Enhancement of chemotherapy using oncolytic virotherapy: Mathematical and optimal control analysis

Oncolytic virotherapy (OV) has been emerging as a promising novel cancer treatment that may be further combined with the existing therapeutic modalities to enhance their effects. To investigate how OV could enhance chemotherapy, we propose an ODE based model describing the interactions between tumour cells, the immune response, and a treatment combination with chemotherapy and oncolytic viruses. Stability analysis of the model with constant chemotherapy treatment rates shows that without any form of treatment, a tumour would grow to its maximum size. It also demonstrates that chemotherapy alone is capable of clearing tumour cells provided that the drug efficacy is greater than the intrinsic tumour growth rate. Furthermore, OV alone may not be able to clear tumour cells from body tissue but would rather enhance chemotherapy if viruses with high viral potency are used. To assess the combined effect of OV and chemotherapy we use the forward sensitivity index to perform a sensitivity analysis, with respect to chemotherapy key parameters, of the virus basic reproductive number and the tumour endemic equilibrium. The results from this sensitivity analysis indicate the existence of a critical dose of chemotherapy above which no further significant reduction in the tumour population can be observed. Numerical simulations show that a successful combinational therapy of the chemotherapeutic drugs and viruses depends mostly on the virus burst size, infection rate, and the amount of drugs supplied. Optimal control analysis was performed, by means of Pontryagin's principle, to further refine predictions of the model with constant treatment rates by accounting for the treatment costs and sides effects.Comment: This is a preprint of a paper whose final and definite form is with 'Mathematical Biosciences and Engineering', ISSN 1551-0018 (print), ISSN 1547-1063 (online), available at [http://www.aimsciences.org/journal/1551-0018]. Submitted 27-March-2018; revised 04-July-2018; accepted for publication 10-July-201

Stochastic Norton-Simon-Massagu\ue9 Tumor Growth Modeling: Controlled and Mixed-Effects Uncontrolled Analysis

Tumorigenesis is a complex process that is heterogeneous and affected by numerous sources of variability. This study presents a stochastic extension of a biologically grounded tumor growth model, referred to as the Norton-Simon-Massagu\ue9 (NSM) tumor growth model. We first study the uncontrolled version of the model where the effect of chemotherapeutic drug agent is absent. Conditions on the model\u2019s parameters are derived to guarantee the positivity of the tumor volume and hence the validity of the proposed stochastic NSM model. To calibrate the proposed model we utilize a maximum likelihood- based estimation algorithm and population mixed-effect modeling formulation. The algorithm is tested by fitting previously published tumor volume mice data. Then, we study the controlled version of the model which includes the effect of chemotherapy treatment. Analysis of the influence of adding the control drug agent into the model and how sensitive it is to the stochastic parameters is performed both in open-loop and closed-loop viewpoints through different numerical simulations

Model--Based Design of Cancer Chemotherapy Treatment Schedules

Cancer is the name given to a class of diseases characterized by an imbalance in cell proliferation and apoptosis, or programmed cell death. Once cancer has reached detectable sizes ($10^{6}$ cells or 1 mm$^3$), it is assumed to have spread throughout the body, and a systemic form of treatment is needed. Chemotherapy treatment is commonly used, and it effects both healthy and diseased tissue. This creates a dichotomy for clinicians who need develop treatment schedules which balance toxic side effects with treatment efficacy. Nominally, the optimal treatment schedule --- where schedule is defined as the amount and frequency of drug delivered --- is the one found to be the most efficacious from the set evaluated during clinical trials. In this work, a model based approach for developing drug treatment schedules was developed. Cancer chemotherapy modeling is typically segregated into drug pharmacokinetics (PK), describing drug distribution throughout an organism, and pharmacodynamics (PD), which delineates cellular proliferation, and drug effects on the organism. This work considers two case studies: (i) a preclinical study of the oral administration of the antitumor agent 9-nitrocamptothecin (9NC) to severe combined immunodeficient (SCID) mice bearing subcutaneously implanted HT29 human colon xenografts; and (ii) a theoretical study of intravenous chemotherapy from the engineering literature.Metabolism of 9NC yields the active metabolite 9-aminocamptothecin (9AC). Both 9NC and 9AC exist in active lactone and inactive carboxylate forms. Four different PK model structures are presented to describe the plasma disposition of 9NC and 9AC: three linear models at a single dose level (0.67 mg/kg 9NC); and a nonlinear model for the dosing range 0.44 -- 1.0 mg/kg 9NC. Untreated tumor growth was modeled using two approaches: (i) exponential growth; and (ii) a switched exponential model transitioning between two different rates of exponential growth at a critical size. All of the PK/PD models considered here have bilinear kill terms which decrease tumor sizes at rates proportional to the effective drug concentration and the current tumor size. The PK/PD model combining the best linear PK model with exponential tumor growth accurately characterized tumor responses in ten experimental mice administered 0.67 mg/kg of 9NC myschedule (Monday-Friday for two weeks repeated every four weeks). The nonlinear PK model of 9NC coupled to the switched exponential PD model accurately captured the tumor response data at multiple dose levels. Each dosing problem was formulated as a mixed--integer linear programming problem (MILP), which guarantees globally optimal solutions. When minimizing the tumor volume at a specified final time, the MILP algorithm delivered as much drug as possible at the end of the treatment window (up to the cumulative toxicity constraint). While numerically optimal, it was found that an exponentially growing tumor, with bilinear kill driven by linear PK would experience the same decrease in tumor volume at a final time regardless of when the drug was administered as long as the {it same amount} was administered. An alternate objective function was selected to minimize tumor volume along a trajectory. This is more clinically relevant in that it better represents the objective of the clinician (eliminate the diseased tissue as rapidly as possible). This resulted in a treatment schedule which eliminated the tumor burden more rapidly, and this schedule can be evaluated recursively at the end of each cycle for efficacy and toxicity, as per current clinical practice.The second case study consists of an intravenously administered drug with first order elimination treating a tumor under Gompertzian growth. This system was also formulated as a MILP, and the two different objectives above were considered. The first objective was minimizing the tumor volume at a final time --- the objective the original authors considered. The MILP solution was qualitatively similar to the solutions originally found using control vector parameterization techniques. This solution also attempted to administer as much drug as possible at the end of the treatment interval. The problem was then posed as a receding horizon trajectory tracking problem. Once again, a more clinically relevant objective returned promising results; the tumor burden was rapidly eliminated

