The long-term efficacy of targeted therapeutics for cancer treatment can be significantly limited by the type of therapy and development of drug resistance, inter alia. Experimental studies indicate that the factors enhancing acquisition of drug resistance in cancer cells include cell heterogeneity, drug target alteration, drug inactivation, DNA damage repair, drug efflux, cell death inhibition, as well as microenvironmental adaptations to targeted therapy, among others. Combination cancer therapies (CCTs) are employed to overcome these molecular and pathophysiological bottlenecks and improve the overall survival of cancer patients. CCTs often utilize multiple combinatorial modes of action and thus potentially constitute a promising approach to overcome drug resistance. Considering the colossal cost, human effort, time and ethical issues involved in clinical drug trials and basic medical research, mathematical modeling and analysis can potentially contribute immensely to the discovery of better cancer treatment regimens. In this article, we review mathematical models on CCTs developed thus far for cancer management. Open questions are highlighted and plausible combinations are discussed based on the level of toxicity, drug resistance, survival benefits, preclinical trials and other side effects.
The role of continua has been clear since antiquity in the mathematical approaches to physics, while discrete manifolds were brought to the limelight mostly by Quantum and Information Theories, in the XX century. We first recall how theorizing and measuring radically change in physics when using discrete vs. continuous mathematical manifolds. It will follow that the reference to discrete structures and digital information is far from neutral in knowledge construction. In biology, in particular, the introduction of information as a new observable on discrete data types has been promoting a dram…
Simple ideas, endowed from the mathematical theory of control, are used in order to analyze in general grounds the human immune system. The general principles are minimization of the pathogen load and economy of resources. They should constrain the parameters describing the immune system. In the simplest linear model, for example, where the response is proportional to the load, the annihilation rate of pathogens in any tissue should be greater than the pathogen's average rate of growth. When nonlinearities are added, a reference value for the number of pathogens is set, and a stability conditi…
Hypoxia and acidity act as environmental stressors promoting selection for cancer cells with a more aggressive phenotype. As a result, a deeper theoretical understanding of the spatio-temporal processes that drive the adaptation of tumour cells to hypoxic and acidic microenvironments may open up new avenues of research in oncology and cancer treatment. We present a mathematical model to study the influence of hypoxia and acidity on the evolutionary dynamics of cancer cells in vascularised tumours. The model is formulated as a system of partial integro-differential equations that describe the p…
Breast Cancer is a major public health problem and the most common diagnosed malignancy in woman. There have been significant developments in clinical approaches and theoretical experimental to understand the interactions of cancer cells dynamics with the immune system, also developments on analytical and computational models to help provide insights into clinical observations for a better understanding of cancer cells, but more are needed, especially at the genetic and molecular levels mathematically. Treatments such as immunotherapy, chemotherapy, hormone therapy, radiotherapy, and gene ther…
While the use of combination therapy is increasing in prevalence for cancer treatment, it is often difficult to predict the exact interactions between different treatment forms, and their synergistic/antagonistic effects on patient health and therapy outcome. In this research, a system of ordinary differential equations is constructed to model nonlinear dynamics between tumor cells, immune cells, and three forms of therapy: chemotherapy, immunotherapy, and radiotherapy. This model is then used to generate optimized combination therapy plans using optimal control theory. In-silico experiments a…
We present a mathematical model, based on ordinary differential equations, for the evolution of solid tumors and their response to treatment. Specifically the effects of a cytotoxic agent and a monoclonal antibody are included as control term in the equations. The variables considered here are: the number of cancerous cells sensitive to chemotherapy, the number of cancerous cells resistant to chemotherapy, the degree of angiogenesis and the average intensity of VEGF. The rules that govern the quantities mentioned above are based on a geometrical argument: we approximate the tumor mass as a sph…
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 …
The goal of cancer immunotherapy is to boost a patient's immune response to a tumor. Yet, the design of an effective immunotherapy is complicated by various factors, including a potentially immunosuppressive tumor microenvironment, immune-modulating effects of conventional treatments, and therapy-related toxicities. These complexities can be incorporated into mathematical and computational models of cancer immunotherapy that can then be used to aid in rational therapy design. In this review, we survey modeling approaches under the umbrella of the major challenges facing immunotherapy developme…
