Cell population heterogeneity and evolution towards drug resistance in cancer: Biological and mathematical assessment, theoretical treatment optimisation

Background Drug-induced drug resistance in cancer has been attributed to diverse biological mechanisms at the individual cell or cell population scale, relying on stochastically or epigenetically varying expression of phenotypes at the single cell level, and on the adaptability of tumours at the cell population level. Scope of review We focus on intra-tumour heterogeneity, namely between-cell variability within cancer cell populations, to account for drug resistance. To shed light on such heterogeneity, we review evolutionary mechanisms that encompass the great evolution that has designed multicellular organisms, as well as smaller windows of evolution on the time scale of human disease. We also present mathematical models used to predict drug resistance in cancer and optimal control methods that can circumvent it in combined therapeutic strategies. Major conclusions Plasticity in cancer cells, i.e., partial reversal to a stem-like status in individual cells and resulting adaptability of cancer cell populations, may be viewed as backward evolution making cancer cell populations resistant to drug insult. This reversible plasticity is captured by mathematical models that incorporate between-cell heterogeneity through continuous phenotypic variables. Such models have the benefit of being compatible with optimal control methods for the design of optimised therapeutic protocols involving combinations of cytotoxic and cytostatic treatments with epigenetic drugs and immunotherapies. General significance Gathering knowledge from cancer and evolutionary biology with physiologically based mathematical models of cell population dynamics should provide oncologists with a rationale to design optimised therapeutic strategies to circumvent drug resistance, that still remains a major pitfall of cancer therapeutics. This article is part of a Special Issue entitled “System Genetics” Guest Editor: Dr. Yudong Cai and Dr. Tao Huang

Optimal distributed control of a diffuse interface model of tumor growth

In this paper, a distributed optimal control problem is studied for a diffuse interface model of tumor growth which was proposed in [A. Hawkins-Daruud, K.G. van der Zee, J.T. Oden, Numerical simulation of a thermodynamically consistent four-species tumor growth model, Int. J. Numer. Math. Biomed. Engng. 28 (2011), 3-24]. The model consists of a Cahn-Hilliard equation for the tumor cell fraction coupled to a reaction-diffusion equation for a variable representing the nutrient-rich extracellular water volume fraction. The distributed control monitors as a right-hand side the reaction-diffusion equation and can be interpreted as a nutrient supply or a medication, while the cost function, which is of standard tracking type, is meant to keep the tumor cell fraction under control during the evolution. We show that the control-to-state operator is Frechet differentiable between appropriate Banach spaces and derive the first-order necessary optimality conditions in terms of a variational inequality involving the adjoint state variables.Comment: A revised version of the paper has been published on Nonlinearity 30 (2017), 2518-2546. Let us point out that in this arXiv:1601.04567 [math.AP] version there is something missing in assumption (H3) at page 6: the first initial value in (H6) must also satisfy a Neumann homogeneous condition at the boundary of the domai

Optimal distributed control of a diffuse interface model of tumor growth

In this paper, a distributed optimal control problem is studied for a diffuse interface model of tumor growth which was proposed by Hawkins--Daruud et al. in \cite{HZO}. The model consists of a Cahn--Hilliard equation for the tumor cell fraction \vp coupled to a reaction-diffusion equation for a function \s representing the nutrient-rich extracellular water volume fraction. The distributed control $u$ monitors as a right-hand side the equation for \s and can be interpreted as a nutrient supply or a medication, while the cost function, which is of standard tracking type, is meant to keep the tumor cell fraction under control during the evolution. We show that the control-to-state operator is Fr\'echet differentiable between appropriate Banach spaces and derive the first-order necessary optimality conditions in terms of a variational inequality involving the adjoint state variables

Mathematical modelling of drug resistance in malignant tumour treatment.

M. Sc. Univeristy of KwaZulu-Natal, Pietermaritzburg 2014.Resistance to conventional chemotherapies, especially to anti-cancer agents, is rapidly becoming a global pandemic. Mutations, in combination with genetic instabilities, play an important role in the molecular heterogeneity of cancerous cells that display resistance to chemotherapeutic drugs. Currently, mechanisms involved in drug resistance phenotype resulting from the interaction of a tumour and anti-cancer agents are not fully understood. In this dissertation, we propose two new dynamical models for the interaction between a tumour and a chemotherapeutic drug. Our focus is only on resistance which is caused by random genetic point mutations. The models consist of coupled systems of ordinary and partial differential equations. Tumour cells are divided into two classes, namely; sensitive and resistant cells. We determine the equilibrium points of the model equations and investigate their stability. In the frst instance, after reviewing the basic modelling assumptions and main results found in the mathematical modelling literature on drug resistance, we present the ordinary differential equation (ODE) model. To account for spatial growth effects, we then extend the model to a partial differential equation (PDE) model that describes the local interaction of the tumour with the anti-cancer agent through convection, reaction and diffsion processes. Some analytical solutions of the PDE model that are comparable to those found in the literature are obtained. One novel outcome of the models in this dissertation is the qualitative demonstration of the possible success of the therapy for certain initial conditions, number of sensitive cells and their interaction with the chemotherapeutic drug. Parameter sensitivity analysis is carried out to determine the influence of each individual parameter in the model. For all the models, numerical solutions which showed the effct of therapeutic agents on the growth and spread of the tumour cells, subject to evolving drug resistance phenomenon, were attained and presented here

