Modelling Radiation Cancer Treatment with a Death-Rate Term in Ordinary and Fractional Differential Equations

Fractional calculus has recently been applied to the mathematical modelling of tumour growth, but its use introduces complexities that may not be warranted. Mathematical modelling with differential equations is a standard approach to study and predict treatment outcomes for population-level and patient-specific responses. Here, we use patient data of radiation-treated tumours to discuss the benefits and limitations of introducing fractional derivatives into three standard models of tumour growth. The fractional derivative introduces a history-dependence into the growth function, which requires a continuous death-rate term for radiation treatment. This newly proposed radiation-induced death-rate term improves computational efficiency in both ordinary and fractional derivative models. This computational speed-up will benefit common simulation tasks such as model parameterization and the construction and running of virtual clinical trials

Mathematical modeling of the metastatic process

Mathematical modeling in cancer has been growing in popularity and impact since its inception in 1932. The first theoretical mathematical modeling in cancer research was focused on understanding tumor growth laws and has grown to include the competition between healthy and normal tissue, carcinogenesis, therapy and metastasis. It is the latter topic, metastasis, on which we will focus this short review, specifically discussing various computational and mathematical models of different portions of the metastatic process, including: the emergence of the metastatic phenotype, the timing and size distribution of metastases, the factors that influence the dormancy of micrometastases and patterns of spread from a given primary tumor.Comment: 24 pages, 6 figures, Revie

Biological and clinical significance of cancer stem cell plasticity

In the past decade, the traditional view of cancers as a homogeneous collection of malignant cells is being replaced by a model of ever increasing complexity suggesting that cancers are complex tissues composed of multiple cell types. This complex model of tumorigenesis has been well supported by a growing body of evidence indicating that most cancers including those derived from blood and solid tissues display a hierarchical organization of tumor cells with phenotypic and functional heterogeneity and at the apex of this hierarchy are cells capable of self-renewal. These “tumor imitating cells” or “cancer stem cells” drive tumorigenesis and contribute to metastasis, treatment resistance and tumor relapse. Although tumor stem cells themselves may display both genetic and phenotypic heterogeneity, recent studies have demonstrated that cancer stem cells maintain plasticity to transition between mesenchymal-like (EMT) and epithelial-like (MET) states, which may be regulated by the tumor microenvironment. These stem cell state transitions may play a fundamental role in tumor progression and treatment resistance. In this review, we discuss the emerging knowledge regarding the plasticity of cancer stem cells with an emphasis on the signaling pathways and noncoding RNAs including microRNAs (miRNA) and long non-coding RNAs (lncRNAs) in regulation of this plasticity during tumor growth and metastasis. Lastly, we point out the importance of targeting both the EMT and MET states of CSCs in order to eliminate these lethal seeds of cancers. ELECTRONIC SUPPLEMENTARY MATERIAL: The online version of this article (doi:10.1186/s40169-014-0032-3) contains supplementary material, which is available to authorized users

Number and Size Distribution of Colorectal Adenomas under the Multistage Clonal Expansion Model of Cancer

Colorectal cancer (CRC) is believed to arise from mutant stem cells in colonic crypts that undergo a well-characterized progression involving benign adenoma, the precursor to invasive carcinoma. Although a number of (epi)genetic events have been identified as drivers of this process, little is known about the dynamics involved in the stage-wise progression from the first appearance of an adenoma to its ultimate conversion to malignant cancer. By the time adenomas become endoscopically detectable (i.e., are in the range of 1–2 mm in diameter), adenomas are already comprised of hundreds of thousands of cells and may have been in existence for several years if not decades. Thus, a large fraction of adenomas may actually remain undetected during endoscopic screening and, at least in principle, could give rise to cancer before they are detected. It is therefore of importance to establish what fraction of adenomas is detectable, both as a function of when the colon is screened for neoplasia and as a function of the achievable detection limit. To this end, we have derived mathematical expressions for the detectable adenoma number and size distributions based on a recently developed stochastic model of CRC. Our results and illustrations using these expressions suggest (1) that screening efficacy is critically dependent on the detection threshold and implicit knowledge of the relevant stem cell fraction in adenomas, (2) that a large fraction of non-extinct adenomas remains likely undetected assuming plausible detection thresholds and cell division rates, and (3), under a realistic description of adenoma initiation, growth and progression to CRC, the empirical prevalence of adenomas is likely inflated with lesions that are not on the pathway to cancer

Bcl-2 inhibits apoptosis by increasing the time-to-death and intrinsic cell-to-cell variations in the mitochondrial pathway of cell death

BH3 mimetics have been proposed as new anticancer therapeutics. They target anti-apoptotic Bcl-2 proteins, up-regulation of which has been implicated in the resistance of many cancer cells, particularly leukemia and lymphoma cells, to apoptosis. Using probabilistic computational modeling of the mitochondrial pathway of apoptosis, verified by single-cell experimental observations, we develop a model of Bcl-2 inhibition of apoptosis. Our results clarify how Bcl-2 imparts its anti-apoptotic role by increasing the time-to-death and cell-to-cell variability. We also show that although the commitment to death is highly impacted by differences in protein levels at the time of stimulation, inherent stochastic fluctuations in apoptotic signaling are sufficient to induce cell-to-cell variability and to allow single cells to escape death. This study suggests that intrinsic cell-to-cell stochastic variability in apoptotic signaling is sufficient to cause fractional killing of cancer cells after exposure to BH3 mimetics. This is an unanticipated facet of cancer chemoresistance.Comment: 11 pages, In pres

