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Fan L, Wang H, Kassab GS, Lee LC. Review of cardiac-coronary interaction and insights from mathematical modeling. WIREs Mech Dis 2024; 16:e1642. [PMID: 38316634 PMCID: PMC11081852 DOI: 10.1002/wsbm.1642] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Key Words] [MESH Headings] [Grants] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 09/13/2023] [Revised: 12/10/2023] [Accepted: 01/08/2024] [Indexed: 02/07/2024]
Abstract
Cardiac-coronary interaction is fundamental to the function of the heart. As one of the highest metabolic organs in the body, the cardiac oxygen demand is met by blood perfusion through the coronary vasculature. The coronary vasculature is largely embedded within the myocardial tissue which is continually contracting and hence squeezing the blood vessels. The myocardium-coronary vessel interaction is two-ways and complex. Here, we review the different types of cardiac-coronary interactions with a focus on insights gained from mathematical models. Specifically, we will consider the following: (1) myocardial-vessel mechanical interaction; (2) metabolic-flow interaction and regulation; (3) perfusion-contraction matching, and (4) chronic interactions between the myocardium and coronary vasculature. We also provide a discussion of the relevant experimental and clinical studies of different types of cardiac-coronary interactions. Finally, we highlight knowledge gaps, key challenges, and limitations of existing mathematical models along with future research directions to understand the unique myocardium-coronary coupling in the heart. This article is categorized under: Cardiovascular Diseases > Computational Models Cardiovascular Diseases > Biomedical Engineering Cardiovascular Diseases > Molecular and Cellular Physiology.
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Affiliation(s)
- Lei Fan
- Joint Department of Biomedical Engineering, Marquette University and Medical College of Wisconsin, Milwaukee, Wisconsin, USA
| | - Haifeng Wang
- Department of Mechanical Engineering, Michigan State University, East Lansing, Michigan, USA
| | - Ghassan S Kassab
- California Medical Innovations Institute, San Diego, California, USA
| | - Lik Chuan Lee
- Department of Mechanical Engineering, Michigan State University, East Lansing, Michigan, USA
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2
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Wang Y, Yin X. Modelling coronary flow and myocardial perfusion by integrating a structured-tree coronary flow model and a hyperelastic left ventricle model. COMPUTER METHODS AND PROGRAMS IN BIOMEDICINE 2024; 243:107928. [PMID: 38000321 DOI: 10.1016/j.cmpb.2023.107928] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Key Words] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 09/26/2023] [Revised: 11/02/2023] [Accepted: 11/13/2023] [Indexed: 11/26/2023]
Abstract
BACKGROUND AND OBJECTIVE There is an increasing demand to establish integrated computational models that facilitate the exploration of coronary circulation in physiological and pathological contexts, particularly concerning interactions between coronary flow dynamics and myocardial motion. The field of cardiology has also demonstrated a trend toward personalised medicine, where these integrated models can be instrumental in integrating patient-specific data to improve therapeutic outcomes. Notably, incorporating a structured-tree model into such integrated models is currently absent in the literature, which presents a promising prospect. Thus, the goal here is to develop a novel computational framework that combines a 1D structured-tree model of coronary flow in human coronary vasculature with a 3D left ventricle model utilising a hyperelastic constitutive law, enabling the physiologically accurate simulation of coronary flow dynamics. METHODS We adopted detailed geometric information from previous studies of both coronary vasculature and left ventricle to construct the coronary flow model and the left ventricle model. The structured-tree model for coronary flow was expanded to encompass the effect of time-varying intramyocardial pressure on intramyocardial blood vessels. Simultaneously, the left ventricle model served as a robust foundation for the calculation of intramyocardial pressure and subsequent quantitative evaluation of myocardial perfusion. A one-way coupling framework between the two models was established to enable the evaluation and examination of coronary flow dynamics and myocardial perfusion. RESULTS Our predicted coronary flow waveforms aligned well with published experimental data. Our model precisely captured the phasic pattern of coronary flow, including impeded or even reversed flow during systole. Moreover, our assessment of coronary flow, considering both globally and regionally averaged intramyocardial pressure, demonstrated that elevated intramyocardial pressure corresponds to increased impeding effects on coronary flow. Furthermore, myocardial blood flow simulated from our model was comparable with MRI perfusion data at rest, showcasing the capability of our model to predict myocardial perfusion. CONCLUSIONS The integrated model introduced in this study presents a novel approach to achieving physiologically accurate simulations of coronary flow and myocardial perfusion. It holds promise for its clinical applicability in diagnosing insufficient myocardial perfusion.
