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Mathematical Modelling of the Human Cardiovascular System

Alfio Quarteroni, Luca Dede', Andrea Manzoni, and Christian Vergara
Cambridge University Press
Publication Date: 
Number of Pages: 
Cambridge Monographs on Applied and Computational Mathematics
[Reviewed by
Sarah Patterson
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The detailed monograph, Mathematical Modelling of the Human Cardiovascular System, is an expansion of the review paper by Quarteroni, Manzoni, and Vergara (2017). This monograph provides an entry point to modeling the cardiovascular system to those who have experience with computational fluid dynamics. Some numerical results are presented that use the finite element method and the software package LifeV. The authors provide the reader with various frameworks for obtaining clinical, patient-specific data by discussing factors such as availability, invasiveness, and cost. For those interested in learning even more about modeling the cardiovascular system, there is an abundance of references to consider. The book is divided into three main parts: (1) arterial circulation, (2) heart function, and (3) optimization, control, uncertainty, and complexity reduction.
The structure of the first two parts of the text is the same. The basic physiology is introduced, methods for obtaining clinical data are reviewed, and then detailed treatments of the numerical methods are presented. When introducing the basic physiology, typical parameter ranges like flow rates, vessel sizes, Reynolds numbers, and Womersley numbers are given. The book also introduces some pathological conditions of interest and indicates how these conditions can cause some parameters to be outside of the normal ranges. 
This book includes methods for generating the fluid computational domain from patient-specific data which requires the acquisition of clinical data, image enhancement, image segmentation, and building the computational mesh. It mentions many methods used for each step with resources for interested readers. Additionally, general boundary conditions are explored for the fluid-structure interaction (FSI) problem and the inlet and outlets of the vessel. Techniques for obtaining boundary conditions from patient data are given.  Since modeling the full circulatory system in 3D is intractable, the monograph includes information about the implementation of multiscale models where the problem is reduced to lower-dimensional models in the lower arterial tree. Details are given about the choice of the numerical scheme since stability is a major issue in FSI problems.
This monograph includes a rich discussion on the variability of input data, such as geometric features of the vessel, boundary conditions, and physical coefficients. It is vital to detect the most relevant parameters and address the impact of the input values on the output. Techniques in (1) optimal control and optimal design, (2) parameter identification and data assimilation, and (3) uncertainty quantification are presented in the latter third of the monograph. 
Overall, there is a nice interplay between the basic biology and physiology needed to understand the model, the pros and cons of different techniques of obtaining clinical data, and the implementation of the numerical methods. Each section includes beautifully colored schematic representations of the cardiovascular system or figures that show quantities of interest on grids generated from patient-specific images including several figures that were not included in the review paper. 


Sarah Patterson is an assistant professor of applied mathematics at the Virginia Military Institute. She received her PhD from Duke University in 2019 under the direction of Anita Layton. Her thesis focused on renal hemodynamic models and the immersed interface method for open interfaces. Sarah’s other research interests include computational fluid dynamics, fluid-structure interactions, numerical analysis, and mathematical modeling.