtÞ is also time dependent. Hence, we must have CLbody [ 0 and
consequently a \ 1, or else ða À 1Þ=t is not a source term (i.e., the sign of the
numerator would be positive). Finally since we want positive MRT and volumes of
distribution, V, Eqs. 16 and 17 tell us that a [ 0 (b is explicitly [0). So ﬁnally
0 \ a \ 1. As presented below, OLS GV ﬁtting produces occasional b \ 0 and
a [ 1 values. This problem is largely solved by adaptively Tikhonov regularizing
the GV ﬁt as follows, next.

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Introduction to Tikhonov regularization
At the heart of the Tk-GV technique is Tikhonov regularization. Tikhonov
regularization is used in a variety of applications to remove solution ambiguity in
ill-posed problems [53–55]. The Tk-GV model implements regularization as an
adaptive regularizing penalty function that rewards smoother ﬁts to the data. Tk-GV
is adaptive because the amount of smoothing is optimized using a controlling
variable factor, k, often called the shrinkage factor. Values for the shrinkage factor
can be selected in a variety of ways. However, the goal in this paper is to measure
CL. Consequently, the shrinkage factor will be adjusted to minimize the relative
error of measuring CL, which error is expressed as the coefﬁcient of variation (CV)
of CL, i.e., CVCL ¼ sCL =CL, calculated from the propagation of small errors (see the
Appendix, Eq. A3).
Tk-GV ﬁtting, unlike OLS ﬁtting, does not minimize the smallest residual
interpolation of the concentration data. The Tk-GV ﬁt is biased. In effect, for
decreasing renal function, increased regularization (i.e., a higher value of the
shrinkage factor) is applied as the relative importance of un-modeled effects extend
later and later into the sample measurement times. However, as the shrinkage factor
increases, the data becomes increasingly unimportant and eventually is not included
in the ﬁt. So results with k ) 1 are suspect. Thus, despite the many expected
beneﬁts of the Tk-GV model, it is prudent to examine all aspects of the procedure.

Methods
Methods, data
Forty-one patient studies used here were made available by Dr. Charles D. Russell
from a published series [56]. These data are thought to be accurate to within 3%
pipetting error [57], contain 8 plasma sample concentrations per case drawn from
adults with a wide range of CL-values from very severely impaired to normal. For
these studies, 1.85 MBq of 169Yb-DTPA were injected after which the syringe was
ﬂushed with blood. Residual activity in the syringe was less than 2% of the dose.
Standards were prepared by dilution from duplicate syringes. Eight blood samples
were drawn into standard EDTA anticoagulated vacuum sample tubes at 10, 20, 30,
45, 60, 120, 180, and 240 min after injection, using a vein other than that used for
injection. A week after centrifugation, duplicate samples of plasma and standard
were pipetted, counted, and the results averaged. The aqueous standard solution of
169
Yb-DTPA was pipetted into counting tubes within 8 h of preparation.
For contrast and completeness, ﬁve additional 99mTc-DTPA patient studies were
included. These studies were made available by Barbara Y. Croft, Ph.D. of the
Cancer Imaging Program of the National Cancer Institute of the U.S.A. National
Institute of Health (private communication, 2007). These patients had 9 blood
samples drawn at 5, 10, 15, 20, 60, 70, 80, 90 and 180 min. The CL for these adults
ranged from moderate renal failure to normal. Patient weight ranged from 40.6 to
119.5 kg and height from 142 to 188 cm over both data sets.

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Methods, numerical methods
In addition to reparametrization as described above, in order to always obtain E2
SET results, it proved necessary to use constrained simulated annealing and random
search, i.e., regression methods with guaranteed convergence [46–48]. Use of these
powerful methods provided converged E2-ﬁts under conditions beyond those
usually recommended. To ensure that the ﬁt results had converged to the global
minimum, two different sets of starting conditions and annealing rates were used for
each data set. In addition, constrained random search ﬁts were performed and the
least-residual-summed-squares values obtained from the three trials were chosen as
the solution. Mathematica ver. 6 global minimization of sum-squared residuals with
the simulated annealing option and perturbation scale set to 3, was found to
converge fairly rapidly.
The GV ﬁts converged using Mathematica software version 7 and 20 repeat
(serial) random applications of the Nelder-Mead algorithm [58] to obtain the global
minima for our OLS solutions both with and without constraints. To conﬁrm this,
the same results were obtained using repeated applications of simulated annealing
[48]. GV ﬁtting can be performed using Multiple Linear Regression (MLR). Writing
the logarithm of Eq. 13 yields.
ln C ¼ ln K þ ða À 1Þ ln t À bt:

ð19Þ

Making the substitutions Y ¼ ln C, X1 ¼ ln t and X2 ¼ t, one can see that
Y ¼ ln K þ ða À 1ÞX1 À bX2 ;

ð20Þ

is linear in Y. This is used for GV and Tk-GV ﬁtting but not for En ! 2 SET models,
the latter of which cannot be usefully transformed into a form for which MLR can
be applied.
Tikhonov regularization (Tk) is widely used in ridge regression in statistics [59],
and is a standard feature of many statistical packages including SPSS, R and
Matlab. The Tk-GV application in Mathematica version 7 has a run time of several
seconds for convergence to 16 decimal places. The algorithm was checked against
SPSS version 15. Global-optimization-search numerical techniques can enforce
convergence [46, 48, 60]. While these methods should ﬁnd the global minimum (of
Eq. A3 of the Appendix), practical implementation may only ﬁnd a local minimum.
To gain conﬁdence that the results presented here are global minima, regressions
with multiple random starting conditions were obtained for each sample combination, and this process was carried out for several different regression methods. In
difﬁcult cases, i.e., noisy cases from leaving out four samples (L4O), 70,000
regressions were performed with each method. Using L4O, there were 3500
selections of different subsets of the data and each was regressed 20 times to ﬁnd the
best regressions using each of three methods: Nelder-Mead, simulated annealing,
and differential evolution. In no single case out of 3500 were the results of any of
the three ﬁts methods for a given set of samples different from the other two
methods’ ﬁt results to 16 decimal places. To obtain Tk-GV converged solutions to
agree within 16 decimal places required a computational precision of more than 32
decimal places, and techniques internally accurate to 40 decimal places were used.

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Methods, details of the tests
Only testing can establish that a given functional form ﬁts concentration curves
appropriately. The testing performed for SET functions was also applied to the GV
function ﬁt using standard OLS methods. However, the OLS-GV ﬁts proved
numerically more stable than SET ﬁts. Accordingly, more advanced versions of the
tests were used for the OLS-GV model. Testing of the Tk-GV model was reﬁned yet
again. The Tk-GV model intrinsically calculates the error in the resulting CL value.
The accuracy of the error determination was double-checked with leave one out
(LOO), a data holdout method not applicable to less robust models. Four different
tests of the suitability of curve ﬁtting were performed.
Test 1, variability of parameters
For the SET and OLS-GV models, Test 1 used bootstrap [16] analysis of variance
(ANOVA) to determine whether the parameters of the models are statistically
warranted, and whether the values of CL and V (in ml) calculated from those
parameters are physical. When ﬁtting the data, when ﬁt parameters take negative
values the value of the ﬁt function becomes non-physical, suggesting that the ﬁt
quality is not appreciably altered by discarding that variable, i.e., the parameter is
unwarranted when the 95% conﬁdence intervals (CIs) include zero. Parameters that
have zero magnitude can occur in two scenarios. The ﬁrst is that the true value of
the parameter could be zero (unlikely here). The second is when the parameter is
statistically unwarranted. Test 1 used bootstrap resampling and OLS regression to
create 1000 values for each parameter for each of the 46 patients. Bootstrap for
regression takes the converged model and constructs residuals as the difference
between the model and the observed data, in this case the logarithm of
concentrations. These residuals are then resampled with replacement, added to
the ﬁt curve to form a new synthetic data set, and a new model ﬁt to the new data. In
this way, after many resamplings of residuals, a distribution of ﬁt parameters can be
constructed. When an equation is ﬁt to the data, the resulting values of the ﬁt
parameters may contain zero within the 90% or 95% CIs. Usually, if a model
contains parameters that have P(0) [ 0.05, those parameters do not contribute to the
quality of the ﬁt, and can be removed (set equal to zero) without signiﬁcantly
affecting the ﬁt quality. Bootstrap was done with SPSS 15 Sequential Quadratic
Nonlinear Algorithm, SQNA. The 95% CIs were estimated twice. The ﬁrst estimate
pﬃﬃﬃ
was from the SEM (standard error of the mean) of parameters, where SEM =r= n,
r is the standard deviation and n is 1000 simulations. Even one excessively large
value outlier (e.g., 106) would inﬂate r. Thus, the mean ± 1.96 SEM, 95% CI is
inﬂated by very far outliers. The second, better CI estimate is to construct 95% CIs
using trimming [16]. A trimmed 95% CI is derived by using the 2.5 and the 97.5
percentiles of the bootstrap distribution as its limits.
The bootstrap of GV inclusive of early time data is useful because it can be
compared directly to bootstrap of E2 SET models for assessment of outliers and CVs
for both CL and V using Wilcoxon signed-ranks sums. Outliers were deﬁned as

