Introduction of Hypermatrix and Operator Notation into a Discrete Mathematics Simulation Model of Malignant Tumour Response to Therapeutic SchemesIn 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.

Integrating Multiscale Modeling with Drug Effects for Cancer Treatment

“…5 An extension of the work will arm researchers with mathematical and systematic ways of understanding the mechanisms of drug action and adverse responses in a computational multiscale context, thereby improving the probability of success in new drug discovery. 38 The ContraCancrum project paradigm 5,[74][75][76][77][78][79][80][81][82] has been focused on clinically driven design of multiscale cancer models. The project has been seeking to develop a multiscale cancer models that contribute to the clinical adaptations of predictive in silico oncology.…”

“…The ContraCancrum project paradigm 74–82 has been focused on clinically driven design of multiscale cancer models. The project has been seeking to develop a multiscale cancer models that contribute to the clinical adaptations of predictive in silico oncology.…”

Top-Down Multiscale Simulation of Tumor Response to Treatment in the Context of In Silico Oncology. The Notion of Oncosimulator

“…Discrete time is used. An important discretization aspect of the method is the mean time spent in the phase of an equivalence class by the biological cells belonging to the equivalence class (Stamatakos et al 2001c(Stamatakos et al , 2002Dionysiou et al 2004Dionysiou et al , 2006Stamatakos and Dionysiou 2009b). In order to allow for spatiotemporal perturbations of critical parameter values throughout the tumor and also avoid artificial cell synchronizations due to quantization, use of pseudo-random numbers is extensively made (Monte Carlo technique).…”

Model-Supported Radiotherapy Personalization: In silico Test of Hyper- and Hypo-Fractionation Effects

“…In the light of this clinical context, mathematical models of tumor evolution and response to treatment could play an important role allowing the customization of radiotherapy simulating different irradiation protocols and thus supporting the selection of the most effective ( Stamatakos and Dionysiou, 2009 ). Mathematical models can range from statistical methods to calculate the tumor control probability (TCP) and normal tissue complication probability (NTCP) ( Thames et al, 1983 ; Lyman, 1985 ) to multiscale 3D models of cellular and subcellular mechanisms that regulate tumor dynamics ( Bellomo et al, 2004 ; Stamatakos and Dionysiou, 2009 ; Boondirek et al, 2010 ). Radiobiological models that evaluate the effects of radiotherapy are routinely used in clinical practice.…”