In standard chemotherapy protocols, drugs are given at maximum tolerated doses
(MTD) with rest periods in between. In this paper, we briefly discuss the
rationale behind this therapy approach and, using as example
multi-drug 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.
We analyze non cell-cycle specific mathematical models for drug resistance in
cancer chemotherapy. In each model developing drug resistance is inevitable
and the issue is how to prolong its onset. Distinguishing between sensitive
and resistant cells we consider a model which includes interactions of two
killing agents which generate separate resistant populations. We formulate an
associated optimal control problem for chemotherapy and analyze the
qualitative structure of corresponding optimal controls.
Mathematical models for cancer chemotherapy as optimal control problems are considered. Results on scheduling optimal therapies when the controls represent the effectiveness of chemotherapeutic agents, or, equivalently, when the simplifying assumption is made that drugs act instantaneously, are compared with more realistic models that include pharmacokinetic (PK) equations modelling the drug's plasma concentration and various pharmacodynamic (PD) models for the effect the concentrations have on cells.
The editors of this special issue of Mathematical Biosciences and Engineering
were the main organizers for the First International Workshop
Mathematical Methods in System Biology,
that took place on January 4-7, 2010 at Tel Aviv University in Tel
Aviv, Israel. The workshop, initially planned as a small meeting,
was an overwhelming success with 170 participants from Israel, the
US, Canada and Europe. It included about 100 presentations: invited
talks, special sessions dedicated to application of mathematical
tools to various areas in biology and poster sessions which gave
graduate students and young scientists a stage to present their
research. We managed to attract a good mix of mathematicians working
on biological and medical applications with biologists and medical
doctors interested to present their challenging problems and to find
mathematical tools for their solution. We would like to take the
opportunity to thank the Office of International Science and
Engineering of the National Science Foundation and the Society for
Mathematical Biology for their support in bringing US participants
to this event. Thanks are also due to the Vice-President for
Research and Development of Tel Aviv University, the Faculty of
Exact Sciences and its Dean Prof. Haim Wolfson, and the School of
Mathematical Sciences for their help with covering local expenses.
Special thanks are coming to the supporting team of students,
postdocs and administrative staff for their incredible contribution
to the success of the workshop.
For more information please click the “Full Text” above.
The editors of this Special Issue of Mathematical Biosciences and
Engineering were the organizers for the Third International
Workshop "Mathematical Methods in
System Biology" that took place on June 15-18, 2015 at the University College Dublin in Ireland. As stated in the workshop goals, we managed to attract a good mix of mathematicians and statisticians working on biological and medical applications with biologists and clinicians interested in presenting
their challenging problems and looking to find mathematical and statistical tools for their solutions.
For more information please click the “Full Text” above.
Anti-angiogenesis is a novel cancer treatment that targets the
vasculature of a growing tumor. In this paper a metasystem is
formulated and analyzed that describes the dynamics of the primary
tumor volume and its vascular support under anti-angiogenic
treatment. The system is based on a biologically validated model by
Hahnfeldt et al. and encompasses several versions of this model
considered in the literature. The problem how to schedule an a
priori given amount of angiogenic inhibitors in order to achieve the
maximum tumor reduction possible is formulated as an optimal control
problem with the dosage of inhibitors playing the role of the
control. It is investigated how properties of the functions defining
the growth of the tumor and the vasculature in the general system
affect the qualitative structure of the solution of the problem. In
particular, the presence and optimality of singular controls is
determined for various special cases. If optimal, singular arcs are
the central part of a regular synthesis of optimal trajectories
providing a full solution to the problem. Two specific examples of a
regular synthesis including optimal singular arcs are given.
In this paper, we use the method of characteristics to study
singularities in the flow of a parameterized family of extremals for
an optimal control problem. By means of the Lyapunov--Schmidt
reduction a characterization of fold and cusp points
is given. Examples illustrate the local behaviors of the flow near
these singular points. Singularities of fold type correspond to the
typical conjugate points as they arise for the classical problem of
minimum surfaces of revolution in the calculus of variations and
local optimality of trajectories ceases at fold points. Simple cusp
points, on the other hand, generate a cut-locus that limits the
optimality of close-by trajectories globally to times prior to the
This paper analyzes a mathematical model for the growth of bone
marrow cells under cell-cycle-specic cancer chemotherapy originally proposed
by Fister and Panetta . The model is formulated as an optimal control
problem with control representing the drug dosage (respectively its effect)
and objective of Bolza type depending on the control linearly, a so-called $L^1$-objective. We apply the Maximum Principle, followed by high-order necessary
conditions for optimality of singular arcs and give sufficient conditions for optimality based on the method of characteristics. Singular controls are eliminated
as candidates for optimality, and easily veriable conditions for strong local
optimality of bang-bang controls are formulated in the form of transversality
conditions at switching surfaces. Numerical simulations are given.
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.
Oncolytic viruses are genetically altered replication-competent vi-ruses which upon death of a cancer cell produce many new viruses that then infect neighboring tumor cells. A mathematical model for virotherapy of glioma is analyzed as a dynamical system for the case of constant viral infusions and TNF-α inhibitors. Aside from a tumor free equilibrium point, the system also has positive equilibrium point solutions. We investigate the number of equilibrium point solutions depending on the burst number, i.e., depending on the number of new viruses that are released from a dead cancer cell and then infect neighboring tumor cells. After a transcritical bifurcation with a positive equilibrium point solution, the tumor free equilibrium point becomes asymptotically stable and if the average viral load in the system lies above a threshold value related to the transcritical bifurcation parameter, the tumor size shrinks to zero exponentially. Other bifurcation events such as saddle-node and Hopf bifurcations are explored numerically.