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parameter values that lie further than three interquartile ranges away from the
median of the population of interest, and are often called far outliers.
Characterization of parameters from Tk-GV ﬁts. Tk ﬁt parameter calculations
have a native estimate of the error in CL; the standard deviation, SD, with notation
SDðxÞ ¼ sx , as part of the Tk regression process, itself. However, since Tk-GV
attempts to minimize sCL =CL, i.e., an error, it is prudent to crosscheck the intrinsic
Tk-GV error calculations. Note that bootstrap regression, which randomizes
residuals in time, is not compatible with time-based adaptive ﬁtting. Instead, the
Tk-GV parameter errors were crosschecked with LOO (Leave One Out, also called
jackknife) analysis of variance for CL and V (373 trials total, resulting from 8 trials
for each of the 41, 8-sample patients plus 9 trials for each of 5, 9-sample patients).
The resulting jackknife variances are corrected for leaving data out under highly
correlated resampling conditions. It is also possible to use L2O (leave two out),
L3O, and so forth. To calculate the SD of any parameter of interest, one leaves out
data and scales the SD for highly correlated resampling conditions as per [61]. LOO
is used only once in Results, Test 1, below. However, leaving out data was also used
for testing algorithms, and ﬁnding extremes of parameter ranges, as presented next.
Test 2, effects of sample-subset selection on model parameter values
When data is excluded from the ﬁt, parameter values are affected and the errors can
become so large as to make a particular procedure unuseful. To explore such effects,
a variation of holdout sampling was used that groups the samples into those from the
earliest times, all times (the complete set of sample), and samples from the latest
times, i.e., (i) ﬁtting the earliest sample times and progressively adding samples
from later times until all samples are used, (ii) then dropping samples from early
times from the selection in sequence, until only the latest samples remain to be
ﬁtted. To still have a statistically independent residual from ﬁtting, the smallest
number of samples that can be used for a model with np parameters is np þ 1
samples. There are several motives for using this holdout testing-scheme. One
advantage of the holdout technique over bootstrap is not assuming homoscedasticity
(uniform variance) of residuals during resampling. Instead, it only uses the actual
data (with its native data errors) for testing. Another advantage is that one wants to
see what using all samples produces, because that is the usual method of calculating
values for an E2. Further, one wants to know what happens to the model if the ﬁrst
or last samples are discarded, or the ﬁrst two or last two samples are discarded, and
so forth. Also, this approach includes the recommended times for E1 sampling, i.e.,
120 to 240 min. Finally, this test examines robustness. These subsets of data were
chosen using temporally consecutive samples to span the broadest possible range of
mean sampling times. In this way, one can quickly visualize when problems with
ﬁtting arise.
A robust model is one that frequently and easily converges to a proper solution.
For this test, physically plausible E1-ﬁts to the measured data and to simulated data
(used as a control) were easy to obtain and E1-ﬁts are robust. However, some
À
Á
E2 ¼ k aeÀk1 t þ eÀk2 t ﬁts were problematic, i.e., not robust. Thus, constraints

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were used to ﬁnd physically plausible solutions. In speciﬁc, the constraints used
were 0 a 5 and 0 k2 k1 2. Test 2 also examined unconstrained GV ﬁts,
GV ¼ KtaÀ1 eÀbt , to the various subsets of sample times. The OLS-GV ﬁtting
proved not to be robust. To ﬁx this the constraint b C 0 was imposed. The behavior
of OLS-GV and Tk-GV models with respect to the extremes of mean sample times
were plotted as the frequency of out of bounds values of a and b for increasing mean
sample time. Also of interest is the number of cases for which Tk-GV ﬁts predict
vanishingly small CL values. This was examined by leaving out four or fewer
samples of the 8 or 9 available for each dilution study.
One can estimate the noise level and the amount of interpolative error contained
in ﬁts by plotting an observed variance called the Mean Square Error, MSE. For
P
each patient study, the MSE ¼ ðm À pÞÀ1 m R2 , where Ri is the ith residual (data
i¼1 i
minus the model value at the ith sample time), m is the number of samples, p is the
number of parameters in the ﬁt equation, and m - p adjusts for the number of
degrees of freedom. We also performed E1 ﬁts to data generated by Monte Carlo
simulation of marker concentration data produced by a E1 model with a MSE of
0.0009, i.e., from a 3%, 1 SD Gaussian noise error in measuring concentration.
Three percent was used as a generous estimate of the SD of the noise, and is larger
than the European guidelines suggestion of 3% as an upper limit for error [11].
Finally, one can examine the spread between the MSE from all samples (with
possibly the worst interpolative error) with that from ﬁtting the earliest and latest
samples (which is closer to noise variance, as any interpolative error present is less
severe over shorter ranges). A model that successfully predicts marker concentration
over time would have a plot of MSE for temporally increasing mean times of sample
time-groupings that is relatively ﬂat and especially should not show systematic
trends such as a maximum variance when all of the samples times are used.
Test 3, signiﬁcance of interpolative error
In Test 3, goodness-of-ﬁt testing of ﬁts to the data was performed for those models
that use OLS ﬁtting (i.e., the probability that the error of interpolating marker
concentration ﬁtting arose from chance alone). As an average value, the residuals
for unconstrained ﬁts should be smaller than for constrained ﬁts as occasionally an
unconstrained ﬁt will ﬁnd a smaller minimum with ﬁt parameters that are outside
the region of constraint. Hence, testing the ability of unconstrained ﬁts to ﬁt the data
is generally more conservative than testing the constrained ﬁts. Test 3 examines the
quality of the interpolative ﬁt, i.e., the middle integral of Eq. 6. This test was done
in two ways. The ﬁrst evaluates the magnitude of the mean residuals for all samples
occurring at each sample time-grouping using a one-sample t-test. In general, the
one-sample t-test determines the signiﬁcance of difference in the mean of a sample
and a hypothesized mean. A bad ﬁt will have large magnitude t-values and
associated probabilities less than 5% for each sample time-group.
The second part of the test measures the goodness of ﬁt of all the samples from
the entire study as a group, using Chi-squared. InÀthis case, Á probability is the
 the
regularized upper incomplete gamma function, C n=2; v2 2 Cðn=2Þ, and is the

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probability that all the residuals arose merely by chance. A bad ﬁt has a Chi-squared
probability of less than 5%.
Characterization of Tk-GV residuals. This examines the structure or temporal
trend of the residuals (the differences between the data and the Tk-GV ﬁts) for
various Tk smoothings, k, and renal rate constants, b. In general, one would only
expect the mean residuals from ordinary least squares (OLS) regression to be zero
for a perfect match between a model and the data. On the other hand, to ﬁnd a more
precise estimate of CL, Tk-GV ﬁtting introduces bias to the otherwise unbiased OLS
solution. In other words, for the Tk-GV method, biased residuals from ﬁtting early
times are desired given that the terminal GV behavior sought is inappropriate to
earlier times. Thus, inspection of these residuals is revealing, and the residuals are
examined in some detail in Results, Test 3.
Test 4, extrapolative error
Test 4 evaluates how well the E2 model extrapolates beyond the range of the data
points. The test used here is the ability of the ﬁt equation to predict future
concentration. Extrapolative error, eextrap , is calculated as follows. Both the ﬁts to
the ﬁrst m À 1 and all m samples were performed and evaluated at the time of the
mth sample and the concentration difference E2 ðm; tm Þ À E2 ðm À 1; tm Þ is called
eextrap . One could as easily have compared E2 ðm À 1; tm Þ directly with Cobs ðtm Þ, but
this would include any systemic interpolative error that exists between E2 ðm; tm Þ
and Cobs ðtm Þ and would be a less accurate measure of extrapolative error alone. (See
Test 4 for a more graphic presentation.) The extrapolation test yields signed error.
The sign of the median error corresponds to a systematic over- or under-estimation
of the third integral of Eq. 6. The differences between the logarithms of
concentrations from all 46 pairs of ﬁts (to m - 1 and to m samples) were tested
for signiﬁcance with the Wilcoxon signed-ranks sums test. A model with a
Wilcoxon Pðeextrap Þ\0:05, is unlikely to extrapolate properly and should be
discarded. On the other hand, models with a Wilcoxon Pðeextrap Þ [ 0:05 might be
acceptable, and models with higher P-values correspond to models that are more
plausible. The other error measure is precision. The precision for predicting CL and
V are measured as the standard deviations of differences between the appropriate
m and m - 1 sample functions extrapolated over the interval of t equals 0 to ?.

Results
Test 1, parameter stability
For OLS E2 and GV ﬁts to all data, Test 1 looks at parameter stability using 1000
bootstrap resamplings of each of 46 patient studies. Table 1 shows the breakdown of
results for each of the ﬁt parameters from bootstrap as the percentage of parameters
that contain zero within their 95% conﬁdence intervals (CIs). Note that for proper
parametrization one expects ln k to be quasi-normal and indeed no range estimation

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Table 1 Test 1. Results for two curve ﬁt models: two exponential terms, E2 SET, and gamma variate,
GV
À
Á
Model, logarithm of:
GV: C ¼ KtaÀ1 eÀbt
E2 SET: C ¼ k aeÀk1 t þ eÀk2 t
Parameter of model

lnk (%)

a (%)

k1 (%)

k2 (%)

lnK (%)

a (%)

b (%)

Frequency SEMa

0

98

80

52

0

0

26

Frequency trimminga

0

0

0

41

0

0

24

Frequency of problematic model parameter 95% CIs (conﬁdence intervals) containing zeros within that
parameter’s CI range where the 95% CIs are from (1) SEM (Standard Error of the Mean) of boot-strap
distribution, and (2) 5% trimming of left and right, 2.5% tails, of each bootstrap distribution. Forty-six
distributions, where each distribution is from 1000 bootstrap samples
a

The percentage of 46 cases having a zero parameter value within the speciﬁed 95% CIs

problems arose for the constant parameter ðln K or ln kÞ for any of the models.
However, for the E2 parameters a; k1 and k2 ; [50% of the 46 (one per patient
study) bootstrap distributions that were ﬁt contained zero within the 95% CI’s as
calculated from the SEM (Standard Error of the Mean) for at least one of these
parameters. Using SEM to estimate conﬁdence intervals, however, probably overestimates the frequency for which parameters should be eliminated from the E2
model without affecting ﬁt quality. The better, trimmed 95% CI approach suggests
a problem only with the distribution of E2 renal elimination rate parameter, k2 , in
19/46 cases. Negative k2 implies that concentration increases at late times and is
nonphysical. The meaning of this is that the smaller, slower rate constant, k2 , is the
most difﬁcult parameter to detect in a E2 model. Perhaps a more direct question
would be to ask how frequently one ﬁnds CL 0 from bootstrap, that result being
6.0% of the time or P ¼ 0:12, two-tailed. When a parameter does not contribute to
the quality of ﬁt of a regression equation, the model is too complex and one would
usually discard that parameter (i.e., k2 ). Alternatively, one should constrain k2 [ 0.
As it happens, there is also a problem with the 2.5% trimmed upper tail values of
E2 ’s a. The median value of the 2.5% upper tail of a is 7.95. However, the average
upper-tail a is 1915, a large number. The frequency of occurrences of a [ 100 is 12/
46 in the 2.5% upper tail. This means that a should be constrained above (i.e.,
a B 5, see Test 2), or we will allow some models to have a ) 100, i.e., much more
initial ﬂow into the deep interstitium than into the central compartment containing
the kidneys. This is implausible—see Test 2 Results.
GV’s parameters from bootstrap regression have similar SEM and 5% trimmed
ranges. This contrasts to the dissimilarity of the SEM and 5% trimming ranges of the
parameters for E2 —see Table 1. The greater disparity between these measurements
in the E2 ﬁts results is from a signiﬁcantly increased number of far outliers for the
E2 models compared to the GV models. Further, this holds for all of their respective
parameters, including CL and V, (P \ 0.0001, Table 2). Moreover, CL and
V estimated from GV ﬁts are signiﬁcantly more precise than CL and V estimated
from E2 (CV, P \ 0.0001, Table 2). Note that the CVs for CL and V for E2 SET
models appear exaggerated compared to the CVs of the GV models. Bootstrap can
exaggerate an estimate of CV (just as it does for SEM) due to far outliers and E2

