On the Inclusion of Short-distance Bystander Effects into a Logistic Tumor Control Probability Model

Currently, interactions between voxels are neglected in the tumor control probability (TCP) models used in biologically-driven intensity-modulated radiotherapy treatment planning. However, experimental data suggests that this may not always be justified when bystander effects are important. We propose a model inspired by the Ising model, a short-range interaction model, to investigate if and when it is important to include voxel to voxel interactions in biologically-driven treatment planning. This Ising-like model for TCP is derived by first showing that the logistic model of tumor control is mathematically equivalent to a non-interacting Ising model. Using this correspondence, the parameters of the logistic model are mapped to the parameters of an Ising-like model and bystander interactions are introduced as a short-range interaction as is the case for the Ising model. As an example, we apply the model to study the effect of bystander interactions in the case of radiation therapy for prostate cancer. The model shows that it is adequate to neglect bystander interactions for dose distributions that completely cover the treatment target and yield TCP estimates that lie in the shoulder of the dose response curve. However, for dose distributions that yield TCP estimates that lie on the steep part of the dose response curve or for inhomogeneous dose distributions having significant hot and/or cold regions, bystander effects may be important. Furthermore, the proposed model highlights a previously unexplored and potentially fruitful connection between the fields of statistical mechanics and tumor control probability/normal tissue complication probability modeling.