Modeling and Analysis of Tumor Growth Inhibition for Combination Therapy using Tumor Static Concentration Curves

T. Cardilin, A. Sostelly, J. Gabrielsson, S. El Bawab, C. Amendt, M. Jirstrand. Proceedings of the 24th Annual meeting of the Population Approach Group in Europe, 2-5 June 2015, Crete, Greece


Objectives: To develop and analyze a Tumor Growth Inhibition (TGI) model for combination therapy based on experimental data using Tumor Static Concentration (TSC) curves.

Methods: Patient-Derived xenograft data on Erbitux-Cisplatin combinations were obtained from mice. Drug exposure profiles were generated based on literature data. Time series of efficacy data were modelled based on the in vivo TGI model with simultaneous cytostatic and cytotoxic drug action. Model parameters were estimated using a mixed-effects approach implemented in Mathematica 10 [1]. The models were then investigated using an analytical approach to obtain Tumor Static Concentration [2] curves, which should be compared with the established concept of isobolograms [3].

Results: The TSC condition for the combination of a cytostatic (A) and a cytotoxic (B) drug can be expressed mathematically as

kgrowth I(CA) = kkill S(CB),

where CA  and CB are the plasma concentrations of drugs A and B, respectively. and S are an inhibitory and a stimulatory function acting on the proliferating cell compartment. kgrowth and  kkill are the cell growth and kill rates. This can be visualized as a curve in the CACB-plane. Keeping the concentrations above this curve gives tumor shrinkage, while falling below it gives tumor growth.

The Erbitux-Cisplatin combination data were adequately modelled with Erbitux as the cytostatic and Cisplatin as the cytotoxic compound, under the assumption of independent action. TSC curves were generated and compared with the exposure profiles of all test compounds. This provided visualization of when and to what extent the concentrations were at a sufficiently high level for tumor shrinkage and helped to suggest times when either a higher or additional dose would be necessary.

Conclusions: The graphical TSC presentation of two compounds proved to be a useful tool for presentation of drug combinations tumor growth/kill interventions.


[1] Almquist J, Leander J, Jirstrand M. Using sensitivity equations for computing gradients of the FOCE and FOCEI approximations to the population likelihood. (In press) J Pharmacokinet Pharmacodyn (2015).

[2] Jumbe NL, Xin Y, Leipold DD, Crocker L, Dugger D, Mai E, Sliwkowski MX, Fielder PJ, Tibbitts J. Modeling the efficacy of transtuzumab-DM1, an antibody drug conjugate, in mice.  J Pharmacokinet Pharmacodyn (2010) 37:221-242.

[3] Tallarida RJ. An Overview of Drug Combination Analysis with Isobolograms. J Pharm Exp Ther (2006) 319:1-7.

Photo credits: Nic McPhee