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Table 2 Test 1 results
CL
GV (%)

V
E2 (%)

a

P(GV [ E2)

GV (%)

E2 (%)

P(GV [ E2)a

Outliers

2.60

6.65

\0.0001

Outliers

2.25

6.10

\0.0001

CV

9.80

61.4

\0.0001

CV

11.3

320

\0.0001

Median percentage outliers (±3 interquartile ranges) and coefﬁcients of variation (CVs) of plasma
clearance (CL) and volume of distribution (V) from GV and E2 SET ﬁt-parameters for 1000 bootstrap
realizations each for 46 patient studies (46,000 simulations)
The right most column shows the vanishingly small probabilities for CL and V that E2 SETs have fewer
outliers (or smaller CVs) than GV as determined by the Wilcoxon signed-ranks sum test

SET ﬁtting had signiﬁcantly more outliers than GV ﬁtting. However, the GV results,
which are improved relative to the E2 model performance, still yielded b parameter
values (renal rate elimination rate) that were statistically indistinguishable from zero
one-quarter of the time (Table 1). This is sufﬁcient to reject the OLS GV model as
offering a viable procedure with statistically warranted b when all samples are ﬁt.
The frequency of inappropriate a and b, GV values changes for different time
ranges and this is examined in Test 2.
Characterization of parameters from Tk-GV ﬁts. As per the Methods section,
bootstrap is not applicable to the Tk-GV model. Tk-GV’s parameters’ errors are
calculated during Tk regularization and were further examined by leaving data out.
The most important observation from Table 3 is that the Tk-GV parameters have
no silly, nonphysical values and have small measurement errors. For example, the
Tk-GV a parameter varied only from 0.59 to 0.99, and no negative (or vanishingly
small) b values occurred. Note that ln K is probably normally distributed, P = 0.97,
which agrees with an error of measuring K, proportional to K.
Table 3 does not show a comparison between parameters and for that covariance
is examined. The values CL and b covary, and, a closest to 1 and the values of b
closest to zero are correlated. This limiting behavior is strong, and otherwise, a and
b are not especially related. When the value of CL becomes small, unmodeled
dilution increasingly predominates, and it takes more and more regularization to
produce a reliable CL estimate (also see Results, Test 3, Characterization of Tk-GV
residuals.). Figure 1 shows Tk-GV and E2 SETs ﬁts for four cases and a range of
CL-values. Tests of these ﬁndings under more strenuous conditions are performed in
Results, Test 2, Effects of sample-subset selection on Tk-GV model parameters.
Test 2, effects of sample-subset selection on model parameters
Test 2 uses subsets of actual sample concentrations to build up histograms of
parameter values (or derived quantities such as CL and V) when all, or less than all,
of the samples are used for ﬁtting. Figure 2 shows results for the E1 MSE (mean
square error) from 451 trials for the Russell et al. data. Figure 2 also shows MSE
when the E1 function is ﬁt to Monte Carlo E1 simulations with a noise of 0.0009.
i.e., a worst case with 3% SD Gaussian noise. This synthetic control data set shows

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0

0.2610

2.197

50th

75th

100th

0.970

-3.386

-3.991

-4.280

-4.537

-5.364

ln K

0.158

0.9895

0.8649

0.7749

0.7140

0.5945

a

0.123

0.009080

0.004556

0.003452

0.002169

0.000106

b (min-1)

0.162

157.6

105.5

74.28

44.39

1.242

CL (ml/min)

0.166

31124

18490

16275

13127

7404

V (ml)

0.116

6.222

3.703

2.510

1.404

0.1876

sCL (ml/min)

0.113

1344

690.6

474.5

288.0

63.02

sV (ml)

\0.0001

10.51

2.708

1.696

0.8542

0.1075

sCL (ml/min)

Jackknife LOOa

0.004

1283

531.9

384.8

200.6

18.71

sV (ml)

b

a

Shapiro–Wilk probability is a method of testing for a Normal Distribution (ND). Note that lnK is an ND

The second method uses jackknife of 373 leave one out (LOO) Tk-GV trials (m - 1 samples)

The 46 patient studies parameters are listed in percentiles from smallest to largest case from ﬁtting all 8 or occasionally 9 samples. Parameters for the ﬁt equation
C ðtÞ ¼ K taÀ1 eÀbt were regressed by Tikhonov regularization with the dimensionless shrinkage factor, k, a.k.a., the Tikhonov ‘‘smoothing’’ parameter. a is also
dimensionless. K taÀ1 has the units of concentration. The resulting plasma–clearance rates and volumes are CL and V. The standard deviations, sCL and sV are calculated by
two independent methods. The ﬁrst is directly from Tikhonov regularization of m samples, using the standard formula for propagation of small errors

\0.0001

0.09308

25th

Probabilityb

0.01216

0th

Percentile

k

Parameters from Tk-GV ﬁts

Table 3 Test 1, summary of Tk-GV results showing no out-of-bounds, i.e., nonphysical, parameter values. Parameter values do not correspond in adjacent row cells!

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Fig. 1 This shows four cases with a range of CL-values. The concentration is in 100/ml or percent (dose)
per ml. This allows for comparison between cases. Each sequential case with progressively lower CL is
offset to the right in time by a factor of 10i. For the i ¼ 0 case, there is no offset (times 1), and the
smoothing, k, is 0. The i ¼ 1; 2; 3 cases have k ¼ 0:0038; 0:23; 1:6, respectively, that is, the smoothing
increases as CL decreases. Note that for each case AUCE2 \AUCTk-GV and CLE2 [ CLTk-GV

how well the ﬁtting technique perform under ideal conditions, i.e., the underlying
truth is a E1 and the noise is random and Gaussian. The E1 ﬁts to the synthetic data
showed an MSE plot that is almost ﬂat for increasing mean ﬁt time and with a
recovered MSE that matches (recovers) the known variance of the injected noise.
This provides assurance that the ﬁtting techniques and the testing of the variance of
ﬁt residuals is working properly. In contrast to this, when 8 samples are included,
the E1 ﬁts to the actual data (n = 41) from the Russell et al. series have an MSE that
is 20 times greater than the simulation, and 2 or 3 times that of the control data when
only early- or late-time samples are included. Consequently, the measured variance
of E1 ﬁts to early- or late-time samples for a large number of cases is not explained
by the noise in the data. That is, Fig. 2 shows that E1 misﬁts the data by multiple
times the value of a worst-case expected measurement noise.
Interestingly, there were no degenerate ﬁts for either the E1 ﬁts to the data or the
E1 simulations. However, plotting MSE for the E2 ﬁts would be misleading given
the high frequency of problems with the E2 ﬁts (Table 4). For testing the ability of
E2 models to provide reliable results, we formed a total of 332 subsets from the 46
patient data. These subset included 5 or more samples from each patient study
(because E2 has 4 parameters).
For E2 model ﬁt attempts, Test 2 used the constraints 0 a 5 and
0 k2 k1 2. The constraint k2 ! 0 prevented inﬁnite concentration as a limit
as time goes to inﬁnity by prohibiting negative k2 . However, this constraint many
times pinned the k2 parameter at zero suggesting that this variable should be
discarded. That is, the constraint converted the 4.5% negative k2 results into
À
Á
E2 ðk2 ¼ 0Þ¼ k aeÀk1 t þ 1 . In these ﬁfteen E2 ðk2 ¼ 0Þ out of 332 ﬁts, CL is also

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E1, E1 simulation for various sample subsets
E1 fit to data
E1 simulation

E1 and E1 simulation MSE

0.016
0.014
0.012
0.010
0.008
0.006
0.004
0.002
0.000

8
to
6 8
to
5 8
to
4 8
to
3
8
to
2 8
to
1
7
to
1
6
to
1
5
to
1 4
to
1 3
to
1

Samples selected for fit

Fig. 2 Test 2 from the Russell et al. series of 41 patients ﬁt with the logarithm of the SET equations.
À
Á
Behavior of Mean Square Error, MSE, for E1 SET (left), and E2 SETs, k aeÀk1 t þ eÀk2 t , a and k1
parameter behavior (right). Both panels show what happens when data is ﬁt to temporally sequential
samples. For example, 1 to 5 on the (left) x-axis means the ﬁrst through ﬁfth sampling times from 10, 20,
30, 45, 60, 120, 180 and 240 min, i.e., 10 to 60 min. This creates a hump for MSE. The hump is due to a
misﬁt between model and data as illustrated by the circles, which are from 41 Monte Carlo simulations of
E1 functions with a constant noise of (logarithm) concentration of 0.0009. The histogram (right) from
Test 2 (n = 332) shows two populations for the parameters a and k1 obtained from constrained
regression. One population contained a values ranging from 0.302 to 4.80 with k1 values from 0.00276 to
0.312. The second population contained 55 k1 ¼ 5 with a values ranging from 0.00707 to 2
À
Á
Table 4 Test 2, results of E2 ¼ k aeÀk1 t þ eÀk2 t ﬁt attempts using the constraints 0
0 k2 k1 2
Problem

E2 Formulae
À

k2 = 0
k1 = 2
k1 = k2
a=5

Á
k aeÀk1 t þ eÀk2 t
À Àk1 t
Á
þ1
k ae
À À2t
Á
k 5e þ eÀk2 t

5 and

n
261
15
11

Àk2 t

1

kða þ 1Þe
À
Á
k 5eÀk1 t þ eÀk2 t

Total

a

}

44

a

27

}

b

71

332

Breakdown of 332, E2 OLS ﬁts by frequency in each problem category
a

Degenerate (i.e., 27 non-E2) forms

b

Suspicious forms; 71 at constraint boundaries

zero, i.e., poorly estimated. When this occurred, the magnitude of k1 was smaller
than usual and closer in magnitude to the usual values for k2 .
The values of 5 for a and 2 for k1 were the upperÀ limit constraint values for
Á
those parameters. The degenerate form, E2 ðk1 ¼ 2Þ ¼ k 5eÀ2t þ eÀk2 t , occurred in
11/332 or 3.3% of the ﬁts. All of these occurred when the earliest-time samples
included in the data set being ﬁt were the 3rd, 4th or 5th temporal samples. In these