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Affiliation(s)
- Yingjie Wang
- School of Mathematics and Statistics, University of Glasgow, Glasgow, United Kingdom.
| | - Xueqing Yin
- School of Mathematics and Statistics, University of Glasgow, Glasgow, United Kingdom
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3
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Thekkethil N, Rossi S, Gao H, Richardson SIH, Griffith BE, Luo X. A stabilized linear finite element method for anisotropic poroelastodynamics with application to cardiac perfusion. COMPUTER METHODS IN APPLIED MECHANICS AND ENGINEERING 2023; 405:115877. [PMID: 37600475 PMCID: PMC10438829 DOI: 10.1016/j.cma.2022.115877] [Citation(s) in RCA: 1] [Impact Index Per Article: 1.0] [Reference Citation Analysis] [Abstract] [Key Words] [Grants] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 08/22/2023]
Abstract
We propose a variational multiscale method stabilization of a linear finite element method for nonlinear poroelasticity. Our approach is suitable for the implicit time integration of poroelastic formulations in which the solid skeleton is anisotropic and incompressible. A detailed numerical methodology is presented for a monolithic formulation that includes both structural dynamics and Darcy flow. Our implementation of this methodology is verified using several hyperelastic and poroelastic benchmark cases, and excellent agreement is obtained with the literature. Grid convergence studies for both anisotropic hyperelastodynamics and poroelastodynamics demonstrate that the method is second-order accurate. The capabilities of our approach are demonstrated using a model of the left ventricle (LV) of the heart derived from human imaging data. Simulations using this model indicate that the anisotropicity of the myocardium has a substantial influence on the pore pressure. Furthermore, the temporal variations of the various components of the pore pressure (hydrostatic pressure and pressure resulting from changes in the volume of the pore fluid) are correlated with the variation of the added mass and dynamics of the LV, with maximum pore pressure being obtained at peak systole. The order of magnitude and the temporal variation of the pore pressure are in good agreement with the literature.
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Affiliation(s)
- Namshad Thekkethil
- School of Mathematics and Statistics, University of Glasgow, Glasgow, UK
| | - Simone Rossi
- Department of Mathematics, University of North Carolina, Chapel Hill, NC, USA
| | - Hao Gao
- School of Mathematics and Statistics, University of Glasgow, Glasgow, UK
| | | | - Boyce E. Griffith
- Departments of Mathematics, Applied Physical Sciences, and Biomedical Engineering, University of North Carolina, Chapel Hill, NC, USA
| | - Xiaoyu Luo
- School of Mathematics and Statistics, University of Glasgow, Glasgow, UK
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Qohar UNA, Zanna Munthe-Kaas A, Nordbotten JM, Hanson EA. A nonlinear multi-scale model for blood circulation in a realistic vascular system. ROYAL SOCIETY OPEN SCIENCE 2021; 8:201949. [PMID: 34966547 PMCID: PMC8633777 DOI: 10.1098/rsos.201949] [Citation(s) in RCA: 3] [Impact Index Per Article: 1.0] [Reference Citation Analysis] [Abstract] [Key Words] [Grants] [Track Full Text] [Figures] [Subscribe] [Scholar Register] [Received: 10/31/2020] [Accepted: 10/28/2021] [Indexed: 06/14/2023]
Abstract
In the last decade, numerical models have become an increasingly important tool in biological and medical science. Numerical simulations contribute to a deeper understanding of physiology and are a powerful tool for better diagnostics and treatment. In this paper, a nonlinear multi-scale model framework is developed for blood flow distribution in the full vascular system of an organ. We couple a quasi one-dimensional vascular graph model to represent blood flow in larger vessels and a porous media model to describe flow in smaller vessels and capillary bed. The vascular model is based on Poiseuille's Law, with pressure correction by elasticity and pressure drop estimation at vessels' junctions. The porous capillary bed is modelled as a two-compartment domain (artery and venous) using Darcy's Law. The fluid exchange between the artery and venous capillary bed compartments is defined as blood perfusion. The numerical experiments show that the proposed model for blood circulation: (i) is closely dependent on the structure and parameters of both the larger vessels and of the capillary bed, and (ii) provides a realistic blood circulation in the organ. The advantage of the proposed model is that it is complex enough to reliably capture the main underlying physiological function, yet highly flexible as it offers the possibility of incorporating various local effects. Furthermore, the numerical implementation of the model is straightforward and allows for simulations on a regular desktop computer.
