Determining optimal combination regimens for patients with multiple myeloma.

While many novel therapies have been approved in recent years for treating patients with multiple myeloma, there is still no established curative regimen, especially for patients with high-risk disease. In this work, we use a mathematical modeling approach to determine combination therapy regimens that maximize healthy lifespan for patients with multiple myeloma. We start with a mathematical model for the underlying disease and immune dynamics, which was presented and analyzed previously. We add the effects of three therapies to the model: pomalidomide, dexamethasone, and elotuzumab. We consider multiple approaches to optimizing combinations of these therapies. We find that optimal control combined with approximation outperforms other methods, in that it can quickly produce a combination regimen that is clinically-feasible and near-optimal. Implications of this work can be used to optimize doses and advance the scheduling of drugs.

Optimal dosage protocols for mathematical models of synergy of chemo- and immunotherapy.

The release of tumor antigens during traditional cancer treatments such as radio- or chemotherapy leads to a stimulation of the immune response which provides synergistic effects these treatments have when combined with immunotherapies. A low-dimensional mathematical model is formulated which, depending on the values of its parameters, encompasses the 3 E's (elimination, equilibrium, escape) of tumor immune system interactions. For the escape situation, optimal control problems are formulated which aim to revert the process to the equilibrium scenario. Some numerical results are included.

On the Role of the Objective in the Optimization of Compartmental Models for Biomedical Therapies.

We review and discuss results obtained through an application of tools of nonlinear optimal control to biomedical problems. We discuss various aspects of the modeling of the dynamics (such as growth and interaction terms), modeling of treatment (including pharmacometrics of the drugs), and give special attention to the choice of the objective functional to be minimized. Indeed, many properties of optimal solutions are predestined by this choice which often is only made casually using some simple ad hoc heuristics. We discuss means to improve this choice by taking into account the underlying biology of the problem.

Optimization of combination therapy for chronic myeloid leukemia with dosing constraints.

In this work, we demonstrate a mathematical technique for optimizing combination regimens with constraints. We apply the technique to a mathematical model for treatment of patients with chronic myeloid leukemia. The in-host model includes leukemic cell and immune system dynamics during treatment with tyrosine kinase inhibitors and immunomodulatory compounds. The model is minimal (semi-mechanistic) with just enough detail that all relevant therapeutic effects can be represented. The regimens are optimized to yield the highest possible reduction in disease burden, taking into account dosing constraints and side effect risks due to drug exposure. We compare the following three types of regimens: (1) regimens that are restricted to certain discrete dose levels, which can only change every three months; (2) optimal regimens determined using optimal control; and (3) regimens that are piecewise-constant like the first type of regimen, but are obtained as approximations to the optimal control regimens. All three types of regimens result in similar outcomes, but the last one is easy to compute in addition to being clinically feasible.

Application of mathematical models to metronomic chemotherapy: What can be inferred from minimal parameterized models?

Metronomic chemotherapy refers to the frequent administration of chemotherapy at relatively low, minimally toxic doses without prolonged treatment interruptions. Different from conventional or maximum-tolerated-dose chemotherapy which aims at an eradication of all malignant cells, in a metronomic dosing the goal often lies in the long-term management of the disease when eradication proves elusive. Mathematical modeling and subsequent analysis (theoretical as well as numerical) have become an increasingly more valuable tool (in silico) both for determining conditions under which specific treatment strategies should be preferred and for numerically optimizing treatment regimens. While elaborate, computationally-driven patient specific schemes that would optimize the timing and drug dose levels are still a part of the future, such procedures may become instrumental in making chemotherapy effective in situations where it currently fails. Ideally, mathematical modeling and analysis will develop into an additional decision making tool in the complicated process that is the determination of efficient chemotherapy regimens. In this article, we review some of the results that have been obtained about metronomic chemotherapy from mathematical models and what they infer about the structure of optimal treatment regimens.

Preface.

