Mathematical Techniques for Circulatory Systems

This article provides an overview on how human circulation is computationally modeled. The discussion presented explains all the latest development in the field and emphasizes the areas that need research focus. The article consists of fundamentals of the circulation, mathematical models, dealing with arterial and venous systems, organs, microcirculation, and parameters involved. The discussion also provides details on why certain parameters are difficult to obtain. The article also accounts in detail the computational models used to solve flow circulation. Although interesting to understand the modeling of circulation, the models should be useful and insightful. This article discusses the usefulness of circulation models and their limitations.

[1]  J. Alastruey,et al.  A systematic comparison between 1‐D and 3‐D hemodynamics in compliant arterial models , 2014, International journal for numerical methods in biomedical engineering.

[2]  Patricia V Lawford,et al.  Virtual fractional flow reserve from coronary angiography: modeling the significance of coronary lesions: results from the VIRTU-1 (VIRTUal Fractional Flow Reserve From Coronary Angiography) study. , 2013, JACC. Cardiovascular interventions.

[3]  Jonathan P Mynard,et al.  Scalability and in vivo validation of a multiscale numerical model of the left coronary circulation. , 2014, American journal of physiology. Heart and circulatory physiology.

[4]  J. Meister,et al.  On the wave transmission and reflection properties of stenoses. , 1996, Journal of biomechanics.

[5]  Charles A. Taylor,et al.  Uncertainty quantification in coronary blood flow simulations: Impact of geometry, boundary conditions and blood viscosity. , 2016, Journal of biomechanics.

[6]  Andrea Arnold,et al.  Uncertainty Quantification in a Patient-Specific One-Dimensional Arterial Network Model: EnKF-Based Inflow Estimator. , 2017, Journal of verification, validation, and uncertainty quantification.

[7]  P. Nithiarasu,et al.  Estimating the accuracy of a reduced‐order model for the calculation of fractional flow reserve (FFR) , 2018, International journal for numerical methods in biomedical engineering.

[8]  H. Tran,et al.  Blood pressure and blood flow variation during postural change from sitting to standing: model development and validation. , 2005, Journal of applied physiology.

[9]  Dongbin Xiu,et al.  Parametric uncertainty analysis of pulse wave propagation in a model of a human arterial network , 2007, J. Comput. Phys..

[10]  Y. Tardy,et al.  Nonlinear separation of forward and backward running waves in elastic conduits. , 1993, Journal of biomechanics.

[11]  P. Nithiarasu,et al.  An advanced computational bioheat transfer model for a human body with an embedded systemic circulation , 2015, Biomechanics and modeling in mechanobiology.

[12]  Tao Zhang,et al.  A study of noninvasive fractional flow reserve derived from a simplified method based on coronary computed tomography angiography in suspected coronary artery disease , 2017, Biomedical engineering online.

[13]  Wouter Huberts,et al.  A Numerical Method of Reduced Complexity for Simulating Vascular Hemodynamics Using Coupled 0D Lumped and 1D Wave Propagation Models , 2012, Comput. Math. Methods Medicine.

[14]  A. Dunning,et al.  Diagnosis of ischemia-causing coronary stenoses by noninvasive fractional flow reserve computed from coronary computed tomographic angiograms. Results from the prospective multicenter DISCOVER-FLOW (Diagnosis of Ischemia-Causing Stenoses Obtained Via Noninvasive Fractional Flow Reserve) study. , 2011, Journal of the American College of Cardiology.

[15]  F. N. Vosse,et al.  A wave propagation model of blood flow in large vessels using an approximate velocity profile function , 2007, Journal of Fluid Mechanics.

[16]  I. Vignon-Clementel,et al.  Data assimilation and modelling of patient-specific single-ventricle physiology with and without valve regurgitation. , 2016, Journal of biomechanics.

[17]  U. Schoepf,et al.  Diagnostic value of quantitative stenosis predictors with coronary CT angiography compared to invasive fractional flow reserve. , 2015, European journal of radiology.

[18]  C. A. Figueroa,et al.  Simulation of short-term pressure regulation during the tilt test in a coupled 3D–0D closed-loop model of the circulation , 2015, Biomechanics and modeling in mechanobiology.

[19]  Perumal Nithiarasu,et al.  A novel method for non-invasively detecting the severity and location of aortic aneurysms , 2017, Biomechanics and Modeling in Mechanobiology.

[20]  Kim H. Parker,et al.  What stops the flow of blood from the heart? , 2005, Heart and Vessels.

[21]  J P Mynard,et al.  A simple, versatile valve model for use in lumped parameter and one‐dimensional cardiovascular models , 2012, International journal for numerical methods in biomedical engineering.