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Affiliation(s)
- Ulin Nuha A. Qohar
- Department of Mathematics, University of Bergen, Allegaten 41, Bergen 5008, Norway
| | | | | | - Erik Andreas Hanson
- Department of Mathematics, University of Bergen, Allegaten 41, Bergen 5008, Norway
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5
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Cai L, Zhang R, Li Y, Zhu G, Ma X, Wang Y, Luo X, Gao H. The Comparison of Different Constitutive Laws and Fiber Architectures for the Aortic Valve on Fluid-Structure Interaction Simulation. Front Physiol 2021; 12:682893. [PMID: 34248670 PMCID: PMC8266211 DOI: 10.3389/fphys.2021.682893] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.7] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 03/19/2021] [Accepted: 04/27/2021] [Indexed: 12/15/2022] Open
Abstract
Built on the hybrid immersed boundary/finite element (IB/FE) method, fluid-structure interaction (FSI) simulations of aortic valve (AV) dynamics are performed with three different constitutive laws and two different fiber architectures for the AV leaflets. An idealized AV model is used and mounted in a straight tube, and a three-element Windkessel model is further attached to the aorta. After obtaining ex vivo biaxial tensile testing of porcine AV leaflets, we first determine the constitutive parameters of the selected three constitutive laws by matching the analytical stretch-stress relations derived from constitutive laws to the experimentally measured data. Both the average error and relevant R-squared value reveal that the anisotropic non-linear constitutive law with exponential terms for both the fiber and cross-fiber directions could be more suitable for characterizing the mechanical behaviors of the AV leaflets. We then thoroughly compare the simulation results from both structural mechanics and hemodynamics. Compared to the other two constitutive laws, the anisotropic non-linear constitutive law with exponential terms for both the fiber and cross-fiber directions shows the larger leaflet displacements at the opened state, the largest forward jet flow, the smaller regurgitant flow. We further analyze hemodynamic parameters of the six different cases, including the regurgitant fraction, the mean transvalvular pressure gradient, the effective orifice area, and the energy loss of the left ventricle. We find that the fiber architecture with body-fitted orientation shows better dynamic behaviors in the leaflets, especially with the constitutive law using exponential terms for both the fiber and cross-fiber directions. In conclusion, both constitutive laws and fiber architectures can affect AV dynamics. Our results further suggest that the strain energy function with exponential terms for both the fiber and cross-fiber directions could be more suitable for describing the AV leaflet mechanical behaviors. Future experimental studies are needed to identify competent constitutive laws for the AV leaflets and their associated fiber orientations with controlled experiments. Although limitations exist in the present AV model, our results provide important information for selecting appropriate constitutive laws and fiber architectures when modeling AV dynamics.
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Affiliation(s)
- Li Cai
- NPU-UoG International Cooperative Lab for Computation and Application in Cardiology, Northwestern Polytechnical University, Xi'an, China
- Xi'an Key Laboratory of Scientific Computation and Applied Statistics, Xi'an, China
| | - Ruihang Zhang
- NPU-UoG International Cooperative Lab for Computation and Application in Cardiology, Northwestern Polytechnical University, Xi'an, China
| | - Yiqiang Li
- NPU-UoG International Cooperative Lab for Computation and Application in Cardiology, Northwestern Polytechnical University, Xi'an, China
| | - Guangyu Zhu
- School of Energy and Power Engineering, Xi'an Jiaotong University, Xi'an, China
| | - Xingshuang Ma
- College of Bioengineering, Chongqing University, Chongqing, China
| | - Yongheng Wang
- NPU-UoG International Cooperative Lab for Computation and Application in Cardiology, Northwestern Polytechnical University, Xi'an, China
| | - Xiaoyu Luo
- School of Mathematics and Statistics, University of Glasgow, Glasgow, United Kingdom
| | - Hao Gao
- School of Mathematics and Statistics, University of Glasgow, Glasgow, United Kingdom
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6
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Richardson SIH, Gao H, Cox J, Janiczek R, Griffith BE, Berry C, Luo X. A poroelastic immersed finite element framework for modelling cardiac perfusion and fluid-structure interaction. INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN BIOMEDICAL ENGINEERING 2021; 37:e3446. [PMID: 33559359 PMCID: PMC8274593 DOI: 10.1002/cnm.3446] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [Key Words] [MESH Headings] [Grants] [Track Full Text] [Subscribe] [Scholar Register] [Received: 09/17/2020] [Revised: 12/22/2020] [Accepted: 01/20/2021] [Indexed: 05/11/2023]
Abstract
Modern approaches to modelling cardiac perfusion now commonly describe the myocardium using the framework of poroelasticity. Cardiac tissue can be described as a saturated porous medium composed of the pore fluid (blood) and the skeleton (myocytes and collagen scaffold). In previous studies fluid-structure interaction in the heart has been treated in a variety of ways, but in most cases, the myocardium is assumed to be a hyperelastic fibre-reinforced material. Conversely, models that treat the myocardium as a poroelastic material typically neglect interactions between the myocardium and intracardiac blood flow. This work presents a poroelastic immersed finite element framework to model left ventricular dynamics in a three-phase poroelastic system composed of the pore blood fluid, the skeleton, and the chamber fluid. We benchmark our approach by examining a pair of prototypical poroelastic formations using a simple cubic geometry considered in the prior work by Chapelle et al (2010). This cubic model also enables us to compare the differences between system behaviour when using isotropic and anisotropic material models for the skeleton. With this framework, we also simulate the poroelastic dynamics of a three-dimensional left ventricle, in which the myocardium is described by the Holzapfel-Ogden law. Results obtained using the poroelastic model are compared to those of a corresponding hyperelastic model studied previously. We find that the poroelastic LV behaves differently from the hyperelastic LV model. For example, accounting for perfusion results in a smaller diastolic chamber volume, agreeing well with the well-known wall-stiffening effect under perfusion reported previously. Meanwhile differences in systolic function, such as fibre strain in the basal and middle ventricle, are found to be comparatively minor.