This volume was inspired by the topics presented at the international conference "Micro and Macro Systems in Life Sciences" which was held on Jun 8-12, 2015 in Bedlewo, Poland. System biology is an approach which tries to understand how micro systems, at the molecular and cellular levels, affect macro systems such as organs, tissue and populations. Thus it is not surprising that a major theme of this volume evolves around cancer and its treatment. Articles on this topic include models for tumor induced angiogenesis, without and with delays, metastatic niche of the bone marrow, drug resistance and metronomic chemotherapy, and virotherapy of glioma. Methods range from dynamical systems to optimal control. Another well represented topic of this volume is mathematical modeling in epidemiology. Mathematical approaches to modeling and control of more specific diseases like malaria, Ebola or human papillomavirus are discussed as well as a more general approaches to the SEIR, and even more general class of models in epidemiology, by using the tools of optimal control and optimization. The volume also brings up challenges in mathematical modeling of other diseases such as tuberculosis. Partial differential equations combined with numerical approaches are becoming important tools in modeling not only tumor growth and treatment, but also other diseases, such as fibrosis of the liver, and atherosclerosis and its associated blood flow dynamics, and our volume presents a state of the art approach on these topics. Understanding mathematics behind the cell motion, appearance of the special patterns in various cell populations, and age structured mutations are among topics addressed inour volume. A spatio-temporal models of synthetic genetic oscillators brings the analysis to the gene level which is the focus of much of current biological research. Mathematics can help biologists to explain the collective behavior of bacterial, a topic that is also presented here. Finally some more across the discipline topics are being addresses, which can appear as a challenge in studying problems in systems biology on all, macro, meso and micro levels. They include numerical approaches to stochastic wave equation arising in modeling Brownian motion, discrete velocity models, many particle approximations as well as very important aspect on the connection between discrete measurement and the construction of the models for various phenomena, particularly the one involving delays. With the variety of biological topics and their mathematical approaches we very much hope that the reader of the Mathematical Biosciences and Engineering will find this volume interesting and inspirational for their own research.

On drug resistance and metronomic chemotherapy: A mathematical modeling and optimal control approach.

Effects that tumor heterogeneity and drug resistance have on the structure of chemotherapy protocols are discussed from a mathematical modeling and optimal control point of view. In the case when two compartments consisting of sensitive and resistant cells are considered, optimal protocols consist of full dose chemotherapy as long as the relative proportion of sensitive cells is high. When resistant cells become more dominant, optimal controls switch to lower dose regimens defined by so-called singular controls. The role that singular controls play in the structure of optimal therapy protocols for cell populations with a large number of traits is explored in mathematical models.

The role of TNF-α inhibitor in glioma virotherapy: A mathematical model.

Virotherapy, using herpes simplex virus, represents a promising therapy of glioma. But the innate immune response, which includes TNF-α produced by macrophages, reduces the effectiveness of the treatment. Hence treatment with TNF-α inhibitor may increase the effectiveness of the virotherapy. In the present paper we develop a mathematical model that includes continuous infusion of the virus in combination with TNF-α inhibitor. We study the efficacy of the treatment under different combinations of the two drugs for different scenarios of the burst size of newly formed virus emerging from dying infected cancer cells. The model may serve as a first step toward developing an optimal strategy for the treatment of glioma by the combination of TNF-α inhibitor and oncolytic virus injection.

Optimizing Chemotherapeutic Anti-cancer Treatment and the Tumor Microenvironment: An Analysis of Mathematical Models.

We review results about the structure of administration of chemotherapeutic anti-cancer treatment that we have obtained from an analysis of minimally parameterized mathematical models using methods of optimal control. This is a branch of continuous-time optimization that studies the minimization of a performance criterion imposed on an underlying dynamical system subject to constraints. The scheduling of anti-cancer treatments has all the features of such a problem: treatments are administered in time and the interactions of the drugs with the tumor and its microenvironment determine the efficacy of therapy. At the same time, constraints on the toxicity of the treatments need to be taken into account. The models we consider are low-dimensional and do not include more refined details, but they capture the essence of the underlying biology and our results give robust and rather conclusive qualitative information about the administration of optimal treatment protocols that strongly correlate with approaches taken in medical practice. We describe the changes that arise in optimal administration schedules as the mathematical models are increasingly refined to progress from models that only consider the cancerous cells to models that include the major components of the tumor microenvironment, namely the tumor vasculature and tumor-immune system interactions.

