Introduction of hypermatrix and operator notation into a discrete mathematics simulation model of malignant tumour response to therapeutic schemes in vivo. Some operator properties

Abstract

The tremendous rate of accumulation of experimental and clinical knowledge pertaining to cancer dictates the development of a theoretical framework for the meaningful integration of such knowledge at all levels of biocomplexity. In this context our research group has developed and partly validated a number of spatiotemporal simulation models of in vivo tumour growth and in particular tumour response to several therapeutic schemes. Most of the modeling modules have been based on discrete mathematics and therefore have been formulated in terms of rather complex algorithms (e.g. in pseudocode and actual computer code). However, such lengthy algorithmic descriptions, although sufficient from the mathematical point of view, may render it difficult for an interested reader to readily identify the sequence of the very basic simulation operations that lie at the heart of the entire model. In order to both alleviate this problem and at the same time provide a bridge to symbolic mathematics, we propose the introduction of the notion of hypermatrix in conjunction with that of a discrete operator into the already developed models. Using a radiotherapy response simulation example we demonstrate how the entire model can be considered as the sequential application of a number of discrete operators to a hypermatrix corresponding to the dynamics of the anatomic area of interest. Subsequently, we investigate the operators' commutativity and outline the "summarize and jump" strategy aiming at efficiently and realistically address multilevel biological problems such as cancer. In order to clarify the actual effect of the composite discrete operator we present further simulation results which are in agreement with the outcome of the clinical study RTOG 83-02, thus strengthening the reliability of the model developed.

Keywords: cancer multiscale modeling; computer models; hypermatrix; in silico oncology; oncosimulator; operator notation; radiobiology; tumour growth; tumour response to treatment.

Figure 1

A schematic representation of equation 1 showing the location of an indicative hypermatrix element ā (x_i, y_j, z_k, p_l, t_n) and its physically inhomogeneous and multidimensional content (g^ijkln, N_p^ijkln, t_p^ijkln, h_p^ijkln, h̃_p^ijkln) [see text for symbols; ~h_G2 ≡ h̃_G]. The proposed five dimensional discrete abstract space of tumour dynamics (corresponding to the localization of each hypermatrix element) is shown on the bottom right of the diagram. Three dimensions (corresponding to variables x_i, y_j, z_k) represent space, another one (corresponding to variable t_n) represents time and the fifth one (corresponding to variable p_l) represents the cell phase within or out of the cell cycle in which a biological cell or a set of biological cells within a geometrical cell of the discretization mesh is found at a given instant.

Figure 2

The proposed ten levels of biocomplexity.

Figure 3

Number of surviving tumour cells as a function of time for the glioblastoma tumour with mutant p53 gene (see section 6). The radiotherapeutic schemes correspond to schemes considered by the rTOg 83-02 clinical study. Abbreviations: AhF, accelerated hyperfractionation; hF, hyperfractionation.

Figure 4

Number of surviving tumour cells as a function of time for the glioblastoma tumour with wild-type p53 gene (see section 6). The radiotherapeutic schemes correspond to schemes considered by the rTOg 83-02 clinical study. The curves on the right correspond to the hF schedules whereas the curves on the left correspond to the AhF schedules. Abbreviations: AhF, accelerated hyperfractionation; hF, hyperfractionation.

Figure 5

Number of surviving tumour cells as a function of time for the glioblastoma tumour with intermediately adjusted radiosensitivity parameters (see section 6). The radiotherapeutic schemes correspond to schemes considered by the rTOg 83-02 clinical study. Abbreviations: AhF, accelerated hyperfractionation; hF, hyperfractionation.