[22]  U. Schoepf,et al.  Coronary CT angiography-derived fractional flow reserve correlated with invasive fractional flow reserve measurements – initial experience with a novel physician-driven algorithm , 2015, European Radiology.

[23]  Lucas O Müller,et al.  A global multiscale mathematical model for the human circulation with emphasis on the venous system , 2014, International journal for numerical methods in biomedical engineering.

[24]  D. F. Young,et al.  Computer simulation of arterial flow with applications to arterial and aortic stenoses. , 1992, Journal of biomechanics.

[25]  Nikos Stergiopulos,et al.  Pulse Wave Propagation in the Arterial Tree , 2011 .

[26]  N Westerhof,et al.  Evaluation of methods for estimation of total arterial compliance. , 1995, The American journal of physiology.

[27]  Kim H. Parker,et al.  An introduction to wave intensity analysis , 2009, Medical & Biological Engineering & Computing.

[28]  P. Nithiarasu,et al.  A 1D arterial blood flow model incorporating ventricular pressure, aortic valve and regional coronary flow using the locally conservative Galerkin (LCG) method , 2008 .

[29]  S. Tsangaris,et al.  A Computer Model for the Prediction of Left Epicardial Coronary Blood Flow in Normal, Stenotic and Bypassed Coronary Arteries, by Single or Sequential Grafting , 1998, Cardiovascular surgery.

[30]  Juan R Cebral,et al.  Patient-specific computational modeling of cerebral aneurysms with multiple avenues of flow from 3D rotational angiography images. , 2006, Academic radiology.

[31]  J. P. Mynard,et al.  A unified method for estimating pressure losses at vascular junctions , 2015, International journal for numerical methods in biomedical engineering.

[32]  Spencer J. Sherwin,et al.  Computational modelling of 1D blood flow with variable mechanical properties and its application to the simulation of wave propagation in the human arterial system , 2003 .

[33]  Mette S Olufsen,et al.  Numerical simulation of blood flow and pressure drop in the pulmonary arterial and venous circulation , 2014, Biomechanics and modeling in mechanobiology.

[34]  Michail I. Papafaklis,et al.  Fast virtual functional assessment of intermediate coronary lesions using routine angiographic data and blood flow simulation in humans: comparison with pressure wire - fractional flow reserve. , 2014, EuroIntervention : journal of EuroPCR in collaboration with the Working Group on Interventional Cardiology of the European Society of Cardiology.

[35]  R. Van Loon,et al.  An implicit solver for 1D arterial network models , 2017, International journal for numerical methods in biomedical engineering.

[36]  K. Parker,et al.  Forward and backward running waves in the arteries: analysis using the method of characteristics. , 1990, Journal of biomechanical engineering.

[37]  Dorin Comaniciu,et al.  Comparison of Fractional Flow Reserve Based on Computational Fluid Dynamics Modeling Using Coronary Angiographic Vessel Morphology Versus Invasively Measured Fractional Flow Reserve. , 2016, The American journal of cardiology.

[38]  A. Kono,et al.  Fractional flow reserve computed from noninvasive CT angiography data: diagnostic performance of an on-site clinician-operated computational fluid dynamics algorithm. , 2015, Radiology.

[39]  Alfio Quarteroni,et al.  Multiscale modelling of the circulatory system: a preliminary analysis , 1999 .

[40]  William Wijns,et al.  Fractional flow reserve calculation from 3-dimensional quantitative coronary angiography and TIMI frame count: a fast computer model to quantify the functional significance of moderately obstructed coronary arteries. , 2014, JACC. Cardiovascular interventions.

[41]  Liang Zhong,et al.  Simplified Models of Non-Invasive Fractional Flow Reserve Based on CT Images , 2016, PloS one.

[42]  R. Armentano,et al.  Linear and Nonlinear Viscoelastic Arterial Wall Models: Application on Animals. , 2016, Journal of biomechanical engineering.

[43]  I. Meredith,et al.  Noninvasive CT-Derived FFR Based on Structural and Fluid Analysis: A Comparison With Invasive FFR for Detection of Functionally Significant Stenosis. , 2017, JACC. Cardiovascular imaging.

[44]  W Huberts,et al.  A pulse wave propagation model to support decision-making in vascular access planning in the clinic. , 2012, Medical engineering & physics.

[45]  J. Meister,et al.  Forward and backward waves in the arterial system: nonlinear separation using Riemann invariants. , 1995, Technology and health care : official journal of the European Society for Engineering and Medicine.

[46]  Perumal Nithiarasu,et al.  One-Dimensional Modelling of the Coronary Circulation. Application to Noninvasive Quantification of Fractional Flow Reserve (FFR) , 2015 .

