Motivation: Neurodegenerative diseases of the central nervous system (CNS) affect more than millions of people worldwide. Effective treatment of a vast majority of such disorders often involves large molecular weight drugs such as proteins. The blood brain barrier (BBB), a network of endothelial cells with selective permeability prevents large molecules entering the brain. One of the effective ways of circumventing this barrier is by deploying invasive techniques in which the drugs are injected directly using a catheter. Despite recent advances in designing drug infusion policies based on empirical data, challenges posed by invasive techniques remain open. This presentation proposes a rigorous mathematical optimization approach for the systematic design of drug infusion policies and targeted macromolecule delivery to the human brain by closing the gap between experimental data and the mechanisms underlying it.

Methodology: In this poster, we present a novel transport and kinetic inversion problem (TKIP) to rigorously determine unknown transport and reaction properties by minimizing the squared error between the experimental imaging data and the mathematical model. The rigorous TKIP methodology uses patient specific brain geometry with partial differential equations set as constraints. A first principles transport model quantifies the biotransport of macromolecules as a function of the effective diffusion and hydraulic conductivity tensors, the convective flow field and associated metabolic reaction rates from in-vivo imaging data obtained as a function of space and time, and the calculated concentration field.

Determination of directional diffusivities of the drug diffusion tensor: The brain composed of gray and white matter. The white matter contains bundles of axons with myelin sheaths, thus causing the transport of substances to be anisotropic and heterogeneous. We will determine using the TKIP approach, unknown directional drug diffusivities of the effective diffusion tensor in highly anisotropic brain regions such as corpus callosum, coronal radiata and internal capsule given the experimentally observed concentration field from in-vivo subject-specific diffusion tensor imaging (DTI).

We also present rigorous mathematical programming techniques to determine optimal catheter location and placement for chemotherapy of tumors occurring in the midbrain.

Optimal catheter placement problem: Which injection site is best for maximizing the drug distribution in the target area without causing toxic effects or excessive tissue strain? Rigorous determination of optimal catheter placement is a challenging distributed optimization problem. We use mathematical programming techniques to determine optimal parameters of invasive drug therapy. Mathematically, optimal catheter-placement can be formulated as a distributed optimization problem with partial differential equation constraints. . This optimal design problem is expected to impact the design of chemo-therapy, drug targeting for Parkinson's or Alzheimer's diseases.

Conclusion: The methodology of the design of drug delivery applications integrates interdisciplinary expertise from medical imaging and diagnosis, systems biology and engineering optimization. The approach will allow physicians and scientists to design and optimize invasive drug delivery therapies in a systematic fashion, thus reducing the need for trial-and-error animal experiments.

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