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Modeling of tumor growth and anticancer effects of combination therapy


Combination therapies are widely used in the treatment of patients with cancer. Selecting synergistic combination strategies is a great challenge during early drug development. Here, we present a pharmacokinetic/pharmacodynamic (PK/PD) model with a smooth nonlinear growth function to characterize and quantify anticancer effect of combination therapies using time-dependent data. To describe the pharmacological effect of combination therapy, an interaction term was introduced into a semi-mechanistic anticancer PK/PD model. This approach enables testing of a pharmacological hypothesis with respect to an anticipated pharmacological synergy of drug combinations, such as an assumed pharmacological synergy of complementary inhibition of a particular signaling pathway. The model was applied to three real data sets derived from preclinical screening experiments using xenograft mice. The suggested model fitted well the observed data from mono- to combination-therapy and allowed physiologically meaningful interpretation of the experiments. The tested drug combinations were assessed for their ability to act as synergistic modulators of tumor growth inhibition by the interaction parameter ψ. The presented approach has practical implications because it can be applied to standard xenograft experiments and may assist in the selection of both optimal drug combinations and administration schedules. The unique feature of the presented approach is the ability to characterize the nature of combined drug interaction as well as to quantify the intensity of such interactions by assessing the time course of combined drug effect.

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  1. Sun YN, Jusko WJ (1998) Transit compartments versus gamma distribution function to model signal transduction processes in pharmacodynamics. J Pharm Sci 87:732–737

    PubMed  Article  CAS  Google Scholar 

  2. Mager DE, Wyska E, Jusko WJ (2003) Diversity of mechanism-based pharmacodynamic models. Drug Metab Dispos 31:510–518

    PubMed  Article  CAS  Google Scholar 

  3. Lobo ED, Balthasar JP (2002) Pharmacodynamic modeling of chemotherapeutic effects: application of a transit compartment model to characterize methotrexate effects in vitro. AAPS Pharm Sci 4:E42

    Article  Google Scholar 

  4. Simeoni M, Magni P, Cammia C, De Nicolao G, Croci V, Pesenti E, Germani M, Poggesi I, Rocchetti M (2004) Predictive pharmacokinetic–pharmacodynamic modeling of tumor growth kinetics in xenograft models after administration of anticancer agents. Cancer Res 64:1094–1101

    PubMed  Article  CAS  Google Scholar 

  5. Rocchetti M, Poggesi I, Germani M, Fiorentini F, Pellizzoni C, Zugnoni P, Pesenti E, Simeoni M, De Nicolao G (2005) A pharmacokinetic–pharmacodynamic model for predicting tumour growth inhibition in mice: a useful tool in oncology drug development. Basic Clin Pharmacol Toxicol 96:265–268

    PubMed  Article  CAS  Google Scholar 

  6. Magni P, Simeoni M, Poggesi I, Rocchetti M, De Nicolao G (2006) A mathematical model to study the effects of drugs administration on tumor growth dynamics. Math Biosci 200:127–151

    PubMed  Article  CAS  Google Scholar 

  7. Rocchetti M, Simeoni M, Pesenti E, De Nicolao G, Poggesi I (2007) Predicting the active doses in humans from animal studies: a novel approach in oncology. Eur J Cancer 43:1862–1868

    PubMed  Article  CAS  Google Scholar 

  8. Berenbaum MC (1989) What is synergy? Pharmacol Rev 41:93–141

    PubMed  CAS  Google Scholar 

  9. Greco WR, Bravo G, Parsons JC (1995) The search for synergy: a critical review from a response surface perspective. Pharmacol Rev 47:331–385

    PubMed  CAS  Google Scholar 

  10. Jusko WJ (1971) Pharmacodynamics of chemotherapeutic effects: dose-time-response relationships for phase-nonspecific agents. J Pharm Sci 60:892–895

    PubMed  Article  CAS  Google Scholar 

  11. Magni P, Germani M, De Nicolao G, Bianchini G, Simeoni M, Poggesi I, Rochetti M (2008) A Minimal Model of Tumor Growth Inhibition. IEEE Biomedical Eng 55(12):2683–2690

    Article  Google Scholar 

  12. Chakraborty A, Jusko WJ (2002) Pharmacodynamic interaction of recombinant human interleukin-10 and prednisolone using in vitro whole blood lymphocyte proliferation. J Pharm Sci 91:1334–1342

    PubMed  Article  CAS  Google Scholar 

  13. D’Argenio DZ, Schumitzky A (1997) ADAPT II user’s guide: pharmacokinetic/pharmacodynamic systems analysis software. Biomedica simulation resuource, Los Angeles

    Google Scholar 

  14. Chou TC, Talalay P (1984) Quantitative analysis of dose-effect relationships: the combined effects of multiple drugs or enzyme inhibitors. Adv Enzyme Regul 22:27–55

    PubMed  Article  CAS  Google Scholar 

  15. Drewinko B, Loo TL, Brown B, Gottlieb JA, Freireich EJ (1976) Combination chemotherapy in vitro with adriamycin. Observations of additive, antagonistic, and synergistic effects when used in two-drug combinations on cultured human lymphoma cells. Cancer Biochem Biophys 1:187–195