Dynamical properties of a minimally parameterized mathematical model for metronomic chemotherapy.

A minimally parameterized mathematical model for low-dose metronomic chemotherapy is formulated that takes into account angiogenic signaling between the tumor and its vasculature and tumor inhibiting effects of tumor-immune system interactions. The dynamical equations combine a model for tumor development under angiogenic signaling formulated by Hahnfeldt et al. with a model for tumor-immune system interactions by Stepanova. The dynamical properties of the model are analyzed. Depending on the parameter values, the system encompasses a variety of medically realistic scenarios that range from cases when (i) low-dose metronomic chemotherapy is able to eradicate the tumor (all trajectories converge to a tumor-free equilibrium point) to situations when (ii) tumor dormancy is induced (a unique, globally asymptotically stable benign equilibrium point exists) to (iii) multi-stable situations that have both persistent benign and malignant behaviors separated by the stable manifold of an unstable equilibrium point and finally to (iv) situations when tumor growth cannot be overcome by low-dose metronomic chemotherapy. The model forms a basis for a more general study of chemotherapy when the main components of a tumor's microenvironment are taken into account.

Dynamics and control of a mathematical model for metronomic chemotherapy.

A 3-compartment model for metronomic chemotherapy that takes into account cancerous cells, the tumor vasculature and tumor immune-system interactions is considered as an optimal control problem. Metronomic chemo-therapy is the regular, almost continuous administration of chemotherapeutic agents at low dose, possibly with small interruptions to increase the efficacy of the drugs. There exists medical evidence that such administrations of specific cytotoxic agents (e.g., cyclophosphamide) have both antiangiogenic and immune stimulatory effects. A mathematical model for angiogenic signaling formulated by Hahnfeldt et al. is combined with the classical equations for tumor immune system interactions by Stepanova to form a minimally parameterized model to capture these effects of low dose chemotherapy. The model exhibits bistable behavior with the existence of both benign and malignant locally asymptotically stable equilibrium points. In this paper, the transfer of states from the malignant into the benign regions is used as a motivation for the construction of an objective functional that induces this process and the analysis of the corresponding optimal control problem is initiated.

Application of ecological and mathematical theory to cancer: New challenges.

According to the World Health Organization, cancer is among the leading causes of morbidity and mortality worldwide. Despite enormous efforts of cancer researchers all around the world, the mechanisms underlying its origin, formation, progression, therapeutic cure or control are still not fully understood. Cancer is a complex, multi-scale process, in which genetic mutations occurring at a sub-cellular level manifest themselves as functional changes at the cellular and tissue scale.

Lessons from the Fourth Metronomic and Anti-angiogenic Therapy Meeting, 24-25 June 2014, Milan.

The Fourth Metronomic and Anti-angiogenic Therapy Meeting was held in Milan 24-25 June 2014. The meeting was a true translational meeting where researchers and clinicians shared their results, experiences, and insights in order to continue gathering useful evidence on metronomic approaches. Several speakers emphasised that exact mechanisms of action, best timing, and optimal dosage are still not well understood and that the field would learn a lot from ancillary studies performed during the clinical trials of metronomic chemotherapies. From the pre-clinical side, new research findings indicate additional possible mechanisms of actions of metronomic schedule on the immune and blood vessel compartments of the tumour micro-environment. New clinical results of metronomic chemotherapy were presented in particular in paediatric cancers [especially neuroblastoma and central nervous system (CNS) tumours], in angiosarcoma (together with beta-blockers), in hepatocellular carcinoma, in prostate cancer, and in breast cancer. The use of repurposed drugs such as metformin, celecoxib, or valproic acid in the metronomic regimen was reported and highlighted the potential of other candidate drugs to be repurposed. The clinical experiences from low- and middle-income countries with affordable regimens gave very encouraging results which will allow more patients to be effectively treated in economies where new drugs are not accessible. Looking at the impact of metronomic approaches that have been shown to be effective, it was admitted that those approaches were rarely used in clinical practice, in part because of the absence of commercial interest for companies. However, performing well-designed clinical trials of metronomic and repurposing approaches demonstrating substantial improvement, especially in populations with the greatest unmet needs, may be an easier solution than addressing the financial issue. Metronomics should always be seen as a chance to come up with new innovative affordable approaches and not as a cheap rescue strategy.