One of the mechanisms that ensure cancer robustness is tumor heterogeneity, and its effects on tumor cells dynamics have to be taken into account when studying cancer progression. There is no unifying theoretical framework in mathematical modeling of carcinogenesis that would account for parametric heterogeneity. Here we formulate a modeling approach that naturally takes stock of inherent cancer cell heterogeneity and illustrate it with a model of interaction between a tumor and an oncolytic virus. We show that several phenomena that are absent in homogeneous models, such as cancer recurrence,…
In this paper, we study a phase-space analysis of a mathematical model of tumor growth with an immune response. Mathematical analysis of the model equations with multipoint initial condition, regarding to dissipativity, boundedness of solutions, invariance of non-negativity, nature of equilibria, local and global stability will be investigated. We study some features of behavior of one three-dimensional tumor growth model with dynamics described in terms of densities of three cells populations: tumor cells, healthy host cells and effector immune cells. We find the upper and lower bounds for th…
Colorectal cancers are the third most common type of cancer. They originate from intestinal crypts, glands that descend from the intestinal lumen into the underlying connective tissue. Normal crypts are thought to exist in a dynamic equilibrium where the rate of cell production at the base of a crypt is matched by that of loss at the top. Understanding how genetic alterations accumulate and proceed to disrupt this dynamic equilibrium is fundamental to understanding the origins of colorectal cancer. Colorectal cancer emerges from the interaction of biological processes that span several spatial…
Cytotoxic chemotherapy is a common treatment for advanced prostate cancer. These tumors are also known to rely on angiogenesis, i.e., the growth of local microvasculature via chemical signaling produced by the tumor. Thus, several clinical studies have been investigating antiangiogenic therapy for advanced prostate cancer, either as monotherapy or combined with standard cytotoxic protocols. However, the complex genetic alterations promoting prostate cancer growth complicate the selection of the best chemotherapeutic approach for each patient's tumor. Here, we present a mathematical model of pr…
Oncolytic virotherapy is an experimental cancer treatment that uses genetically engineered viruses to target and kill cancer cells. One major limitation of this treatment is that virus particles are rapidly cleared by the immune system, preventing them from arriving at the tumour site. To improve virus survival and infectivity modified virus particles with the polymer polyethylene glycol (PEG) and the monoclonal antibody herceptin. While PEG modification appeared to improve plasma retention and initial infectivity it also increased the virus particle arrival time. We derive a mathematical mode…
Background: Radiotherapy outcomes are usually predicted using the Linear Quadratic model. However, this model does not integrate complex features of tumor growth, in particular cell cycle regulation. Methods: In this paper, we propose a multiscale model of cancer growth based on the genetic and molecular features of the evolution of colorectal cancer. The model includes key genes, cellular kinetics, tissue dynamics, macroscopic tumor evolution and radiosensitivity dependence on the cell cycle phase. We investigate the role of gene-dependent cell cycle regulation in the response of tumors to th…
In this paper we consider continuous mathematical models of tumour growth and invasion based on the model introduced by Chaplain and Lolas \cite{Chaplain&Lolas2006}, for the case of one space dimension. The models consist of a system of three coupled nonlinear reaction-diffusion-taxis partial differential equations describing the interactions between cancer cells, the matrix degrading enzyme and the tissue. For these models under certain conditions on the model parameters we obtain exact analytical solutions in terms of traveling wave variables. These solutions are smooth positive definite fun…
In this work we propose and investigate a family of models, which admits as particular cases some well known mathematical models of tumor-immune system interaction, with the additional assumption that the influx of immune system cells may be a function of the number of cancer cells. Constant, periodic and impulsive therapies (as well as the non-perturbed system) are investigated both analytically for the general family and, by using the model by Kuznetsov et al. (V. A. Kuznetsov, I. A. Makalkin, M. A. Taylor and A. S. Perelson. Nonlinear dynamics of immunogenic tumors: Parameter estimation and…
