Optimal Control of Medical Drug in a Nonlocal Model of Solid Tumor Growth

TL;DR

This work develops a nonlocal tumor growth model coupled to chemotherapy, where tumor density $p(t,x)$ and drug concentration $d(t,x)$ interact through a nonlocal velocity field $\mathrm{V}[w_p]$ with $w_p(t,x)=\int_\Omega K(x,y)p(t,y)\,dy$. An optimal control problem is posed to minimize tumor burden while limiting drug toxicity, with a convex admissible control set $U_{ad}$ and a cost functional $J(I)$ that blends tumor mass at final time, cumulative tumor load, and drug usage. The authors establish well-posedness of the coupled system, demonstrate existence of an optimal control, and derive first-order optimality conditions via a linearized system and an adjoint system, yielding a gradient-based characterization of optimal dosing. Theoretical results are complemented by numerical simulations showing substantial tumor reduction and a decreasing, practically interpretable dosing schedule, validating the approach and illustrating potential clinical relevance for chemotherapy planning in spatially heterogeneous tumors. The framework is poised for extensions to immune interactions and tumor heterogeneity, with implications for nonlocal control strategies in oncology and related biological systems.

Abstract

This paper presents a mathematical framework for optimizing drug delivery in cancer treatment using a nonlocal model of solid tumor growth. We present a coupled system of partial differential equations that incorporate long-range cellular interactions through integral terms and drug-induced cell death. The model accounts for spatial heterogeneity in both tumor cell density and drug concentration while capturing the complex dynamics of drug resistance development. We first establish the well-posedness of the coupled system by proving the existence and uniqueness of a solution under appropriate regularity conditions. The optimal control problem is then formulated to minimize tumor size while accounting for drug toxicity constraints. Using variational methods, we derive the necessary optimality conditions and characterize the optimal control through an adjoint system. Theoretical results can help to design effective chemotherapy schedules that balance treatment efficacy with adverse effects.

Figures (9)

• Figure 1: Optimal control profile $I^*(t)$ over the treatment interval $[0,T]$.
• Figure 2: Three-dimensional visualization of tumor density evolution $p(t,x)$.
• Figure 3: Tumor density profiles $p(t,x)$ at selected time points comparing controlled (solid blue) and uncontrolled (dashed red) cases.
• Figure 4: Contour plots of tumor density $p(t,x)$ in the $(x,t)$ plane.
• Figure 5: Three-dimensional visualization of drug concentration $d(t,x)$.

Theorems & Definitions (19)

• Remark
• Theorem 3.1: Well-posedness of the coupled system
• proof
• Lemma 3.1: Non-negativity of the solutions
• proof
• Theorem 3.2: Existence of an Optimal Control
• proof
• Lemma 4.1
• proof
• Lemma 4.2: Gâteaux Differentiability of the Control-to-State Mapping