Deep Brain Stimulation (DBS) consists of sending mild electric stimuli to the brain via a chronically implanted lead. The therapy is used to alleviate the symptoms of different neurological diseases, such as Parkinson's Disease. However, its underlying biological mechanism is currently unknown. DBS patients undergo a lengthy trial-and-error procedure in order to tune the stimuli so that the treatment achieves maximal therapeutic benefits while limiting side effects that are often present with large stimulation values.

The present licentiate thesis deals with mathematical modeling for DBS, extending it towards optimization. Mathematical modeling is motivated by the difficulty of obtaining in vivo measurements from the brain, especially in humans. It is expected to facilitate the optimization of the stimuli delivered to the brain and be instrumental in evaluating the performance of novel lead designs. Both topics are discussed in this thesis.

First, an analysis of numerical accuracy is presented in order to verify the DBS models utilized in this study. Then a performance comparison between a state-of-the-art lead and a novel field-steering lead using clinical settings is provided. Afterwards, optimization schemes using intersection of volumes and electric field control are described, together with some simplification tools, in order to speed up the computations involved in the modeling.

Deep Brain Stimulation is an established therapy that consists of sending mild electrical pulses to the brain via a surgically implanted electrode. It is used to treat the symptoms of neurodegenerative diseases such as Parkinson's Disease and Essential Tremor. The stimulation effect is however patient-specific and difficult to predict. Further, impedance measurements indicate changes in the medium around the lead over time that are generally challenging to account for. Although mathematical models to characterize the extent of the stimulation have been developed in recent years, it is imperative that the model parameters are realistic enough to produce correct results. This study aims at developing a system identification approach to capture the changes in the medium around the lead by measuring the potential on non-active contacts during stimulation. Due to infeasibility of assigning properties to the medium around the lead in vivo, synthetic data are used for analysis instead. Results suggest that a transfer function with one zero and one pole that corresponds to a circuit composed of two RC contours in cascade is sufficient to describe the measurements in silico. Furthermore, the identified parameters show an injective relation to the properties of the medium. Continuous Least Squares and Laguerre domain identification are utilized for obtaining parameter estimates.

Deep Brain Stimulation (DBS) is an established treatment, in e.g. Parkinson's Disease, whose underlying biological mechanisms are unknown. In DBS, electrical stimulation is delivered through electrodes surgically implanted into certain regions of the brain of the patient. Mathematical models aiming at a better understanding of DBS and optimization of its therapeutical effect through the simulation of the electrical field propagating in the brain tissue have been developed in the past decade. The contribution of the present study is twofold: First, an analytical approximation of the electric field produced by an emitting contact is suggested and compared to the numerical solution given by a Finite Element Method (FEM) solver. Second, the optimal stimulation settings are evaluated by fitting the field distribution to a target one to control the spread of the stimulation. Optimization results are compared to those of a geometric approach, maximizing the intersection between the target and the activated volume in the brain tissue and reducing the stimulated area beyond said target. Both methods exhibit similar performance with respect to the optimal stimuli, with the electric field control approach being faster and more versatile.