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455

cases, the fast exponential term was \5 Â 10À13 in magnitude, i.e., not reliably
detected without the earliest-time data.
The constraint a \ 5 was useful in preventing implausible outliers, and this
yielded better extrapolation results (Test 4) than the unconstrained alternative. The
lowest a-value detected was a = 0.302, see Fig. 2. However, in agreement with the
bootstrap results, there were a number of cases in which high values of a occurred.
Figure 2 shows that the a and k1 values can be viewed as grouped into two separate
populations. These groupings may not really be separate populations. It is just that if
a were not tightly constrained from above, there would be numerous very large
outlier values of a, and, Eq. 10 would provide poor quality estimates for CL. As per
Table 4, including the single k1 ¼ k2 ﬁt (in effect, an E1 ﬁt result), we encountered a
grand total of 8.1% or 27/332 problematic E2 ﬁts. If one were also to consider all the
a = 5 values at the constraint boundary to be implausible, 71/332 or 21.4% of
solutions are at least somewhat disturbing. Fortunately, for ﬁts to data with all time
samples, or for all time samples save the latest time sample, unconstrained ﬁts
converged to physically permitted values. Moreover, for these sample choices, the
constrained regressions did not have implausible k1 -value solutions at the constraint
boundaries. Thus, Tests 3 and 4 could be performed without discarding any E2 ﬁts
for being degenerate.
Test 2, Effects of sample-subset selection on OLS GV and Tk-GV model
parameters. Figure 3 shows the frequency of occurrence of problematic values for
the parameters a and b from OLS-GV and Tk-GV ﬁts to subsets of samples taken
from the 41 cases from the 8 sample Russell et al. data (The 5 cases with 9 samples
are not shown to keep it simple.) Each plot presents from left to right, the earliest- to
latest-time subsets. For OLS-GV ﬁtting, there are no subsets that do not contain a

25

Number of α > 1 and β ≈ 0 for subsets (n = 41)

β=0
α>1

15

Number

Number

20

Number of β = 0 and α > 1 cases for temporally
sequential subsets of samples (n = 41)

10
5
0
1 to 1 to 1 to 1 to 1 to 2 to 3 to 4 to 5 to
4
5
8
8
8
8
7
8
6

Subsets of 8 samples

10
9
8
7
6
5
4
3
2
1
0

α >1
β ≈0

1 to

4

1 to

5

1 to

6

1 to

7

1 to

8

2 to 3 to 4 to 5 to
8
8
8
8

Subsets of 8 samples

Fig. 3 Test 2 results from OLS GV function (left) and Tk-GV regression (right) ﬁts to 41 patients from
the Russell et al. series of concentration samples. OLS-GV (left) constrained by b ! 0 gradually ﬁnds
fewer b ¼ 0 (and CL = 0) solutions as the selections are chosen later. This is a maximum of 21/41 or
51% CL = 0 solutions for ﬁts to the 1st through 4th samples and a minimum of 1/41 at the 5th through
8th samples. Also plotted are the numbers of times that a [ 1 are encountered. Note that a [ 1 does not
occur when all samples are included, or when a maximum of two last samples are left out. From Tk-GV
ﬁts (right) the open circles with dashed lines show the frequency of a being out of bounds ða [ 1Þ for
subsets of all samples. Problems with detection of CL are shown with solid circles and solid lines, i.e., for
b close to zero ðb\1 Á 10À7 ; CL\0:001Þ. Note that there are no questionable results for a or b when only
one sample or no samples are left out, i.e., samples sets 1 to 7, 1 to 8, and 2 to 8

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model with b pinned to the b C 0 boundary or have a [ 1. The b = 0 results from
OLS ﬁtting constrained by b C 0, would have produced negative b values without
the constraint, and are obviously incorrect. As per the Introduction, a [ 1 is a nonphysical result. Moreover, Fig. 3 shows that the a [ 1 frequency increases for OLS
ﬁts to late-time data only, just when b = 0 becomes infrequent, which is also
problematic. Figure 3 suggests a major improvement in conditioning of the Tk-GV
versus the OLS GV models. It is clear that b & 0 occurrences are signiﬁcantly less
frequent for Tk-GV (1/369) than for OLS GV constrained ﬁtting (62/369). From
Fig. 3, the Tk-GV versus OLS GV ﬁtting, the frequency of a [ 1 (11/369 versus 26/
369) is also less for leaving out samples by this method. For Tk-GV ﬁtting, from
Fig. 3, the three earliest and the three latest sample subsets contain problematic
a [ 1 solutions and in one instance a b & 0 solution. If an a B 1 constraint were
used this would produce a E1 , a single exponential term (i.e., low quality, inﬂated
CL estimating) ﬁt would result. Consequently, the best strategy is to include enough
sampling time between the ﬁrst and last samples to avoid an appreciable likelihood
of producing an a [ 1 ﬁt or, when inappropriate, a b & 0 result.
The trend noted in Test 1 above for the Tk-GV model was that as a ! 1, b ! 0.
Indeed, a ! 1 quickly, and V ! CL=b becomes a constant ratio. To see how this
arises, as Cð1Þ ¼ 1, by using the form of Eq. 14 employing j, one obtains that for
low function (a % 1), CL=b % D=j % V. Now since concentration is relatively
static for low function, and since j, the concentration constant, is even tamer, then
both V and vanishingly small CL should be accurately and simultaneously
measurable. This limiting behavior is a result of minimizing the relative error in CL
given by Eq. A3, which then effectively acts as an additional constraint equation. So
for Tk-GV, if lim b ! 0, then one should be able to use the Tk-GV method to
a!1

measure CL and V for patients with very low CL as then CL=b % D=j % V are
constants.
When one has fewer samples to ﬁt, one begins to ﬁnd b % 0 solutions, which
show lim b ! 0. Amongst the 7963 combinations for leaving out 4 or fewer
a!1

samples, we encountered trivially small values for b and CL only 9 times (0.11%)
and all for the same patient. These occurred for patient 20 when at least the 7th and
8th samples (i.e., the last two) are left out. Using all samples, patient 20 has the
smallest CL (1.24 ml/min) of the 46 patients, and a V of 11631 ml, found with a
relatively high k = 1.61 and high a = 0.9895. The median CL for patient 20 with
L4O (i.e., from 4 samples) is 1.69 ml/min, (compared to the all–8 sample data set
result of 1.24 ml/min.) However, of the 70 L4O trials for patient 20, there are 5
sample combinations (7.14%) with nearly zero renal function. These examples
show how remarkably stable the determination of V is for the Tk-GV method. For
patient 20, if one leaves out samples 1, 4, 7, and 8 the resulting Tk-GV parameters
become a = 1 (exactly to 40 decimal places), k ¼ 7:34 Á 1041 (very high regularization), CL ¼ 1:71 Á 10À43 ml/min, b ¼ 1:53 Á 10À47 min-1, and V = 11164 ml.
This strongly demonstrates that the ratio of CL to b, i.e., V, is preserved by the
Tk-GV method even when renal function is vanishingly small. An upper limit of
a = 1 was not found in prior works which performed OLS GV ﬁts [20]. However,
a B 1 is consistent with CLtotal [ CLurine , i.e., total plasma clearance is greater than

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457

renal clearance from urinary collection. When a & 1, yielding vanishingly small
CL-values, the Tk-GV method found a GV ﬁt with a constant concentration,
C(t) & K. So leaving out data can make the data noisier, and cause a problem in
detecting already small CL. In that case, the residuals, Cobs(t) - C(t), become
Cobs ðtÞ À K. This is further explored in Results, Test 3.
In practice, it seems that one needs to include both early and late-time samples
to provide good conditioning to the ﬁt in the sense of avoiding a [ 1 solutions.
For the leave out 4 or fewer samples, there are 7963 different subsets of the data,
having a total of 99 ﬁts that produce a value of a greater than 1 (1.24% of all of
the trials). For the 4874 trials that included the ﬁrst sample, there were only 6
with a [ 1 (0.12% of the 4874 trials). One should also note the good result that as
long as both the 10 and 240 min samples are included, only 4 samples total are
needed to obtain a Tk-GV solution. These solutions are not much different than
using twice as many samples; see Results, Test 4, Comparison with published
values, below.
Test 3, signiﬁcance of interpolative error
Test 3 examines the goodness of ﬁt of the mean of residuals relative to an
expected residual of zero. Test 3 also examined the Chi-squared statistics for the
joint probability of a zero mean of residuals for the Russell et al. data of
Table 5. For the E1 model, the Chi-squared probability would accept the model
as plausible (P = 0.2) so long as the recommended 2 to 4 h sampling times are
used. However, the E1 model had a low probability of ﬁtting each sample
correctly as the extreme residuals (at early- and late-times) were signiﬁcantly
positive and the middle residual(s) signiﬁcantly negative (Sample groups 6
through 8, t-statistics: 2.22, -2.25, 2.25). Not shown in Table 5 is that this
curvature mismatch was more pronounced between 60 and 240 min (Sample
groups 5 through 8, t-statistics: 5.59, -4.98, -3.77, 5.48), and for all other ﬁts
to samples with more sampling times.
Unconstrained E2 models were applied to all of the available samples. The
unconstrained ﬁts used here often have similar, but sometimes (2/46 with
a slightly greater than 5) smaller magnitude residuals than the corresponding
constrained ﬁts. Fortunately, using all samples, the resulting unconstrained ﬁts are
not unphysical. Thus, it is (slightly) more conservative to use unconstrained,
rather than constrained, ﬁtting to generate residuals for this test. Table 3 shows
this for the 8 time-sample data from Russell et al., the E2 residuals are usually
larger than is expected on the basis of random noise. The low probabilities that
mean residuals this large (or larger) occur at random shows a signiﬁcant failure of
E2 to ﬁt adequately most (6/8) of the data at single sample times, especially the
samples with the two earliest times (probability that this occurred randomly is
P B 0.0002). The joint probability for the residuals at all sample times was then
calculated from the Chi-squared goodness-of-ﬁt statistic, where n is the number of
samples times the number of cases or 8 9 41 = 328 degrees of freedom. This
probability, as per Table 5, is 0.025, making it unlikely that the misﬁts of E2
models to the data are due to noise alone. Moreover, when the Chi-squared

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Table 5 Test 3 results
Fit
Probabilities from
type
Student’s-t

