2677 - Mathematical Modeling of Skin Toxicity in Proton Therapy

Purpose/Objective(s): Proton treatment often results in higher skin dose compared to conventional photon therapy, which leads to acute skin toxicity. Clinical observations frequently reveal flareups. We propose a mathematical framework using a dynamic system to model and explain the non-monotonic, patient-specific skin reactions.

Materials/Methods: Skin response is quantified by the grade of radiation dermatitis. We model the skin response using a second-order nonlinear ordinary differential equation that has 4 terms : linear term represents natural recovery ; quadratic term is time-dependent suppression representing a nonlinear, severity-dependent recovery mechanism that strengthens over time, especially after radiation ends; third term is a function of accumulated dose; and the last term is oscillatory term. Dose function is modeled by stepwise accumulated dose until finishing treatment then constant. Every solution curve under a parameter combination, which is a trajectory in dynamic system, represents a patient-specific skin reaction. The equation is numerically solved by Runge-Kutta 4^th order method. We use the limited-memory Broyden-Fletcher-Goldfarb-Shanno with box constraints optimization process to minimize the cost function that is the square difference between observation and model skin response to find the optimal parameters. A prescription of head neck case 2.2 Gy per fraction, 25 initial fractions and 7 boost fractions in 49 days is used. Three patient-specific types with different parameters are simulated with same oscillatory amplitude 0.3.

Results: The nonlinear dynamical system models flare-ups emerging from the interplay between dose-induced damage, biological recovery rates, and an intrinsic oscillatory inflammatory response. The model simulated skin toxicity trajectory that has delayed peak toxicity, recurrent flares and patient-specific variability. Highly radiosensitive patients with low recovery rates show larger oscillations and prolonged toxicity. The simulated 3 patient-specific types are shown in the table below.

Conclusion: The nonlinear dynamic system model captures non-monotonic skin toxicity in proton therapy through a biologically motivated oscillatory term. It will have potential applications in individualized toxicity prediction and patient counselling and adaptive planning.

Table: Simulated patient-specific type skin toxicity reaction using dynamic system

Parameter fitting	Patient-specific type
	Type 1	Type 2	Type 3
Recover rate	0.120	0.150	0.180
Radiosensitivity coefficient	0.020	0.018	0.015
Oscillatory frequency	0.100	0.300	0.500
Simulated phenomenon	Sharp rise, recover slow	Median response	Rapid recovery, less reaction, frequent fluctuation