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Affiliation(s)
| | - Hao Gao
- School of Mathematics and Statistics, University of
Glasgow, Glasgow, UK
| | | | | | - Boyce E. Griffith
- Departments of Mathematics, Applied Physical Sciences, and
Biomedical Engineering, University of North Carolina, Chapel Hill, North Carolina,
USA
| | - Colin Berry
- British Heart Foundation Glasgow Cardiovascular Research
Centre, University of Glasgow, Glasgow, UK
| | - Xiaoyu Luo
- School of Mathematics and Statistics, University of
Glasgow, Glasgow, UK
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7
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Myocardial Perfusion Simulation for Coronary Artery Disease: A Coupled Patient-Specific Multiscale Model. Ann Biomed Eng 2020; 49:1432-1447. [PMID: 33263155 PMCID: PMC8057976 DOI: 10.1007/s10439-020-02681-z] [Citation(s) in RCA: 8] [Impact Index Per Article: 2.0] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 05/07/2020] [Accepted: 10/25/2020] [Indexed: 11/26/2022]
Abstract
Patient-specific models of blood flow are being used clinically to diagnose and plan treatment for coronary artery disease. A remaining challenge is bridging scales from flow in arteries to the micro-circulation supplying the myocardium. Previously proposed models are descriptive rather than predictive and have not been applied to human data. The goal here is to develop a multiscale patient-specific model enabling blood flow simulation from large coronary arteries to myocardial tissue. Patient vasculatures are segmented from coronary computed tomography angiography data and extended from the image-based model down to the arteriole level using a space-filling forest of synthetic trees. Blood flow is modeled by coupling a 1D model of the coronary arteries to a single-compartment Darcy myocardium model. Simulated results on five patients with non-obstructive coronary artery disease compare overall well to [\documentclass[12pt]{minimal}
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\begin{document}$$\text {H}_{{2}}$$\end{document}H2O PET exam data for both resting and hyperemic conditions. Results on a patient with severe obstructive disease link coronary artery narrowing with impaired myocardial blood flow, demonstrating the model’s ability to predict myocardial regions with perfusion deficit. This is the first report of a computational model for simulating blood flow from the epicardial coronary arteries to the left ventricle myocardium applied to and validated on human data.
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8
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Mynard JP, Kondiboyina A, Kowalski R, Cheung MMH, Smolich JJ. Measurement, Analysis and Interpretation of Pressure/Flow Waves in Blood Vessels. Front Physiol 2020; 11:1085. [PMID: 32973569 PMCID: PMC7481457 DOI: 10.3389/fphys.2020.01085] [Citation(s) in RCA: 40] [Impact Index Per Article: 10.0] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 05/21/2020] [Accepted: 08/06/2020] [Indexed: 01/10/2023] Open
Abstract
The optimal performance of the cardiovascular system, as well as the break-down of this performance with disease, both involve complex biomechanical interactions between the heart, conduit vascular networks and microvascular beds. ‘Wave analysis’ refers to a group of techniques that provide valuable insight into these interactions by scrutinizing the shape of blood pressure and flow/velocity waveforms. The aim of this review paper is to provide a comprehensive introduction to wave analysis, with a focus on key concepts and practical application rather than mathematical derivations. We begin with an overview of invasive and non-invasive measurement techniques that can be used to obtain the signals required for wave analysis. We then review the most widely used wave analysis techniques—pulse wave analysis, wave separation and wave intensity analysis—and associated methods for estimating local wave speed or characteristic impedance that are required for decomposing waveforms into forward and backward wave components. This is followed by a discussion of the biomechanical phenomena that generate waves and the processes that modulate wave amplitude, both of which are critical for interpreting measured wave patterns. Finally, we provide a brief update on several emerging techniques/concepts in the wave analysis field, namely wave potential and the reservoir-excess pressure approach.
