# A Hybrid Model for Cardiac Perfusion: Coupling a Discrete Coronary Arterial Tree Model with a Continuous Porous-Media Flow Model of the Myocardium

^{1}

^{2}

^{3}

^{4}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Mathematical Modeling

#### 2.1. The Hybrid Model

#### 2.2. Generating Optimized Arterial Tree Models

- (A1)
- CCO trees grown under different optimization target functions show differences in structure that can be quantified by appropriately chosen numerical indexes [19]. In order to quantify the optimality of the CCO tree, the total intravascular volume is chosen for the optimization target function as recommended by other authors [16,17,23], according to the following equation:$$V=\pi \sum _{s=1}^{{k}_{tot}}{l}_{s}{r}_{s}^{2},$$
- (A2)
- The piece of tissue to be perfused is geometrically represented by a perfusion area or a perfusion volume.
- (A3)
- The arterial tree is modeled as a dichotomously branching system (binary tree) of straight cylindrical tubes representing the vessel segments, which are assumed to be rigid.
- (A4)
- The blood is modeled as an incompressible, homogeneous Newtonian fluid at steady state and laminar flow conditions.
- (A5)
- The flow resistance ${R}_{s}$ of each segment of the tree is assumed to follow Poiseuille’s law [24]:$${R}_{s}=\left(\frac{8\eta}{\pi}\right)\frac{{l}_{s}}{{r}_{s}^{4}},$$
- (A6)
- The pressure drop $\Delta {p}_{s}$ along segment s is given by$$\Delta {p}_{s}={R}_{s}{Q}_{s},$$

- (C1)
- Each terminal segment supplies an individual amount of blood flow ${Q}_{term}$ into the microcirculatory network, which is not modeled in detail.
- (C2)
- All terminal segments drain against a given, unique terminal pressure, ${p}_{term}$.
- (C3)
- The resistance of the resulting model tree induces a prespecified perfusion flow ${Q}_{perf}$ across the overall pressure gradient:$$\Delta p={p}_{perf}-{p}_{term},$$
- (C4)
- At bifurcations, the radii of parent (${r}_{0}$) and daughter segments (${r}_{1}$, ${r}_{2}$) are forced to exactly fulfill a bifurcation law derived from real coronary trees [25]:$${r}_{0}^{\gamma}={r}_{1}^{\gamma}+{r}_{2}^{\gamma},$$