Sample
No. of
selection samples
Chi-squared
divided
by 41

E1

0.032 0.030 0.030a 0.215

6 to 8

E2

0.000 0.000 0.031 0.028 0.040 0.214 0.066 0.037a 0.025

1 to 8

8

GV

0.007 0.027 0.019 0.004

1 to 4

4

GV

0.016 0.043 0.015 0.201 0.249

0.221

1 to 5

5

GV

0.020 0.030 0.012 0.722 0.420 0.257

0.259

1 to 6

6

GV

0.025 0.023 0.008 0.915 0.703 0.476 0.582

0.290

1 to 7

7

GV

0.035 0.024 0.013 0.982 0.884 0.700 0.750 0.658

0.348

1 to 8

8

GV

0.499 0.172 0.617 0.994 0.666 0.757 0.437

0.546

2 to 8

7

GV

0.578 0.414 0.907 0.518 0.664 0.309

0.548

3 to 8

6

GV

0.899 0.982 0.867 0.877 0.643

0.580

4 to 8

5

0.910 0.989 0.923 0.870a 0.573

5 to 8

4

GV

0.090

3

From groups with sample number
1

2
3
Having " (min)b
t

4

5

6

10.9

45.9

60.4

121.7 180.7 242.3

20.3

30.3

7

8

Probabilities of goodness-of-ﬁt using Student’s t and Chi-squared for comparison of three ﬁt functions for
the Russell et al. data (n = 41 cases)
a

The best results for E1, E2 and GV models are in bold. For each ﬁt method and sample time subset, each
sample time group is associated with a Student’s t probability of the mean being zero, i.e., unbiased. Each
ﬁt method and sample time subset corresponds to a Chi-squared probability of the sum variance regularized squared residuals being zero
b
Listed below the sample number, 1, 2, 3. . .8, each of which has 41 samples, are the mean times, ", for
t
each group of samples

goodness of ﬁt is extended to all 46 patients in both data sets (with 373 degrees of
freedom), P = 0.039, or an insigniﬁcant chance that the mean residuals at each
group of sample times arose from noise alone. These goodness-of-ﬁt test results
provide evidence that E2 does not interpolate properly. Summarizing both E1 and
E2 SET model results, the Student’s-t probabilities for each group of sample times
shows improper interpolation. How can one then expect SETs to do the harder job
of extrapolating properly?
Test 3 also examined the GV function goodness-of-ﬁt t-testing and Chi-squared
testing for the Russell et al. data of Table 5. Furthermore, for GV ﬁtting, the added
disadvantage of using b C 0 constrained ﬁtting was compared to the more liberal
unconstrained E1 and E2 SET ﬁtting to construct Table 5. Despite this additional
disadvantage, the GV model clearly outperformed the two SET models. As shown
in Table 5, there were no good ﬁts from the E1 and E2 models, as either the
Student’s-t or Chi-squared probabilities or both were insigniﬁcant. For the GV
model, as long as the data before 20 min was not used, all the t-statistics and Chisquared values indicated a signiﬁcantly good ﬁt. Indeed, when only the last four

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and sample number

0.0014
0.0012
0.0010
0.0008
0.0006
0.0004
0.0002
0.0000
-0.0002

Residuals versus

Residuals

Residuals

Residuals versus

459

1
2

3
4

Sample number

5

6
7
8

1.180
0.327
0.156
0.092
0.052
0.021
0.007
0.001

and sample number

0.0014
0.0012
0.0010
0.0008
0.0006
0.0004
0.0002
0.0000
-0.0002
1

2
3

4

Sample number

5
6

7
8

0.0003
0.0011
0.0024
0.0031
0.0040
0.0046
0.0060
0.0078

Fig. 4 From Tk-GV applied to the Russell et al. data, residual concentrations (100/ml, not as logarithms)
for each of 8 samples from 328 LOO trials are grouped into 8 octiles. Sorted (left) into octiles of
increasing shrinkage, k, are mean residuals and sample number. Note that the residual concentrations are
unbiased for low shrinkage, and increasingly biased for progressively larger shrinkage. Sorted (right) into
octiles of decreasing renal rate constant b are mean residuals versus b and sample number. Note that the
residuals are unbiased for high elimination rates, and more biased for low renal function

samples times were used, the t-statistics were small, i.e., jt - statisticj\0:2,
indicating a very good ﬁt.
Characterization of Tk-GV residuals. Figure 4 shows the mean residuals
from Tk-GV regressions of the 41 Russell et al. cases. These residuals are
plotted for LOO ﬁts in 8 equal octiles of increasing shrinkage values having 41
regressions in each octile or 328 in total. When k = 0, the Tk-GV solution is
unbiased and identical to the OLS solution. As can be seen in Fig. 4, when the
shrinkage, k, is very small, the residual function, or difference between the ﬁt
and the concentrations, is also small (mean ﬁrst octile k = 0.001, and more
generally for all 46 LOO series, k = 0, 17/373 times or 4.6%). This is
consistent with the smallest relative errors for measuring CL corresponding to
the smallest k, and the largest CL values. That is, for high normal renal
function, the GV model sometimes ﬁts the data without the need for Tk
smoothing. Figure 4 conﬁrms that for high renal elimination rates, b, the
residuals are small. As the k values increase in Fig. 4, and the b values
decrease, the residuals especially for the earliest sample(s) increase so that the
ﬁt then underestimates the concentrations. Moreover, for large k or low b
values, the ﬁt overestimates the concentrations of the late samples. In summary,
as the shrinkage increases, so too does the regression bias needed to ﬁnd the
Tk-GV ﬁt with the correct late-time behavior and minimum error in CL.
Figure 4 shows graphically how large the bias becomes when estimating very
low renal function. In effect, when there is zero renal function, the Tk-GV ﬁt is
a ﬂat-line, and the residual concentration reﬂects unmodeled dilution, with high
initial concentration, that decreases with time. On the other hand, when k = 0
and there is high CL, the Tk-GV ﬁt solutions becomes OLS GV regressions and
ﬁt the concentration curves well.

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Test 4, extrapolative error
Test 4 examined whether the E2 SET, GV and Tk-GV models accurately extrapolate
beyond the range of the measurement times. For noisy data and good models,
extrapolations will produce half of their predictions in the extrapolated range above
the actual value and half below. Models whose median extrapolated values do not
coincide with the expected value will produce a biased value for Est-AUC.
Extrapolation results are summarized in Table 6. The worst extrapolations were
for E2 SET models having 31 of 46 extrapolations under-predict the interpolative
values from the ﬁt to all the data at the latest sampling time. The Wilcoxon signedranks sum test gives this a two-tailed probability of P = 0.0046 of the extrapolated
values being an unbiased predictor of the median for unconstrained ﬁts (with
less powerful sign test P = 0.026, 2-tailed), and a Wilcoxon P = 0.0071 for
constrained ﬁts. Figure 5 shows a typical E2 result for the 169Yb-DTPA data. This is
Table 6 Test 4 results from extrapolation testing using the Wilcoxon signed-ranks sum test.a Over- or
underestimation of extrapolated ln C(t) values for ﬁve ﬁt methods. Note that Tk-GV extrapolated best
Fit type

Wilcoxon test
probability

Median
differencea

Median difference
95% conﬁdence intervals

E2

0.0046

-0.0383

-0.0634 to -0.0140

E2b

0.0071

-0.0373

GV

0.2446

0.0131

-0.0088 to 0.0491

GVc

0.7638

0.0039

-0.0158 to 0.0334

Tk-GV

0.9087

-0.0011

-0.0192 to 0.0217

b
c

This does not always agree in sign with a sign difference
À
Á
Constrained ﬁt: 5 ! a ! 0, 2 ! k1 ! k2 ! 0 for C ðtÞ ¼ k aeÀk1 t þ eÀk2 t
Constrained ﬁt: b ! 0 for C ðtÞ ¼ KtaÀ1 eÀbt

Fig. 5 Test 4 example. The
y-axis units are ln(100/ml), i.e.,
logarithm of percentage (of total
dose) per ml. Shown is the
serious problem of extrapolation
to values that are in general
lower than the expected
concentration. Shown for the 8th
data point is the interpolation
error (in err) and extrapolation
error (ex err). At 240 min, the
extrapolation of the ﬁt to the ﬁrst
7 samples (dashed line)
noticeably underestimates both
the ﬁt to all samples (solid line)
and the 8th sample
concentration itself

lnE2

-4.5

lnE2 fit to points 1 through 8

Patient 12, lnC [ln(100/ml)]

a

-0.0629 to -0.0118

-5.0

lnE2 fit to points 1 through 7
Data

-5.5

-6.0

-6.5
in err
ex err
-7.0
0

30

60

90 120 150 180 210 240 270

t (min)

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461

a middle-of-the-road case with midrange renal function, and typical underestimation of future concentration using E2. Note that the last data point to the right is also
slightly above the interpolative values from the ﬁt to all of the temporal samples.
This is the typical result in our ﬁts and is consistent with signiﬁcant interpolative
error at that point (in Table 5, P = 0.037, n = 41). From this test, we learn that E2
SETs is a poor model for marker concentration because it too frequently (31/46,
sign test P = 0.026) underestimates the last sample value, which in turn suggests
that the Est-AUC calculated from E2 models will be systematically too small.
Test 4 also examined the GV function’s performance for extrapolation. The
difference between the concentrations at the time of the mth sample between the ﬁt
to m samples and m-1 samples was positive 22 times and negative 24 times. This
shows that proper extrapolation cannot be ruled out by the sign test (P = 0.88,
2-tailed) or the Wilcoxon test (P = 0.24, two-tailed). Using b C 0 constrained
GV ﬁtting, the Wilcoxon probability becomes P = 0.76 (two-tailed), with no
change in the sign test result. In other words, one cannot discard the GV as a
properly extrapolating function for the data set used here. (OLS-GV ﬁtting fails
Tests 1 and 2.)
Test 4 examined extrapolation of the Tk-GV model. The best extrapolation, from
Tk-GV, has a Wilcoxon test probability of 0.91 of the errors being from random
noise. Hence, the Tk-GV method offers better assurance that AUC will be correctly
estimated than by using the other methods of Table 6. Calculation of CL uses
extrapolation of the concentration curve to inﬁnity to ﬁnd the AUC estimate. Since
the Tk-GV method extrapolates better than other methods, the Tk-GV method’s
value for CL should be more accurate as well. Fortunately, this can be tested.
CL values were calculated for the ﬁrst m - 1 samples versus all m samples, and
the effects of extrapolation on CL values examined. Table 7 shows this for the
Tk-GV model and for the constrained E2 SET model, which provided the best
SET-model performance. From Table 7, one can see that the beneﬁt of waiting
another 65 min after the next to last sample to take a last sample is to reduce
the value of CLTk-GV by about 0.5 ml/min and to reduce constrained CLE2 by from 1
to 4 ml/min. Also note that the change in mean CL, i.e., the DCL values, the sCL
(SD of CL) and CV of CL are all improved for the Tk-GV model versus the
Table 7 Test 4 difference between CL (ml/min) values without the last sample (m-1) and with all
m samples for the Tk-GV and constrained E2 SET models
CLm-1

