Abstract
In order to model the immune response in tumor of the patients under radiotherapy for cancer. Characteristics of the numerical model and preliminary application are presented.
Immune response was modelled by 4 set of ordinary difference equations (ODE) as a function of biomedical variables including the amount of tumor antigen naturally released by tumor, the activation of cytotoxic T-lymphocytes (CTL) by radiation, immune suppression by tumor volume and the use of immunotherapy drugs. The effect of heterogeneous radiation dose distribution is also considered by hyperbolic tangent function to account for the immunogenic response of tissue under highly heterogeneous dose distribution intentionally modulated in spatially fractionated radiotherapy. Boundary behaviors of the model were investigated for tumors with different biomedical characteristics and under different treatment conditions. The developed model was applied to the tumor volume change in a mouse with 67NR tumors after radiation of 10 Gy to full or half volume of tumor and a clinical patient treated for sarcoma three times over 4 years.
Tumor growth is exponential at early phase, slow down over time with increasing immune response and eventually reaches an equilibrium condition (known as terminal tumor volume) for tumors with little to no immune suppression capability (ISC) even in the lack of radiation treatment. Breaking-through the equilibrium for tumor to grow exponentially happens when ISC is larger than the bifurcation threshold, analytically calculated from the proposed model. Tumor with ISC close to the bifurcation threshold can show complex growth behavior depending on the treatment condition and it should be carefully considered for the optimal treatment. Tumor volume change over 30 days period on mouse was modeled well with this model. Full dose irradiation reduced the tumor volume faster in the first 10 days but half volume irradiation reduced the tumor volume faster at later stage due to the improved immune response which cannot be modelled with traditional linear quadratic model. Tumor volume on a patient retreated three times over 4 years was also accurately estimated.
The proposed immune response model is capable to estimate tumor volume response under full or partial volume irradiation considering the complex immune characteristics of the tissue.