    PubMed  CAS  Google Scholar 

  16. Greco WR, Park HS, Rustum YM (1990) Application of a new approach for the quantitation of drug synergism to the combination of cis-diamminedichloroplatinum and 1-beta-D-arabinofuranosylcytosine. Cancer Res 50:5318–5327

    PubMed  CAS  Google Scholar 

  17. Berenbaum MC (1990) Re: W. R. Greco et al., application of a new approach for the quantitation of drug synergism to the combination of cis-diamminedichloroplatinum and 1-beta-d-arabinofuranosylcytosine. Cancer Res 50:5318–5327. Cancer Res 52:4558–4560 (1992)

  18. Loewe S, Muischnek H (1926) Effect of combinations: mathematical basis of problem. Arch Exp Pathol Pharmakol 114:313–326

    Article  CAS  Google Scholar 

  19. Hairer E, Nørsett SP, Wanner G (1993) Solving ordinary differential equations I, Chapter I. vol 14, 2nd edn. Springer, USA

    Google Scholar 

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The authors are grateful to WJ Jusko for motivating and valuable discussion on this modeling approach and to Karl Zech who continuously supported the project. The authors thank the scientists Petra Gimmnich, Astrid Zimmermann, Ragna Hussong, and Rolf Herzog for their dedicated involvement during the generation of pharmacological and pharmacokinetic data, and Kathy B. Thomas for helpful suggestions during the preparation of the manuscript. This study was sponsored by Nycomed GmbH, Konstanz, Germany.

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Correspondence to Antje Walz.



Appendix 1: The unperturbed growth function

In the Simeoni model, the unperturbed tumor growth function is modeled by a linear and a constant part

$$ g\left( w \right) = \left\{ \begin{gathered} \lambda_{0} w,\quad w \le w_{th} \hfill \\ \lambda_{1}, \quad w > w_{th} \hfill \\ \end{gathered} \right.,\quad \quad w_{th} = \frac{{\lambda_{1} }}{{\lambda_{0} }}, $$

which are simply glued together. Here \( \lambda_{0} > 0 \) and \( \lambda_{1} > 0 \) describe the rate of exponential and linear growth of tumor weight w. Moreover the growth function g suffers at a lack of differentiability for \( w = w_{th} . \)

Alternatively, we model the unperturbed tumor growth by the term

$$ f\left( {x_{1} } \right) = \frac{{ax_{1} }}{{b + x_{1} }}\quad a,b > 0. $$

To achieve physiological meaningful parameters with the same meaning as in Simeoni’s model we adjust actual values of a and b according to

$$ \mathop {\lim }\limits_{{x_{1} \to \infty }} f\left( {x_{1} } \right) = \lambda_{1} \,\quad {\text{and}}\,\quad f\left( {\frac{{\lambda_{1} }}{{2\lambda_{0} }}} \right) = \frac{a}{2}. $$

These two conditions ensure that the maximal growth rate and the half maximum value of the Simeoni growth function g and f coincide.

We obtain \( a = \lambda_{1} ,b = \frac{{\lambda_{1} }}{{2\lambda_{0} }} \) and end up with the expression

$$ f\left( {x_{1} } \right) = \frac{{\lambda_{1} x_{1} }}{{\frac{{\lambda_{1} }}{{2\lambda_{0} }} + x_{1} }} = \frac{{2\lambda_{0} \lambda_{1} x_{1} }}{{\lambda_{1} + 2\lambda_{0} x_{1} }}. $$

Remark: The growth function g from Simeoni et al. (Eq. 7) is not differentiable at the switch w th therefore they suggested an approximation

$$ g_{a} \left( w \right) = \frac{{\lambda_{0} w}}{{\left( {1 + \left( {\frac{{\lambda_{0} }}{{\lambda_{1} }}w} \right)^{\psi } } \right)^{{\frac{1}{\psi }}} }} $$

with ψ = 20 fixed. A comparison of the growth functions is shown in Fig. 6.

Fig. 6
figure 6

The characteristics of the Simeoni growth function g, the approximation g a and the nonlinear growth function f for λ 0 = 0.2 and λ 1 = 0.4 are shown. The inlet is showing a zoom of g and g a at the kink

Appendix 2: The threshold concentration

In this section we give a proof of (Eq. 4) and calculate a similar result for combination therapy. Let the drug flow be constant for all t ≥ 0. The stationary equations of the presented model (Eq. 3) and (Eq. 6) read:

$$ 0 = \frac{{2\lambda_{0} \lambda_{1} x_{1}^{2} }}{{\left( {\lambda_{1} + 2\lambda_{0} x_{1} } \right)w}} - \gamma x_{1} , $$
$$ 0 = \gamma x_{1} - k_{1} x_{2} , $$
$$ 0 = k_{1} \left( {x_{i - 1} - x_{i} } \right),\quad i = 3, \ldots ,N $$


$$ \gamma = \left\{ \begin{gathered} \begin{array}{*{20}l} {k_{2} c} & {{\text{for}}\,\,{\text{monotherapy,}}} \\ \end{array} \hfill \\ \begin{array}{*{20}c} {k_{2}^{a} c_{1} + k_{2}^{b} c_{2} \psi } & {{\text{in}}\,\,{\text{case}}\,\,{\text{of}}\,{\text{combination}}\,\,{\text{therapy}}} \\ \end{array} \hfill \\ \end{gathered} \right.. $$