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Affiliation(s)
- Jonathan P Mynard
- Heart Research, Murdoch Children's Research Institute, Melbourne, VIC, Australia.,Department of Paediatrics, The University of Melbourne, Melbourne, VIC, Australia.,Department of Biomedical Engineering, The University of Melbourne, Melbourne, VIC, Australia.,Department of Cardiology, The Royal Children's Hospital, Parkville, VIC, Australia
| | - Avinash Kondiboyina
- Heart Research, Murdoch Children's Research Institute, Melbourne, VIC, Australia.,Department of Paediatrics, The University of Melbourne, Melbourne, VIC, Australia
| | - Remi Kowalski
- Heart Research, Murdoch Children's Research Institute, Melbourne, VIC, Australia.,Department of Paediatrics, The University of Melbourne, Melbourne, VIC, Australia.,Department of Cardiology, The Royal Children's Hospital, Parkville, VIC, Australia
| | - Michael M H Cheung
- Heart Research, Murdoch Children's Research Institute, Melbourne, VIC, Australia.,Department of Paediatrics, The University of Melbourne, Melbourne, VIC, Australia.,Department of Cardiology, The Royal Children's Hospital, Parkville, VIC, Australia
| | - Joseph J Smolich
- Heart Research, Murdoch Children's Research Institute, Melbourne, VIC, Australia.,Department of Paediatrics, The University of Melbourne, Melbourne, VIC, Australia
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9
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Fan L, Namani R, Choy JS, Kassab GS, Lee LC. Effects of Mechanical Dyssynchrony on Coronary Flow: Insights From a Computational Model of Coupled Coronary Perfusion With Systemic Circulation. Front Physiol 2020; 11:915. [PMID: 32922304 PMCID: PMC7457036 DOI: 10.3389/fphys.2020.00915] [Citation(s) in RCA: 7] [Impact Index Per Article: 1.8] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 04/30/2020] [Accepted: 07/08/2020] [Indexed: 01/01/2023] Open
Abstract
Mechanical dyssynchrony affects left ventricular (LV) mechanics and coronary perfusion. Due to the confounding effects of their bi-directional interactions, the mechanisms behind these changes are difficult to isolate from experimental and clinical studies alone. Here, we develop and calibrate a closed-loop computational model that couples the systemic circulation, LV mechanics, and coronary perfusion. The model is applied to simulate the impact of mechanical dyssynchrony on coronary flow in the left anterior descending artery (LAD) and left circumflex artery (LCX) territories caused by regional alterations in perfusion pressure and intramyocardial pressure (IMP). We also investigate the effects of regional coronary flow alterations on regional LV contractility in mechanical dyssynchrony based on prescribed contractility-flow relationships without considering autoregulation. The model predicts that LCX and LAD flows are reduced by 7.2%, and increased by 17.1%, respectively, in mechanical dyssynchrony with a systolic dyssynchrony index of 10% when the LAD's IMP is synchronous with the arterial pressure. The LAD flow is reduced by 11.6% only when its IMP is delayed with respect to the arterial pressure by 0.07 s. When contractility is sensitive to coronary flow, mechanical dyssynchrony can affect global LV mechanics, IMPs and contractility that in turn, further affect the coronary flow in a feedback loop that results in a substantial reduction of dPLV/dt, indicative of ischemia. Taken together, these findings imply that regional IMPs play a significant role in affecting regional coronary flows in mechanical dyssynchrony and the changes in regional coronary flow may produce ischemia when contractility is sensitive to the changes in coronary flow.
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Affiliation(s)
- Lei Fan
- Department of Mechanical Engineering, Michigan State University, East Lansing, MI, United States
| | - Ravi Namani
- Department of Mechanical Engineering, Michigan State University, East Lansing, MI, United States
| | - Jenny S Choy
- California Medical Innovation Institute, San Diego, CA, United States
| | - Ghassan S Kassab
- California Medical Innovation Institute, San Diego, CA, United States
| | - Lik Chuan Lee
- Department of Mechanical Engineering, Michigan State University, East Lansing, MI, United States
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10
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Zhang W, Capilnasiu A, Sommer G, Holzapfel GA, Nordsletten DA. An efficient and accurate method for modeling nonlinear fractional viscoelastic biomaterials. COMPUTER METHODS IN APPLIED MECHANICS AND ENGINEERING 2020; 362:112834. [PMID: 34136022 PMCID: PMC7610983 DOI: 10.1016/j.cma.2020.112834] [Citation(s) in RCA: 3] [Impact Index Per Article: 0.8] [Reference Citation Analysis] [Abstract] [Key Words] [Grants] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 05/12/2023]
Abstract
Computational biomechanics plays an important role in biomedical engineering: using modeling to understand pathophysiology, treatment and device design. While experimental evidence indicates that the mechanical response of most tissues is viscoelastic, current biomechanical models in the computational community often assume hyperelastic material models. Fractional viscoelastic constitutive models have been successfully used in literature to capture viscoelastic material response; however, the translation of these models into computational platforms remains limited. Many experimentally derived viscoelastic constitutive models are not suitable for three-dimensional simulations. Furthermore, the use of fractional derivatives can be computationally prohibitive, with a number of current numerical approximations having a computational cost that is 𝒪 ( N T 2 ) and a storage cost that is 𝒪(NT ) (NT denotes the number of time steps). In this paper, we present a novel numerical approximation to the Caputo derivative which exploits a recurrence relation similar to those used to discretize classic temporal derivatives, giving a computational cost that is 𝒪(NT ) and a storage cost that is fixed over time. The approximation is optimized for numerical applications, and an error estimate is presented to demonstrate the efficacy of the method. The method, integrated into a finite element solid mechanics framework, is shown to be unconditionally stable in the linear viscoelastic case. It was then integrated into a computational biomechanical framework, with several numerical examples verifying the accuracy and computational efficiency of the method, including in an analytic test, in an analytic fractional differential equation, as well as in a computational biomechanical model problem.
