Models For Cancer Chemotherapy Research Articles

New asymptotic stability theory for real order systems and applications

Metzler asymptotic stability of real order systems gives hidden paths of nontrivial solutions to geometric constant solutions as time tends to ∞ in positive and negative orthants of Euclidean space when associated with random initial-time. Suppose that there is a governing law that describes the flow of geometric variables described by a real-order system in Euclidean space and a geometric solution zero vector that satisfies such a system. We introduce the concepts of orthant Metzler asymptotic stability that give rise to the notion of so-called new L-asymptotic stability. We prove that there exist order-dependent conditions that give a new L-asymptotic stability. The main results of this paper are comparison theorems that give L-asymptotic stability, which is essential in finding convergence of non-negative and non-positive solutions to geometric constant solutions. We establish upper and lower theoretical α-order Mittag-Leffler bounds of the Euclidean norm measure of non-trivial solutions to such systems, which characterize the Mittag-Leffler rate of decay. As applications of some theoretical results, we provide three examples that include a feedback method to control cancer in a cancer chemotherapy model to illustrate the effectiveness.

Optimization of drug scheduling for cancer chemotherapy with considering reducing cumulative drug toxicity

An improved optimal drug scheduling model with considering two control drugs is proposed and the Gauss pseudospectral-based optimization method is studied to decrease the tumor size and drug toxicity in this work. Firstly, the Dexrazoxane drug, which has significant clinical effect to reduce the toxicity of the anticancer drug, is introduced. By analyzing the growth kinetics model of cancer chemotherapy, the toxicity reduction drug is regarded as the second input in the cancer dynamic equations. Correspondingly, the drug scheduling optimization problem with particular optimization goal and necessary constraints is established. Next, a model transformation technique is proposed to reduce the complexity of dynamic equations. With deriving the Gaussian time grid discretization detailly, the Gauss pseudospectral method (GPM)-based cancer chemotherapy drug scheduling algorithm is presented to test the performance of the proposed model within different rates. Finally, the implementation structure of drug scheduling optimization is given in detail. To test and validate the performance of proposed chemotherapy model, extensive simulation results and comparative evaluation are carried out on a specific mathematical model. Simulation results show that the improved optimization model is superior to other literature studies, resulting in the average improvement of performance index by 66.54% and revealing the significant guiding property for cancer chemotherapy.

The Structure of Optimal Protocols for a Mathematical Model of Chemotherapy with Antiangiogenic Effects

We analyze a mathematical model for cancer chemotherapy which includes antiangiogenic effects of the cytotoxic agent. Assuming that the total amount of agents to be given has been determined a priori based on a medical assessment of its side effects, we consider the problem how to best administer this amount. The model assumes a homogenous tumor and if the aim is to minimize the tumor volume, then optimal controls administer the total dose in a single maximum dose session. As, however, angiogenic effects of the agent are taken into account, this no longer is optimal. Lower dose strategies determined by an optimal singular arc with significantly reduced dose rates give a better response over time. In this paper, for the medically realistic domain of the mathematical model, the concatenation structure of optimal controlled trajectories as segments of bang and singular arcs is determined. This leads to simple numerical minimization procedures for the computation of globally optimal controls.

Control Theory and Cancer Chemotherapy: How They Interact.

Control theory arises in most modern real-life applications, not least in biological and medical applications. In particular, in biological and medical contexts, the role of control theory began to take shape in the early 1980s when the first works appeared on the application of control theory in models of pharmacokinetics and pharmacodynamics for antitumor therapies. Forty years after those first works, the theory of control continues to be considered a mathematical analysis tool of extreme importance and usefulness, but the challenges it must overcome in order to manage the complexity of biological processes are in fact not yet overcome. In this article, we introduce the reader to the basic ideas of control theory, its aims and its mathematical formalization, and we review its use in cell phase-specific models for cancer chemotherapy. We discuss strengths and limitations of the control theory approach to the analysis pharmacokinetics and pharmacodynamics models, and we will see that most of them are strongly related to data availability and mathematical form of the model. We propose some future research directions that could prove useful in overcoming the these limitations and we indicate the crucial steps preliminary to a useful and informative application of control theory to cancer chemotherapy modeling.

Bilinear state systems on an unbounded time scale

We demonstrate the existence and uniqueness of solutions to a bilinear state system with locally essentially bounded coefficients on an unbounded time scale. We obtain a Volterra series representation for these solutions which is norm convergent and uniformly convergent on compact subsets of the time scale. We show the associated state transition matrix has a similarly convergent Peano-Baker series representation and identify a necessary and sufficient condition for its invertibility. Finally, we offer numerical applications for dynamic bilinear systems – a frequency modulated signal model and a two-compartment cancer chemotherapy model.