On the MTD paradigm and optimal control for multi-drug cancer chemotherapy.

In standard chemotherapy protocols, drugs are given at maximum tolerated doses (MTD) with rest periods in between. In this paper, we briey discuss the rationale behind this therapy approach and, using as example multidrug cancer chemotherapy with a cytotoxic and cytostatic agent, show that these types of protocols are optimal in the sense of minimizing a weighted average of the number of tumor cells (taken both at the end of therapy and at intermediate times) and the total dose given if it is assumed that the tumor consists of a homogeneous population of chemotherapeutically sensitive cells. A 2-compartment linear model is used to model the pharmacokinetic equations for the drugs.

On optimal chemotherapy with a strongly targeted agent for a model of tumor-immune system interactions with generalized logistic growth.

In this paper, a mathematical model for chemotherapy that takes tumor immune-system interactions into account is considered for a strongly targeted agent. We use a classical model originally formulated by Stepanova, but replace exponential tumor growth with a generalised logistic growth model function depending on a parameter v. This growth function interpolates between a Gompertzian model (in the limit v → 0) and an exponential model (in the limit v → ∞). The dynamics is multi-stable and equilibria and their stability will be investigated depending on the parameter v. Except for small values of v, the system has both an asymptotically stable microscopic (benign) equilibrium point and an asymptotically stable macroscopic (malignant) equilibrium point. The corresponding regions of attraction are separated by the stable manifold of a saddle. The optimal control problem of moving an initial condition that lies in the malignant region into the benign region is formulated and the structure of optimal singular controls is determined.

Optimal response to chemotherapy for a mathematical model of tumor-immune dynamics.

An optimal control problem for cancer chemotherapy is considered that includes immunological activity. In the objective a weighted average of several quantities that describe the effectiveness of treatment is minimized. These terms include (i) the number of cancer cells at the terminal time, (ii) a measure for the immunocompetent cell densities at the terminal point (included as a negative term), (iii) the overall amount of cytotoxic agents given as a measure for the side effects of treatment and (iv) a small penalty on the terminal time that limits the overall therapy horizon which is assumed to be free. This last term is essential in obtaining a well-posed problem formulation. Employing a Gompertzian growth model for the cancer cells, for various scenarios optimal controls and corresponding responses of the system are calculated. Solutions initially follow a full dose treatment, but then at one point switch to a singular regimen that only applies partial dosages. This structure is consistent with protocols that apply an initial burst to reduce the tumor volume and then maintain a small volume through lower dosages. Optimal controls end with either a prolonged period of no dose treatment or, in a small number of scenarios, this no dose interval is still followed by one more short burst of full dose treatment.

Optimal and suboptimal protocols for a mathematical model for tumor anti-angiogenesis in combination with chemotherapy.

We consider the problem of minimizing the tumor volume with a priori given amounts of anti-angiogenic and cytotoxic agents. For one underlying mathematical model, optimal and suboptimal solutions are given for four versions of this problem: the case when only anti-angiogenic agents are administered, combination treatment with a cytotoxic agent, and when a standard linear pharmacokinetic equation for the anti-angiogenic agent is added to each of these models. It is shown that the solutions to the more complex models naturally build upon the simplified versions. This gives credence to a modeling approach that starts with the analysis of simplified models and then adds increasingly more complex and medically relevant features. Furthermore, for each of the problem formulations considered here, there exist excellent simple piecewise constant controls with a small number of switchings that virtually replicate the optimal values for the objective.