Abstract
Laser photocoagulation is a widely used treatment for a variety of retinal diseases. Temperature-controlled irradiation is a promising approach to enable uniform heating, reduce the risks of over- or undertreatment, and unburden the ophthalmologists from a time consuming manual power titration. In this paper, an approach is proposed for the development of models with different levels of detail, which serve as a basis for improved, more accurate observer and control designs. To this end, we employ a heat diffusion model and propose a suitable discretization and subsequent model reduction procedures. Since the absorption of the laser light can vary strongly at each irradiation site, a method for identifying the absorption coefficient is presented. To identify a parameter in a reduced order model, an optimal interpolatory projection method for parametric systems is used. In order to provide an online identification of the absorption coefficient, we prove and exploit monotonicity of the parameter influence.
Zusammenfassung
Die Laser-Photokoagulation ist eine weit verbreitete Behandlungsmethode verschiedener Netzhauterkrankungen. Die temperaturgeregelte Bestrahlung ist ein vielversprechender Ansatz, um eine gleichmäßige Erwärmung zu ermöglichen, die Risiken einer Über- oder Unterbehandlung zu verringern und die Augenärzte von einer zeitintensiven, manuellen Laserleistungstitration zu entlasten. In diesem Artikel wird ein alternativer Ansatz für die Entwicklung von Modellen mit unterschiedlichem Detaillierungsgrad vorgeschlagen, die als Grundlage für verbesserte, genauere Beobachter- und Regelungsdesigns dienen. Zu diesem Zweck verwenden wir ein Wärmeleitungsmodell und schlagen ein geeignetes Verfahren zur Diskretisierung und anschließender Modellreduktion vor. Da die Absorption des Laserlichts abhängig von jedem Bestrahlungsort stark variieren kann, wird ein Verfahren zur Identifikation des Absorptionskoeffizienten vorgestellt. Um Parameter in einem ordnungsreduzierten Modell zu identifizieren, wird eine optimale interpolatorische Projektionsmethode für parametrische Systeme verwendet. Die Monotonie des Parametereinflusses wird bewiesen und ausgenutzt, um den Absorptionskoeffizienten online zu identifizieren.
Funding source: Deutsche Forschungsgemeinschaft
Award Identifier / Grant number: MU3929/3-1
Award Identifier / Grant number: WO2056/7-1
Award Identifier / Grant number: BR 1349/6-1
Award Identifier / Grant number: WO2056/1-1
Award Identifier / Grant number: WO2056/6-1
Funding statement: The collaborative project “Temperature controlled retinal laser treatment” is funded by the German Research Foundation (DFG) under the project number 430154635 (MU3929/3-1, WO2056/7-1, BR 1349/6-1). H. Gernandt and K. Worthmann gratefully acknowledge the support of the German Research Foundation (DFG) via grants WO2056/1-1, WO2056/6-1. H. S. Abbas is funded by the German Research Foundation (DFG), project number 419290163.
About the authors
Viktoria Kleyman received a bachelor’s degree in nanotechnology and a master’s degree in mechatronics from Leibniz University Hannover, Germany. Since 2018 she is a Ph.D. candidate at the Institute of Automatic Control, Leibniz University Hannover. Her research interests include identification, estimation, model predictive control and robotics.
Hannes Gernandt studied mathematics at TU Ilmenau, where he is currently completing his doctoral thesis. His research focuses on differential-algebraic equations and optimization of electrical circuits.
Karl Worthmann received the Diploma degree in business mathematics and the Ph.D. degree in mathematics from the University of Bayreuth, Germany. At Technische Universität Ilmenau, Germany, he became full professor for “Optimization-based Control” in 2019 after being appointed assistant professor for “Differential Equations” in 2014. His current research interests include systems and control theory with a particular focus on nonlinear model predictive control and sampled-data systems. He was the recipient of the Ph.D. Award from the City of Bayreuth, Germany, and the German National Academic Foundation Scholarship, Germany. He has been appointed Junior Fellow of the Society of Applied Mathematics and Mechanics (GAMM) in 2013 where he served as speaker in 2014 and 2015. Currently, Karl Worthmann is one of three chairmen of the GAMM activity group “Dynamics and Control Theory”. Moreover, he received a Heisenberg-professorship by the German Research Foundation (DFG) in 2018.
Hossam Seddik Abbas completed his B.Sc. and M.Sc. degrees at Assiut University in 1997 and 2001, respectively, and his Ph.D. at Hamburg University of Technology, Germany, in 2010. He is currently a Senior Scientist with the Institute for Electrical Engineering in Medicine, Universität zu Lübeck, Germany. In 2019 he worked in the Medical Laser Centre Lübeck, Germany, as a Senior Scientist. During the academic years of 2017 to 2019, he was an Alexander-von-Humboldt Fellow (Georg Forster Research Fellowship for Experienced Researchers) in Institute for Electrical Engineering in Medicine, Universität zu Lübeck, Germany. He was a research fellow in Hamburg University of Technology, and Eindhoven University of Technology, The Netherlands, in 2011 and 2013, respectively. He is also affiliated with the Electrical Engineering Department, Faculty of Engineering, Assiut University, Egypt since 2010. His research interests lie in the area of control systems engineering and its applications. His focus includes linear parameter varying systems, optimal and robust control, model predicative control, distributed control and system identification.
Ralf Brinkmann studied physics at the University of Hannover, Germany, with a focus on quantum optics and lasers. After a 5-year industrial interim period, he joined the Medical Laser Center in Lübeck, Germany, in 1993, and received his Ph.D. from the University of Lübeck. Since 2005 he has been holding a permanent position as a faculty member at the University’s Institute of Biomedical Optics, focusing on biophotonics and laser applications in medicine. His strong interest in technology transfer of basic research results to industry is expressed by leading the Medical Laser Center Lübeck, a non-profit R & D company for optics and biophotonics on the BioMedTec Science campus Lübeck.
Matthias A. Müller received a Diploma degree in Engineering Cybernetics from the University of Stuttgart, Germany, and an M.S. in Electrical and Computer Engineering from the University of Illinois at Urbana-Champaign, US, both in 2009. In 2014, he obtained a Ph.D. in Mechanical Engineering, also from the University of Stuttgart, Germany, for which he received the 2015 European Ph.D. award on control for complex and heterogeneous systems. Since 2019, he is director of the Institute of Automatic Control and full professor at the Leibniz University Hannover, Germany. In 2020, he was recipient of the inaugural Brockett-Willems Outstanding Paper Award for the best paper published in Systems & Control Letters in the period 2014–2018. His research interests include nonlinear control and estimation, model predictive control, and data-/learning-based control, with application in different fields including biomedical engineering.
Acknowledgment
We would like to thank the anonymous referees for their careful reading and their valuable suggestions.
In this section, we extend Theorem 3.1 to several independent parameters in the matrices B and C. To this end, we replace the assumption (12) by the more general assumption
where the sums are taken over all multi-indices
Using this notation enables us to generalize Theorem 3.1 as follows.
Theorem 1.
Given
Proof.
Observe that the proof is essentially the same as the proof of Theorem 3.1. The only adjustments are described below. First, one has to replace the matrices on the right hand side of (19) by the matrices that appear as integrands in the definition of
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