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Towards holistic mathematical models of human fluid physiology and associated diseases

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Abstract In this talk, Prof Toro will present some recent advances in the construction of mathematical models of the global circulation in the entire human body, coupled to a refined description of the cerebrospinal fluid dynamics in the craniospinal cavity [1]. The present model represents a substantially revised version of the original Müller-Toro mathematical model [2]. The geometric multi-scale mathematical model presented here includes one-dimensional non-linear systems for 323 major blood vessels (arteries and veins) and 85 zero-dimensional, differential-algebraic systems for the remaining components of the complete model. Highlights include the myogenic mechanism of cerebral blood regulation; refined vasculature for the inner ear; refined vasculature for the brainstem and cerebellum, and viscoelastic, rather than purely elastic, models for all blood vessels. The derived one-dimensional parabolic systems for all major vessels are approximated by hyperbolic systems with stiff source terms following a relaxation approach [3]. Sophisticated numerical methodology is deployed to solve the equations [3]. The full model is validated through comparison of computational results against published data and bespoke MRI measurements. Prof Toro will present results for a medical application that concerns transverse sinus stenoses and their relation to Idiopathic Intracranial Hypertension [4].

References [1] E F Toro, M Celant, Q Zhang, C Contarino, N Agarwal, A A Linninger, L O M üller. Cerebrospinal fluid dynamics coupled to the global circulation in holistic setting: mathematical models, numerical methods and applications. International Journal for Numerical Methods in Biomedical Engineering. May 2021. [2] Lucas O. Müller and Eleuterio F. Toro. A global multi-scale model for the human circulation with emphasis on the venous system. International Journal for Numerical Methods in Biomedical Engineering. Vol. 30, Issue 7, pp: 681-725, July 2014. [3] Eleuterio F. Toro and Gino I. Montecinos. Advection-diffusion-reaction equations: hyperbolisation and high-order ADER discretizations. SIAM Journal of Scientific Computing. Vol. 36, No. 5, pp: A2423 -A2457, 2014. [4] N. Agarwal, C. Contarino, L. Bertazzi, N. Limbucci, G. Rossi and Eleuterio F. Toro. Intracranial fluid dynamics changes in idiopathic intracranial hypertension: pre and post lumbar puncture, medical and stent placement. Current Neurovascular Research. Vol. 15, pp: 1-9, 2018.

About the speaker

Eleuterio Toro is currently an Emeritus Professor of Mathematics in the University of Trento, Italy, and formerly a full Professor (Professore Chiara Fama) of Numerical Analysis at the Laboratory of Applied Mathematics, in the same university. He holds an Honours BSc in pure mathematics (University of Warwick, UK, 1977), an MSc in functional analysis and differential equations (University of Dundee, UK, 1978) and a PhD in computational mathematics (Teesside University, UK 1982 ). Previously held academic appointments include several institutions, such as the University of Leeds (UK), Cranfield University (UK), Manchester Metropolitan University (UK) and Cambridge University (UK). Professor Toro has received several honours and distinctions, which include the honorary title OBE from Queen Elizabeth II (UK, 2000); Life Fellow, Claire Hall, University of Cambridge (UK, 2003); Fellow of the Indian Society for Shock Wave Research (Bangalore, 2005); Doctor Honoris Causa (Universidad de Santiago de Chile, 2008); William Penney Fellow, University of Cambridge (UK, 2010); Doctor Honoris Causa (Universidad de la Frontera, Chile, 2012) and Honorary Professor, Moscow Institute of Physics and Technology, Russia. Professor Toro’s research has for many years focused on the design of computational methods for solving partial differential equations, with particular emphasis on hyperbolic balance laws and applications to industrial, aerospace, and environmental problems. Contribution highlights include: WAF (1989), HLLC (1992), FORCE (1996), ADER (2001), MUSTA (2004), TV flux splitting (2012).

For the last ten years Professor Toro’s research has been focused on the mathematical modelling of the fluid physiology of neurological diseases, with particular attention the role of anatomical malformations in disturbing the dynamical interaction of fluid compartments in the central nervous system. The approach taken is holistic, including all major fluid compartments (arterial blood, venous blood, microvasculature, interstitial fluid, cerebrospinal fluid, the brain parenchyma) in the entire human body. In the last decade or so Prof. Toro has been an invited keynote speaker in more than 100 international scientific events. Professor Toro has held many visiting appointments round the world, which include several European countries, Japan, China and USA . He is author of more than 300 research works, including books, articles and technical reports, with more than 26000 citations on GoogleScholar (


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This talk is part of the GAPSTI Science & Technology Distinguished Seminars Series series.

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