[47]  K Low,et al.  An improved baseline model for a human arterial network to study the impact of aneurysms on pressure‐flow waveforms , 2012, International journal for numerical methods in biomedical engineering.

[48]  Andrew J. Pullan,et al.  An Anatomically Based Model of Transient Coronary Blood Flow in the Heart , 2002, SIAM J. Appl. Math..

[49]  D Liepsch,et al.  Experimental and CFD flow studies in an intracranial aneurysm model with Newtonian and non-Newtonian fluids. , 2016, Technology and health care : official journal of the European Society for Engineering and Medicine.

[50]  Mette S Olufsen,et al.  Structured tree outflow condition for blood flow in larger systemic arteries. , 1999, American journal of physiology. Heart and circulatory physiology.

[51]  Timothy J. Pedley,et al.  Numerical solutions for unsteady gravity-driven flows in collapsible tubes: evolution and roll-wave instability of a steady state , 1999, Journal of Fluid Mechanics.

[52]  Lucas O. Müller,et al.  A benchmark study of numerical schemes for one‐dimensional arterial blood flow modelling , 2015, International journal for numerical methods in biomedical engineering.

[53]  Sergey Simakov,et al.  Virtual fractional flow reserve assessment in patient-specific coronary networks by 1D hemodynamic model , 2015 .

[54]  H. Langtangen,et al.  Direct numerical simulation of transitional flow in a patient-specific intracranial aneurysm. , 2011, Journal of biomechanics.

[55]  Hiroshi Ito,et al.  Diagnostic performance of noninvasive fractional flow reserve derived from coronary computed tomography angiography in suspected coronary artery disease: the NXT trial (Analysis of Coronary Blood Flow Using CT Angiography: Next Steps). , 2014, Journal of the American College of Cardiology.

[56]  Stefan Baumann,et al.  Comparison of diagnostic value of a novel noninvasive coronary computed tomography angiography method versus standard coronary angiography for assessing fractional flow reserve. , 2014, The American journal of cardiology.

[57]  Perumal Nithiarasu,et al.  A Robust Finite Element Modeling Approach to Conjugate Heat Transfer in Flexible Elastic Tubes and Tube Networks , 2015 .

[58]  Lucas O. Müller,et al.  Enhanced global mathematical model for studying cerebral venous blood flow. , 2014, Journal of biomechanics.

[59]  W Huberts,et al.  A 1D pulse wave propagation model of the hemodynamics of calf muscle pump function , 2015, International journal for numerical methods in biomedical engineering.

[60]  T. Korakianitis,et al.  Numerical simulation of cardiovascular dynamics with healthy and diseased heart valves. , 2006, Journal of biomechanics.

[61]  D. Comaniciu,et al.  A machine-learning approach for computation of fractional flow reserve from coronary computed tomography. , 2016, Journal of applied physiology.

[62]  Sunčica Čanić,et al.  Blood flow through compliant vessels after endovascular repair: wall deformations induced by the discontinuous wall properties , 2002 .

[63]  Michael J Pencina,et al.  Diagnostic accuracy of fractional flow reserve from anatomic CT angiography. , 2012, JAMA.

[64]  Mette S. Olufsen,et al.  Modeling the Afferent Dynamics of the Baroreflex Control System , 2013, PLoS Comput. Biol..

[65]  J. Mynard,et al.  One-Dimensional Haemodynamic Modeling and Wave Dynamics in the Entire Adult Circulation , 2015, Annals of Biomedical Engineering.

[66]  Jordi Alastruey,et al.  Arterial Pressure and Flow Wave Analysis Using Time-Domain 1-D Hemodynamics , 2014, Annals of Biomedical Engineering.

[67]  S. Sherwin,et al.  Pulse wave propagation in a model human arterial network: Assessment of 1-D visco-elastic simulations against in vitro measurements , 2011, Journal of biomechanics.

[68]  L. Formaggia,et al.  Numerical modeling of 1D arterial networks coupled with a lumped parameters description of the heart , 2006, Computer methods in biomechanics and biomedical engineering.

[69]  A. Quarteroni,et al.  One-dimensional models for blood flow in arteries , 2003 .

[70]  J K Raines,et al.  A computer simulation of arterial dynamics in the human leg. , 1974, Journal of biomechanics.

[71]  Pablo J. Blanco,et al.  Multidimensional modelling for the carotid artery blood flow , 2006 .

[72]  S. Sherwin,et al.  Reduced modelling of blood flow in the cerebral circulation: Coupling 1‐D, 0‐D and cerebral auto‐regulation models , 2008 .

[73]  B J B M Wolters,et al.  A patient-specific computational model of fluid-structure interaction in abdominal aortic aneurysms. , 2005, Medical engineering & physics.