What is it about?
Optimal control theory applied to BCG immunotherapy for developing a treatment against superficial bladder cancer.
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Why is it important?
The main objective of this work is to show the importance of taking a quadratic "nonlinear form" control in place of a linear control in the integrand of an objective function we aim to minimize. Mathematically speaking, quadratic costs in such optimization problems where a control function is involved, have a direct relationship with the classical form of linear-quadratic optimal control problems. However, when it comes to biology or medicine, it seems not clear to many scholars we met, how could that form of cost be interpreted. In this paper, we discussed from different sides, the mathematical and biological meanings of such choices when they are subjects to optimization problems related to cancer therapies.
Perspectives
A quadratic control or linear control, both verify optimality conditions related to a chosen criterion of maximization (resp. minimization), and these two forms of control in the objective function, both allow us to reach our main goals which are summarized in the minimization of the dose of BCG for lesser side-effects. On the other hand, we observe that the final numerical results associated with the choice of the quadratic cost of treatment, give better therapeutic recommendations. Read the paper for more information about that conclusion.
Ilias Elmouki
Read the Original
This page is a summary of: Quadratic and linear controls developing an optimal treatment for the use of BCG immunotherapy in superficial bladder cancer, Optimal Control Applications and Methods, February 2015, Wiley,
DOI: 10.1002/oca.2161.
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Resources
BCG immunotherapy optimization on an isoperimetric optimal control problem for the treatment of superficial bladder cancer
The Bacillus Calmette-Guerin immunotherapy (BCG) is a clinical procedure used as the treatment by success for superficial bladder cancer. However, the toxicity of BCG, its maintenance schedule, optimal amount of dosage sufficient for the destruction of cancerous cells and its efficiency remain all unclear due to the lack of published data. We serve in this paper the optimization of BCG treatment by seeking the optimal dose we inject in the bladder during the intravesical therapy for a hypothetical patient. We outline the different steps of resolution of an optimal control problem resolved using the Pontryagin’s maximum principle. The theoretical approach leads us to use the forward-backward sweep method and the secant-method as the appropriate numerical technique to solve a two-point boundary value problem with an isoperimetric constraint on the control process function representing the optimal concentration suggested to use in each instillation of BCG.
Impulsive control dosing BCG immunotherapy for non-muscle invasive bladder cancer
This article provides a solution for a control system derived from a mathematical model of four ordinary differential equations that describe the dynamics between the bacillus Calmette-Guérin (BCG) vaccine concentration, immune-system and tumor cells in non-muscle invasive bladder cancer. Generally, in cancer treatments, such as immunotherapies, the problems of administration procedures, do not take place through continuous injections of clinical agents in the diseased organs, but are often referred to therapeutic optimization problems with pulse vaccinations. For our study, we discuss the advantages of BCG immunotherapy when it is administered as a sequence of pulsed instillations in the bladder. We include numerical simulations based on the variational equation method resolved using a fourth-order iterative Runge–Kutta scheme combined with an optimization technique that computes the gradient of the objective function to find the optimal vaccination times and BCG dosage amounts.
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