Mathematical analysis and simulation study of a phase-field model of prostate cancer growth with chemotherapy and antiangiogenic therapy effects

Chemotherapy is a common treatment for advanced prostate cancer. The standard approach relies on cytotoxic drugs, which aim at inhibiting proliferation and promoting cell death. Advanced prostatic tumors are 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 in combination with standard cytotoxic protocols. However, the complex genetic alterations that originate and sustain prostate cancer growth complicate the selection of the best chemotherapeutic approach for each patient’s tumor. Here, we present a mathematical model of prostate cancer growth and chemotherapy that may enable physicians to test and design personalized chemotherapeutic protocols in silico. We use the phase-field method to describe tumor growth, which we assume to be driven by a generic nutrient following reaction–diffusion dynamics. Tumor proliferation and apoptosis (i.e. programmed cell death) can be parameterized with experimentally-determined values. Cytotoxic chemotherapy is included as a term downregulating tumor net proliferation, while antiangiogenic therapy is modeled as a reduction in intratumoral nutrient supply. An additional equation couples the tumor phase field with the production of prostate-specific antigen, which is a prostate cancer biomarker that is extensively used in the clinical management of the disease. We prove the well posedness of our model and we run a series of representative simulations leveraging an isogeometric method to explore untreated tumor growth as well as the effects of cytotoxic chemotherapy and antiangiogenic therapy, both alone and combined. Our simulations show that our model captures the growth morphologies of prostate cancer as well as common outcomes of cytotoxic and antiangiogenic mono therapy and combined therapy. Additionally, our model also reproduces the usual temporal trends in tumor volume and prostate-specific antigen evolution observed in experimental and clinical studies.

Preliminary Study for Development of an Experimental Model of Cancer Chemotherapy Induced Diarrhea

Preliminary Study for Development of an Experimental Model of Cancer Chemotherapy Induced Diarrhea

Optimal control for a mathematical model for chemotherapy with pharmacometrics

An optimal control problem for an abstract mathematical model for cancer chemotherapy is considered. The dynamics is for a single drug and includes pharmacodynamic (PD) and pharmacokinetic (PK) models. The aim is to point out qualitative changes in the structures of optimal controls that occur as these pharmacometric models are varied. This concerns (i) changes in the PD-model for the effectiveness of the drug (e.g., between a linear log-kill term and a non-linear Michaelis-Menten type Emax-model) and (ii) the question how the incorporation of a mathematical model for the pharmacokinetics of the drug effects optimal controls. The general results will be illustrated and discussed in the framework of a mathematical model for anti-angiogenic therapy.

Optimal control for cancer chemotherapy under tumor heterogeneity with Michealis-Menten pharmacodynamics

We analyze mathematical models for cancer chemotherapy under tumor heterogeneity as optimal control problems. Tumor heterogeneity is incorporated into the model through a potentially large number of states which represent sub-populations of tumor cells with different chemotherapeutic sensitivities. In this paper, a Michaelis-Menten type functional form is used to model the pharmacodynamic effects of the drug concentrations. In the objective, a weighted average of the tumor volume and the total amounts of drugs administered (taken as an indirect measurement for the side effects) is minimized. Mathematically, incorporating a Michaelis-Menten form creates a nonlinear structure in the controls with partial convexity properties in the Hamiltonian function for the optimal control problem. As a result, optimal controls are continuous and this fact can be utilized to set up an efficient numerical computation of extremals. Second order Jacobi type necessary and sufficient conditions for optimality are formulated that allow to verify the strong local optimality of numerically computed extremal controlled trajectories. Examples which illustrate the cases of both strong locally optimal and non-optimal extremals are given.

Iterative method for estimating the robust domains of attraction of non-linear systems: Application to cancer chemotherapy model with parametric uncertainties

In this paper, we present an iterative procedure method for estimating the robust domains of attraction of non-linear systems. This method is based on the approximation of the uncertain non-linear system with a parameters-dependent Convex Difference Inclusions (CDI) system and the classical iterative methods for linear systems, which are introduced in this paper. A robust one-step operator computing a sequence of convex sets is derived, and the polyhedral case is discussed. An algorithm summarizing the iterative procedure based on the robust one-step operator is given, which is the theoretical contribution of this paper. This method is applied to cancer chemotherapy model considering parametric uncertainties and it is shown that drastic reduction of the robust domain of attraction of the cancer chemotherapy model has happened and this is caused by the presence of parametric uncertainties. It is also proved that the aggressive chemotherapy is not the effective treatment for all the patients.

