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The mathematical study of cancer received great attention in recent years. The theoretical and numerical investigations of models employed for studying the growth of cancer cells and treatment of cancer can help the clinical practitioners in adapting new strategies to face the challenges posed by the deadly disease. Chemotherapy is the most common method of treatment for cancer. The resistance of tumor cells toward the administered drug and the toxic effect of anti-cancer agents on healthy cells are major hurdles to the success of therapy. In this paper, we study this as an optimal control problem in which the amount of drug injected is taken as control. The evolution of different types of cells — sensitive tumor cells, resistant tumor cells and normal cells — under treatment are analyzed. The pharmacokinetics of the drug is also incorporated into the mathematical model. We propose a treatment protocol that assures the death of a maximum number of tumor cells but also manages to keep the normal cells at a sufficient level. We also validated our theoretical results numerically. [ABSTRACT FROM AUTHOR]

The main purpose of this paper is to propose an optimal finite duration treatment method for preventing tumor growth. The obtained results show that changing the dynamics of the cancer model is essential for a finite duration treatment. Therefore, vaccine therapy is used for changing the parameters of the system and chemotherapy is applied for pushing the system to the domain of attraction of the healthy state. The state-dependent Riccati equation (SDRE)-based optimal control method is used for optimal chemotherapy. In this method, the special conditions of the patients could be considered by choosing suitable weighting matrices in the cost function and restricting the drug dosage. Also, there are infinite ways to choose these state-dependent matrices. In this paper, these interesting features of this method are used for each patient. Since measuring the states of the system is impossible at each time for states feedback; an extended Kalman filter (EKF) is designed as an observer in the nonlinear system. So, the SDRE method is employed just by measuring the population of normal cells. Numerical simulations show the flexibility and effectiveness of this treatment method. [ABSTRACT FROM AUTHOR]

In this paper, I continue the study of the mathematical models presented in [J. C. Larsen, Models of cancer growth, J. Appl. Math. Comput. 53(1-2) (2015) 613-645] and [J. C. Larsen, The bistability theorem in a model of metastatic cancer, to appear in Appl. Math.]. I shall prove the bistability theorem for the ODE model from [Larsen, 2015]. It is a mass action kinetic system in the variables cancer, GF growth factor and GI growth inhibitor. This theorem says that for some values of the parameters, there exist two positive singular points , of the vector field. Here and is stable and is unstable, see Sec. 2. There is also a discrete model in [Larsen, 2015], it is a linear map () on three-dimensional Euclidean vector space with variables where these variables have the same meaning as in the ODE model above. In [Larsen, 2015], I showed that one can sometimes find affine vector fields on three-dimensional Euclidean vector space whose time one map is . I shall also show this in the present paper in a more general setting than in [Larsen, 2015]. This enables me to find an expression for the rate of change of cancer growth on the coordinate hyperplane in Euclidean vector space. I also present an ODE model of cancer metastasis with variables where is primary cancer and is metastatic cancer and GF, GI are growth factors and growth inhibitors, respectively. [ABSTRACT FROM AUTHOR]

This paper deals with a control strategy through impulsive drug treatment to study the effect of Chemotherapy and CTL activator drug to eradicate the cervical cancer. CTL kills the HPV infected cells and helps to prevent cervical cancer. Dendritic cell (DC) plays a crucial role in this disease dynamics in enhancing the activity of CTL. To explore these effects, we have formulated a mathematical model consisting of six compartments to describe the interactions between HPV and three classes of cervical cells (susceptible, infected, cancerous), with DC and CTL. Calculating the basic reproduction ratio, we find that there exists a disease-free equilibrium point and an endemic equilibrium point. In our study, we have shown that though DC enhances the activity of CTL, it is not sufficient to eradicate the disease totally. How we could restrict the cancer development by using double drug in an impulsive way with particular drug doses is being reflected in the results. To illustrate our results, numerical simulations are also presented. [ABSTRACT FROM AUTHOR]

This article concentrates on the study of delay effect on a model of schistosomiasis transmission with control measures such as predation or harvesting and chemotherapy. In the presence of predation or harvesting and chemotherapy, system admits multiple endemic equilibria. Mathematical analysis shows that they are opposite in nature regarding stability. One may observe switching phenomena for the unstable equilibrium by incorporating delay. The disease may be highly endemic if there is no control measure, which is obvious from the model analysis. Results obtained in this paper are also verified through numerical simulations. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

