Data fitting model for the kernel of integral operator from radiation therapy

An acute problem in radiation therapy is to set the treatment fields and to calculate the intensities of the fields that will provide a prescribed dose distribution in patient. The commonly accepted and used mathematical model consists of the Fredholm integral equation of the first kind. The integral equation includes a kernel, the so-called composed average energy deposition kernel, which is composed of several average energy deposition kernels corresponding to different field settings. The analytical expression of the kernel(s) is unknown. This paper considers some data fitting techniques to get a continuous approximation for the kernel. The aim of the study is to get a continuous kernel which can be computed in practice. An inverse problem which enables the optimal settings of fields and field intensities (under some limitations) is stated but not simulated here. Some simulations which confirm the suitability of the fitting method are presented.