A LQ-based kinetic model formulation for exploring dynamics of treatment response of tumours in patients

Abstract

A kinetic bio-mathematical, linear-quadratic (LQ) based model description for clonogenic survival is presented. In contrast to widely used formulations of models, a dynamic approach based on ordinary differential equations for coupling a repair model with a tumour growth model is used to allow analysis of intercellular process dynamics and submodel interference. The purpose of the model formulation is to find a quantitative framework for investigation of tumour response to radiotherapy in vivo. It is not the intention of the proposed model formulation to give a mechanistic explanation for cellular repair processes. This article addresses bio-mathematical aspects of the simplistic kinetic approach used for description of repair. The model formulation includes processes for cellular death, repopulation and cellular repair. The explicit use of the population size in the model facilitates the coupling of the sub-models including aspects of tissue dynamics (competition, oxygenation). The cellular repair is summarized by using a kinetic model for a dose equivalent Γ describing production and elimination of sublethal lesions. This dose equivalent replaces the absorbed dose used in the common LQ- model. Therefore, this approach is called the Γ- LQ- formulation. A comparison with two kinetic radiobiological models (the LPL model of Curtis and the compartmental model of Carlone) is carried out. The resulting differential equations are solved by numerical integration using a Runge-Kutta algorithm. The comparison reveals a good agreement between the Γ- LQ- formulation and the models of Curtis and Carlone under certain, defined conditions: The proposed formulation leads to results which are identical to the model of Carlone over a wide range of investigated biological parameters and different fractionation schemes when using first order repair kinetics. The comparison with experimental data and the LPL- model of Curtis shows a good agreement of the Γ- LQ- formulation using second order repair kinetics over a wide range of dose rate. Over a limited range, the use of second order repair in the Γ- LQ- formulation approximates the same dose rate dependency of clonogenic survival using only one additional parameter to those of the common LQ model. Within the investigated range of parameters, the presented Γ-LQ- formulation may be used to describe the in-vivo tumour response to radiation. The influence of repopulation, oxygenation and other aspects of tissue dynamics may override the differences between the intrinsic radiosensitivity yielded by each of the models. The proposed model formulation can be extended with additional static and dynamic tissue behaviours. This may be useful for the understanding of the reaction of tissues to heat (hyperthermia) or combined anti-cancer treatments (chemo-radiotherapy).