Optimal radiotherapy treatment planning using minimum entropy models

We study the problem of finding an optimal radiotherapy treatment plan. A time-dependent Boltzmann particle transport model is used to model the interaction between radiative particles with tissue. This model allows for the modeling of inhomogeneities in the body and allows for anisotropic sources modeling distributed radiation—as in brachytherapy—and external beam sources—as in teletherapy. We study two optimization problems: minimizing the deviation from a spatially-dependent prescribed dose through a quadratic tracking functional; and minimizing the survival of tumor cells through the use of the linear-quadratic model of radiobiological cell response. For each problem, we derive the optimality systems. In order to solve the state and adjoint equations, we use the minimum entropy approximation; the advantages of this method are discussed. Numerical results are then presented.

💡 Research Summary

This paper formulates radiotherapy treatment planning as an optimal control problem governed by a time‑dependent Boltzmann particle transport equation. The authors argue that conventional dose‑calculation methods such as the Fermi‑Eyges theory inadequately handle tissue heterogeneities and anisotropic radiation sources, especially in brachytherapy and intensity‑modulated external beam therapy. By employing the full Boltzmann equation, the model captures absorption, scattering, and directional dependence of photon transport, with material properties described through spatially varying total and scattering cross‑sections and a Henyey‑Greenstein scattering kernel.

Two distinct objective functionals are considered. The first is a quadratic tracking functional that penalizes the L² deviation between the delivered total dose (the time‑integrated angular flux) and a prescribed spatial dose distribution, with spatially varying weights for tumor, risk‑organ, and normal tissue regions, plus a regularization term on the control magnitude. The second functional incorporates the linear‑quadratic (LQ) radiobiological model: the surviving fraction of cells is expressed as S = exp(−αD − βD²). By integrating this expression over the domain with tissue‑specific cell densities and weighting factors, the authors obtain a non‑convex cost that simultaneously minimizes tumor cell survival and normal‑tissue damage, again regularized by a control‑norm term.

Direct solution of the Boltzmann equation is computationally prohibitive because it depends on time, three spatial coordinates, and two angular variables. To overcome this, the authors adopt a minimum‑entropy (M₁) closure, which retains only the zeroth and first angular moments (energy density and flux) while reconstructing higher moments by minimizing the angular entropy. This closure preserves essential physical properties such as positivity, energy conservation, and a finite propagation speed, yet reduces the dimensionality dramatically, making the state and adjoint equations tractable.

The optimality system is derived via the Lagrange‑multiplier (adjoint) method. The state equation (Boltzmann with control source q) and the adjoint equation (the same transport operator with a source term given by the derivative of the objective with respect to the state) are coupled. The optimal control is expressed as a projection onto the admissible interval