A mathematical model of vascular tumor treatment by chemotherapy

Chemoimmunotherapy Administration Protocol Design for the Treatment of Leukemia through Mathematical Modeling and In Silico Experimentation

Cancer with all its more than 200 variants continues to be a major health problem around the world with nearly 10 million deaths recorded in 2020, and leukemia accounted for more than 300,000 cases according to the Global Cancer Observatory. Although new treatment strategies are currently being developed in several ongoing clinical trials, the high complexity of cancer evolution and its survival mechanisms remain as an open problem that needs to be addressed to further enhanced the application of therapies. In this work, we aim to explore cancer growth, particularly chronic lymphocytic leukemia, under the combined application of CAR-T cells and chlorambucil as a nonlinear dynamical system in the form of first-order Ordinary Differential Equations. Therefore, by means of nonlinear theories, sufficient conditions are established for the eradication of leukemia cells, as well as necessary conditions for the long-term persistence of both CAR-T and cancer cells. Persistence conditions are important in treatment protocol design as these provide a threshold below which the dose will not be enough to produce a cytotoxic effect in the tumour population. In silico experimentations allowed us to design therapy administration protocols to ensure the complete eradication of leukemia cells in the system under study when considering only the infusion of CAR-T cells and for the combined application of chemoimmunotherapy. All results are illustrated through numerical simulations. Further, equations to estimate cytotoxicity of chlorambucil and CAR-T cells to leukemia cancer cells were formulated and thoroughly discussed with a 95% confidence interval for the parameters involved in each formula.

A caution for oncologists: chemotherapy can cause chaotic dynamics

BACKGROUND AND OBJECTIVE

The effect of chemotherapy in cancer models is mostly handled by using a separate equation for chemotherapeutic agent. In this study, we do not consider a separate equation for drug but rather introduce its effect in terms of a parameter m representing the fraction of tumor cells killed by chemotherapeutic drug module. The main objective of this study is to provide conditions on model parameters which when fulfilled the grave consequences of cancer can be avoided. This study also shows that chemotherapy at times can produce unexpected results.

METHODS

Linearization method to study the stability of model equilibria.

RESULTS

The results obtained in this study are governed by the trichotomy law on the number 1-a₁₂-d₁, where a₁₂ represents the negative effect on the growth of cancer cells due to their competition with host cells for resources and d₁ is rate of annihilation of cancer cells due to chemotherapy. It is seen that in case of under-dose drug module when d₁<1-a₁₂, the complete eradication of cancer is not possible. When d₁=1-a₁₂, the model suggests occurrence of chaotic dynamics. When the drug dose is properly adjusted so that d₁>1-a₁₂, the complete eradication of cancer is guaranteed.

CONCLUSION

The results of the model of this paper given for the post vascular stages of tumor suggest criteria to select a particular drug module (a single drug or a combination of drugs) that the chemotherapy procedure should adapt to eradicate cancer. This study injects a note of caution for oncologists that chemotherapy as cancer treatment can also cause chaotic dynamics in certain situations. This study also presents a plausible explanation to the question why sometimes a tumor grows in the body and then gets cured without any medical intervention.

Optimal control methods for drug delivery in cancerous tumour by anti-angiogenic therapy and chemotherapy

There are numerous mathematical models simulating the behaviour of cancer by considering variety of states in different treatment strategies, such as chemotherapy. Among the models, one is developed which is able to consider the blood vessel‐production (angiogenesis) in the vicinity of the tumour and the effect of anti‐angiogenic therapy. In the mentioned‐model, normal cells, cancer cells, endothelial cells, chemotherapy and anti‐angiogenic agents are taking into account as state variables, and the rate of injection of the last two are considered as control inputs. Since controlling the cancerous tumour growth is a challenging matter for patient's life, the time schedule design of drug injection is very significant. Two optimal control strategies, an open‐loop (calculus of variations) and a closed‐loop (state‐dependent Riccati equation), are applied on the system in order to find an optimal time scheduling for each drug injection. By defining a proper cost function, an optimal control signal is designed for each one. Both obtained control inputs have reasonable answers, and the system is controlled eventually, but by comparing them, it is concluded that both methods have their own benefits which will be discussed in details in the conclusion section.

A mixed therapy minimal model: Some strategies for eradication or minimization of cancer

BACKGROUND AND OBJECTIVE

In most of the cancer therapeutic models separate equations for consumption of drugs are used, we however use parameters m and s to see the effect of chemotherapy and immunotherapy respectively. The main objective of this theoretical study is to develop strategies for eradication or minimization of cancer.

METHODS

Linearization method to study the local stability of model equilibria.

RESULTS

The results obtained in this study provide thresholds on m-fraction of cancer cells killed by chemotherapy and s-fraction of immune cells stimulated by immunotherapy.