The role of Allee effect in modelling post resection recurrence of glioblastoma

Resection of the bulk of a tumour often cannot eliminate all cancer cells, due to their infiltration into the surrounding healthy tissue. This may lead to recurrence of the tumour at a later time. We use a reaction-diffusion equation based model of tumour growth to investigate how the invasion front is delayed by resection, and how this depends on the density and behaviour of the remaining cancer cells. We show that the delay time is highly sensitive to qualitative details of the proliferation dynamics of the cancer cell population. The typically assumed logistic type proliferation leads to unrealistic results, predicting immediate recurrence. We find that in glioblastoma cell cultures the cell proliferation rate is an increasing function of the density at small cell densities. Our analysis suggests that cooperative behaviour of cancer cells, analogous to the Allee effect in ecology, can play a critical role in determining the time until tumour recurrence

Quantitative Interpretation of a Genetic Model of Carcinogenesis Using Computer Simulations

The genetic model of tumorigenesis by Vogelstein et al. (V theory) and the molecular definition of cancer hallmarks by Hanahan and Weinberg (W theory) represent two of the most comprehensive and systemic understandings of cancer. Here, we develop a mathematical model that quantitatively interprets these seminal cancer theories, starting from a set of equations describing the short life cycle of an individual cell in uterine epithelium during tissue regeneration. The process of malignant transformation of an individual cell is followed and the tissue (or tumor) is described as a composite of individual cells in order to quantitatively account for intra-tumor heterogeneity. Our model describes normal tissue regeneration, malignant transformation, cancer incidence including dormant/transient tumors, and tumor evolution. Further, a novel mechanism for the initiation of metastasis resulting from substantial cell death is proposed. Finally, model simulations suggest two different mechanisms of metastatic inefficiency for aggressive and less aggressive cancer cells. Our work suggests that cellular de-differentiation is one major oncogenic pathway, a hypothesis based on a numerical description of a cell's differentiation status that can effectively and mathematically interpret some major concepts in V/W theories such as progressive transformation of normal cells, tumor evolution, and cancer hallmarks. Our model is a mathematical interpretation of cancer phenotypes that complements the well developed V/W theories based upon description of causal biological and molecular events. It is possible that further developments incorporating patient- and tissue-specific variables may build an even more comprehensive model to explain clinical observations and provide some novel insights for understanding cancer

Towards Predicting the Response of a Solid Tumour to Chemotherapy and Radiotherapy Treatments: Clinical Insights from a Computational Model

In this paper we use a hybrid multiscale mathematical model that incorporates both individual cell behaviour through the cell-cycle and the effects of the changing microenvironment through oxygen dynamics to study the multiple effects of radiation therapy. The oxygenation status of the cells is considered as one of the important prognostic markers for determining radiation therapy, as hypoxic cells are less radiosensitive. Another factor that critically affects radiation sensitivity is cell-cycle regulation. The effects of radiation therapy are included in the model using a modified linear quadratic model for the radiation damage, incorporating the effects of hypoxia and cell-cycle in determining the cell-cycle phase-specific radiosensitivity. Furthermore, after irradiation, an individual cell's cell-cycle dynamics are intrinsically modified through the activation of pathways responsible for repair mechanisms, often resulting in a delay/arrest in the cell-cycle. The model is then used to study various combinations of multiple doses of cell-cycle dependent chemotherapies and radiation therapy, as radiation may work better by the partial synchronisation of cells in the most radiosensitive phase of the cell-cycle. Moreover, using this multi-scale model, we investigate the optimum sequencing and scheduling of these multi-modality treatments, and the impact of internal and external heterogeneity on the spatio-temporal patterning of the distribution of tumour cells and their response to different treatment schedules

Promotion of variant human mammary epithelial cell outgrowth by ionizing radiation: an agent-based model supported by in vitro studies

IntroductionMost human mammary epithelial cells (HMEC) cultured from histologically normal breast tissues enter a senescent state termed stasis after 5 to 20 population doublings. These senescent cells display increased size, contain senescence associated beta-galactosidase activity, and express cyclin-dependent kinase inhibitor, p16INK4A (CDKN2A; p16). However, HMEC grown in a serum-free medium, spontaneously yield, at low frequency, variant (v) HMEC that are capable of long-term growth and are susceptible to genomic instability. We investigated whether ionizing radiation, which increases breast cancer risk in women, affects the rate of vHMEC outgrowth.MethodsPre-stasis HMEC cultures were exposed to 5 to 200 cGy of sparsely (X- or gamma-rays) or densely (1 GeV/amu 56Fe) ionizing radiation. Proliferation (bromodeoxyuridine incorporation), senescence (senescence-associated beta-galactosidase activity), and p16 expression were assayed in subcultured irradiated or unirradiated populations four to six weeks following radiation exposure, when patches of vHMEC became apparent. Long-term growth potential and p16 promoter methylation in subsequent passages were also monitored. Agent-based modeling, incorporating a simple set of rules and underlying assumptions, was used to simulate vHMEC outgrowth and evaluate mechanistic hypotheses.ResultsCultures derived from irradiated cells contained significantly more vHMEC, lacking senescence associated beta-galactosidase or p16 expression, than cultures derived from unirradiated cells. As expected, post-stasis vHMEC cultures derived from both unirradiated and irradiated cells exhibited more extensive methylation of the p16 gene than pre-stasis HMEC cultures. However, the extent of methylation of individual CpG sites in vHMEC samples did not correlate with passage number or treatment. Exposure to sparsely or densely ionizing radiation elicited similar increases in the numbers of vHMEC compared to unirradiated controls. Agent-based modeling indicated that radiation-induced premature senescence of normal HMEC most likely accelerated vHMEC outgrowth through alleviation of spatial constraints. Subsequent experiments using defined co-cultures of vHMEC and senescent cells supported this mechanism.ConclusionsOur studies indicate that ionizing radiation can promote the outgrowth of epigenetically altered cells with pre-malignant potential