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:
In thisSupplementary 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 todemonstrate the generality of the methodology proposed in the paper, the derivation of the characteristic curvefor a log cell kill model withlogistic-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 profileswere obtained using aone compartment model with sequential zero-first order input and first orderoutput.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 concentrationswere180 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 IC50valueismuch higher (set 1) or much lower (set 2) than theexperimental drug concentrations. This corresponds to biomarkers eitherweaklyor 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 PanelsA and B, drug concentration profiles and the corresponding biomarker ones are displayed.In PanelsC 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 modelsexhibit substantially different performanceswhen applied toset 2.While thematchedmodel(B2-Simeoni)fits more than satisfactorily alsoset 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-Simeoniare apparent when drug concentrations are much higher than the IC50value.For reader's convenience, the drug-to-biomarker characteristic curves of IRMsused to simulateset 1 and set 2 are depicted in Panels A and B of Fig. S3, respectively. When drug concentrationremains 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 modelsare interchangeable. Conversely, when drug concentrationsare in the nonlinear saturation portion of the drug-to-biomarker characteristic curve, the fitting capabilities depart from one another. Indeed, the B1-Simeoni 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 strength of a reverse engineered models such as the B2-Simeoni one, is its generalvalidity, irrespective of the levels of drug concentration. Therefore, the B2-Simeoni model describes equally well tumor growth modulation induced by biomarkersthat are either weakly or strongly inhibited by the drug concentrations.
A B
C D
Fig. S1Comparison 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 orderoutput. 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 Simeonimodel 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.
A B
C D
Fig. S2Comparison 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 orderoutput.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 Simeonimodel 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.
A B
Fig. S3 Characteristic curves of drug-to-biomarkerIndirect Response models. Panel A: set 1.Panel B:set 2. The vertical dashed lines represent: IC50value (red), steady-state concentration for arm1 (blue), steady-state 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 log-cell 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:
(1)
whereλBdescribes a logistic growth characterized by the parametersλ2and b (the so-called carrying capacity of the logistic model):
Moreover, according to the log cell kill hypothesis,
Following the procedure described in the "Drug-to-Tumor Characteristic Curve" Section of the manuscript, the characteristic curve, displayedin 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
As long as i.e, the nonnull solution is the stable one. Otherwise the stable steady state is .
Therefore, the formula for the drug-to-tumor characteristic curve is
As in the standard TGI Simeoni model, also in this case the threshold concentrationcT 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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