An application of genetic algorithms to chemotherapy treatment.

The present work investigates methods for optimising cancer chemotherapy within the bounds of clinical acceptability and making this optimisation easily accessible to oncologists. Clinical oncologists wish to be able to improve existing treatment regimens in a systematic, effective and reliable way. In order to satisfy these requirements a novel approach to chemotherapy optimisation has been developed, which utilises Genetic Algorithms in an intelligent search process for good chemotherapy treatments. The following chapters consequently address various issues related to this approach. Chapter 1 gives some biomedical background to the problem of cancer and its treatment. The complexity of the cancer phenomenon, as well as the multi-variable and multi-constrained nature of chemotherapy treatment, strongly support the use of mathematical modelling for predicting and controlling the development of cancer. Some existing mathematical models, which describe the proliferation process of cancerous cells and the effect of anti-cancer drugs on this process, are presented in Chapter 2. Having mentioned the control of cancer development, the relevance of optimisation and optimal control theory becomes evident for achieving the optimal treatment outcome subject to the constraints of cancer chemotherapy. A survey of traditional optimisation methods applicable to the problem under investigation is given in Chapter 3 with the conclusion that the constraints imposed on cancer chemotherapy and general non-linearity of the optimisation functionals associated with the objectives of cancer treatment often make these methods of optimisation ineffective. Contrariwise, Genetic Algorithms (GAs), featuring the methods of evolutionary search and optimisation, have recently demonstrated in many practical situations an ability to quickly discover useful solutions to highly-constrained, irregular and discontinuous problems that have been difficult to solve by traditional optimisation methods. Chapter 4 presents the essence of Genetic Algorithms, as well as their salient features and properties, and prepares the ground for the utilisation of Genetic Algorithms for optimising cancer chemotherapy treatment. The particulars of chemotherapy optimisation using Genetic Algorithms are given in Chapter 5 and Chapter 6, which present the original work of this thesis. In Chapter 5 the optimisation problem of single-drug chemotherapy is formulated as a search task and solved by several numerical methods. The results obtained from different optimisation methods are used to assess the quality of the GA solution and the effectiveness of Genetic Algorithms as a whole. Also, in Chapter 5 a new approach to tuning GA factors is developed, whereby the optimisation performance of Genetic Algorithms can be significantly improved. This approach is based on statistical inference about the significance of GA factors and on regression analysis of the GA performance. Being less computationally intensive compared to the existing methods of GA factor adjusting, the newly developed approach often gives better tuning results. Chapter 6 deals with the optimisation of multi-drug chemotherapy, which is a more practical and challenging problem. Its practicality can be explained by oncologists' preferences to administer anti-cancer drugs in various combinations in order to better cope with the occurrence of drug resistant cells. However, the imposition of strict toxicity constraints on combining various anticancer drugs together, makes the optimisation problem of multi-drug chemotherapy very difficult to solve, especially when complex treatment objectives are considered. Nevertheless, the experimental results of Chapter 6 demonstrate that this problem is tractable to Genetic Algorithms, which are capable of finding good chemotherapeutic regimens in different treatment situations. On the basis of these results a decision has been made to encapsulate Genetic Algorithms into an independent optimisation module and to embed this module into a more general and user-oriented environment - the Oncology Workbench. The particulars of this encapsulation and embedding are also given in Chapter 6. Finally, Chapter 7 concludes the present work by summarising the contributions made to the knowledge of the subject treated and by outlining the directions for further investigations. The main contributions are: (1) a novel application of the Genetic Algorithm technique in the field of cancer chemotherapy optimisation, (2) the development of a statistical method for tuning the values of GA factors, and (3) the development of a robust and versatile optimisation utility for a clinically usable decision support system. The latter contribution of this thesis creates an opportunity to widen the application domain of Genetic Algorithms within the field of drug treatments and to allow more clinicians to benefit from utilising the GA optimisation