Optimisation of cancer drug treatments using cell population dynamics

International audienceCancer is primarily a disease of the physiological control on cell population proliferation. Tissue proliferation relies on the cell division cycle: one cell becomes two after a sequence of molecular events that are physiologically controlled at each step of the cycle at so-called checkpoints, in particular at transitions between phases of the cycle [105]. Tissue proliferation is the main physiological process occurring in development and later in maintaining the permanence of the organism in adults, at that late stage mainly in fast renewing tissues such as bone marrow, gut and skin

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

Combination of Chemotherapy and Antiangiogenic Therapies: A Mathematical Modelling Approach

A brief introduction to cancer biology and treatment is presented with a focus on current clinical advances in the delivery of chemotherapy and antiangiogenic therapies. Mathematical oncology is then surveyed with summaries of various models of tumor growth, tumor angiogenesis and other relevant biological entities such as angiogenic growth factors. Both strictly time-dependent ordinary differential equation (ODE)-based and spatial partial differential equation (PDE)-based models are considered. These biological models are first developed into an ODE model where various treatment options can be compared including different combinations of drugs and dosage schedules. This model gives way to a PDE model that includes the spatially heterogeneous blood vessel distribution found in tumors, as well as angiogenic growth factor imbalances. This model is similarly analyzed and implications are summarized. Finally, including the effects of interstitial fluid pressure into an angiogenic activity model is performed. This model displays the importance of factor convection on the angiogenic behaviour of tumours

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

Molecular Imaging

The present book gives an exceptional overview of molecular imaging. Practical approach represents the red thread through the whole book, covering at the same time detailed background information that goes very deep into molecular as well as cellular level. Ideas how molecular imaging will develop in the near future present a special delicacy. This should be of special interest as the contributors are members of leading research groups from all over the world

Molecular and Cellular Mechanisms of Preeclampsia

This Special Issue on the “Molecular and Cellular Mechanisms of Preeclampsia” belongs to the section “Molecular Pathology, Diagnostics, and Therapeutics” of the International Journal of Molecular Sciences. It was a very successful Special Issue as it contains 20 published papers, including one editorial, nine original research papers, and ten reviews on the topic. The original publications cover a wide spectrum of topics, including alterations and involvement of specific factors during preeclampsia, new non-invasive technologies to identify changes, new treatment options, animal models, gender aspects, and effects of the pregnancy pathology later in life. The review publications again cover a wide spectrum of topics, including factors and pathways involved in preeclampsia, effects on the maternal vascular and immune systems, effects on the placenta and the trophoblast, epigenetic changes, new preventive strategies, and new views on the current hypotheses on preeclampsia. Taken together, this Special Issue gives a fantastic overview on a broad spectrum of topics, all of which are important to identify the real etiology of preeclampsia and to finally develop real treatment options

Optimization-based finite element methods for evolving interfaces

This thesis is concerned with the development of new approaches to redistancing and conservation of mass in finite element methods for the level set transport equation. The first proposed method is a PDE- and optimization-based redistancing scheme. In contrast to many other PDE-based redistancing techniques, the variational formulation derived from the minimization problem is elliptic and can be solved efficiently using a simple fixed-point iteration method. Artificial displacements are effectively prevented by introducing a penalty term. The objective functional can easily be extended so as to satisfy further geometric properties. The second redistancing method is based on an optimal control problem. The objective functional is defined in terms of a suitable potential function and aims at minimizing the residual of the Eikonal equation under the constraint of an augmented level set equation. As an inherent property of this approach, the interface cannot be displaced on a continuous level and numerical instabilities are prevented. The third numerical method under investigation is an optimal control approach designed to enforce conservation of mass. A numerical solution to the level set equation is corrected so as to satisfy a conservation law for the corresponding Heaviside function. Two different control approaches are investigated. The potential of the proposed methods is illustrated by a wide range of numerical examples and by numerical studies for the well-known rising bubble benchmark