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Transfer of Drug Resistance Characteristics Between Cancer Cell Subpopulations: A Study Using Simple Mathematical Models

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Abstract

Resistance to chemotherapy is a major cause of cancer treatment failure. The processes of resistance induction and selection of resistant cells (due to the over-expression of the membrane transporter P-glycoprotein, P-gp) are well documented in the literature, and a number of mathematical models have been developed. However, another process of transfer of resistant characteristics is less well known and has received little attention in the mathematical literature. In this paper, we discuss the potential of simple mathematical models to describe the process of resistance transfer, specifically P-gp transfer, in mixtures of resistant and sensitive tumor cell populations. Two different biological hypotheses for P-gp transfer are explored: (1) exchange through physical cell–cell connections and (2) through microvessicles released to the culture medium. Two models are developed which fit very well the observed population growth dynamics. The potential and limitations of these simple “global” models to describe the aforementioned biological processes involved are discussed on the basis of the results obtained.

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Acknowledgments

We would like to acknowledge Gabriel F. Calvo (Universidad de Castilla-La Mancha) for discussions. The authors would also like to thank the anonymous referees for valuable suggestions. This research has been partially supported by Grants MTM2015-71200-R (Ministerio de Economía y Competitividad/FEDER) and PEII-2014-031P (Junta de Comunidades de Castilla-La Mancha and FEDER).

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Correspondence to Arturo Álvarez-Arenas.

Appendix

Appendix

1.1 Basic Properties of Model Eq. (1)

First we state the following:

Theorem 1

For any nonnegative initial data \((S_{0},R_{0},S_{R0})\) and \(\tau _{s}> 0\), \(\tau _{r}> 0\), the solutions to Eq. (1) exist for \(t>0\), are nonnegative, bounded and unique.

Proof

Let \(N=S+R+S_{R}\); then, defining \(N=S+R+S_{R}\) we can write Eq. (1) as

$$\begin{aligned} \frac{\hbox {d}N}{\hbox {d}t}\le \gamma N\left( 1-\frac{N}{K}\right) , \end{aligned}$$
(5)

with \(\gamma = 1/\min \{ \tau _{s}, \tau _{r}\}\). Therefore N(t) is bounded and thus S, R and \(S_{R}\) are bounded as well. Since Eq. (1) has bounded coefficients, and the right-hand side of Eq. (5) is a continuous function in \((S,R,S_{R})\), it is straightforward to obtain the existence of solutions of Eq. (1) (as detailed in Perko (1991)). As the partial derivatives of the right-hand side of the system are continuous and bounded, uniqueness follows from the Picard–Lindelof theorem.

Representing the solutions of Eq. (1) in integral form we get:

$$\begin{aligned} R(t)=R(0)\exp \left( \frac{1}{\tau _r}\int \limits _{0}^{t} \left[ 1-\frac{S(s)+R(s)+S_{R}(s)}{K}\right] ds\right) \ge 0. \end{aligned}$$
(6)

Similarly, it is straightforward to prove the same inequalities for S(t) and \(S_{R}(t)\). Thus, if the initial conditions are nonnegative, the components of the solution are nonnegative for all finite time t. \(\square \)

In order to understand better the dynamics of Eq. (1), we will first calculate the fixed points and determine their stability. These are the isolated point \(P_{1}=(0,0,0)\) and the family of points of the intersection between the plane and the surface

$$\begin{aligned} R+S+S_R= & {} K, \end{aligned}$$
(7a)
$$\begin{aligned} S_R= & {} \frac{\tau _*}{\tau _c}SR, \end{aligned}$$
(7b)

which can be written in parametric form as

$$\begin{aligned} P_{2}(\lambda )=(S,R,S_{R})=\left( \lambda , \frac{K-\lambda }{1+\frac{\tau _*}{\tau _c}\lambda }, \frac{\lambda (K-\lambda )}{\lambda +\frac{\tau _c}{\tau _*}}\right) ,\quad 0\le \lambda \le K. \end{aligned}$$
(8)

To analyze the stability of these points, we calculate the Jacobian matrix of Eq. (1):

$$\begin{aligned} J= \left( \begin{array}{ccc} \frac{1}{\tau _{s}}\left( 1-\frac{S+R+S_{R}}{K}\right) -\frac{S}{\tau _{s}K}-\frac{R}{\tau _{c}}&{} -\frac{S}{\tau _{s}K} - \frac{S}{\tau _{c}} &{} -\frac{S}{\tau _{s}K}+\frac{1}{\tau _{*}} \\ -\frac{R}{\tau _{r}K} &{} \frac{1}{\tau _{r}} \left( 1-\frac{S+R+S_{R}}{K}\right) - \frac{R}{\tau _{r}K}&{} -\frac{R}{\tau _{r}K} \\ -\frac{S_{R}}{\tau _{r}K}+\frac{R}{\tau _{c}} &{} -\frac{S_{R}}{\tau _{r}K}+\frac{S}{\tau _{c}} &{} \frac{1}{\tau _{r}} \left( 1-\frac{S+R+S_{R}}{K}\right) -\frac{S_{R}}{\tau _{r}K} -\frac{1}{\tau _{*}} \end{array}\right) \end{aligned}$$

