Optimal control of drug delivery to brain tumors for a distributed parameters model

Chakrabarty, Siddhartha P.; Hanson, Floyd B.

doi:10.1109/acc.2005.1470086

Cited by 19 publications

(24 citation statements)

References 9 publications

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“…In [2], [3] the Crank-Nicolson implicit method combined with predictor-corrector methods are used to study the problem numerically. However, using finite difference methods like Crank-Nicolson implicit method alone and alternating directions implicit method have serious drawbacks.…”

Section: Galerkin Finite Element Methodsmentioning

confidence: 99%

“…In the distributed parameter control model of Chakrabarty and Hanson [2], [3], the tumor cell and normal cell density and the drug concentration at any position vector x and time t ∈ [0, t f ], in the interior Ω of the domain, denoted by n 1 (x, t), n 2 (x, t) and c(x, t) respectively, are taken as the state variables. Defining the global state vector as…”

Section: Mathematical Modelmentioning

confidence: 99%

“…Also the functional dependence of the higher derivatives in time and state of the extended state perturbations must be eliminated on lower order terms by one or two integrations by parts, (using Green's formula [7]). Rearranging inner products and collecting terms, the extended state equations yields an intermediate form (see [1], [2], [3] for details). The resulting coefficients of independent variations are set to zero to obtain the state, control and co-state equations.…”

Section: Mathematical Modelmentioning

confidence: 99%

“…The two other co-state vectors should not be needed, but satisfy rather simple equations [2], [3]. However, in general, there are 6 vector PDEs in 3 dimensional space, leading to a very high computational complexity when converted to numerical methods.…”

Section: Co-state Equationsmentioning

confidence: 99%

“…The tumor-drug model here is from Chakrabarty and Hanson [2], [3], which was influenced by Gatenby et al [5] and Mansuri [9]. Murray's fine text [10] is a good source of different types of growth mechanisms.…”

Section: Introductionmentioning

confidence: 99%

See 4 more Smart Citations

Cancer Drug Delivery in Three Dimensions For a Distributed Parameter Control Model Using Finite Elements

Chakrabarty

Hanson

2006

Proceedings of the 45th IEEE Conference on Decision and Control

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Abstract-Optimal distributed parameter control and a Galerkin finite element method are used to develop procedures for an ideal model of optimal drug delivery to brain tumors. The mathematical model comprises of a system of three coupled reaction diffusion models, involving the density of tumor cells, normal tissue and also the drug concentration. An optimal control problem is formulated with the goal of minimizing the tumor cell density and reducing the side effects of the drug. A distributed parameter method based on a simple application of variational calculus is used on a pseudo-Hamiltonian, which is then used to obtain a coupled system of forward state equations and backward co-state equations. The Galerkin finite element method is used to realistically represent the brain structure. Finally, an ideal three dimensional test case is considered and partitioned into a set of brick finite elements in spherical coordinates, with tri-linear basis functions. Non-uniqueness of nodes in spherical coordinates is removed by combining like nodes, such as at the origin, at the poles and at the polar angle discontinuity. The Galerkin ODEs are solved by a combination of Crank-Nicolson and predictor-corrector methods.

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Section: Galerkin Finite Element Methodsmentioning

confidence: 99%

Section: Mathematical Modelmentioning

confidence: 99%