Existence, uniqueness and asymptotic analysis of optimal control problems for a model of groundwater pollution

International audienceOn considère un problème de contrôle optimal de contamination des eaux souterraines. L'objectiféconomique prend en compte le nécessaire compromis entre l'utilisation du polluant-par exemple de l'engrais-et les coûts de dépollution. Il est soumisà la contrainte d'un modèle hy-drogéologique pour la propagation de la pollution dans l'aquifère. On considère une large gamme de réactions cinétiques. L'objectif de cet article est double. D'une part, nous construisons rigoureusement, par analyse asymptotique, le problème de contrôle optimal e↵ectif pour des contaminants présents en faible concentration dans l'aquifère. D'autre part, nous analysons le problème de contrôle optimal et nous montrons en particulier que le problème e↵ectif est bien posé. De plus, nous démontrons une pro-priété de stabilité du processus de contrôle optimal : toute solution optimale du problème adimensionné converge vers la solution optimale du problème e↵ectif lorsque l'ordre de grandeur de la concentration du polluant décroît.An optimal control problem of contaminated underground water is considered. The spatio-temporal objective takes into account the economic trade o↵ between the pollutant use-for instance fertilizer-and the cleaning costs. It is constrained by a hydrogeological model for the spread of the pollution in the aquifer. We consider a broad range of reaction kinetics. The aim of the paper is twofold. On the one hand, we rigorously derive, by asymptotic analysis, the e↵ective optimal control problem for contaminant species that are slightly concentrated in the aquifer. On the other hand, the mathematical analysis of the optimal control problems is performed and we prove in particular that the latter e↵ective problem is well-posed. Furthermore, a stability property of the optimal control process is provided: any optimal solution of the properly scaled problem tends to the optimal solution of the e↵ective problem as the characteristic pollutant concentration decreases

The Effect of Hyperthermia on Doxorubicin Therapy and Nanoparticle Penetration in Multicellular Ovarian Cancer Spheroids

The efficient treatment of cancer with chemotherapy is challenged by the limited penetration of drugs into the tumor. Nanoparticles (10 – 100 nanometers) have emerged as a logical choice to specifically deliver chemotherapeutics to tumors, however, their transport into the tumor is also impeded owing to their bigger size compared to free drug moieties. Currently, monolayer cell cultures, as models for drug testing, cannot recapitulate the structural and functional complexity of in-vivo tumors. Furthermore, strategies to improve drug distribution in tumor tissues are also required. In this study, we hypothesized that hyperthermia (43°C) will improve the distribution of silica nanoparticles in three-dimensional multicellular tumor spheroids. Tumor spheroids mimic the functional and histomorphological complexity of in-vivo avascular tumors and are therefore valuable tools to study drug distribution. Ovarian cancer (Skov3) and uterine sarcoma (MES-SA/Dx5) spheroids were generated using the liquid overlay method. The growth ratio and cytotoxicity assays showed that the application of adjuvant hyperthermia with Doxorubicin (DOX) did not yield higher cell killing compared to DOX therapy alone. These results illustrated the role of spheroids in resistance to heat and DOX. In order to study the cellular uptake kinetics of nanoparticles under hyperthermia conditions, the experimental measurements of silica nanoparticle uptake by cells were fitted using a novel inverse estimation method based on Bayesian estimation. This was coupled with advection reaction transport to model nanoparticle transport in spheroids. The model predicted an increase in Area Under the Curve (AUC) and penetration distance (W1/2) that were validated with in-vitro experiments in spheroids. Based on these observations, a novel multifunctional theranostic nanoparticle probe was created for generating highly localized hyperthermia by encapsulating a Near Infrared (NIR) dye, IR820 (for imaging and hyperthermia) and DOX in Organically modified silica nanoparticles (Ormosil). Pegylated Ormosil nanoparticles had an average diameter of 58.2±3.1 nm, zeta potential of -6.9 ± 0.1 mV and high colloidal stability in physiological buffers. Exposure of the IR820 within the nanoparticles to NIR laser led to the generation of hyperthermia as well as release of DOX which translated to higher cell killing in Skov3 cells, deeper penetration of DOX into spheroids and complete destruction of the spheroids. In-vivo bio-distribution studies showed higher fluorescence from organs and increased plasma elimination life of IR820 compared to free IR820. However, possible aggregation of particles on laser exposure and accumulation in lungs still remain a concern

