Supplementary material
Article title: Biomarker- vs Drug- driven Tumor Growth Inhibition Models: an equivalence analysis
Journal name: Journal of Pharmacokinetics and Pharmacodynamics
Authors: M.L. Sardu (1), I. Poggesi (2), G. De Nicolao (1)
Affiliations: (1)Dipartimento di Ingegneria Industriale e dell'Informazione, Università di Pavia
(2) Model Based Drug Development, Janssen Pharmaceutica NV (Beerse)
E-mail: marialuisa.sardu@unipv.it
In this Supplementary Material two issues are addressed. The first section refers to the comparison of
the fitting performances of B1 and B2 Simeoni models. In the second section, in order to demonstrate
the generality of the methodology proposed in the paper, the derivation of the characteristic curve for a
log cell kill model with logistic-type unperturbed growth is detailed.
A.1Fitting performances of B2 and B1-Simeoni models.
In order to compare the fitting performances of B1 and B2 Simeoni models, a simulation approach was
applied. In particular, to generate the benchmark dataset, the standard Simeoni model was adopted.
Drug concentration profiles were obtained using a one compartment model with sequential zero-first
order input and first order output. Two treatment arms were simulated, where the drug was
administered following an infusion in the depot compartment with duration of 700 hours and amounts of
250 mg (arm 1) and 1000 mg (arm 2), respectively. The steady-state drug concentrations were 180
ng/ml for arm 1 and 720 ng/ml for arm 2. In order to fit these data with the biomarker-to-tumor models,
also simulated biomarker profiles were made available using type I Indirect Response Models (IRMs).
In particular, two sets of IRM parameters, differing in terms of IC50, were used: set 1: Baseline=100,
kOUT=7 h-1, IC50=5000 ng/ml; set 2: Baseline=100, kOUT=7 h-1, IC50=50 ng/ml. These two datasets allow
to compare the fitting performance of the biomarker-to-tumor models when the IC50 valueis much
higher (set 1) or much lower (set 2) than the experimental drug concentrations. This corresponds to
biomarkers either weakly or strongly inhibited from the drug concentration. To evaluate the fitting
performance, the simulated tumor growth data were fitted with either the B1 or B2 Simeoni models.
The results are displayed in Fig. S1 (set 1) and S2 (set 2). In Panels A and B, drug concentration
profiles and the corresponding biomarker ones are displayed. In Panels C and D, tumor volumes fitted
with the B1 and B2 Simeoni models are displayed. The comparison highlights significant differences
between the two sets. In particular, the B1 and B2 Simeoni models perform equally well on set 1.
Conversely, the two models exhibit substantially different performances when applied to set 2.While the
matched model (B2-Simeoni) fits more than satisfactorily also set 2, the B1 Simeoni model (which does
not meet the matching condition) fails to show acceptable predictive capabilities on this second dataset.
In particular, the poor fitting performances of B1-Simeoni are apparent when drug concentrations are
much higher than the IC50value. For reader's convenience, the drug-to-biomarker characteristic curves
of IRMs used to simulate set 1 and set 2 are depicted in Panels A and B of Fig. S3, respectively. When
drug concentration remains within the linear portion of the drug-to-biomarker characteristic curve (IC50
much greater than drug concentration), the fitting capabilities of the B1 and B2 models are
interchangeable. Conversely, when drug concentrations are in the nonlinear saturation portion of the
drug-to-biomarker characteristic curve, the fitting capabilities depart from one another. Indeed, the B1Simeoni predicts lower effects due to the saturation of the biomarker inhibition I, which enters linearly
the growth inhibition model, rather than being nonlinearly modulated as in the B2-Simeoni one. The key