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Affiliation(s)
- Will Zhang
- Department of Biomedical Engineering, University of Michigan, Ann Arbor, USA
| | - Adela Capilnasiu
- School of Biomedical Engineering and Imaging Sciences, King’s College London, London, UK
| | - Gerhard Sommer
- Institute of Biomechanics, Graz University of Technology, AT, Austria
| | - Gerhard A. Holzapfel
- Institute of Biomechanics, Graz University of Technology, AT, Austria
- Department of Structural Engineering, Norwegian University of Science and Technology, Trondheim, NO, Norway
| | - David A. Nordsletten
- Department of Biomedical Engineering, University of Michigan, Ann Arbor, USA
- Department of Cardiac Surgery, University of Michigan, Ann Arbor, USA
- School of Biomedical Engineering and Imaging Sciences, King’s College London, London, UK
- Corresponding author at: B20 212W, NCRC, 2800 Plymouth Rd, Ann Arbor, MI, USA, 48109. (D.A. Nordsletten)
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11
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Duanmu Z, Chen W, Gao H, Yang X, Luo X, Hill NA. A One-Dimensional Hemodynamic Model of the Coronary Arterial Tree. Front Physiol 2019; 10:853. [PMID: 31338038 PMCID: PMC6629789 DOI: 10.3389/fphys.2019.00853] [Citation(s) in RCA: 15] [Impact Index Per Article: 3.0] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 02/11/2019] [Accepted: 06/20/2019] [Indexed: 01/28/2023] Open
Abstract
One-dimensional (1D) hemodynamic models of arteries have increasingly been applied to coronary circulation. In this study, we have adopted flow and pressure profiles in Olufsen's 1D structured tree as coronary boundary conditions, with terminals coupled to the dynamic pressure feedback resulting from the intra-myocardial stress because of ventricular contraction. We model a trifurcation structure of the example coronary tree as two adjacent bifurcations. The estimated results of blood pressure and flow rate from our simulation agree well with the clinical measurements and published data. Furthermore, the 1D model enables us to use wave intensity analysis to simulate blood flow in the developed coronary model. Six characteristic waves are observed in both left and right coronary flows, though the waves' magnitudes differ from each other. We study the effects of arterial wall stiffness on coronary blood flow in the left circumflex artery (LCX). Different diseased cases indicate that distinct pathological reactions of the cardiovascular system can be better distinguished through Wave Intensity analysis, which shows agreement with clinical observations. Finally, the feedback pressure in terminal vessels and measurement deviation are also investigated by changing parameters in the LCX. We find that larger feedback pressure increases the backward wave and decreases the forward one. Although simplified, this 1D model provides new insight into coronary hemodynamics in healthy and diseased conditions. We believe that this approach offers reference resources for studies on coronary circulation disease diagnosis, treatment and simulation.
