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A mathematical model to predict growth and treatment for UPS cancer

Sumit Roy

Abstract

We propose a mathematical model for the growth and treatment dynamics of Undifferentiated Pleomorphic Sarcoma (UPS) based on a system of nonlinear differential equations. The model integrates Gompertz-type tumor growth with surface-area dependent necrotic loss, surgical resection with residual disease, postoperative recovery, tumor-immune interaction, and an optimal radiation treatment component. We analyze the resulting dynamical system and obtain several properties of the model. The growth equation exhibits a threshold below which tumor volume cannot be sustained. The postoperative phase shows transient dynamics prior to proliferative recovery. For the tumor-immune subsystem, equilibrium states and local stability conditions are identified. The radiation treatment problem is formulated as an optimal control problem, and the optimal strategy is shown to be of bang-bang type. Numerical simulations illustrate the influence of biological and treatment parameters on tumor evolution, and the results are qualitatively consistent with clinical patterns reported for UPS.

A mathematical model to predict growth and treatment for UPS cancer

Abstract

We propose a mathematical model for the growth and treatment dynamics of Undifferentiated Pleomorphic Sarcoma (UPS) based on a system of nonlinear differential equations. The model integrates Gompertz-type tumor growth with surface-area dependent necrotic loss, surgical resection with residual disease, postoperative recovery, tumor-immune interaction, and an optimal radiation treatment component. We analyze the resulting dynamical system and obtain several properties of the model. The growth equation exhibits a threshold below which tumor volume cannot be sustained. The postoperative phase shows transient dynamics prior to proliferative recovery. For the tumor-immune subsystem, equilibrium states and local stability conditions are identified. The radiation treatment problem is formulated as an optimal control problem, and the optimal strategy is shown to be of bang-bang type. Numerical simulations illustrate the influence of biological and treatment parameters on tumor evolution, and the results are qualitatively consistent with clinical patterns reported for UPS.
Paper Structure (14 sections, 12 theorems, 106 equations, 1 figure, 3 tables)

This paper contains 14 sections, 12 theorems, 106 equations, 1 figure, 3 tables.

Table of Contents

  1. Introduction
  2. Tumor Growth Dynamics
  3. Surgical Resection Module
  4. Mathematical Formulation and Analysis
  5. Therapeutic Implications
  6. Postoperative Recovery Dynamics in UPS
  7. Inflammatory Phase Model
  8. Proliferative Phase Analysis
  9. Phase Transition Analysis
  10. Optimal Radiation Timing
  11. Immunotherapy Dynamics
  12. Outcome Projections
  13. Model Validation and Clinical Correlation
  14. Conclusion

Key Result

Theorem 2.1

Consider the tumor growth model eq:growth with $\lambda > 0$.

Figures (1)

  • Figure 1: Comparison of tumor growth trajectories predicted by the modified Gompertz--necrosis model for representative tumor types.

Theorems & Definitions (27)

  • Theorem 2.1
  • proof
  • Remark 2.1
  • Theorem 2.2
  • proof
  • Remark 2.2
  • Theorem 3.1
  • proof
  • Theorem 3.2
  • proof
  • ...and 17 more