In this paper, we present a mathematical model that describes tumor-normal cells interaction dynamics focusing on role of drugs in treatment of brain tumors. The goal is to predict distribution and necessary quantity of drugs delivered in drug-therapy by using optimal control framework. The model describes interactions of tumor and normal cells using a system of reactions-diffusion equations involving the drug concentration, tumor cells and normal tissues. The control estimates simultaneously blood perfusion rate, reabsorption rate of drug and drug dosage administered, which affect the effects of brain tumor chemotherapy. First, we develop mathematical framework which models the competition between tumor and normal cells under chemotherapy constraints. Then, existence, uniqueness and regularity of solution of state equations are proved as well as stability results. Afterwards, optimal control problems are formulated in order to minimize the drug delivery and tumor cell burden in different situations. We show existence and uniqueness of optimal solution, and we derive necessary conditions for optimality. Finally, to solve numerically optimal control and optimization problems, we propose and investigate an adjoint multiple-relaxation-time lattice Boltzmann method for a general nonlinear coupled anisotropic convection-diffusion system (which includes the developed model for brain tumor targeted drug delivery system). [ABSTRACT FROM AUTHOR]

We present in this paper a new mathematical model for the scheduling of angiogenic inhibitors in combination with a chemotherapeutic agent for a tumor. Our model takes into account the process of angiogenesis and the quality of the vasculature discriminating between stable blood vessels and unstable blood vessels. We characterize theoretically the optimal controls on drug distribution to minimize the number of cancer cells at the end of the treatment in a free horizon time problem with restrictions on the total amount of drug doses. Finally, we solve the optimal control problem by using numerical simulations, obtaining as a result that, despite the number of the tumor cells decrease with anti-angiogenic treatment, the best results are reached at the end of the chemotherapy treatment. [ABSTRACT FROM AUTHOR]

Monomethyl auristatin E (MMAE) is a derivative of the marine peptide Dolastatin 10, which has therapeutic effects against various cancers according to its antimitotic activity in multiple clinical trials. The antibody drug conjugate (ADC) of MMAE is currently used in clinical practice. However, the safety issues of MMAE-based ADC, such as high drug toxicity and poor bioavailability, still exist when using it for anticancer therapy. A sustained release of drug delivery approach should be used to reduce toxicity and achieve sufficient anticancer effects. Herein, PLGA-b-PEG 2 0 0 0 with excellent biocompatibility and slow degradation ability was adopted to construct MMAE-loaded nanoparticles for safe and effective chemotherapy. The sustained release effect and the immunogenic cell death (ICD) effect of PLGA-MMAE nanoparticles were assessed by in vitro experiments. The PLGA-MMAE nanoparticles effectively accumulated in the tumor through the enhanced permeability and retention (EPR) effect, inducing cell apoptosis and causing a certain degree of immune response. The sustained drug release of PLGA-MMAE improved the bioavailability and effectively reduced the toxicity and development of the tumor compared to the effect of free MMAE or ADC. Overall, this study provides a safe and effective chemotherapeutic approach, as well as a simple and effective synthetic process for MMAE-based nanoparticles, improving their therapeutic efficacy and safety. [ABSTRACT FROM AUTHOR]

Asian ginseng, American ginseng, and notoginseng are three major species in the ginseng family. Notoginseng is a Chinese herbal medicine with a long history of use in many Oriental countries. This botanical has a distinct ginsenoside profile compared to other ginseng herbs. As a saponin-rich plant, notoginseng could be a good candidate for cancer chemoprevention. However, to date, only relatively limited anticancer studies have been conducted on notoginseng. In this paper, after reviewing its anticancer data, phytochemical isolation and analysis of notoginseng is presented in comparison with Asian ginseng and American ginseng. Over 80 dammarane saponins have been isolated and elucidated from different plant parts of notoginseng, most of them belonging to protopanaxadiol or protopanaxatriol groups. The role of the enteric microbiome in mediating notoginseng metabolism, bioavailability, and pharmacological actions are discussed. Emphasis has been placed on the identification and isolation of enteric microbiome-generated notoginseng metabolites. Future investigations should provide key insights into notoginseng's bioactive metabolites as clinically valuable anticancer compounds. [ABSTRACT FROM AUTHOR]

In this paper, a new mathematical model of the interactions between a growing tumor and an immune system is presented by incorporating the danger model. The populations involved are tumor cells, CD8+ T-cells, natural killer cells (NK-cells), dendritic cells (DCs) and cytokine interleukin-12 (IL-12). A key feature of this work is the inclusion of the danger model into the dynamics of the immune system, which is rarely considered by previous works. Regarding the constructed mathematical model, both the location of equilibria and their stability properties are discussed, which are useful not only to gain a broad understanding of the specific system dynamics, but also to help guide the development of therapies. Moreover, numerical simulations of the system with chemotherapy and immunotherapy by using specific parameters are presented to illustrate that proper therapy is able to eliminate the entire tumor. In addition, we illustrate cases for which neither chemotherapy nor immunotherapy alone are able to control tumor growth, but a combination treatment is sufficient to eliminate the tumor cells. [ABSTRACT FROM AUTHOR]