#### 2.3. Coupling Discrete and Continuous Models

#### 2.4. Extravascular Model

#### 2.5. Contrast Agent Adsorption

#### 2.6. Recirculation of the Contrast Agent

#### 2.7. Initial and Boundary Conditions

## 3. Numerical Methods

#### 3.1. Discretization of the Continuous Model

#### 3.2. Discretization of the Discrete Coronary Arterial Tree Model

## 4. Numerical Experiments and Results

#### 4.1. Setup of the Numerical Experiments

#### 4.2. Grid Independence Test

#### 4.3. Profiles of the CCO Arterial Trees

#### 4.4. Simulation Results

## 5. Discussion, Conclusions and Future Works

#### 5.1. Validation of the Results

#### 5.2. Limitations and Future Works

#### 5.3. Conclusions

## Supplementary Materials

## Author Contributions

## Funding

## Conflicts of Interest

## References

- Mendis, S.; Puska, P.; Norrving, B. Global Atlas on Cardiovascular Disease Prevention and Control; World Health Organization: Geneva, Switzerland, 2011.
- Mortality, G. Causes of Death C. Causes of Death C. Global, regional, and national age-sex specific all-cause and cause-specific mortality for 240 causes of death, 1990–2013: A systematic analysis for the global burden of disease study 2013. Lancet
**2015**, 385, 117–171. [Google Scholar] - Bassingthwaighte, J.; Wang, C.; Chan, I. Blood-tissue exchange via transport and transformation by capillary endothelial cells. Circ. Res.
**1989**, 65, 997–1020. [Google Scholar] [CrossRef] [PubMed] - Michler, C.; Cookson, A.N.; Chabiniok, R.; Hyde, E.; Lee, J.; Sinclair, M.; Sochi, T.; Goyal, A.; Vigueras, G.; Nordsletten, D.A.; et al. A computationally efficient framework for the simulation of cardiac perfusion using a multi-compartment Darcy porous-media flow model. Int. J. Numer. Methods Biomed. Eng.
**2013**, 29, 217–232. [Google Scholar] [CrossRef] [PubMed] - Cookson, A.N.; Lee, J.; Michler, C.; Chabiniok, R.; Hyde, E.; Nordsletten, D.; Smith, N. A spatially-distributed computational model to quantify behaviour of contrast agents in MR perfusion imaging. Med. Image Anal.
**2014**, 18, 1200–1216. [Google Scholar] [CrossRef] - Alves, J.R.; Queiroz, R.A.B.; Santos, R.W. Simulation of cardiac perfusion by contrast in the myocardium using a formulation of flow in porous media. J. Comput. Appl. Math.
**2016**, 295, 13–24. [Google Scholar] [CrossRef] - Alves, J.R.; de Queiroz, R.A.; Bär, M.; dos Santos, R.W. Simulation of the Perfusion of Contrast Agent Used in Cardiac Magnetic Resonance: A Step Toward Non-invasive Cardiac Perfusion Quantification. Front. Physiol.
**2019**, 10, 177. [Google Scholar] [CrossRef] - De Bruyne, B.; Sarma, J. Fractional flow reserve: A review. Heart
**2008**, 94, 949–959. [Google Scholar] [CrossRef] - Hlatky, M.A.; De Bruyne, B.; Pontone, G.; Patel, M.R.; Norgaard, B.L.; Byrne, R.A.; Curzen, N.; Purcell, I.; Gutberlet, M.; Rioufol, G.; et al. Quality-of-life and economic outcomes of assessing fractional flow reserve with computed tomography angiography: PLATFORM. J. Am. Coll. Cardiol.
**2015**, 66, 2315–2323. [Google Scholar] [CrossRef] - Zhou, L.; Liu, Y.; Sun, H.; Li, H.; Zhang, Z.; Hao, P. Usefulness of enzyme-free and enzyme-resistant detection of complement component 5 to evaluate acute myocardial infarction. Sens. Actuators B Chem.
**2022**, 369, 132315. [Google Scholar] [CrossRef] - Hao, P.; Li, H.; Zhou, L.; Sun, H.; Han, J.; Zhang, Z. Serum metal ion-induced cross-linking of photoelectrochemical peptides and circulating proteins for evaluating cardiac ischemia/reperfusion. ACS Sens.
**2022**, 7, 775–783. [Google Scholar] [CrossRef] - Xue, F.; Cheng, J.; Liu, Y.; Cheng, C.; Zhang, M.; Sui, W.; Chen, W.; Hao, P.; Zhang, Y.; Zhang, C. Cardiomyocyte-specific knockout of ADAM17 ameliorates left ventricular remodeling and function in diabetic cardiomyopathy of mice. Signal Transduct. Target. Ther.
**2022**, 7, 259. [Google Scholar] [CrossRef] [PubMed] - Tian, Z.; Zhang, Y.; Zheng, Z.; Zhang, M.; Zhang, T.; Jin, J.; Zhang, X.; Yao, G.; Kong, D.; Zhang, C.; et al. Gut microbiome dysbiosis contributes to abdominal aortic aneurysm by promoting neutrophil extracellular trap formation. Cell Host Microbe
**2022**, 30, 1450–1463. [Google Scholar] [CrossRef] [PubMed] - Arai, A.E. The cardiac magnetic resonance CMR approach to assessing myocardial viability. J. Nucl. Cardiol.
**2011**, 18, 1095–1102. [Google Scholar] [CrossRef] - Taylor, C.A.; Fonte, T.A.; Min, J.K. Computational Fluid Dynamics Applied to Cardiac Computed Tomography for Noninvasive Quantification of Fractional Flow Reserve: Scientific Basis. J. Am. Coll. Cardiol.
**2013**, 61, 2233–2241. [Google Scholar] [CrossRef] - Karch, R.; Neumann, F.; Neumann, M.; Schreiner, W. A tree-dimensional model for arterial tree representation, generated by constrained constructive optimization. Comput. Biol. Med.
**1999**, 29, 19–38. [Google Scholar] [CrossRef] [PubMed] - Schreiner, W.; Buxbaum, P. Computer-optimization of vascular trees. IEEE Trans. Biomed. Eng.
**1993**, 40, 482–491. [Google Scholar] [CrossRef] [PubMed] - Schreiner, W.; Neumann, M.; Neumann, F.; Roedler, S.; End, A.; Buxbaum, P.; Muller, M.; Spieckermann, P. The branching angles in computer-generated optimized models of arterial trees. J. Gen. Physiol.