(Eq. 8b) and (Eq. 8c) directly imply

$$ x_{2} = \ldots = x_{N} = \frac{{\gamma x_{1} }}{{k_{1} }}. $$

Inserting the relation (Eq. 9) into (Eq. 8a) leads to the quadratic equation

$$ \gamma^{2} \left( {\left( {N - 1} \right)\frac{{\lambda_{1} }}{{k_{1} }} + 2\left( {N - 1} \right)\frac{{\lambda_{0} x_{1} }}{{k_{1} }}} \right) + \gamma \left( {\lambda_{1} + 2\lambda_{0} x_{1} } \right) - 2\lambda_{0} \lambda_{1} = 0 $$

for γ. We set x 1 = 0 in (Eq. 10) due to the eradication of the tumor and find the unique positive solution

$$ \gamma = \frac{{ - k_{1} + k_{1} \sqrt {1 + 8\left( {N - 1} \right)\frac{{\lambda_{0} }}{{k_{1} }}} }}{{2\left( {N - 1} \right)}} $$

of (Eq. 10). Thus, with \( \tau = \frac{N - 1}{{k_{1} }} \) we obtain for the monotherapy model

$$ k_{2} C_{T} = \frac{1}{2\tau }\left( { - 1 + \sqrt {1 + 8\tau \lambda_{0} } } \right) $$

and for the combination therapy system

$$ k_{2}^{a} C_{T}^{a} + k_{2}^{b} C_{T}^{b} \psi = \frac{1}{2\tau }\left( { - 1 + \sqrt {1 + 8\tau \lambda_{0} } } \right). $$

Moreover, it is confirmed in both cases by numerical simulation that the domain of attraction of the steady state (0,…,0) covers \( {\mathbb{R}}_{ + }^{N} . \)

Appendix 3: Derivatives with respect to parameters

In this section we present the concept of variations, that is, an alternative approach to compute partial derivatives of solutions of ordinary differential equations with respect to parameters. With this approach the user can balance accuracy of the computations versus computational effort. Please keep in mind that the accurate computation of these partial derivatives affects PK/PD modeling for two reasons. First it opens the route to use more powerful numerical algorithms to solve nonlinear least squares problems and, secondly, it can be useful to reliably compute the covariance matrix, which is the basis for every a posteriori stochastic analysis.

The model equations of tumor development in PK/PD analysis have the general form

$$ x^{\prime } \left( t \right) = f\left( {t,x\left( t \right),\theta } \right),\quad x\left( 0 \right) = x_{0} = \left( {w_{0} ,0, \ldots ,0} \right)^{T} \in {\mathbb{R}}^{N} $$

with \( \theta = \left( {\theta_{1} ,\theta_{2} ,\theta_{3} ,\theta_{4} } \right) \) and with the differentiable solution \( \overline{x} \left( {t,x_{0} ,\theta } \right). \) To calculate the derivatives \( \frac{{\partial \overline{x} }}{{\partial \theta_{k} }}\left( {t,x_{0} ,\theta } \right),k = 1, \ldots ,4 \) due to Hairer [19] one has to solve the matrix differential equation

$$ v^{\prime } \left( t \right) = \frac{\partial f}{\partial x}\left( {t,\overline{x} \left( {t,x_{0} ,\theta } \right),\theta } \right)v\left( t \right) + \frac{\partial f}{\partial \theta }\left( {t,\overline{x} \left( {t,x_{0} ,\theta } \right),\theta } \right),\,\,\,\,\,\,v\left( 0 \right) = 0 \in {\mathbb{R}}^{N,4} $$

and \( \frac{{\partial \overline{x} }}{{\partial w_{0} }}\left( {t,x_{0} ,\theta } \right) \) is the solution of

$$ z^{\prime } \left( t \right) = \frac{\partial f}{\partial x}\left( {t,\overline{x} \left( {t,x_{0} ,\theta } \right),\theta } \right)z\left( t \right),\;\;z\left( 0 \right) = \left( {1,0, \ldots ,0} \right)^{T} \in {\mathbb{R}}^{N}. $$

In (Eq. 12), \( {\mathbb{R}}^{N,4} \) stands for the space of matrices with N rows and 4 columns. To actually compute the derivatives one has to solve (Eq. 11), (Eq. 12) and (Eq. 13) simultaneously with an appropriate numerical scheme. Note that (Eq. 12) has to be solved column by column.

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Koch, G., Walz, A., Lahu, G. et al. Modeling of tumor growth and anticancer effects of combination therapy. J Pharmacokinet Pharmacodyn 36, 179–197 (2009).

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  • Anticancer PK/PD model
  • Drug combination
  • Nature of interaction
  • Synergistic modulator
  • Quantification of combined drug effect