A tumor growth system with immune response and chemotherapy is put in a nonlinear dynamical system whose solutions are relative to the initial data. This study presents a phase space analysis of the system. Here, the basin of equilibrium points attraction is determined for a particular class of systems and is subjected to input and state constraints in which all points in phase space would be close to the equilibrium points according to the region of attraction it starts. The addition of a drug term to the system can move the solution trajectory to the desirable basin of attraction. The proposed method gives static output feedback controllers that guarantee the convergence of the generic solutions. Although such a set-point regulation problem is too challenging for general nonlinear systems, the standard surface is found by the proposed approach, which is called separatrix for the controller. This criterion of separating border can perform well even when the mentioned system has limited change parameters. The control is set by separatrix in which the output feedback controller therapy can take all solutions to the healthy state through a constrained chemotherapy protocol. Moreover, this protocol can enable globalization of healthy equilibria. [ABSTRACT FROM AUTHOR]

The development of more accurate cancer mathematical models leads to present more realistic treatment protocols, especially in model-based treatment protocol design. Hence, a cancer mathematical model is presented by considering tumor cells, immune cells, interleukins, macrophages polarization, and chemotherapy based on biological concepts. Both local and global sensitivity analyses are done to examine the effect of changing the parameters on the final tumor population. Then, the tumor-free equilibrium points of the system are derived, and their stabilities are studied. The main target of chemotherapy is to eliminate the tumor while limiting drug toxicity. The SDRE method is used to construct a sub-optimal control strategy by using the developed nonlinear cancer model. For simulation, three patients are considered: a young patient, an old patient, and a pregnant patient. These cases have different immune system strengths. Also, three initial tumor sizes are regarded for each case. So, different treatment strategies are suggested. Eradication of tumor cells in a finite duration with a desired amount of chemo-drug is shown in the simulation results. The results confirmed that the immune system's ability plays an important role in treatment success. It is shown that there are different treatment protocols for different patients, and the SDRE method is more flexible and effective than the others. [ABSTRACT FROM AUTHOR]

The role of traditional Chinese medicine (TCM) on treatment of metastatic colorectal cancer (mCRC) remains controversial, and its active components and potential targets are still unclear. This study mainly aimed to assess the efficacy and safety of TCM in mCRC treatment through meta-analysis and explore the effective components and potential targets based on the network pharmacology method. We systematically searched PubMed, EMBASE, Cochrane, CBM, WanFang, and CNKI database for randomized controlled trials (RCTs) comparing the treatment of mCRC patients with and without TCM. A meta-analysis using RevMan 5.4 was conducted. In total, 25 clinical trials were analyzed, and the result demonstrated that TCM was closely correlated with the improved OS (HR: 0.63; 95% CI: 0.52–0.76; p < 0.00001) and PFS (HR: 0.73; 95% CI: 0.61–0.88; p = 0.0010). Then, high-frequency Chinese herbs from the prescriptions extracted from the trails included in the OS meta-analysis were counted to construct a core-effective prescription. The TCMSP database was used to retrieve the active chemical components and predict herb targets. The Genecards, OMIM, Disgenet, DrugBank, and TTD database were searched for colorectal cancer targets. R-package was used to construct the Component-Target (C-T) network based on the intersection genes. Further, we extracted hub genes from C-T network and performed functional enrichment and pathway analysis. Finally, the C-T network showed 120 herb and disease co-target genes, and the most important top 10 active components were: Quercetin, Luteolin, Wogonin, Kaempferol, Nobiletin, Baicalein, Licochalcone A, Naringenin, Isorhamnetin, and Acacetin. The first 20 hub genes were extracted: CDKN1A, CDK1, CDK2, E2F1, CDK4, PCNA, RB1, CCNA2, MAPK3, CCND1, CCNB1, JUN, MAPK1, RELA, FOS, MAPK8, STAT3, MAPK14, NR3C1, and MYC. Thus, effective Chinese herb components may inhibit the mCRC by targeting multiple biological processes of the above hub genes. [ABSTRACT FROM AUTHOR]

