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A Mathematical Model for Chemotherapy, Immunotherapy and Virotherapy Treatments of Cancer

Tarini Kumar Dutta, Silmera A Sangma, Janice Moore, Meir Shillor

TL;DR

The analysis shows that the model solutions exist, are bounded, and nonnegative on each finite time interval, thus biologically feasible, and once validated in the field, the model can be used to design treatment schedules of combinations of the three modalities for improved outcomes.

Abstract

We continue our study of a model for cancer treatment, constructed in Dutta et. al., 2025, by adding Virotherapy to the Chemotherapy and Immunotherapy studied there. It is a dynamical system model for the spread of cancer in healthy tissue. It allows computer experiments of various combinations of the three modalities, which cannot be performed in the laboratory or experimentally. The novelty is the addition of Virotherapy. The analysis shows that the model solutions exist, are bounded, and nonnegative on each finite time interval, thus biologically feasible. A time-stepping algorithm is constructed and implemented, and computer simulations are presented. The simulations show the development of the disease under various treatment options, including a baseline case without treatment, cases for each of the three treatments separately, and some combinations of the three treatments. These simulations indicate that combinations of treatments are more effective. However, we do not consider any limitation or incompatibilities of the joint application of the three modalities, that may exist in practice. Once validated in the field, the model can be used to design treatment schedules of combinations of the three modalities for improved outcomes.

A Mathematical Model for Chemotherapy, Immunotherapy and Virotherapy Treatments of Cancer

TL;DR

The analysis shows that the model solutions exist, are bounded, and nonnegative on each finite time interval, thus biologically feasible, and once validated in the field, the model can be used to design treatment schedules of combinations of the three modalities for improved outcomes.

Abstract

We continue our study of a model for cancer treatment, constructed in Dutta et. al., 2025, by adding Virotherapy to the Chemotherapy and Immunotherapy studied there. It is a dynamical system model for the spread of cancer in healthy tissue. It allows computer experiments of various combinations of the three modalities, which cannot be performed in the laboratory or experimentally. The novelty is the addition of Virotherapy. The analysis shows that the model solutions exist, are bounded, and nonnegative on each finite time interval, thus biologically feasible. A time-stepping algorithm is constructed and implemented, and computer simulations are presented. The simulations show the development of the disease under various treatment options, including a baseline case without treatment, cases for each of the three treatments separately, and some combinations of the three treatments. These simulations indicate that combinations of treatments are more effective. However, we do not consider any limitation or incompatibilities of the joint application of the three modalities, that may exist in practice. Once validated in the field, the model can be used to design treatment schedules of combinations of the three modalities for improved outcomes.
Paper Structure (12 sections, 3 theorems, 62 equations, 7 figures, 2 tables, 1 algorithm)

This paper contains 12 sections, 3 theorems, 62 equations, 7 figures, 2 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Mathematical Model
  3. Existence, positive invariance and boundedness
  4. Equilibrium points and their stability
  5. Disease free equilibrium
  6. Endemic equilibrium
  7. Numerical algorithm and simulations
  8. Simulations
  9. Combinations of treatments
  10. Numerical Convergence
  11. Conclusions
  12. Appendix

Key Result

Proposition 3.1

Assume that the initial conditions satisfy (27) and $q,\varphi_D, \varphi_V$ and $\psi$ are Lipschitz, nonnegative and bounded functions, say $0\leq q, \varphi_D, \varphi_V, \psi \leq M$. Then, every solution of system (21)--(26) satisfies the estimates: for $0<T<\infty$. Here,

Figures (7)

  • Figure 1: No Treatment. The behavior of the system without any treatment. Data values are shown in Table 1. Cancer grows until it reaches its carrying capacity. Healthy and immune cells decline as the cancer grows, and without treatment, these cells eventually die out completely.
  • Figure 2: Chemotherapy Weekly. Only chemotherapy is applied in a weekly dose, where $\varphi_D = 2000$ for 1 hour of treatment on days 7, 14, and 21. The graph indicates that after three rounds of chemotherapy cancer is eliminated on day 28.
  • Figure 3: Immunotherapy Weekly. Only immunotherapy is applied in a weekly dose where $\psi = 302.5$ for 1 hour of treatment on days 7, 14, and 21. Following three rounds of immunotherapy cancer is eliminated on day 26.
  • Figure 4: Virotherapy Weekly. Only virotherapy is applied in a weekly dose where $\varphi_V = 350$ for 1 hour of treatment on days 7, 14, 21 and 28. It is seen that after four rounds of virotherapy cancer disappears on day 34.
  • Figure 5: Three Treatments at Once. Weekly administered chemotherapy, immunotherapy, and virotherapy. Cancer (black -- uninfected cells) vanishes ($K_U<1$) after the first round, on day 11; then the healthy and immune cell populations increase monotonically towards their steady states.
  • ...and 2 more figures

Theorems & Definitions (5)

  • Remark 2.2
  • Proposition 3.1
  • proof
  • Theorem 3.2: Global Existence
  • Proposition 4.1