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strength of a reverse engineered models such as the B2-Simeoni one, is its general validity,
irrespective of the levels of drug concentration. Therefore, the B2-Simeoni model describes equally well
tumor growth modulation induced by biomarkers that are either weakly or strongly inhibited by the drug
concentrations.
A
B
C
D
Fig. S1 Comparison of fitting performances of B1- and B2-Simeoni models on data from set 1. Panel A: drug concentration
profiles simulated with a one compartment model with sequential zero-first order input and first order output. Administration
route: infusion; duration: 700 h, amounts: 250 mg (arm 1, blue continuous line), and 1000 mg (arm 2, magenta continuous line ).
PK parameters: ka=0.0807 h-1, ke=6.01 h-1, V=0.331∙10-4L/kg. Panel B: biomarker profiles. Panels C and D : tumor volume (circles)
simulated with standard Simeoni model and fitting obtained with B1-Simeoni (panel C) and B2-Simeoni model (panel D).
Parameters: w0=188 mm3, λ0=0.0127mm3/h, λ1=9.04 h-1, k2=2.24∙10-5h-1ml/ng, k1=0.0416 h-1.
2
A
B
C
D
Fig. S2 Comparison of fitting performances of B1- and B2-Simeoni models on data from set 2. Panel A: drug concentration profiles
simulated with one compartment model with sequential zero-first order input and first order output. Administration route:
infusion; duration: 700 h, amounts: 250 mg (arm 1, blue continuous line), and 1000 mg (arm 2, magenta continuous line ). PK
parameters: ka=0.0807 h-1, ke=6.01 h-1, V=0.331∙10-4L/kg. Panel B: biomarker profiles. Panels C and D: tumor volume (circles)
simulated with standard Simeoni model and fitting obtained with B1-Simeoni (panel C) and B2-Simeoni model (panel D).
Parameters: w0=188 mm3, λ0=0.0127mm3/h, λ1=9.04 h-1, k2=2.24∙10-5 h-1ml/ng, k1=0.0416 h-1.
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A
B
Fig. S3 Characteristic curves of drug-to-biomarker Indirect Response models. Panel A: set 1. Panel B: set 2.
The vertical dashed lines represent: IC50 value (red), steady-state concentration for arm1 (blue), steadystate concentration for arm 2 (magenta).
2. Characteristic curve of log cell kill model with logistic unperturbed growth
In order to demonstrate the general applicability of the notion of characteristic curve, its extension to a logcell kill model with a logistic growth function is detailed in the following text. Using the general notation
similar to that of Eq. (2), (3) in the manuscript, the log cell kill with logistic growth function can be written
as:
x1 (t )  (B ( x(t ))  D (c(t ))) x1 (t )
w(t )  x1 (t )
(1)
where λB describes a logistic growth characterized by the parameters λ2 and b (the so-called carrying
capacity of the logistic model):
B (x)  2(b  x1(t))
Moreover, according to the log cell kill hypothesis,

 (c)  k2c

Following the procedure described in the "Drug-to-Tumor Characteristic Curve" Section of the manuscript,
the characteristic curve, displayed in Fig. S4, can be derived by finding the equilibrium point of the system,
i.e. by zeroing the derivative in (1) and solving the linear system for the equilibrium state, yielding
D
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
 0
x1  b  k 2c

 2
b
kc

2
2
As long as b    0 i.e c  k  cT , the nonnull solution is the stable one. Otherwise the stable steady
2
2
state is x  0 .
Therefore, the formula for the drug-to-tumor characteristic curve is
 k2
b  c
 DT (c )   2
 0
c  cT
c  cT
As in the standard TGI Simeoni model, also in this case the threshold concentration cT indicates the value
above which tumor regression is achieved, leading to a tumor volume of zero at steady-state.
Fig. S4Characteristic curve of Simeoni model with logistic growth. PK parameters: Infusion rate=0.5,
V=0.23∙10-3, ka=0.0807, k02=1.31, Logistic growth: b=3000, λ2=5.27∙10-6. Other Simeoni parameters: k1=
0.0416, k2=3.36∙10-6,ψ=20, cT =4705.
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