On the role of tumor heterogeneity for optimal cancer chemotherapy

Urszula Ledzewicz; Heinz Schättler; Shuo Wang

doi:10.3934/nhm.2019007

Article Contents

2019, Volume 14, Issue 1: 131-147. Doi: 10.3934/nhm.2019007

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On the role of tumor heterogeneity for optimal cancer chemotherapy

1.
Institute of Mathematics, Lodz University of Technology, 90-924 Lodz, Poland
2.
Dept. of Mathematics and Statistics, Southern Illinois University Edwardsville, Edwardsville, Il, 62026-1653, USA
3.
Dept. of Electrical and Systems Engineering, Washington University, St. Louis, Mo, 63130, USA
4.
Department of Mechanical and Aerospace Engineering, University of Texas at Arlington, Arlington, TX, 76010, USA

^* Corresponding author: Urszula Ledzewicz
^* Corresponding author: Urszula Ledzewicz

Received: April 2018

Revised: October 2018

Published: January 2019

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Abstract

We review results about the influence tumor heterogeneity has on optimal chemotherapy protocols (relative to timing, dosing and sequencing of the agents) that can be inferred from mathematical models. If a tumor consists of a homogeneous population of chemotherapeutically sensitive cells, then optimal protocols consist of upfront dosing of cytotoxic agents at maximum tolerated doses (MTD) followed by rest periods. This structure agrees with the MTD paradigm in medical practice where drug holidays limit the overall toxicity. As tumor heterogeneity becomes prevalent and sub-populations with resistant traits emerge, this structure no longer needs to be optimal. Depending on conditions relating to the growth rates of the sub-populations and whether drug resistance is intrinsic or acquired, various mathematical models point to administrations at lower than maximum dose rates as being superior. Such results are mirrored in the medical literature in the emergence of adaptive chemotherapy strategies. If conditions are unfavorable, however, it becomes difficult, if not impossible, to limit a resistant population from eventually becoming dominant. On the other hand, increased heterogeneity of tumor cell populations increases a tumor's immunogenicity and immunotherapies may provide a viable and novel alternative for such cases.

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Mathematics Subject Classification: Primary: 92C50, 49K15; Secondary: 93C15.

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Figure 1. Example of locally optimal controls for a $ 3 $-compartment model with cytotoxic ($ u $) and cytostatic ($ v $) agents. The initial condition is the normalized (in terms of percentages [44]) steady-state solution of the uncontrolled system and an objective of the type (2) has been minimized

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Figure 2. Example of an extremal control and associated states for a bang-singular controlled trajectory

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Figure 3. Extremal controls (top), evolution of the total tumor $ \bar{N} $ (middle) and profiles $ n(20,x) $ at the terminal time $ T = 20 $ for different mutation rates $ \theta $

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Figure 4. Example of the phase portraits for the system (19)-(20) with a Gompertzian growth function $ F(x) = - \xi \ln \left( \frac{x}{K} \right) $ with tumor growth rate $ \xi $ and carrying capacity $ K $. The benign equilibrium point is marked with a green star and the malignant one with a red star

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Figure 5. Example of numerically computed optimal controls for the system (19)-(20) with a Gompertzian growth function

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