Adaptive therapy is a dynamic cancer treatment protocol that updates (or "adapts") treatment decisions in anticipation of evolving tumor dynamics. This broad term encompasses many possible dynamic treatment protocols of patient-specific dose modulation or dose timing. Adaptive therapy maintains high levels of tumor burden to benefit from the competitive suppression of treatment-sensitive subpopulations on treatment-resistant subpopulations. This evolution-based approach to cancer treatment has been integrated into several ongoing or planned clinical trials, including treatment of metastatic ca…
Leukaemia accounts for around 3% of all cancer types diagnosed in adults, and is the most common type of cancer in children of paediatric age. There is increasing interest in the use of mathematical models in oncology to draw inferences and make predictions, providing a complementary picture to experimental biomedical models. In this paper we recapitulate the state of the art of mathematical modelling of leukaemia growth dynamics, in time and response to treatment. We intend to describe the mathematical methodologies, the biological aspects taken into account in the modelling, and the conclusi…
Immunotherapy has the potential to change the way all cancer types are treated and cured. Cancer immunotherapies use elements of the patient immune system to attack tumor cells. One of the most successful types of immunotherapy is CAR-T cells. This treatment works by extracting patient's T-cells and adding to them an antigen receptor allowing tumor cells to be recognized and targeted. These new cells are called CAR-T cells and are re-infused back into the patient after expansion in-vitro. This approach has been successfully used to treat B-cell malignancies (B-cell leukemias and lymphomas). Ho…
Immunotherapy is currently regarded as the most promising treatment to fight against cancer. This is particularly true in the treatment of chronic lymphocytic leukemia, an indolent neoplastic disease of B-lymphocytes which eventually causes the immune system's failure. In this and other areas of cancer research, mathematical modeling is pointed out as a prominent tool to analyze theoretical and practical issues. Its lack in studies of chemoimmunotherapy of chronic lymphocytic leukemia is what motivates us to come up with a simple ordinary differential equation model. It is based on ideas of de…
Tumors are defined by their intense proliferation, but sometimes cancer cells turn senescent and stop replicating. In the stochastic cancer model in which all cells are tumorigenic, senescence is seen as the result of random mutations, suggesting that it could represent a barrier to tumor growth. In the hierarchical cancer model a subset of the cells, the cancer stem cells, divide indefinitely while other cells eventually turn senescent. Here we formulate cancer growth in mathematical terms and obtain predictions for the evolution of senescence. We perform experiments in human melanoma cells w…
Known as one of the hallmarks of cancer [30], cancer cell invasion of human body tissue is a complicated spatio-temporal multiscale process which enables a localised solid tumour to transform into a systemic, metastatic and fatal disease. This process explores and takes advantage of the reciprocal relation that solid tumours establish with the extracellular matrix (ECM) components and other multiple distinct cell types from the surrounding microenvironment. Through the secretion of various proteolytic enzymes such as matrix metalloproteinases (MMP) or the urokinase plasminogen activator (uPA),…
Cancer pathology is unique to a given individual, and developing personalized diagnostic and treatment protocols are a primary concern. Mathematical modeling and simulation is a promising approach to personalized cancer medicine. Yet, the complexity, heterogeneity and multiscale nature of cancer present severe challenges. One of the major barriers to use mathematical models to predict the outcome of therapeutic regimens in a particular patient lies in their initialization and parameterization in order to reflect individual cancer characteristics accurately. Here we present a study where we use…
A small but growing number of people are finding interesting parallels between ecosystems as studied by ecologists (think of a Savanna or the Amazon rain forest or a Coral reef) and tumours1-3. The idea of viewing cancer from an ecological perspective has many implications but fundamentally, it means that we should not see cancer just as a group of mutated cells. A more useful definition of cancer is to consider it a disruption in the complex balance of many interacting cellular and microenvironmental elements in a specific organ. This perspective means that organs undergoing carcinogenesis sh…
Adoptive T cell based immunotherapy is gaining significant traction in cancer treatment. Despite its limited success, so far, in treating solid cancers, it is increasingly successful, demonstrating to have a broader therapeutic potential. In this paper we develop a mathematical model to study the efficacy of engineered T cell receptor (TCR) T cell therapy targeting the E7 antigen in cervical cancer cell lines. We consider a dynamical system that follows the population of cancer cells, TCR T cells, and IL-2. We demonstrate that there exists a TCR T cell dosage window for a successful cancer eli…