Equilibrium point \(P_{1}=(0,0,0)\). The Jacobian matrix is

$$\begin{aligned} J(P_{1})= \left( \begin{array}{ccc} 1/\tau _s &{} 0 &{} 0 \\ 0 &{} 1/\tau _r &{} 0 \\ 0 &{} 0 &{} 1/\tau _r - 1/\tau _* \end{array} \right) \end{aligned}$$

and the eigenvalues \(\mu _{1}= 1/\tau _s, \mu _{2}=1/\tau _{r}\) and \(\mu _{3}=1/\tau _r-1/\tau _{*}\). As in our model, because of the biological meaning of the parameters, \(1/\tau _r < 1/\tau _*\), \(\mu _1, \mu _2>0\) and \(\mu _3<0\) and point \(P_{1}\) is a saddle point and therefore, an unstable point.

Equilibrium points \(P_{2}(\lambda )= \left( \lambda , \frac{K-\lambda }{1+\tau _*\lambda /\tau _c}, \frac{\lambda (K-\lambda )}{\lambda +\tau _c/\tau _*}\right) \).

The Jacobian matrix for all these points is:

$$\begin{aligned} J(P_{2}(\lambda ))= \left( \begin{array}{ccc} -\frac{\lambda }{K\tau _s} + \frac{K-\lambda }{\tau _c+\tau _*\lambda } &{} -\frac{\lambda }{K\tau _s} - \frac{\lambda }{\tau _c} &{} -\frac{\lambda }{K\tau _s} + \frac{1}{\tau _*} \\ -\frac{1}{K\tau _r} \frac{K-\lambda }{1+\frac{\tau _*}{\tau _c}\lambda } &{} -\frac{1}{K\tau _r} \frac{K-\lambda }{1+\frac{\tau _*}{\tau _c}\lambda } &{} -\frac{1}{K\tau _r} \frac{K-\lambda }{1+\frac{\tau _*}{\tau _c}\lambda }\\ \frac{K-\lambda }{1+\frac{\tau _*}{\tau _c}\lambda } \frac{1}{\tau _c}- \frac{\lambda (K-\lambda )}{\lambda +\frac{\tau _c}{\tau _*}} \frac{1}{K\tau _r} &{}\quad \frac{\lambda }{\tau _c}- \frac{\lambda (K-\lambda )}{\lambda +\frac{\tau _c}{\tau _*}} \frac{1}{K\tau _r} &{}\quad - \frac{\lambda (K-\lambda )}{\lambda +\frac{\tau _c}{\tau _*}} \frac{1}{K\tau _r} - \frac{1}{\tau _*} \end{array} \right) \end{aligned}$$

The eigenvalues of the matrix are

$$\begin{aligned} \mu _{1} = 0, \quad \mu _{2} = -\frac{\tau _{c}+K\tau _{*}}{\lambda \tau _{*}^{2}+\tau _{c}\tau _{*}},\quad \mu _{3}=-\frac{\lambda \tau _{r}-\lambda \tau _{s}+K\tau _{s}}{K\tau _{r}\tau _{s}}. \end{aligned}$$

Since \(\mu _{1}=0\), these points are non-hyperbolic points. Moreover, as \(\mu _{2}\) and \(\mu _{3}\) are negative, for the parameters calculated in the article, the fixed points \(P_{2}(\lambda )\) possess two local stable manifolds (corresponding to the eigenvalues \(\mu _{2}\) and \(\mu _{3}\)) each of dimension 1 and a local center manifold (corresponding to the eigenvalue \(\mu _{1}\)) also of dimension 1.

At this point, since these stationary points are non-hyperbolic, the linear approximation is not valid. As \(\mu _{2}\) and \(\mu _{3}\) are negative, all the orbits starting near these equilibrium points approach the center manifold (see Fig. 11). Since the center manifold is formed by the curve of equilibrium points (8), all the orbits starting near these points tend to them. Thus, these points are stable.

The phase portrait for solutions of system (1) is shown in Fig. 11.

Fig. 11
figure 11

(Color figure online) (Left) Phase portrait of the orbits of Eq. (1). Black curve represents the set of equilibrium points \(P_{2}(\lambda )\), corresponding to the center manifold. Red arrows show the flow of the orbits, which are given by blue lines. (Right) Plot of eigenvalues corresponding to the equilibrium points \(E_2(\alpha )\) as a function of \(\alpha \)

1.2 Basic Properties of Model Eq. (2)

For model Eqs (2), we have:

Theorem 2

For any nonnegative initial data \((S_{0},R_{0},S_{R0}, Q_{0})\) and \(\tau _{s}> 0\), \(\tau _{r}>0\), the solutions to Eq. (2) exist for \(t>0\), are nonnegative and unique.