CONCLUSION

The model considered relates to immune-cancer-normal cell interactions in post vascularization process. The study aims to develop strategies for complete eradication or minimization of cancer in terms of model parameters. This paper presents a minimal immuno-chemotherapeutic cancer model by describing interacting dynamics of cancer, immune and normal cells in a system of three ordinary differential equations. The source of the immune cells is considered outside the sytem given by a constant influx rate, s. The minimality of the model lies in not considering a separate equation for the dynamics of the drug but its overall killing effect on the cancer cells represented by a parameter, m. Thus the parameter m relates to chemotherapy and s to immunotherapy. The analysis of the model yields thresholds on these parameters for therapeutic strategies which guarantee either eradication or minimization of cancer from a patient's body.

Tumor Clearance Analysis on a Cancer Chemo-Immunotherapy Mathematical Model

Mathematical models may allow us to improve our knowledge on tumor evolution and to better comprehend the dynamics between cancer, the immune system and the application of treatments such as chemotherapy and immunotherapy in both short and long term. In this paper, we solve the tumor clearance problem for a six-dimensional mathematical model that describes tumor evolution under immune response and chemo-immunotherapy treatments. First, by means of the localization of compact invariant sets method, we determine lower and upper bounds for all cells populations considered by the model and we use these results to establish sufficient conditions for the existence of a bounded positively invariant domain in the nonnegative orthant by applying LaSalle's invariance principle. Then, by exploiting a candidate Lyapunov function we determine sufficient conditions on the chemotherapy treatment to ensure tumor clearance. Further, we investigate the local stability of the tumor-free equilibrium point and compute conditions for asymptotic stability and tumor persistence. All conditions are given by inequalities in terms of the system parameters, and we perform numerical simulations with different values on the chemotherapy treatment to illustrate our results. Finally, we discuss the biological implications of our work.

Mathematical Analysis of a Mathematical Model of Chemovirotherapy: Effect of Drug Infusion Method

A mathematical model for the treatment of cancer using chemovirotherapy is developed with the aim of determining the efficacy of three drug infusion methods: constant, single bolus, and periodic treatments. The model is in the form of ODEs and is further extended into DDEs to account for delays as a result of the infection of tumor cells by the virus and chemotherapeutic drug responses. Analysis of the model is carried out for each of the three drug infusion methods. Analytic solutions are determined where possible and stability analysis of both steady state solutions for the ODEs and DDEs is presented. The results indicate that constant and periodic drug infusion methods are more efficient compared to a single bolus injection. Numerical simulations show that with a large virus burst size, irrespective of the drug infusion method, chemovirotherapy is highly effective compared to either treatments. The simulations further show that both delays increase the period within which a tumor can be cleared from body tissue.

Mathematical Modelling and Prediction of the Effect of Chemotherapy on Cancer Cells

Cancer is a class of diseases characterized by out-of-control cells' growth which affect DNAs and make them damaged. Many treatment options for cancer exist, with the primary ones including surgery, chemotherapy, radiation therapy, hormonal therapy, targeted therapy and palliative care. Which treatments are used depends on the type, location, and grade of the cancer as well as the person's health and wishes. Chemotherapy is the use of medication (chemicals) to treat disease. More specifically, chemotherapy typically refers to the destruction of cancer cells. Considering the diffusion of drugs in cancer cells and fractality of DNA walks, in this research we worked on modelling and prediction of the effect of chemotherapy on cancer cells using Fractional Diffusion Equation (FDE). The employed methodology is useful not only for analysis of the effect of special drug and cancer considered in this research but can be expanded in case of different drugs and cancers.

Mathematical modeling of tumor therapy with oncolytic viruses: effects of parametric heterogeneity on cell dynamics

Background:

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.

Results:

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, tumor dormancy, and others, appear in heterogeneous setting. We also demonstrate that, within the applied modeling framework, to overcome the adverse effect of tumor cell heterogeneity on the outcome of cancer treatment, a heterogeneous population of an oncolytic virus must be used. Heterogeneity in parameters of the model, such as tumor cell susceptibility to virus infection and the ability of an oncolytic virus to infect tumor cells, can lead to complex, irregular evolution of the tumor. Thus, quasi-chaotic behavior of the tumor-virus system can be caused not only by random perturbations but also by the heterogeneity of the tumor and the virus.

Conclusion:

The modeling approach described here reveals the importance of tumor cell and virus heterogeneity for the outcome of cancer therapy. It should be straightforward to apply these techniques to mathematical modeling of other types of anticancer therapy.

Reviewers:

Leonid Hanin (nominated by Arcady Mushegian), Natalia Komarova (nominated by Orly Alter), and David Krakauer.