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Optimal control of treatment in a free boundary problem modeling multilayered tumor growth

Xinyue Evelyn Zhao, Yixiang Wu, Rachel Leander, Wandi Ding, Suzanne Lenhart

Abstract

We study the optimal control problem of a free boundary PDE model describing the growth of multilayered tumor tissue in vitro. We seek the optimal amount of tumor growth inhibitor that simultaneously minimizes the thickness of the tumor tissue and mitigates side effects. The existence of an optimal control is established, and the uniqueness and characterization of the optimal control are investigated. Numerical simulations are presented for some scenarios, including the steady-state and parabolic cases.

Optimal control of treatment in a free boundary problem modeling multilayered tumor growth

Abstract

We study the optimal control problem of a free boundary PDE model describing the growth of multilayered tumor tissue in vitro. We seek the optimal amount of tumor growth inhibitor that simultaneously minimizes the thickness of the tumor tissue and mitigates side effects. The existence of an optimal control is established, and the uniqueness and characterization of the optimal control are investigated. Numerical simulations are presented for some scenarios, including the steady-state and parabolic cases.

Paper Structure

This paper contains 9 sections, 5 theorems, 147 equations, 5 figures, 2 algorithms.

Table of Contents

  1. Introduction
  2. A special case: steady-state solution
  3. Existence and uniqueness of a positive state solution
  4. Existence of an optimal control
  5. Derivation of the optimality system
  6. Numerical simulations
  7. The steady-state case
  8. Parabolic case
  9. Discussion

Key Result

Theorem 3.1

Suppose that $u_0\in C^2[0,1]$ satisfy $0\le u_0(\xi)\le 1$ for $0<\xi < 1$ and the compatibility condition $($compatible$)$. For any $m\in U_M$, system $($eqn1$)$ -- $($eqn6$)$ has a unique solution $(u,\rho) \in W^{2,1,p}((0,1)\times(0,T)) \times C^1[0,T]$ for any $p>1$. In addition,

Figures (5)

  • Figure 1: The convergence of the optimal control in the steady-state case for $\widetilde{\sigma}=0.25$,$\mu=0.5$,$B=2$,$M=1$,and $TOL=10^{-5}$. The left figure shows the convergence with an initial $m_0=0.8$, and the right figure with an initial $m_0=0.2$.
  • Figure 2: The chosen parameters are $\rho_0=2$, $u_0(\xi)=\frac{\cosh(\rho_0 \xi)}{\cosh(\rho_0)} + 0.1\cos(\frac{7}{2} \pi \xi)$, $\widetilde{\sigma}=0.25$, $B=0.05$, $M=1$, $\mu=0.5$, and $TOL=10^{-3}$. The first row shows the initial $m_0(t)=0.35$ and its corresponding $\rho(t)$, $u(\xi,t)$, and $J=12.1722$. The second row shows the optimized $m^*(t)$ and its corresponding $\rho(t)$, $u(\xi,t)$, and $J=11.5277$. The red dashed curve in the second row represents $\rho(t)$ without any control.
  • Figure 3: The chosen parameters are $\rho_0=2$, $u_0(\xi)=\frac{\cosh(\rho_0 \xi)}{\cosh(\rho_0)} + 0.1\cos(\frac{7}{2} \pi \xi)$, $\widetilde{\sigma}=0.25$, $B=0.05$, $M=1$, $\mu=0.5$, and $TOL=10^{-3}$. The first row shows the initial $m_0(t)=0.35+0.1\cos(4\pi t)$ and its corresponding $\rho(t)$, $u(\xi,t)$, and $J=12.1735$. The second row shows the optimized $m^*(t)$ and its corresponding $\rho(t)$, $u(\xi,t)$, and $J=11.5277$. The red dashed curve in the second row represents $\rho(t)$ without any control.
  • Figure 4: The optimized $m^*(t)$ and its corresponding $\rho(t)$ and $u(\xi,t)$. The chosen parameters are $\rho_0=2$, $u_0(\xi)=\frac{\cosh(\rho_0 \xi)}{\cosh(\rho_0)} + 0.1\cos(\frac{7}{2} \pi \xi)$, $\widetilde{\sigma}=0.75$, $B=0.05$, $M=1$, $\mu=0.5$, and $TOL=10^{-3}$. The red dashed curve in the second figure represents $\rho(t)$ without any control.
  • Figure 5: The optimized $m^*(t)$ and its corresponding $\rho(t)$ and $u(\xi,t)$. The chosen parameters are $\rho_0=2$, $u_0(\xi)=\frac{\cosh(\rho_0 \xi)}{\cosh(\rho_0)} + 0.1\cos(\frac{7}{2} \pi \xi)$, $\widetilde{\sigma}=0.25$, $B=0.5$, $M=1$, $\mu=0.5$, and $TOL=10^{-3}$. The red dashed curve in the second figure represents $\rho(t)$ without any control.

Theorems & Definitions (10)

  • Theorem 3.1
  • proof
  • Theorem 4.1
  • proof
  • Theorem 5.1
  • proof
  • Theorem 5.2
  • proof
  • Theorem 5.3
  • proof