Cancer is a term used to refer to a large set of diseases. The cancerous cells grow and divide and, as a result, they form tumours that grow in size. The immune system recognise the cancerous cells and attack them, though, it can be weakened by the cancer. One type of cancer treatment is chemotherapy, which uses drugs to kill cancer cells. Clinical, experimental, and theoretical research has been developed to understand the dynamics of cancerous cells with chemotherapy treatment, as well as the interaction between tumour growth and immune system. We study a mathematical model that describes th…
Tumors are believed to consist of a heterogeneous population of tumor cells originating from rare cancer stem cells (CSCs). However, emerging evidences show that tumor may also originate from non-CSCs. Here, we give evidences supporting that the number of tumorigenic tumor cells is higher than the number of CSCs and tumor can also derive from non-CSCs. First, we applied an idealized mathematical model and theoretically calculated that non-CSCs could initiate tumor if their proliferation potential was adequate. Next, we demonstrated by experimental studies that 17.7%, 38.6% and 5.2% of tumor ce…
Adoptive Cell Transfer therapy of cancer is currently in full development and mathematical modeling is playing a critical role in this area. We study a stochastic model developed by Baar et al. in 2015 for modeling immunotherapy against melanoma skin cancer. First, we estimate the parameters of the deterministic limit of the model based on biological data of tumor growth in mice. A Nonlinear Mixed Effects Model is estimated by the Stochastic Approximation Expectation Maximization algorithm. With the estimated parameters, we head back to the stochastic model and calculate the probability that t…
Cancer stem cells are controlled by developmental networks that are often topologically indistinguishable from normal, healthy stem cells. The question is why cancer stem cells can be both phenotypically distinct and have morphological effects so different from normal stem cells. The difference between cancer stem cells and normal stem cells lies not in differences their network architecture, but rather in the spatial-temporal locality of their activation in the genome and the resulting expression in the body. The metastatic potential cancer stem cells is not based primarily on their network d…
Recent works have stressed the important role that random mutations have in the development of cancer phenotype. We challenge this current view by means of bioinformatic data analysis and computational modelling approaches. Not all the mutations are equally important for the development of metastasis. The survival of cancer cells from the primary tumour site to the secondary seeding sites depends on the occurrence of very few driver mutations promoting oncogenic cell behaviours and on the order with which these mutations occur. We introduce a model in the framework of Cellular Automata to inve…
Although there are various types of cancer treatments, none of these currently take into account the effect of ageing of the immune system and hence altered responses to cancer. Recent studies have shown that in vitro stimulation of T cells can help in the treatment of patients. There are many factors that have to be considered when simulating an organism's immunocompetence. Our particular interest lies in the study of loss of immunocompetence with age. We are trying to answer questions such as: Given a certain age of a patient, how fit is their immune system to fight cancer? Would an immune b…
Hydrogen peroxide (H2O2) is an important signaling molecule in cancer cells. However, the significant secretion of H2O2 by cancer cells have been rarely observed. Cold atmospheric plasma (CAP) is a near room temperature ionized gas composed of neutral particles, charged particles, reactive species, and electrons. Here, we first demonstrated that breast cancer cells and pancreatic adenocarcinoma cells generated micromolar level H2O2 during just 1 min of direct CAP treatment on these cells. The cell-based H2O2 generation is affected by the medium volume, the cell confluence, as well as the disch…
Current clinical decision-making in oncology relies on averages of large patient populations to both assess tumor status and treatment outcomes. However, cancers exhibit an inherent evolving heterogeneity that requires an individual approach based on rigorous and precise predictions of cancer growth and treatment response. To this end, we advocate the use of quantitative in vivo imaging data to calibrate mathematical models for the personalized forecasting of tumor development. In this chapter, we summarize the main data types available from both common and emerging in vivo medical imaging tec…
Oncolytic virotherapy - the use of viruses that specifically kill tumor cells - is an innovative and highly promising route for treating cancer. However, its therapeutic outcomes are mainly impaired by the host immune response to the viral infection. In the present work, we propose a multiscale mathematical model to study how the immune response interferes with the viral oncolytic activity. The model assumes that cytotoxic T cells can induce apoptosis in infected cancer cells and that free viruses can be inactivated by neutralizing antibodies or cleared at a constant rate by the innate immune …