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Affiliation(s)
- Zheng Duanmu
- Key Laboratory of the Ministry of Education for Optoelectronic Measurement Technology and Instrument, Beijing Information Science and Technology University, Beijing, China
| | - Weiwei Chen
- Guangxi Key Laboratory of Regenerative Medicine, Research Centre for Regenerative Medicine, Guangxi Medical University, Guangxi, China
| | - Hao Gao
- School of Mathematics and Statistics, University of Glasgow, Glasgow, United Kingdom
| | - Xilan Yang
- Second Affiliated Hospital of Nanjing Medical University, Nanjing, China
| | - Xiaoyu Luo
- School of Mathematics and Statistics, University of Glasgow, Glasgow, United Kingdom
| | - Nicholas A Hill
- School of Mathematics and Statistics, University of Glasgow, Glasgow, United Kingdom
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12
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Mynard JP, Penny DJ, Smolich JJ. Major influence of a 'smoke and mirrors' effect caused by wave reflection on early diastolic coronary arterial wave intensity. J Physiol 2018; 596:993-1017. [PMID: 29318640 DOI: 10.1113/jp274710] [Citation(s) in RCA: 4] [Impact Index Per Article: 0.7] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 09/04/2017] [Accepted: 01/02/2018] [Indexed: 01/25/2023] Open
Abstract
KEY POINTS Coronary wave intensity analysis (WIA) is an emerging technique for assessing upstream and downstream influences on myocardial perfusion. It is thought that a dominant backward decompression wave (BDWdia ) is generated by a distal suction effect, while early-diastolic forward decompression (FDWdia ) and compression (FCWdia ) waves originate in the aorta. We show that wave reflection also makes a substantial contribution to FDWdia , FCWdia and BDWdia , as quantified by a novel method. In 18 sheep, wave reflection accounted for ∼70% of BDWdia , whereas distal suction dominated in a computer model representing a hypertensive human. Non-linear addition/subtraction of mechanistically distinct waves (e.g. wave reflection and distal suction) obfuscates the true contribution of upstream and downstream forces on measured waves (the 'smoke and mirrors' effect). The mechanisms underlying coronary WIA are more complex than previously thought and the impact of wave reflection should be considered when interpreting clinical and experimental data. ABSTRACT Coronary arterial wave intensity analysis (WIA) is thought to provide clear insight into upstream and downstream forces on coronary flow, with a large early-diastolic surge in coronary flow accompanied by a prominent backward decompression wave (BDWdia ), as well as a forward decompression wave (FDWdia ) and forward compression wave (FCWdia ). The BDWdia is believed to arise from distal suction due to release of extravascular compression by relaxing myocardium, while FDWdia and FCWdia are thought to be transmitted from the aorta into the coronary arteries. Based on an established multi-scale computational model and high-fidelity measurements from the proximal circumflex artery (Cx) of 18 anaesthetized sheep, we present evidence that wave reflection has a major impact on each of these three waves, with a non-linear addition/subtraction of reflected waves obscuring the true influence of upstream and downstream forces through concealment and exaggeration, i.e. a 'smoke and mirrors' effect. We also describe methods, requiring additional measurement of aortic WIA, for unravelling the separate influences of wave reflection versus active upstream/downstream forces on coronary waves. Distal wave reflection accounted for ∼70% of the BDWdia in sheep, but had a lesser influence (∼25%) in the computer model representing a hypertensive human. Negative reflection of the BDWdia at the coronary-aortic junction attenuated the Cx FDWdia (by ∼40% in sheep) and augmented Cx FCWdia (∼5-fold), relative to the corresponding aortic waves. We conclude that wave reflection has a major influence on early-diastolic WIA, and thus needs to be considered when interpreting coronary WIA profiles.
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Affiliation(s)
- Jonathan P Mynard
- Heart Research, Clinical Sciences, Murdoch Children's Research Institute, Parkville, VIC 3052, Australia.,Department of Paediatrics, University of Melbourne, Parkville, VIC 3052, Australia.,Department of Cardiology, Royal Children's Hospital, Parkville, VIC 3052, Australia
| | - Daniel J Penny
- Heart Research, Clinical Sciences, Murdoch Children's Research Institute, Parkville, VIC 3052, Australia.,Department of Paediatrics, University of Melbourne, Parkville, VIC 3052, Australia.,Department of Cardiology, Royal Children's Hospital, Parkville, VIC 3052, Australia.,Institute of Reproduction and Development, Monash University, Clayton, VIC, Australia
| | - Joseph J Smolich
- Heart Research, Clinical Sciences, Murdoch Children's Research Institute, Parkville, VIC 3052, Australia.,Department of Paediatrics, University of Melbourne, Parkville, VIC 3052, Australia.,Institute of Reproduction and Development, Monash University, Clayton, VIC, Australia
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13
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A numerical study of the hemodynamic effect of the aortic valve on coronary flow. Biomech Model Mechanobiol 2017; 17:319-338. [DOI: 10.1007/s10237-017-0962-y] [Citation(s) in RCA: 10] [Impact Index Per Article: 1.4] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 01/25/2017] [Accepted: 09/05/2017] [Indexed: 01/09/2023]
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14
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Guala A, Leone D, Milan A, Ridolfi L. In silico analysis of the anti-hypertensive drugs impact on myocardial oxygen balance. Biomech Model Mechanobiol 2017; 16:1035-1047. [PMID: 28070737 DOI: 10.1007/s10237-017-0871-0] [Citation(s) in RCA: 6] [Impact Index Per Article: 0.9] [Reference Citation Analysis] [Abstract] [Key Words] [MESH Headings] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 07/28/2016] [Accepted: 01/02/2017] [Indexed: 01/09/2023]
Abstract
Hypertension is a very common pathology, and its clinical treatment largely relies on different drugs. Some of these drugs exhibit specific protective functions in addition to those resulting from blood pressure reduction. In this work, we study the impact of commonly used anti-hypertensive drugs (RAAS, [Formula: see text] and calcium channel blockers) on myocardial oxygen supply-consumption balance, which plays a crucial role in type 2 myocardial infarction. To this aim, 42 wash-out hypertensive patients were selected, a number of measured data were used to set a validated multi-scale cardiovascular model to subject-specific conditions, and the administration of different drugs was suitably simulated. Our results ascribe the well-known major cardioprotective efficiency of [Formula: see text] blockers compared to other drugs to a positive change of myocardial oxygen balance due to the concomitant: (1) reduction in aortic systolic, diastolic and pulse pressures, (2) decrease in left ventricular work, diastolic cavity pressure and oxygen consumption, (3) increase in coronary flow and (4) ejection efficiency improvement. RAAS blockers share several positive outcomes with [Formula: see text] blockers, although to a reduced extent. In contrast, calcium channel blockers seem to induce some potentially negative effects on the myocardial oxygen balance.