Nonlinear cancer chemotherapy: Modelling the Norton-Simon hypothesis

A fundamental model of tumor growth in the presence of cytotoxic chemotherapeutic agents is formulated. The model allows to study the role of the Norton-Simon hypothesis in the context of dose-dense chemotherapy. Dose-dense protocols aim at reducing the period between courses of chemotherapy from three weeks to two weeks, in order to avoid tumor regrowth between cycles. We address the conditions under which these protocols might be more or less beneficial in comparison to less dense settings, depending on the sensitivity of the tumor cells to the cytotoxic drugs. The effects of varying other parameters of the protocol, as for example the duration of each continuous drug infusion, are also inspected. We believe that the present model might serve as a foundation for the development of more sophisticated models for cancer chemotherapy.

A case study-mathematical models for drug tolerance and cancer chemotherapy

Drug tolerance is a contributing factor of drug addiction. The rate depends on the particular drug, dosage and frequency of use. Differential development occurs for different effects of the same drug. Mathematical models of drug tolerance was improved based on the assumption that the decrease of drug effect after repeated administration of a drug is caused by the involved regulations in the organism adapting themselves to the presence of the drug. Cancer can develop fundamentally in any of the body's tissues. Though, every type of cancer has its own unique features, the basic processes that produce cancer are quite similar in all forms of the disease. Chemotherapy works by stopping or slowing the growth of cancer cells, which grow and divide quickly. Mathematical models have become popular everywhere in cancer research. Mathematical modeling is one of the tools that can provide a structure to better understand cancer evolution and its response to chemotherapy. In this paper, we have provided the contributions of mathematical models for drug tolerance and cancer chemotherapy.

Docetaxel does not impair skeletal muscle force production in a murine model of cancer chemotherapy.

Chemotherapy drugs such as docetaxel are commonly used to treat cancer. Cancer patients treated with chemotherapy experience decreased physical fitness, muscle weakness and fatigue. To date, it is unclear whether these symptoms result only from cancer‐derived factors or from the combination of cancer disease and cancer treatments, such as chemotherapy. In this study, we aimed at determining the impact of chemotherapy per se on force production of hind limb muscles from healthy mice treated with docetaxel. We hypothesized that docetaxel will decrease maximal force, exacerbate the force decline during repeated contractions and impair recovery after fatiguing stimulations. We examined the function of soleus and extensor digitorum longus (EDL) muscles 24 h and 72 h after a single injection of docetaxel (acute treatment), and 7 days after the third weekly injection of docetaxel (repeated treatment). Docetaxel was administrated by intravenous injection (20 mg/kg) in female FVB/NRj mice and control mice were injected with saline solution. Our results show that neither acute nor repeated docetaxel treatment significantly alters force production during maximal contractions, repeated contractions or recovery. Only a tendency to decreased peak specific force was observed in soleus muscles 24 h after a single injection of docetaxel (−17%, P = 0.13). In conclusion, docetaxel administered intravenously does not impair force production in hind limb muscles from healthy mice. It remains to be clarified whether docetaxel, or other chemotherapy drugs, affect muscle function in subjects with cancer and whether the side effects associated with chemotherapy (neurotoxicity, central fatigue, decreased physical activity, etc.) are responsible for the experienced muscle weakness and fatigue.

Constrained bilinear control problem: Application to a cancer chemotherapy model

The aim of this paper is to investigate the optimal control problem for finite-dimensional bilinear systems and its application to a chemotherapy model. We characterize an optimal control that minimizes a quadratic cost functional in two cases of constrained admissible controls, then we give sufficient conditions for the uniqueness of such a control, and we derive useful algorithms for the computation of the optimal control. The established results are applied to a cancer chemotherapy bilinear model in order to simulate the optimal treatment protocol using two different approaches: one based on a limited instant toxicity, and the other on a limited cumulative toxicity along the therapy session.

Optimal control of a mathematical model for cancer chemotherapy under tumor heterogeneity

We consider cancer chemotherapy as an optimal control problem with the aim to minimize a combination of the tumor volume and side effects over an a priori specified therapy horizon when the tumor consists of a heterogeneous agglomeration of many subpopulations. The mathematical model, which accounts for different growth and apoptosis rates in the presence of cell densities, is a finite-dimensional approximation of a model originally formulated by Lorz et al. [18,19] and Greene et al. [10,11] with a continuum of possible traits. In spite of an arbitrarily high dimension, for this problem singular controls (which correspond to time-varying administration schedules at less than maximum doses) can be computed explicitly in feedback form. Interestingly, these controls have the property to keep the entire tumor population constant. Numerical computations and simulations that explore the optimality of bang-bang and singular controls are given. These point to the optimality of protocols that combine a full dose therapy segment with a period of lower dose drug administration.

