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Modeling the Dynamics of Heterogeneity of Solid Tumors in Response to Chemotherapy

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Abstract

In this paper, we extend the model of the dynamics of drug resistance in a solid tumor that was introduced by Lorz et al. (Bull Math Biol 77:1–22, 2015). Similarly to the original, radially symmetric model, the quantities we follow depend on a phenotype variable that corresponds to the level of drug resistance. The original model is modified in three ways: (i) We consider a more general growth term that takes into account the sensitivity of resistance level to high drug dosage. (ii) We add a diffusion term in space for the cancer cells and adjust all diffusion terms (for the nutrients and for the drugs) so that the permeability of the resource and drug is limited by the cell concentration. (iii) We add a mutation term with a mutation kernel that corresponds to mutations that occur regularly or rarely. We study the dynamics of the emerging resistance of the cancer cells under continuous infusion and on–off infusion of cytotoxic and cytostatic drugs. While the original Lorz model has an asymptotic profile in which the cancer cells are either fully resistant or fully sensitive, our model allows the emergence of partial resistance levels. We show that increased drug concentrations are correlated with delayed relapse. However, when the cancer relapses, more resistant traits are selected. We further show that an on–off drug infusion also selects for more resistant traits when compared with a continuous drug infusion of identical total drug concentrations. Under certain conditions, our model predicts the emergence of a heterogeneous tumor in which cancer cells of different resistance levels coexist in different areas in space.

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Notes

  1. Since the perturbations in the decay coefficients are small, for instance, \(\gamma _s \gg \int _0^1 p(\theta ) n(t,r,\theta ) \mathrm{d}\theta \), this estimation is similar to our simulation.

  2. The solutions can be explicitly written in terms of a modified Bessel function of the first kind \(I_0(r)\) as \(s(r) = S_1 I_0\left( \sqrt{\lambda _s}r \right) / I_0\left( \sqrt{\lambda _s} \right) \), where \(\lambda _s = {\gamma _s}/{\alpha _s}\), and similarly for \(c_1(r)\) and \(c_2(r)\). Then, \(\hat{R}_k(\theta ) = \frac{p(\theta )}{1+\mu _2 c_2(r_k)} s(r_k)\) and \( \hat{C}_k(\theta ) = \mu _1(\theta ) c_1(r_k)\).

  3. \(R_k^M = \max _{\theta } \left[ \frac{p(\theta )}{1+\mu _2 c_2(r_k)} s(r_k) \right] \), \(C_{k}^m = \min _\theta \left[ \mu _1(\theta ) c_1(r_k)\right] \). Space independent bounds can be given as \(R^M = \max _{\theta } \left[ \frac{p(\theta )}{1+\mu _2 C_2/I_0(\sqrt{\lambda _{c_2}})} S_1 \right] \) and \(C^m = \min _\theta \left[ \mu _1(\theta ) C_1/I_0(\sqrt{\lambda _{c_1}}) \right] \), where \(I_0(r)\) is a modified Bessel function of the first kind, \(\lambda _{c_1} = {\gamma _{c_1}}/{\alpha _{c_1}}\), and \(\lambda _{c_2} = {\gamma _{c_2}}/{\alpha _{c_2}}\).

  4. \(D_{KL}(Q_0||Q_t)\) is the KL divergence from the initial distribution \(Q_0(\theta ) := Q(t=0,\theta )\),

    $$\begin{aligned} D_{KL} ( Q_0 || Q_t ) := \int _0^1 Q_0(\theta ) \log \dfrac{Q_0(\theta )}{Q({t,\theta })} \mathrm{d}\theta , \end{aligned}$$

    that represents the divergence of the phenotype distribution from initial time.

  5. Similar behavior can be modeled using the Gaussian kernel (18) by considering a piecewise continuous correlation length on \(\{\varOmega _i\}\). For instance, \( \ell _u(\theta ,\theta ^{\prime }) = \ell _i( \bar{\theta } )\) on each \(\bar{\theta } \in \varOmega _i,\) where \(\ell _i( \bar{\theta } )\) is a quadratic function that decays as \(\bar{\theta }\) approaches the partition boundary.

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Acknowledgements

The work of DL was supported in part by the National Science Foundation under Grant Number DMS-1713109, by the John Simon Guggenheim Memorial Foundation, by the Simons Foundation, and by the Jayne Koskinas and Ted Giovanis Foundation.

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Authors and Affiliations

  1. Department of Mathematics, University of Maryland, College Park, MD, 20742, USA

    Heyrim Cho

  2. Department of Mathematics and Center for Scientific Computation and Mathematical Modeling (CSCAMM), University of Maryland, College Park, MD, 20742, USA

    Doron Levy

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Correspondence to Doron Levy.

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Cho, H., Levy, D. Modeling the Dynamics of Heterogeneity of Solid Tumors in Response to Chemotherapy. Bull Math Biol 79, 2986–3012 (2017). https://doi.org/10.1007/s11538-017-0359-1

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