Prostate cancer can be lethal in advanced stages, for which chemotherapy may become the only viable therapeutic option. While there is no clear clinical management strategy fitting all patients, cytotoxic chemotherapy with docetaxel is currently regarded as the gold standard. However, tumors may regain activity after treatment conclusion and become resistant to docetaxel. This situation calls for new delivery strategies and drug compounds enabling an improved therapeutic outcome. Combination of docetaxel with antiangiogenic therapy has been considered a promising strategy. Bevacizumab is the most common antiangiogenic drug, but clinical studies have not revealed a clear benefit from its combination with docetaxel. Here, we capitalize on our prior work on mathematical modeling of prostate cancer growth subjected to combined cytotoxic and antiangiogenic therapies, and propose an optimal control framework to robustly compute the drug-independent cytotoxic and antiangiogenic effects enabling an optimal therapeutic control of tumor dynamics. We describe the formulation of the optimal control problem, for which we prove the existence of at least a solution and determine the necessary first-order optimality conditions. We then present numerical algorithms based on isogeometric analysis to run a preliminary simulation study over a single cycle of combined therapy. Our results suggest that only cytotoxic chemotherapy is required to optimize therapeutic performance and we show that our framework can produce superior solutions to combined therapy with docetaxel and bevacizumab. We also illustrate how the optimal drug-naïve cytotoxic effects computed in these simulations may be successfully leveraged to guide drug production and delivery strategies by running a nonlinear least-square fit of protocols involving docetaxel and a new design drug. In the future, we believe that our optimal control framework may contribute to accelerate experimental research on chemotherapeutic drugs for advanced prostate cancer and ultimately provide a means to design and monitor its optimal delivery to patients. [ABSTRACT FROM AUTHOR]

Cancer is a complex, multiscale process involving interactions at intracellular, intercellular and tissue scales that are in turn susceptible to microenvironmental changes. Each individual cancer cell within a cancer cell mass is unique, with its own internal cellular pathways and biochemical interactions. These interactions contribute to the functional changes at the cellular and tissue scale, creating a heterogenous cancer cell population. Anticancer drugs are effective in controlling cancer growth by inflicting damage to various target molecules and thereby triggering multiple cellular and intracellular pathways, leading to cell death or cell-cycle arrest. One of the major impediments in the chemotherapy treatment of cancer is drug resistance driven by multiple mechanisms, including multi-drug and cell-cycle mediated resistance to chemotherapy drugs. In this article, we discuss two hybrid multiscale modelling approaches, incorporating multiple interactions involved in the sub-cellular, cellular and microenvironmental levels to study the effects of cell-cycle, phase-specific chemotherapy on the growth and progression of cancer cells. [ABSTRACT FROM AUTHOR]

We investigate a mathematical population model of tumor-immune interactions. The populations involved are tumor cells, specific and non-specific immune cells, and concentrations of therapeutic treatments. We establish the existence of an optimal control for this model and provide necessary conditions for the optimal control triple for simultaneous application of chemotherapy, tumor infiltrating lymphocyte (TIL) therapy, and interleukin-2 (IL-2) treatment. We discuss numerical results for the combination of the chemo-immunotherapy regimens. We find that the qualitative nature of our results indicates that chemotherapy is the dominant intervention with TIL interacting in a complementary fashion with the chemotherapy. However, within the optimal control context, the interleukin-2 treatment does not become activated for the estimated parameter ranges. [ABSTRACT FROM AUTHOR]

This work assesses the impact of the first line drug regimen on active disease control under the stipulated time of tuberculosis (TB) treatment. In an effort to understand why a robust immune response mechanism sometimes fails to completely control TB infection, we first developed a model that captures the human immune response mechanisms to Mycobacterium tuberculosis (Mtb) infection. We then extended the model to include drug therapy. The drug therapy model is used to assess the potency of the recommended six-month TB drug chemotherapy in infected individuals. The efficacy of each drug was explored and observations show that low drug efficacy values result in extension of treatment period. The numerical results confirm typical clinical disease progression patterns noticed in individuals under TB therapy. The drug model simulations and analysis show that administration of the recommended first line three-drug regimen normally cures the TB infection. Using the model, we established that only Isoniazid monotherapy drug treatment, and any combination therapy of two drugs including Isoniazid are potent enough to resolve the TB infection. [ABSTRACT FROM AUTHOR]

Nausea and vomiting are significant adverse effects of chemotherapeutic agents like cisplatin, and cause significant patient morbidity. Cisplatin treatment results in oxidant gut injury, which is postulated to be the primary cause of nausea and vomiting. We evaluated the effects of two antioxidant herbs, Scutellaria baicalensis and American ginseng berry, on cisplatin-induced nausea and vomiting using a rat model. Rats react to emetic or nausea-producing stimuli, such as cisplatin, with altered feeding habits, manifested by increased kaolin consumption (pica). We measured pica in rats to quantify cisplatin-induced nausea. We observed that pretreatment of rats with S. baicalensis or ginseng berry extracts resulted in a significant reduction in cisplatin-induced pica. The in vitro free radical scavenging ability of the herbal extract observed in the study, further confirmed the antioxidant action of the herb. We conclude that herbal antioxidants may have a role in attenuating cisplatin-induced nausea and vomiting. [ABSTRACT FROM AUTHOR]