A biological optimization method for carbon therapy via iterative Jacobian-based linearization

<i>Objective.</i>Carbon ion radiotherapy (CIRT) can provide higher biological effectiveness and cause more damage to cancer cells compared to photon or proton radiotherapy, especially for radio-resistant tumors. The optimization of biological dose is essential for CIRT, to achieve the desirable tumoricidal dose while mitigating biological damage to normal tissues and organs at risk (OAR). However, the biological optimization for CIRT is mathematically challenging, due to the nonlinear nature of biological dose model, which can lead to computational inaccuracy and inefficiency. This work will develop an accurate and efficient biological optimization method for CIRT.<i>Approach.</i>The proposed method is called iterative Jacobian-based linearization (IJL). In IJL, the biological dose is modeled as the product of the physical dose and relative biological effect, which is based on the linear-quadratic model via the local effect model in this work, and the optimization objective consists of dose-volume histogram based biological dose objectives within clinical target volume and OAR. The optimization algorithm for IJL is through iterative convex relaxation, in which the nonlinear biological dose is iteratively linearized using Jacobian-based approximations and the linear subproblems are solved using alternating direction method of multipliers. To compare with IJL, the limited-memory quasi-Newton (QN) method (limited-memory version) is developed that directly solves the same nonlinear biological optimization problem.<i>Main results.</i>Compared to the QN, IJL demonstrated superior plan accuracy, e.g. better OAR sparing with the reduction of biological dose in the CTV-surrounding volume (PTV1cm) to 89.7%, 95.0%, 88.3% for brain, lung, and abdomen, respectively; IJL also had higher computational efficiency, with approximately 1/10 the computational time per iteration and continuously decreasing objectives (while being stagnated for QN after certain number of iterations).<i>Significance.</i>A novel optimization algorithm, IJL, incorporating iterative linearization of biological dose, is proposed to accurately and efficiently solve the biological optimization problem for CIRT. It demonstrates superior plan accuracy and computational efficiency compared to the direct nonlinear QN optimization method.