CLm

DCL

sCL

Mean

74.99

74.47

0.52

3.90

8.96%

Median

74.73

74.28

0.46

2.65

5.05%

Mean

81.95

80.95

1.00

6.10

11.23%

Median

82.78

78.76

4.02

3.58

5.73%

CV

Tk-GV

E2

DCL is the difference between the CL values calculated using m - 1 and m samples. SCL is the standard
deviation between each calculation, and CV is the coefﬁcient of variation from using m - 1 and
m samples. Note that sCL ðTk-GVÞ [ sCL ðE2 Þ is implausible, (P = 0.02, Wilcoxon, 1-tailed)

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constrained E2 SET model. The sCL were pair-wise tested with Wilcoxon signedranks sum test for improvement in performance of the Tk-GV method as compared
to constrained ﬁts with an E2 SET model. This showed that the precision of
CLTk-GV was signiﬁcantly better than that of CLE2 (P = 0.045, two-tailed). Another
question is whether the 0.5 ml/min drop in CLTk-GV from ﬁts to m rather than
m - 1 samples is signiﬁcant. The Wilcoxon signed-ranks sum test of the 46 paired
differences is P = 0.23, two-tailed, or not signiﬁcant. However, the same
calculation for the constrained CLE2 is signiﬁcant (P = 0.0049, two-tailed).
Reiterating, CLTk-GV was not signiﬁcantly altered and constrained CLE2 lost
signiﬁcant estimated CL by adding another period of an additional average of
65 min to take a last 8th (or 9th) sample. Not shown in Table 7 are the V results for
both models.
As calculated from the Tk-GV ﬁt parameters, V was 16378 ± 644 ml
(mean ± mean sV) with a mean CV of 4.04%. For the constrained E2 SET model,
V was 15281 ± 1589 ml with a mean CV of 9.49%. This suggests that use of the
Tk-GV method represents a major increase in precision in the determination of V
and that this reduction in V’s CV is very signiﬁcant (P = 0.0014, Wilcoxon, twotailed). Summarizing, the Tk-GV model parameters were signiﬁcantly less altered
by varying the number of samples ﬁt than constrained E2 SETs.
Given this evidence that Tk-GV is the superior method, an appropriate question
is how this compares to the results from constant infusion of inulin and AUC with
exponential extrapolation, which latter are often touted as the gold standards for
such measurements. To assess this, one can compare our results to those of methods
in the literature. This is done next.
Comparison with published values
Florijn et al. [39] show that use of E2 overestimates plasma clearance from constant
infusion of inulin and that CLtotal [ CLurine . Florijn et al.’s scaling of CL conversion
between methods was done by the method of Du Bois and Du Bois [62]: EðBSAÞ ¼
0:007184W 0:425 H 0:725 in m2, where W is patient mass in kg, and H is patient crownheal height in cm. In short, Florijn et al. give a 5.1 ml/min/1.73 m2 greater CLE2
than CLtotal . The comparable number from the 46 studies here is 6.1 ml/min/
1.73 m2, see Table 8. Our CLE2 À CLTk-GV value is similar to Florijn et al.’s
CLE2 À CLtotal result, and the difference observed by Florijn et al.’s is within our
95% CI. If one accepts this result at face value, one concludes that CLTk-GV %
CLtotal , which is signiﬁcantly more accurate than CLE2 .
Testing of Florijn et al.’s scaling methods was performed with ANOVA (analysis
of variance). Note that more precision for a regression formula need not be
signiﬁcantly more precision, when applying the ANOVA requirement that the
probability for each model’s parameter(s) partial correlation coefﬁcient achieves
signiﬁcance. Even a comparison of R-values or standard errors adjusted for highly
correlated conditions (rarely used correctly) would yield probabilities that have no
bearing on the ANOVA requirements for signiﬁcance. Our result,

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Table 8 Comparison of CLE2 , CLtotal and CLurine plasma clearances from three sources. Note that
CLTk-GV is more accurate than CLE2 , yet is within the CI for constant infusion and 24 h AUC
Units:

ml/min/1.73 m2

Source:

Florijn et al.

E2 approximation: 5 min injectiona

Percent
This paper

Moore et al. This paper

Constrainedb

4 h AUCc

Constrained

24 h AUC

Tk-GV

CLtotal method:

Constant infusion Tk-GV

CLE2 [ CLtotal

5.1

6.1

4.7 to 7.6 10.0

CLtotal [ CLurine

8.3

*7.3

–

CLE2 [ CLurine

13.4

–

–

95% CI

a

Inulin, Cobs ðtÞ % C ðtÞ ¼ C1 e

b 99m

Àk1 ðtÀ5Þ

95% CI
10.6

8.6 to 12.7

7.6

*7.0

–

17.6

–

–

Àk2 ðtÀ5Þ

þ C2 e
À
Á
Tc-DTPA, Cobs ðtÞ % C ðtÞ ¼ k aeÀk1 t þ eÀk2 t with constraints 0

a

5 and 0

k2

k1

2

c 51

Cr-EDTA, from numerical integration of Cobs ðtÞ, then mono-exponential extrapolation

CLE2 CLTk-GV
¼
þ 6:1 ml=min=1:73 m2 ;
BSA
BSA

ð21Þ

has a large standard error of estimation for CL of 5.0 ml/min/1.73 m2 with
R2 = 0.9869. However, the exponents of W and H of Eq. 21 (in BSA) are not
statistically warranted (P [ 0.1, ANOVA). This is expected as catabolism promotes
CL and BSA has been shown to be spuriously correlated to CL [63, 64]. To calculate
better scaling, one uses the mean CLTk-GV value of 74.47 ml/min and the mean
bTk-GV value of 0.003614 s-1, and obtains




CLTk-GV 0:9972 bTk-GV À0:1155
CLE2 ¼ 1:106 Â 74:47
ml=min;
ð22Þ
74:47
0:003614
which reduces the standard error of estimation to 4.2 ml/min, and increases the R2
to 0.9919. Note that in Eq. 22, the offset has been dropped as being probably not
different from zero (0.1 ml/min, P(offset = 0) = 0.96, two-tailed t-test). Similarly,
the CLTk-GV 0.9972 exponent could be changed to one without changing the
equation signiﬁcantly. The constant multiplier 1.106 suggests a 10.6% higher
CLE2 than weighted mean CLTk-GV value. Compared to their respective standard
methods, the 10% overestimation of CL from 4 h of data from Moore et al. and
the 10.6% (95% CI, 8.6% to 12.7%) overestimation by CLE2 from Table 8
imply comparable exponential ﬁt underestimations of late AUC values. Thus, the
CLTK-GV-values from 4-h of data correspond to 24-h CL-value estimates from
Moore et al. with the difference, 0.6% (0.04 SD), being insigniﬁcant.
Equations 21 and 22 are imprecise because the E2 SET renal elimination rate
parameter is statistically unwarranted (P = 0.12, two-tailed, see Test 1). Using all
samples and the E1 SET model, one ﬁnds a better regression ﬁt,




CLTk-GV 1:038 bTk-GV À0:2102
CLE1 ¼ 1:135 Â 74:47
þ5:967;
ð23Þ
74:47
0:003614

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where the standard error is 3.4 ml/min., and R2 = 0.9947. And, an offset appears
probable as P(offset = 0) = 0.005. Eq. 23 does not fail ANOVA t-testing, and is
more precise than Eqs. 21 and 22, which do fail ANOVA t-testing in one way or
another. However, the E1 ﬁt to the data fails t-testing for goodness of ﬁt (Test 3).
Although one can use a formula like Eq. 21 to compare with published values,
CLE2 is unnecessary for modeling our data. If one has inert marker data enough to
calculate a CLE2 then the best conversion just calculates the more accurate and
precise CLTk-GV for that data. Being robust, CLTk-GV only needs 4 samples for a
solution. For example, if one chooses the 10, 30, 120, and 240 min Russell et al.
samples and compares this to using all 8 samples then
CLTk-GV ð4 samples) ¼ 0:9992 CLTk-GV ð8 samples)

ð24Þ

with a standard error of 2.7 ml/min and R2 = 0.9965, where a non-zero intercept
is unlikely P(offset = 0) = 0.91, and a slope of one is plausible (95% CI,
0.9895–1.0089).

Discussion
The properties of a good model for renal markers are (i) to interpolate Cobs
accurately over short time intervals, and (ii) yield a terminal or limiting function for
the concentration curve with a good ﬁt to the data, i.e., it should extrapolate
correctly so that the Est-AUC accurately predicts Phy-AUC. (iii) It should contain as
few parameters in number as possible, parameters whose non-degenerate values are
physically interpretable and whose errors are small so that plasma clearance and
volume of distribution can be precisely calculated.
In otherwise totally unrelated work in children with normal renal function, some
of us found that CL-values determined from 99mTc-DTPA E1 SET models scale for
body size as approximately proportional to V 2=3 W 1=4 , where V is volume of
distribution, and W is patient mass in kg [63, 64]. In the language of West et al.
[65–67], this V, W scaling formula (with V estimated from E1 ﬁts) suggested an
underlying fractal network as the root of this functional dependency. However, the
two propositions, that of an E1 (i.e., not scale invariant) and that of fractal (i.e.,
scale-free) model are somewhat antithetical and hence paradoxical. This prompted a
search for a scalable model for DTPA radiochelates that led to the GV function. We
are not the ﬁrst authors to postulate a fractal model for dilution curves [30, 31]. Such
structures are scale-free, and must be described by mathematical formulas that are
also scale-free, e.g., power functions. Generally, biological fractal structures show
self-similarity, e.g., branching structures, independent of magniﬁcation down to the
small-scale limit (interstitial terminal spaces, in the context of DTPA chelates) of
those branching structures. How the body introduces a power function of time in the
GV function, which seems to best reﬂect the late-time concentrations of markers, is
a matter of conjecture [e.g., see 20,31]. Moreover, GV functions model require
adaptation for practical usage.

