Mathematical modeling for optimal control of breast cancer
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Date
2019
Authors
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Journal ISSN
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Publisher
University of Zululand
Abstract
Breast cancer, which often occurs in the inner lining of milk ducts, is the deadliest and
most common form of invasive cancer among females according to a 2017 report of the
World Health Organization. The purpose of this study was to develop a four compartmental
mathematical model using a system of nonlinear Ordinary Di erential Equations
(ODEs) which investigates the impact of anti-cancer drugs, ketogenic-diets and immune
boosters on the dynamics of breast cancer. The study focused on the dynamical interaction
of normal and tumor cells as well as the invasion of tumor cells during the metastasis
stage of breast cancer. The systems of ODEs were analytically solved for the equilibria.
Using the next generation matrix method, a threshold quantity called the treatment in-
duced invasion reproduction number (R
i ) was computed. Center manifold theory was used
to investigate the possibility of the bifurcation analysis of R
i being greater than unity.
Using a suitable Lyapunov functions, the global stability of the tumor-free equilibrium
was achieved in conjuction with LaSalle's invariance principle. Uncertainty and sensitivity
analyses were performed on R
i using Latin Hypercube Sampling (LHS) and Partial Rank
Correlation Coe cient (PRCC). R
i was used as the response function while investigating
the most signi cant parameters (such as: 1, 2, 1, d, and 1 ) that a ects disease
progression and cell invasion. Optimal control theory was applied using the Pontryagins'
Maximum Principle to investigate optimal strategies for controlling and eliminating tumor
cells using time dependent controls such as u1(t) (anti-cancer drugs) and u2(t) (ketogenic
diets). Numerical simulation results using a set of parameter values were provided to
validate the analytical results. It was found that the tumor-free equilibrium points for
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breast cancer was locally asymptotically stable when the associated invasion reproduction
number was less than unity and that it was otherwise unstable. The tumor-free equilibrium
was found to be globally asymptotically stable if (Ri) < 1 . Sensitivity analysis
showed that the natural death rate of normal cells has the most positive sensitivity index.
However, increasing the death rate as a control measure is unreasonable biologically. The
level of ketogenic diet rate was found to be most negatively sensitive to Ri. Therefore, the
formulated model showed that reduction of the invasion reproduction number (R
i ) below
unity can be achieved by maintaining the level of ketogenic diet and by reducing tumor
progression rate. It was shown from this study that the breast cancer model exhibited
backward bifurcation with bifurcation parameter 1 which implies that the reduction R
i
below unity alone is not su cient to eradicate tumor cells from the body system while in
the case of forward bifurcation, the reduction of R
i above unity is su cient to eradicate
tumor cells from the body system . The incremental cost-e ectiveness analysis of control
strategies adapted in treating breast cancer has shown that the integration of ketogenic
diet and anti-cancer drugs as intervention strategy is the most cost-e ective in ghting
tumor cells.
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Description
Keywords
Mathematics, Breast cancer