Robust numerical methods to solve differential equations arising in cancer modeling

Philosophiae Doctor - PhDCancer is a complex disease that involves a sequence of gene-environment interactions in a progressive process that cannot occur without dysfunction in multiple systems. From a mathematical point of view, the sequence of gene-environment interactions often leads to mathematical models which are hard to solve analytically. Therefore, this thesis focuses on the design and implementation of reliable numerical methods for nonlinear, first order delay differential equations, second order non-linear time-dependent parabolic partial (integro) differential problems and optimal control problems arising in cancer modeling. The development of cancer modeling is necessitated by the lack of reliable numerical methods, to solve the models arising in the dynamics of this dreadful disease. Our focus is on chemotherapy, biological stoichometry, double infections, micro-environment, vascular and angiogenic signalling dynamics. Therefore, because the existing standard numerical methods fail to capture the solution due to the behaviors of the underlying dynamics. Analysis of the qualitative features of the models with mathematical tools gives clear qualitative descriptions of the dynamics of models which gives a deeper insight of the problems. Hence, enabling us to derive robust numerical methods to solve such models

Nanomedicines in cancer therapy : from long-circulating drug carriers to novel therapeutic concepts

Cancer is still a leading cause of death worldwide. Despite the progress in the molecular understanding of cancer diseases, there’s an urgent need in novel therapeutics and drug delivery strategies. Many novel anti-cancer compounds in early development are characterized by unfavorable physico-chemical properties and lack in drug-like properties. As a result, many of these compounds suffer from insufficient pharmacokinetic properties and show a high accumulation in off- target tissue that can induce dose-limiting side effects. Nanomedicines depict a promising strategy to optimize the pharmacokinetics of such compounds and to deliver them to their site of action: The cancer cell. The goal of this thesis was to develop nanoparticulate drug delivery platforms for passive and active drug targeting. In addition, a novel nanoparticle-based gene therapeutic for the treatment of liver cancer was evaluated. This thesis can be summarized in two main parts as follows: In a first part, a biocompatible and biodegradable polymer was used to prepare micelles for the delivery of small molecular anticancer drugs. These micelles were tested subsequently on in vitro and in vivo models. A highly reproducible protocol for the formulation of doxorubicin-loaded micelles was developed and micelles were characterized extensively for their physico-chemical properties. Cellular uptake of micelles was analyzed and their therapeutic potential was assessed in vitro on human cancer cells. To passively accumulate in solid tumors, nanoparticles need to be long-circulating and must remain in the blood circulation for hours. Therefore, the pharmacokinetic profile and biodistribution of doxorubicin-loaded micelles in rats was analyzed and compared to the gold standard of long-circulating nanoparticles: PEGylated liposomes. In a next step, a protocol for the preparation of so-called gold-nanohybrids was developed. Such nanohybrids are valuable tools to analyze nanoparticle-cell interactions and the intracellular fate of nanoparticles in detail. Further, such nanoparticles can be used as diagnostic tools. In a last step, micelles were functionalized with an antibody for targeted drug delivery. Cellular internalization of these micelles was analyzed using an array of methods. In a second part, a novel therapeutic strategy using the main effector protein of the rat parvovirus (H-1) for the treatment of hepatocellular carcinoma (HCC) was developed. H-1 parvovirus showed promising results in the preclinical setting and was consequently tested in a clinical trial in patients suffering form glioma. Despite this development, viral therapies may be linked with several issues. Therefore, the potential of the viral effector protein NS1 for the treatment of liver cancer was analyzed after non-viral gene delivery. In a first step, the gene-delivery efficiency and the therapeutic effect were analyzed in a panel of human liver cancer-derived cell lines. Various in vitro assays were used to study the NS1-induced cell death in detail. To show that this therapeutic approach is specific for cancer cells, the treatment was furthermore tested on healthy human liver cells. To identify cells that are susceptible for this therapeutic approach, a biomarker for the sensitivity to non-viral NS1 therapy was evaluated. Finally, safety of this therapy was analyzed in mice after single and multiple dosing