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The need to impose constraints to control parameter instability is inappropriate
for sensible modeling. Imposing constraints, i.e., insisting on physical solutions, in
SET modeling is occasionally performed by its proponents. For example Russell
et al. [57] use a Bayesian prior approach to E2 ﬁtting and Khinkis et al. [68] use
constrained optimization techniques. Without constraints, non-physical results may
occur as per Table 1. There are no ideal values for the upper constraints for the fast
rate constant (k1) or the relative ﬁrst to second exponential initial ﬂow (a).
Furthermore, constraints were also imposed on OLS-GV ﬁtting, and the need for
this was blatant. That is, negative renal rate constants (b) sometimes occurred.
The typical argument for the use of SETs is that they ﬁt the plasma disappearance
curve well. SET models arise naturally from separate, linearly-coupled, fast mixing
compartments. The notion of SETs forming a complete basis, i.e., that anything can
be described by SETs, is so ingrained that one often neglects the obvious, that not
every basis is efﬁcient for every problem. Sine and cosine functions also form a
complete basis, and that is insufﬁcient grounds to recommend using that basis to ﬁt
marker concentration curves. Concerning efﬁciency, E1 ﬁts to the data have multiple
times larger residuals than an E1 simulation using exaggerated noise levels (see
Fig. 2 and Results, Test 1). E1 proved to be an arbitrary model in that it much less
efﬁciently ﬁtted the data than expected for a properly matched two-parameter
model/data combination. Test 2 ﬁnds that E2 models are overly complex and that the
increase in ﬁt quality over an E1 model provided by adding another exponential term
is unwarranted P(k2 = 0) [ 0.05. The additional constant multiplier and exponential coefﬁcient of the second exponential term too frequently resulted in degenerate
forms (Results, Test 1, and Table 4). There is a clearly identiﬁable physical model
for SETs, which sets an example for physical modeling. The authors are quick to
point out, that there is nothing wrong with compartmental theory; SETs just do not
ﬁt or span the bolus plasma–dilution curves for DTPA radiochelates properly, and
should not be used for that purpose.
SET models underestimate concentrations at late times and are known to
underestimate Phy-AUC over the entire range of CL-values, necessitating correction
factors to reduce the calculated values for CL for the E1 and E2 SET (or equivalent)
models [10, 11, 24, 39]. Current guidelines suggest that CL from E1 SET models
should be multiplied by 0.87, called the Chantler correction factor, which implies
that AUC is underestimated by a factor of 0.87 [10, 11, 69]. Also, for E2 SETs, we
are not the ﬁrst authors to ﬁnd that they overestimate CL [23, 39]. We may,
however, be the ﬁrst to give a global reason for this occurring. When E2
extrapolation was performed, E2 was also found to overestimate CL, and in general,
En overestimates CL. Indeed, the under-extrapolation of AUC is from the native
shape of exponentials. Exponentials go to zero too quickly and cannot mimic the
true shape of inert marker concentration curves, and this defect cannot easily be
eliminated by increasing the number of exponential terms. After observing slow
redistribution in nephrectomized dogs, i.e., with CLurine ¼ 0, Schloerb warned us to
expect SET models to fail [17]. Indeed, the ﬁndings of improper extrapolation for E2
and mismatched curve shape of E1 ﬁts to late data argue (Table 5) against the notion
of a ‘‘terminal exponential’’ for the concentration curve. In other words, just because
all SETs eventually become a straight line on a plot of logarithm of concentration,

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does not imply that the logarithm of concentration curves of data become linear with
elapsed time. That is, SETs models always converge to zero concentration at a
constant rate of dilution for late times, where, physically speaking, a decreasing rate
of dilution occurs at late times from Eq. 1: CLtotal ¼ CLbody þ CLurine , where CLbody
slowly decreases in time. Larson and Cox [1974, 18] related that an E2 ﬁt to plasma
24
Na concentrations had different coefﬁcients when ﬁt from data decayed to 1% of
the initial value, compared to slower coefﬁcients from a ﬁt containing even more
data, and 72 h whole body 51Cr-EDTA retention has been measured as 4.5% [1969,
41]. So the evidence for CLbody slowly decreasing in time has been available for a
long time.
One workaround for using SET models is to attempt to optimize the ﬁtting of
poorly ﬁtting SET functions (e.g., see [57, 68, 70]). Another workaround for using
SET models when renal function is low is to extend the data collection to include
very late times. Chantler and Barrett [69] cite Maisey et al. [71] relating that
‘‘Though the tracer is substantially equilibrated by 2 h in the normal subject,
complete equilibration probably takes much longer. The fraction not equilibrated by
2 h, however, is so small that it cannot be detected in relation to a fast clearance, but
in the presence of renal failure (GFR [sic, CL], 15 ml/min per 1.73 m2), the
apparent clearance of this small fraction into the tissues becomes more important in
relation to the renal clearance and accurate estimates of GFR (sic, CL) require
plasma sampling for up to 6 to 8 h.’’ This comment is optimistic for the E1 ﬁt being
referred to, and for which we detected a departure from good ﬁtting at 2 h—see
Table 5. However, Chantler and Barrett’s observation does apply to the OLS
GV-model, which ﬁt 1–4 h concentrations well. That is, with one caveat, namely
that these good ﬁts are problematic for ﬁnding CL. So, it would seem, the only
expedient solution is to use Tk-GV, and let the adaptive smoothing ﬁnd the correct
values for CL when CLbody is larger than usual compared to CLurine .
For the clearance problem, the OLS-GV model has less wayward assumptions
than SET models. For example, the GV model can explain that plasma clearance of
a marker is faster than urinary collection of that same marker, because the sense of
the body clearance term, (a - 1)/t of Eq. 18, is a loss from plasma that never
appears in urine, i.e., that 0 \ a B 1. Further, as Table 5 shows, the GV function ﬁts
the late samples precisely, where SETs offer only poor ﬁts.
Is the adaptation of the gamma variate, GV, to the plasma clearance of
radiolabeled chelates incomplete? For example, Wise [20] and later Macheras [31]
merely used power functions, or a GV with b = 0, for those cases or for early data
for each case in which the b rate constant was ‘‘undetectable’’ (sic-unphysically
negative), and claimed that it is only the last few samples that determine this rate
constant. The late samples do not determine detectability. Five of 46 cases had an
incorrect b when all samples were used for ﬁtting. In agreement with Table 5, the
most plausible explanation for this is that the OLS GV ﬁt procedure is less
appropriate for earlier-time data, and thus the GV is an incomplete model. Rather,
the use of only late samples avoids modeling earlier dilution that is not related to the
GV function (or a power function), the inclusion of which would cause detection of
the wrong value of b. Only one of 46 cases (Results; Test 3, above) had an incorrect

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b value when only the last four blood samples were used for GV ﬁtting. However,
as shown in Fig. 3, if only the last four samples are used for OLS GV ﬁtting, then
b = 0 becomes infrequent (1/41 and 1/46), just when a [ 1 becomes frequent
(12/41 and 14/46). Unfortunately, results with a [ 1 are implausible, so the
OLS-GV model is not useful by itself. What then causes occasional a [ 1 results
when the latest four samples of the 4-h DTPA data are ﬁt with a GV? In those cases,
the GV ﬁt is unstable not in the region of the well-ﬁt data, but at the t-axis intercept
where the extrapolated value may have any concentration ranging from 0 to inﬁnity.
ÀR 1
ÁÀ1
In formal language, ﬁnding CL from CLGV ¼ D 0 GVðtÞdt
is challenging
because the integral is ill-posed, i.e., it depends sensitively on the ability of the OLS
GV ﬁt to correctly extrapolate (integrals i and iii) in Eq. 6, and these extrapolations
depend sensitively on the speciﬁcs of the data. The ill-posed nature of CLGV was
solved by introducing CLTk-GV , i.e., regularizing the ﬁtting procedure and hence
stabilizing the method so that it is less sensitive to data speciﬁcs. An additional
strong beneﬁt is that the Tk-GV regularization also adapts to minimize the relative
error of CL. To be useful, this regularization must weight the early- and late-time
data in such a way that a and b exhibit ranges that are physical. Notice, however,
that the GV model, even when regularized properly, is still an incomplete model.
Tk-GV, a solution for the ill-posed clearance problem
The results show (see Fig. 3) that Tk-GV ﬁts have signiﬁcantly better performance
for predicting a and especially for b than from OLS GV ﬁts. Indeed, b illconditioning renders OLS GV ﬁtting useless for determining inert marker CL. As
above, marker is always strictly lost to the interstitium, thus a B 1. Moreover,
a B 1 results were obtained for Tk-GV ﬁtting at least when the sample times chosen
were of sufﬁcient temporal range to allow for good conditioning of those ﬁts.
An alternative view is that since the magnitude of the plasma clearance to tissue,
(a - 1)/t of Eq. 18, is ever decreasing (it is proportional to 1/t), it only represents
loss to parts of the interstitium that are not in rough concentration equilibrium with
the plasma. In such a view, portions of the interstitium that are in more intimate
contact with the plasma, with fast exchange times ( t, readily exchange marker
with the plasma. It would then be the less intimate interstitium, which represents a
pure sink of marker, which in turn explains why the 1/t term decreases in time.
Whatever one’s point of view, inert marker concentration ultimately follows a GV
temporal dependency. Figure 4 shows less discordance between the concentration
curve and the GV ﬁt function from the Tk-GV method when renal function is high
(i.e., when k is small). When the renal function is good, it takes less time to establish
a dynamic-equilibrium concentration than when renal function is poor. The meaning
of this is that the GV describes an ultimate dynamic-equilibrium concentration
balance with plasma having two marker loss processes, one being loss to urine
through the kidneys, i.e., the constant loss rate term dCðtÞ=dt ¼ ÀbC ðtÞ, and the
other being loss to other parts of the body still not in concentration equilibrium with
the plasma, which decreases in time, i.e., dCðtÞ=dt ¼ CðtÞða À 1Þ=t. Thus, the right
hand side of Eq. 18 is consistent with a concentration weighted average value