Effects of vascularization on cancer nanochemotherapy outcomes

Cancer therapy requires anticancer agents capable of efficient and uniform systemic delivery. One promising route to their development is nanotechnology. Here, a previous model for cancer chemotherapy based on a nanosized drug carrier (Paiva et al., 2011) is extended by including tissue vasculature and a three-dimensional growth. We study through computer simulations the therapy against tumors demanding either large or small nutrient supplies growing under different levels of tissue vascularization. Our results indicate that highly vascularized tumors demand more aggressive therapies (larger injected doses administrated at short intervals) than poorly vascularized ones. Furthermore, nanoparticle endocytic rate by tumor cells, not its selectivity, is the major factor that determines the therapeutic success. Finally, our finds indicate that therapies combining cytotoxic agents with antiangiogenic drugs that reduce the abnormal tumor vasculature, instead of angiogenic drugs that normalize it, can lead to successful treatments using feasible endocytic rates and administration intervals.

An integrated cellular and sub-cellular model of cancer chemotherapy and therapies that target cell survival.

Apoptosis resistance is a hallmark of human cancer, and tumor cells often become resistant due to defects in the programmed cell death machinery. Targeting key apoptosis regulators to overcome apoptotic resistance and promote rapid death of tumor cells is an exciting new strategy for cancer treatment, either alone or in combination with traditionally used anti-cancer drugs that target cell division. Here we present a multiscale modeling framework for investigating the synergism between traditional chemotherapy and targeted therapies aimed at critical regulators of apoptosis.

A 3-Compartment Model for Chemotherapy of Heterogeneous Tumor Populations

We consider a mathematical model for cancer chemotherapy with a single agent that distinguishes three levels of sensitivities calling the subpopulations `sensitive', `partially sensitive' and `resistant'. We analyze the dynamic properties of the system under what could be considered metronomic (continuous, low-dose, constant) chemotherapy and, more generally, also consider the optimal control problem of minimizing the tumor burden over a prescribed therapy interval. Interestingly, when several levels of chemotherapeutic sensitivities are taken into account in the model, lower time-varying dose rates as they are given by singular controls become a treatment option. This is only the case once a significant residuum of resistant cells has been created in a simpler 2-compartment model that only considers sensitive and resistant cells. For heterogeneous tumor populations, a more modulated approach that varies the dose rates of the drugs may be more beneficial than the classical maximum tolerated dose approach pursued in medical practice.

ON OPTIMAL CHEMOTHERAPY FOR HETEROGENEOUS TUMORS

A mathematical model for cancer chemotherapy of heterogeneous tumor populations is considered as an optimal control problem with the objective to minimize the tumor burden over a prescribed therapy horizon. While an upfront maximum tolerated dose (MTD) regimen with rest-period has been confirmed as mathematically optimal for models when the tumor population is homogeneous, in the presence of partially sensitive or even resistant cells, protocols that administer the therapeutic agents at lower dose rates described by so-called singular controls become a viable alternative. In this paper, the structure of protocols that follow an initial upfront maximum dose treatment with reduced dose rate singular controls is investigated. Such protocols reflect structures which in the medical literature sometimes are called chemo-switch protocols.

Modeling and Control of Heterogeneous Tumors under Chemotherapy

Tumor cells typically are genetically highly unstable and as a response to mutations, they frequently consist of heterogeneous agglomerations of various cell populations that exhibit a wide range of sensitivities towards particular chemotherapeutic agents. However, in response to different growth and apoptosis rates as well as increasing tumor cell densities, specific traits become dominant. We consider a mathematical model for cancer chemotherapy with a single chemotherapeutic agent for three distinctly separate levels of drug sensitivity and analyze the dynamic properties of the system under metronomic (continuous low-dose) chemotherapy. More generally, the optimal control problem of minimizing the tumor burden over a prescribed therapy interval is considered. Interestingly, when several levels of drug sensitivity are taken into account in the model, lower time-varying dose rates become a viable option. For simpler models that only distinguish between sensitive and resistant subpopulations, this only holds once a significant residuum of resistant cells has developed. For heterogeneous tumor populations, a more modulated approach that varies the dose rates of the drugs may be more beneficial than the classical maximum tolerated dose approach pursued in medical practice.