**1994**, 103, 975–989. [Google Scholar] [CrossRef] [PubMed] - Schreiner, W.; Neumann, F.; Neumann, M.; End, A.; Roedler, S.; Aharinejad, S. The influence of optimization target selection on the structure of arterial tree models generated by constrained constructive optimization. J. Gen. Physiol.
**1995**, 106, 583–599. [Google Scholar] [CrossRef] - Schreiner, W.; Neumann, F.; Neumann, M.; Karch, R.; End, A.; Roedler, S. Limited bifurcation asymmetry in coronary arterial tree models generated by constrained constructive optimization. J. Gen. Physiol.
**1997**, 109, 129–140. [Google Scholar] [CrossRef] - Blanco, P.; de Queiroz, R.B.; Feijóo, R. A computational approach to generate concurrent arterial networks in vascular territories. Int. J. Numer. Methods Biomed. Eng.
**2013**, 29, 601–614. [Google Scholar] [CrossRef] - Talou, G.D.M.; Safaei, S.; Hunter, P.J.; Blanco, P.J. Adaptive constrained constructive optimisation for complex vascularisation processes. Sci. Rep.
**2021**, 11, 6180. [Google Scholar] [CrossRef] - Sherman, T. On connecting large vessels to small: The meaning of Murray’s law. J. Gen. Physiol.
**1981**, 78, 431–453. [Google Scholar] [CrossRef] [PubMed] - Fung, Y. Biomechanics: Circulation; Springer: Berlin/Heidelberg, Germany, 1984. [Google Scholar]
- Zamir, M. Distributing and delivering vessels of the human heart. J. Gen. Physiol.
**1988**, 91, 725–735. [Google Scholar] [CrossRef] - Arts, T.; Kruger, R.; van Gerven, W.; Lambregts, J.; Reneman, R. Propagation velocity and reflection of pressure waves in the canine coronary artery. Am. J. Physiol.
**1979**, 237, H469–H474. [Google Scholar] [CrossRef] - Rodbard, S. Vascular caliber. Cardiology
**1975**, 60, 4–49. [Google Scholar] [CrossRef] - Ferreira, V.; de Queiroz, R.; Lima, G.; Cuenca, R.; Oishi, C.; Azevedo, J.; Mckee, S. A bounded upwinding scheme for computing convection-dominated transport problems. Comput. Fluids
**2012**, 57, 208–224. [Google Scholar] [CrossRef] - Mann, D.L.; Zipes, D.P.; Libby, P.; Bonow, R.O. Braunwald’s Heart Disease E-Book: A Textbook of Cardiovascular Medicine; Elsevier Health Sciences; Elsevier: Amsterdam, The Netherlands, 2014. [Google Scholar]
- Ortiz-Pérez, J.T.; Rodríguez, J.; Meyers, S.N.; Lee, D.C.; Davidson, C.; Wu, E. Correspondence between the 17-segment model and coronary arterial anatomy using contrast-enhanced cardiac magnetic resonance imaging. JACC Cardiovasc. Imaging
**2008**, 1, 282–293. [Google Scholar] [CrossRef] - Alzhanov, N.; Ng, E.Y.; Su, X.; Zhao, Y. CFD Computation of Flow Fractional Reserve (FFR) in Coronary Artery Trees Using a Novel Physiologically Based Algorithm (PBA) Under 3D Steady and Pulsatile Flow Conditions. Bioengineering
**2023**, 10, 309. [Google Scholar] [CrossRef] - Barnafi, N.; Gómez-Vargas, B.; Lourenço, W.d.J.; Reis, R.F.; Rocha, B.M.; Lobosco, M.; Ruiz-Baier, R.; dos Santos, R.W. Finite element methods for large-strain poroelasticity/chemotaxis models simulating the formation of myocardial oedema. J. Sci. Comput.
**2022**, 92, 92. [Google Scholar] [CrossRef] - Reis, R.F.; dos Santos, R.W.; Rocha, B.M.; Lobosco, M. On the mathematical modeling of inflammatory edema formation. Comput. Math. Appl.
**2019**, 78, 2994–3006. [Google Scholar] [CrossRef] - Jarzyńska, M.; Pietruszka, M. The application of the Kedem–Katchalsky equations to membrane transport of ethyl alcohol and glucose. Desalination
**2011**, 280, 14–19. [Google Scholar] [CrossRef] - Smith, A.F.; Shipley, R.J.; Lee, J.; Sands, G.B.; LeGrice, I.J.; Smith, N.P. Transmural variation and anisotropy of microvascular flow conductivity in the rat myocardium. Ann. Biomed. Eng.
**2014**, 42, 1966–1977. [Google Scholar] [CrossRef] [PubMed] - Habib, A.; Lachman, N.; Christensen, K.N.; Asirvatham, S.J. The anatomy of the coronary sinus venous system for the cardiac electrophysiologist. Europace
**2009**, 11, v15–v21. [Google Scholar] [CrossRef] [PubMed] - Sakamoto, S.; Takahashi, S.; Coskun, A.U.; Papafaklis, M.I.; Takahashi, A.; Saito, S.; Stone, P.H.; Feldman, C.L. Relation of distribution of coronary blood flow volume to coronary artery dominance. Am. J. Cardiol.
**2013**, 111, 1420–1424. [Google Scholar] [CrossRef] - Engblom, H.; Xue, H.; Akil, S.; Carlsson, M.; Hindorf, C.; Oddstig, J.; Hedeer, F.; Hansen, M.S.; Aletras, A.H.; Kellman, P.; et al. Fully quantitative cardiovascular magnetic resonance myocardial perfusion ready for clinical use: A comparison between cardiovascular magnetic resonance imaging and positron emission tomography. J. Cardiovasc. Magn. Reson.
**2017**, 19, 78. [Google Scholar] [CrossRef] - Kellman, P.; Hansen, M.S.; Nielles-Vallespin, S.; Nickander, J.; Themudo, R.; Ugander, M.; Xue, H. Myocardial perfusion cardiovascular magnetic resonance: Optimized dual sequence and reconstruction for quantification. J. Cardiovasc. Magn. Reson.
**2017**, 19, 43. [Google Scholar] [CrossRef] - Brown, L.A.; Onciul, S.C.; Broadbent, D.A.; Johnson, K.; Fent, G.J.; Foley, J.R.; Garg, P.; Chew, P.G.; Knott, K.; Dall’Armellina, E.; et al. Fully automated, inline quantification of myocardial blood flow with cardiovascular magnetic resonance: Repeatability of measurements in healthy subjects. J. Cardiovasc. Magn. Reson.
**2018**, 20, 48. [Google Scholar] [CrossRef]