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Affiliation(s)
- A Guala
- DIATI, Politecnico di Torino, Turin, Italy.
- Vall D'Hebron Institute of Research (VHIR), Universitat Autònoma de Barcelona, Barcelona, Spain.
| | - D Leone
- Department of Medical Sciences, Hypertension Unit, University of Torino, Turin, Italy
| | - A Milan
- Department of Medical Sciences, Hypertension Unit, University of Torino, Turin, Italy
| | - L Ridolfi
- DIATI, Politecnico di Torino, Turin, Italy
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15
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Rivolo S, Hadjilucas L, Sinclair M, van Horssen P, van den Wijngaard J, Wesolowski R, Chiribiri A, Siebes M, Smith NP, Lee J. Impact of coronary bifurcation morphology on wave propagation. Am J Physiol Heart Circ Physiol 2016; 311:H855-H870. [PMID: 27402665 PMCID: PMC5114464 DOI: 10.1152/ajpheart.00130.2016] [Citation(s) in RCA: 4] [Impact Index Per Article: 0.5] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Figures] [Journal Information] [Submit a Manuscript] [Subscribe] [Scholar Register] [Received: 02/09/2016] [Accepted: 07/05/2016] [Indexed: 01/09/2023]
Abstract
The branching pattern of the coronary vasculature is a key determinant of its function and plays a crucial role in shaping the pressure and velocity wave forms measured for clinical diagnosis. However, although multiple scaling laws have been proposed to characterize the branching pattern, the implications they have on wave propagation remain unassessed to date. To bridge this gap, we have developed a new theoretical framework by combining the mathematical formulation of scaling laws with the wave propagation theory in the pulsatile flow regime. This framework was then validated in multiple species using high-resolution cryomicrotome images of porcine, canine, and human coronary networks. Results demonstrate that the forward well-matchedness (no reflection for pressure/flow waves traveling from the coronary stem toward the microcirculation) is a salient feature in the coronary vasculature, and this result remains robust under many scenarios of the underlying pulse wave speed distribution assumed in the network. This result also implies a significant damping of the backward traveling waves, especially for smaller vessels (radius, <0.3 mm). Furthermore, the theoretical prediction of increasing area ratios (ratio between the area of the mother and daughter vessels) in more symmetric bifurcations found in the distal circulation was confirmed by experimental measurements. No differences were observed by clustering the vessel segments in terms of transmurality (from epicardium to endocardium) or perfusion territories (left anterior descending, left circumflex, and right coronary artery).
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Affiliation(s)
- Simone Rivolo
- Department of Biomedical Engineering, Division of Imaging Sciences and Biomedical Engineering, King's College London, London, United Kingdom, European Union
| | - Lucas Hadjilucas
- Department of Biomedical Engineering, Division of Imaging Sciences and Biomedical Engineering, King's College London, London, United Kingdom, European Union
| | - Matthew Sinclair
- Department of Biomedical Engineering, Division of Imaging Sciences and Biomedical Engineering, King's College London, London, United Kingdom, European Union
| | - Pepijn van Horssen
- Department of Biomedical Engineering and Physics, Academic Medical Center, University of Amsterdam, Amsterdam, The Netherlands
| | - Jeroen van den Wijngaard
- Department of Biomedical Engineering and Physics, Academic Medical Center, University of Amsterdam, Amsterdam, The Netherlands
| | - Roman Wesolowski
- Department of Cardiovascular Imaging, Division of Imaging Sciences and Biomedical Engineering, King's College London, London, United Kingdom, European Union; and
| | - Amedeo Chiribiri
- Department of Cardiovascular Imaging, Division of Imaging Sciences and Biomedical Engineering, King's College London, London, United Kingdom, European Union; and
| | - Maria Siebes
- Department of Biomedical Engineering and Physics, Academic Medical Center, University of Amsterdam, Amsterdam, The Netherlands
| | - Nicolas P Smith
- Faculty of Engineering, The University of Auckland, Auckland, New Zealand
| | - Jack Lee
- Department of Biomedical Engineering, Division of Imaging Sciences and Biomedical Engineering, King's College London, London, United Kingdom, European Union;
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