Proof

The proof mimics the steps of the proof of theorem 1 with the exception of the boundedness of the solution Q, which is not bounded. The remainder of the steps work in the same way so the repetitive details are omitted. \(\square \)

The equilibrium points of Eq. (2) are grouped in two different sets:

Equilibrium points \(E_1=\left( S,R,S_R, Q\right) \) are given by

$$\begin{aligned} S = 0,\quad R = 0,\quad S_R = 0,\quad Q = \alpha , \quad \text {with} \quad 0 \le \alpha < \infty . \end{aligned}$$

For these points, their corresponding eigenvalues are:

$$\begin{aligned} \mu _{1} = 0,\quad \mu _{2} = \frac{1}{\tau _{r}}, \quad \mu _{3}=f(\alpha ),\quad \mu _{4} = -\mu _{3}, \end{aligned}$$

where \(f(\alpha )\) is a continuous function depending on \(\alpha \). As there is at least one positive eigenvalue, \(\mu _{2}\), these points are all unstable equilibrium points. It has a biological explanation: Without cells, the concentration of MVs will not be affected, and the system will never tend to equilibrium.

Equilibrium points \(E_2(\alpha )=\left( S,R,S_R, Q\right) \), these points satisfy the condition \(S+R+S_{R} = K\) and they can be written as points of the following parametrized curve

$$\begin{aligned} S= & {} \frac{\tau _{n}\lambda _{1}K}{\tau _{n}\left( \lambda _{1}+\lambda _{2}\frac{\alpha }{Q_{th}+\alpha }\right) +\tau _{*}\alpha \lambda _{1}}, \\ R= & {} K-\frac{\lambda _{1}K(\tau _{*}\alpha +\tau _{n})}{\tau _{n}\left( \lambda _{1}+\lambda _{2}\frac{\alpha }{Q_{th}+\alpha }\right) +\tau _{*}\alpha \lambda _{1}},\\ S_R= & {} \frac{\tau _{*}\alpha \lambda _{1}K}{\tau _{n}\left( \lambda _{1}+\lambda _{2}\frac{\alpha }{Q_{th}+\alpha }\right) +\tau _{*}\alpha \lambda _{1}},\\ Q= & {} \alpha , \end{aligned}$$

where \(0 \le \alpha < \infty \). For these points, their corresponding eigenvalues are:

$$\begin{aligned} \beta _{1} = 0, \quad \beta _{2}=g(\alpha ),\quad \beta _{3} =h(\alpha ), \end{aligned}$$

and

$$\begin{aligned} \beta _{4} =-\frac{Q_{th}\lambda _1 \tau _n \tau _r+\lambda _1 \alpha \tau _n \tau _r+ \lambda _2 \alpha \tau _n \tau _s+\lambda _1 \alpha ^2 \tau _{*}\tau _s+Q_{th}\lambda _1 \alpha \tau _{*}\tau _s}{\tau _r\tau _s(\lambda _1 \alpha ^2 \tau _{*}+Q_{th}\lambda _1 \tau _n+\lambda _1 \alpha \tau _n+\lambda _2 \alpha \tau _n +Q_{th}\lambda _1 \alpha \tau _{*})}, \end{aligned}$$

where \(g(\alpha )\) and \(h(\alpha )\) are continuous functions depending on \(\alpha \). A graph of these eigenvalues is shown in Fig. 11, for the values of the parameters previously calculated.

It is straightforward to check that \(\beta _{2}\) and \(\beta _{4}\) do not cross the \(\alpha \)-axis. Thus, we obtain two negative \(\beta _2\), \(\beta _4\) and one null \(\beta _1\) eigenvalues.

Figure 11 shows that the eigenvalue \(\beta _3\) is positive or negative depending on the value of \(\alpha \). If \(\alpha \le 0.67\), all eigenvalues are non-positive, and, similarly, as in Appendix, we conclude that this subset of equilibrium points is stable. On the other hand, if \(\alpha > 0.67\), then the eigenvalue \(\beta _3\) is positive, and the equilibrium points \(E_2(\alpha )\) are unstable.

Therefore, the point \(E_2(0.67)\) is a bifurcation point.

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Rosa Durán, M., Podolski-Renić, A., Álvarez-Arenas, A. et al. Transfer of Drug Resistance Characteristics Between Cancer Cell Subpopulations: A Study Using Simple Mathematical Models. Bull Math Biol 78, 1218–1237 (2016). https://doi.org/10.1007/s11538-016-0182-0

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