Abstract
Stereotactic neurosurgery requires a careful planning of cannulae paths to spare eloquent areas of the brain that, if damaged, will result in loss of essential neurological function such as sensory processing, linguistic ability, vision, or motor function. We present an approach based on modelling, simulation, and optimization to set up a computational assistant tool. Thereby, we focus on the modeling of the brain topology, where we construct ellipsoidal approximations of voxel clouds based on processed MRI data. The outcome is integrated in a path-planning problem either via constraints or by penalization terms in the objective function. The surgical planning problem with obstacle avoidance is solved for different types of stereotactic cannulae using numerical simulations. We illustrate our method with a case study using real MRI data.
Zusammenfassung
Die stereotaktische Neurochirurgie erfordert eine sorgfältige Planung der Kanülenwege, um eloquente Hirnareale, die bei einer Schädigung zum Verlust wesentlicher neurologischer Funktionen wie Wahrnehmung, Sprachfähigkeit, Sehkraft oder Motorik führen, zu umgehen. Wir stellen einen Ansatz vor, der auf Modellierung, Simulation und Optimierung basiert, um ein computergestütztes Assistenzwerkzeug zu erstellen. Dabei konzentrieren wir uns auf die Modellierung der Gehirntopologie, für die die Voxel-Wolken aus den verarbeiteten MRT-Daten durch Ellipsoide approximiert werden. Diese werden in ein Pfadplanungsproblem entweder über Nebenbedingungen oder durch Bestrafungsterme in der Zielfunktion integriert. Das chirurgische Planungsproblem mit Hindernisvermeidung wird für verschiedene Typen von stereotaktischen Kanülen mittels numerischer Simulationen gelöst. Wir illustrieren unsere Methode anhand einer Fallstudie mit realen MRT-Daten.
Funding source: Deutsche Forschungsgemeinschaft
Award Identifier / Grant number: 406141926
Funding statement: Karl Worthmann gratefully acknowledges funding by the German Research Foundation (DFG: Heisenberg professorship Optimization Based Control, project number 406141926).
About the authors
Annika Hackenberg received the Master’s degree in Technomathematik from the University of Bremen, Germany, in 2020. Since 2021, she has been a Ph.D. student and a research assistant in the working group Optimization and Optimal Control under supervision of Prof. Dr. Christof Büskens. Her research interests are in the area of optimization and optimal control.
Karl Worthmann received the Diploma degree in business mathematics and the Ph.D. degree in mathematics from the University of Bayreuth, Germany. 2014 he was appointed assistant professor for “Differential Equations” at Technische Universität Ilmenau (TU Ilmenau), Germany. 2019 he was promoted to full professor after receiving the Heisenberg-professorship “Optimization-based Control” by the German Research Foundation in 2018. Karl Worthmann’s current research interests include systems and control theory with a particular focus on nonlinear model predictive control, stability analysis, and sampled-data systems. He was recipient of the Ph.D. Award from the City of Bayreuth, Germany, and stipend of the German National Academic Foundation. 2013 he has been appointed Junior Fellow of the Society of Applied Mathematics and Mechanics (GAMM), where he served as speaker in 2014 and 2015. Currently, Karl Worthmann is chairman “Mathematical Systems Theory” of the interdisciplinary GAMM activity group “Dynamics and Control Theory”.
Torben Pätz received the Diploma degree in mathematics from the University of Bremen, Germany, in 2009. In 2012 he obtained a Ph.D. in mathematics from Jacobs University Bremen, Germany, with a thesis on the segmentation of stochastic images with stochastic partial differential equations. He is currently Principal Scientist at the Fraunhofer Institutefor Digital Medicine MEVIS, Bremen. His research interests include software support for image-guided interventions and modeling and simulation of biomedical processes.
Dörthe Keiner is a consultant neurosurgeon at the Department of Neurosurgery, Saarland University Medical Center and Saarland University of Medicine since 2014. Her research interests include minimally-invasive and neuroendoscopic surgical techniques and stereotactic and functional neurosurgery.
Joachim Oertel is full Professor of Neurosurgery and Managing Professor of Neurosurgery, Faculty of Medicine at Saarland University. Since 2010 he is the director and head of the Department of Neurosurgery, Saarland University Hospital. Dr. Oertel was trained in Munich, Boston, Greifswald, Hannover and Mainz. His personal major expertise and research interest are the treatment of skull base tumors, vascular malformation and spine diseases. He has a particular expertise in minimally invasive endoscopic techniques.
Kathrin Flaßkamp is full professor for Systems Modeling and Simulation in Systems Engineering at Saarland University since 2020. She received a Diploma degree in Technomathematik (2008) and a PhD in applied mathematics (2013), both from the University of Paderborn. Her research interests include structure-preserving modeling and simulation, structure-exploiting numerical methods for optimal control and system identification, and applications.
Acknowledgment
We gratefully acknowledge Wolfgang Reith, Klinik für Diagnostische und Interventionelle Neuroradiologie des UKS, for providing the MRI data set. Annika Hackenberg and Kathrin Flaßkamp gratefully acknowledge the support from Matthias Knauer (University of Bremen) on the application of TransWORHP.
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