**Figure 1.**Model of two domains (

**a**), used to simulate the scenarios of (1) healthy and (2) ischemic myocardium; and the model with three domains (

**b**). The third domain is used to simulate a scenario of (3) infarction.

**Figure 2.**Communication between intravascular domain, built using CCO method, and the extravascular one; continuum. This communication takes place at the terminal segments of the arterial tree.

**Figure 3.**Data structure used to represent the arterial tree provided by the CCO method. On the left, the figure represents the network of points obtained from the spatial discretization of size $\Delta x$, made upon the arterial tree. The figure on the right shows the graph structure: there is a linked list of nodes, and each node has a list of edges. This way, each node has the information on where the CA is coming from and to which neighbors the flow is going to be directed.

**Figure 4.**Node 1, of the regular type, i.e., has only two neighbours, 0 and 2, and the respective faces used at the FVM, a and b.

**Figure 5.**Node 2, of the type bifurcation, i.e., it has three neighbours, 1, 3 and 4, and the respective faces used at the FVM, a, b and c.

**Figure 6.**Node 3, of the terminal type, i.e., it has only one neighbour, node 2, and the respective faces a and b used at the FVM scheme.

**Figure 7.**Simplified representation of the regions perfused by the trees from coronary arteries: LCX, LAD, and RCA. It is also indicated the position of the myocardial papilary muscles. