Optimization of tumour control probability for heterogeneous tumours in fractionated radiotherapy treatment protocols.

We find the dose distribution that maximizes the tumour control probability (TCP) for a fixed mean tumour dose per fraction. We consider a heterogeneous tumour volume having a radiation response characterized by the linear quadratic model with heterogeneous radiosensitivity and repopulation rate that may vary in time. Using variational calculus methods a general solution is obtained. We demonstrate the spatial dependence of the optimal dose distribution by explicitly evaluating the solution for different functional forms of the tumour properties. For homogeneous radiosensitivity and growth rate, we find that the dose distribution that maximizes TCP is homogeneous when the clonogen cell density is homogeneous, while for a heterogeneous initial tumour density we find that the first dose fraction is inhomogeneous, which homogenizes the clonogen cell density, and subsequent dose fractions are homogeneous. When the tumour properties have explicit spatial dependence, we show that the spatial variation of the optimized dose distribution is insensitive to the functional form. However, the dose distribution and tumour clonogen density are sensitive to the value of the repopulation rate. The optimized dose distribution yields a higher TCP than a typical clinical dose distribution or a homogeneous dose distribution.