Mathematical models of tumor growth inhibition in xenograft mice after administration of anticancer agents given in combination
Nadia Terranova (1), Massimiliano Germani (2), Francesca Del Bene (2), Paolo Magni (1).
(1) Dipartimento di Ingegneria Industriale e dell'Informazione, UniversitÓ degli Studi di Pavia, Italy; (2) PK & Modeling, Accelera srl, Nerviano (MI), Italy.
In clinical oncology, combination treatments are widely used and increasingly preferred over single drug administrations. Therefore, the R&D process is nowadays focused on the development of new compounds that can be successfully administered in combination with drugs already on the market. To this aim, preclinical studies are routinely performed, even if they are only qualitatively analyzed, on xenograft mice for the assessment of new combination therapies. The ability of deriving from single drug experiments a reference response to a joint administration, assuming no interaction, and comparing it to real response would be the key to recognize synergic and antagonist compounds.
This work is aimed at deriving quantitative information from standard experiments. In particular, the definition of no interaction between drug effects has been provided by means of a new mathematical model. On this basis, we have also developed a new combination model able to predict the tumor growth inhibition (TGI) in combination regimens and provide a quantitative measurement of the nature and the strength of the pharmacological drug interaction as well.
The experimental setting is that of a typical in vivo study routinely performed within several drug development projects using human carcinoma cell lines on xenograft mice . The typical combination experiment involves the control arm, the single agent arms and the combination arms. Average data of tumor weight of control and treated groups were considered. The PKs are evaluated in separated studies.
The no interaction model
Starting from a minimal set of basic assumptions at cellular level that include and extend those formulated for the single drug administration , a minimal model able to define and simulate the no interacting behavior of an arbitrary number of co-administered antitumor drugs has been formulated. The tumor growth dynamics is described by an ordinary and several partial differential equations. Under suitable assumptions, the model reduces to a lumped parameter model that represents the extension of the very popular Simeoni TGI model  to the combined administration of two non-interacting drugs.
The TGI minimal model parameters relative to the tumor growth and to the drugs action were estimated from experimental data coming from single-drug administrations and used to simulate combination regimens under the hypothesis of no interaction. Fitting was performed by nonlinear least squares as implemented in the lsqnonlin routine of MATLAB 2007b suite with analytical computation of the Jacobian. Each residual was weighted proportionally to the inverse of the related measurement.
The combination model
Starting from the TGI minimal model, we have also developed a new PK-PD model able to predict tumor growth after the co-administration of two anticancer agents, assessing the nature and the strength of interaction as well. The tumor growth rate assessed in untreated mice is decreased by two terms proportional to drug concentrations and decreased-increased by one interaction term proportional to their product. In order to provide an understandable measure of the strength of the interaction, two indexes (called synergistic/antagonistic combination index) were defined.
PK and PD models were implemented in WinNolin 3.1 and Matlab 2007b for the analysis of several experiments. Model identification was performed by using the nonlinear weighted least squared algorithm (with weights equal to the inverse of the related measurement). As for the minimal model the tumor-related parameters and the drug-related parameters were estimated by fitting the Simeoni TGI model on the single agent arms. Then, fixing these parameters to the estimated values, the new proposed TGI model was fitted against the combination arms to obtain the value of the interaction term.
The no interaction model
The minimal TGI model specialized for the case of two non-interacting drugs has been applied to analyze the study of irinotecan CPT-11 in combination with two different dosages of a novel compound (here call Drug B) on the HT29 human colon adenocarcinoma cell line. The validity of the no interaction hypothesis was then assessed by a suitable statistical test . CPT-11 and Drug B showed a negative interaction, namely a (slight) antagonistic behavior in both combination arms.
The combination model
The model was successfully applied to four novel anticancer candidates, synthesized by Nerviano Medical Sciences, Nerviano, co-administered with four drugs already available on the market for the treatment of three different tumor cell lines. In total, six experiments, testing 11 different combination treatments involving more than 230 mice, were led. The estimation of the interaction term allowed an easy evaluation of the nature of the interaction. The combination indexes were then evaluated for the combination treatment in order to have an absolute measure of the strength of interaction. The model has also shown very good capabilities in predicting different combination regimens in which the same drugs were administered at different doses/schedules.
Starting from a minimal set of assumptions formulated at cellular level, the proposed minimal TGI model has been defined to describe the case of no interaction between co-administered drugs, in order to provide a theoretical definition of interaction. The model defines a general class of models and at least in one of its specialized form, can be used for the evaluation of drug combinations by exploiting simulations, providing a rigorous alternative to the subjective and qualitative visual comparison of experimental data.
Starting from the concepts, a new PK-PD model has been developed and implemented aiming to be an approach of practical use in assessing combination therapy in standard xenograft experiments as well as identifying synergistic drug combinations.
The relevance and applicability of the combination model were demonstrated analyzing several studies. This model can be considered an indispensable tool in the preclinical drug development and a crucial advance in the knowledge as it integrates the previous information to improve the decision making.
This work was supported by the DDMoRe project (www.ddmore.eu).
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