Modeling of tumor growth and anticancer effects of combination therapy

Abstract

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.

Appendix

Appendix 1: The unperturbed growth function

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

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

(7)

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

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:

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

(8a)

$$ 0 = \gamma x_{1} - k_{1} x_{2} , $$

(8b)

$$ 0 = k_{1} \left( {x_{i - 1} - x_{i} } \right),\quad i = 3, \ldots ,N $$

(8c)

with

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

(Eq. 8b) and (Eq. 8c) directly imply

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

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

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

(10)

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

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

(11)

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

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

(12)

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

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

(13)

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.