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interpretation of CL, Eq. 3, in that (i) the rate of CL is variable in time, (ii) the AUC
dose method gives a concentration-weighted, average estimate of this variable CL,
and (iii) for any bolus inert marker CLtotal is faster than CLurine [24, 39].
Concerning the volume of distribution, the most useful deﬁnition is also an
average value, i.e., that based on the concentration-weighted mean residence time as
given in Eqs. 4 and 16. Notably, our results show V values to be remarkably stable
and to give consistent values even for vanishingly small values of CL. This results
from the Tk-GV technique’s minimization of sCL =CL, which prevents zero values of
CL by enforcing CL=b % D=j % V for low function (see the Results, Test 3) where
D/j is a very stable concentration ratio. Furthermore, apart from the Tk-GV method,
we know of no techniques that are capable of giving reliable estimates of V when
CL is very small. In addition, Tk-GV’s a B 1 results, i.e., that plasma deﬁnitely
leaks inert marker into the body, is suggestive of and consistent with the total
plasma clearance being greater than urinary collection of cleared substance, i.e.,
renal clearance. Finally, the Tk-GV model is extremely stable computationally.
Only 4 plasma samples are needed for Tk-GV ﬁt solutions, preferable including one
early, e.g., 10 min and one later, 4-h sample. E2 SET models, on the other hand, are
sometimes degenerate and converge to non-E2 functions with a 6% likelihood of
this happening using 8 plasma samples.
Precision
Results for Tk-GV method precision were calculated by three methods as in Test 1
and 4, and Tables 3 and 8. Tk-GV offered a signiﬁcant, improvement in sCL (SD of
CL) from constrained E2 SET models, P = 0.045, two-tailed, from pair-wise
Wilcoxon signed-ranks sum. Test 4 relates a very signiﬁcant improvement in the
relative precision of VTk-GV compared to VE2 , (Wilcoxon P = 0.0007, one-tailed).
Thus, the Tk-GV model outperformed the constrained E2 SET model with respect to
CL and V precision. Test 4 showed that for CLTk-GV the decrease in CL from
extrapolating 65 min (from *3 to *4 h) was insigniﬁcant P = 0.23, two-tailed.
However, the same calculation for the constrained E2 SET model was signiﬁcant to
the P = 0.0049 level. Thus, 3 h is not enough elapsed time to use an E2 SET model.
But, the same thing cannot be said for the Tk-GV model.
Accuracy
In Table 8 and Results, Comparison with published values, using inulin constant
infusion as CLtotal , Florijn et al.’s calculated difference CLE2 À CLtotal is 5.1 ml/
min/1.73 m2. Moore et al. using 51Cr-EDTA found a 10% difference between CL4h
and CL24h . Since the CLE2 À CLTk-GV difference is 6.1 ml/min/1.73 m2 or 10.6%,
this largely reconciles the differences and suggests agreement amongst three
methods; CLTk-GV and two gold standard CLtotal estimates, i.e., that of total plasmaclearance as CLTk-GV , as CLtotal from inulin constant infusion and as CLtotal from
CL24h . A more complete validation of the proposed Tk-GV method would require
comparison with suitable gold-standard measurements in the same group of
subjects. Nevertheless, the comparison with published data provides strong evidence

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that CLTk-GV is accurate to within small errors, and is consistent with agreement
among the gold standards themselves.
Moreover, the Tk-GV model clearly outperforms its SETs competitors in many
ways. For example, it would be problematic to perform an L2O experiment on E2,
as E2 regression attempts may result in non-E2 models—see Table 4 and [57]. So,
by process of elimination, the L2O experiment can only be performed using more
robust ﬁt-models like Tk-GV or E1 SET. There were no failures to ﬁnd a global
minimum converged Tk-GV ﬁt solution for the leave out B 4 samples for 7963
sample combinations, even though a small percentage of those 7963 ﬁts were illadvised sample choices, i.e., when b & 0 or a [ 1, e.g., see Fig. 3. The Tk-GV
method only needs 4 samples over 4 h for an accurate solution. This is
exceptional performance. Russell, who supplied much of the data used here,
suggests that the minimum practical number of samples for our conditions and an
E2 model is six [72]. Current determinations of CL from bolus techniques are
recommended for use with 4 or 5 h of sampling with 8–13 samples [10, 11], up to
6 or 8 h of sampling in renal failure [69, 71] and 24 h of sampling without using
correction factors [24].
The Tk-GV model correctly estimates the CL and V values robustly, even when
CL is near zero. Even constant infusion cannot achieve this feat, as the infusion is
problematic in renal failure. This gives conﬁdence that the Tk-GV method and GV
models for marker concentrations have a physiological basis, even though some of
the details are unknown. For example, how to diminish inert marker total plasma
clearance to estimate urinary clearance cannot, at present, be calculated directly.
However, the GV differential equation leak term is consistent with this occurring
when a B 1, of which the Tk-GV results provide good assurance, and other models
have no such results. Moreover, the leakage can at least be estimated from prior
work. Moore et al. estimate this to be 7.6% greater CL from total clearance than
urinary clearance, which would correspond to 7% for CLTk-GV from Eq. 22. Finally,
for certain applications, e.g., drug effects for which plasma concentration in time
(AUC of dosage) is the consideration, total plasma-clearance may be more relevant
than urinary collection, e.g., methotrexate, carboplatin [2]. Wise [20] pointed out
that the majority ﬁts with E2 and E3 SET functions to drug clearance curves that he
surveyed were well ﬁt by GV functions, and he suggested the use of GV functions
instead of exponentials to model drug elimination. A GV function is equivalent to a
SET model with an inﬁnite number of terms, but with only three parameters instead
of an inﬁnite number of parameters. The GV functions from the Tk-GV method
afforded curve ﬁts efﬁcient for ﬁnding CL of the inert markers examined here. Thus
the Tk-GV method provides a key step likely missing for implementation of Wise’s
suggestion to use GV functions for CL determinations, namely somewhat
paradoxically to extract an accurate terminal ﬁt of marker concentrations from
early-time data while at the same time obtaining physiological range plasma leak
constants appropriate to inert markers. This is where for i.v. bolused inert markers,
the Tk-GV method bridges the gap by allowing for more robust, more versatile,
more useful, and more credible total CL estimates. The adaptation of the Tk-GV
model for more complex pharmacokinetics, absorption, infusions or multiple-dosing
regimens is left for future work.

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Conclusions
Modeling tests were constructed and used to examine three models as applied to 46
radiolabeled DTPA bolus concentration curves. Sum of exponential term (SET)
models failed the tests for parameter stability, statistically warranted parameters,
curve shape, goodness of ﬁt, accuracy and useful extrapolation. Gamma variate (GV)
functions ﬁt the late (2 to 4 h) plasma samples with high probability of goodness of
ﬁt, but were unreliable for estimating CL. Adaptive Tikhonov (Tk) extraction of the
GV functions (Tk-GV) robustly converged to global minima with good evidence of
precise (Table 1) and accurate CLTk-GV values, which agreed with published
corrections of CL from constant infusion of inulin [39] and 51Cr-EDTA bolus
modeling [24] to within insigniﬁcant errors (Table 8) without the need for correction
factors, constant infusion, overabundant samples, or prolonging the time of data
collection from 4 to 24 h. Generally CLE1 [ CLE2 [ CLTk-GV . By design, the
Tk-GV method produces CL-values with the smallest CV and consistent values for
V for vanishingly small CL, a major achievement. E2 and higher models may be
replaced for labeled DTPA by Tk-GV, which requires only 4 samples to ﬁnd
solutions (compare to 8 samples for an E2 SET). While the Tk-GV method may seem
complicated, its use is simpler and more practical than constant infusion, 24-h AUC
with mono-exponential extrapolation or E2 models. Thus, as far as we can determine,
the Tk-GV model results reﬂect accurate renal function estimates Furthermore, E2
SET models sometimes produce nonphysical CL-values, and when physical values
occur, they are imprecise and biased. In addition, unlike inulin constant infusion,
which is not useful in renal failure and which can cause anaphylaxis [73, 74], or
tedious 24 h AUC with mono-exponential extrapolation, Tk-GV provides precise
volumes of distribution, V, even when CL is vanishingly small.
Acknowledgements The authors wish to express their gratitude to Dr. Charles D. Russell for his
valuable consultation and advice about modeling and to both Dr. Russell and Dr. Barbara Y. Croft for
making patient data available. This work was partially supported by grant HIC-07.49 from the Eastern
Health Foundation.
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Appendix: error adaptive Tikhonov regularization equation section (Next)
For an over-determined system of linear equations, Ax ¼ b, the Tikhonov
regularization (Tk) of this problem introduces the penalty function Cx and seeks
to ﬁnd a solution that minimizes kAx À bk2 þkCxk2 . This latter is the square of a
norm of the residuals, kAx À bk2 , plus the square of a norm of the product of the
Tikhonov matrix, C, with the x ﬁt parameters (unknowns). The more general
CT Cregularizing term is often, as it is here, replaced by kI, where I is the identity
matrix, and k is a Lagrange (i.e., constraint) multiplier, also commonly called the
shrinkage, Tikhonov or damping factor. There are two points of note. First, although

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471

it offers no computational advantage to do so, ridge regression, used here, is
Tikhonov regularization with correlation scaling that standardizes k values. Second,
k = 0 is equivalent to the problem of minimizing the norm kAx À bk2 , which is
most commonly solved using ordinary least squares (OLS).
A constraint on lnK
A most common constraint for regression is to require the ﬁt function to pass
""
through the data mean point (a.k.a., the centroid, x; y). Because the logarithm of
concentrations is the more homoscedastic quantity, it is common to ﬁt the
logarithms of marker concentrations rather than the concentrations themselves.
Thus, for the Tk-GV method, the GV function is written ln C ¼ ln K þ ða À 1Þ
ln t À bt, where the constant term ln K need not be independent, but can be
determined from the other ﬁt parameters using a mean value constraint. Taking
averages over the data
"
ln K ¼ ln C ðtÞ À ða À 1Þln t þ b" ¼ b À ða À 1Þ"1 þ b"2 ;
t
a
a

ðA1Þ

" "
"
"
such that b, a1 and a2 are data constants, where b is the mean value of the logarithms
"
"
of the concentrations, a1 is the mean of the logarithms of the sample times and a2 , is
the mean of the sample times. Then, Eq. A1 is used to remove K from the formula
for CL formula (Eq. 15), and an expression is derived for the errors in CL with only
a and b as independent parameters, as follows
CL ¼

Dba
;
"
expðb À ða À 1Þ"1 þ b"2 ÞCðaÞ
a
a

ðA2Þ

Error propagation
One applies the well known error propagation formula [75] to Eq. A2 with respect
to a and b yielding

2
s 2
a
CL
"
"
À a2
¼ s2 ða1 þ ln b À WðaÞÞ2 þs2
a
b
b
CL
ðA3Þ


a
"
"
À a2 :
þ2sab ða1 þ ln b À WðaÞÞ
b
where WðaÞ is the digamma function of a and WðaÞ ¼ d½ln CðaÞ=da ¼ C0 ðaÞ=CðaÞ,
the subscripted s variables are the standard deviations of the subscripted quantities,
and ðsCL =CLÞ2 is the squared coefﬁcient of variation ðCVÞ2 of CL. Minimizing the
right hand side of Eq. A3 as a function of the shrinkage, k, selects a k value that
produces the CL value with the smallest relative error achievable. Also, minimizing
the relative error in CL is indispensable for making reliable measures of CL when
CL is small.
The variance of V is similarly calculated from application of the propagation of
error formula to the substitution of Eq. 17 into Eq. A2 yielding

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J Pharmacokinet Pharmacodyn (2010) 37:435–474

s 2
V

V


2

2
1
a 1
"
"
þ a1 þ ln b À WðaÞ þs2
À À a2
b
a
b b



1
a 1
"
"
þ a1 þ ln b À WðaÞ
À À a2 :
þ2sab
a
b b

¼ s2
a

ðA4Þ

The square root of Eq. A4 is the coefﬁcient of variation, CV, of the individual
V values, i.e., sV =V.

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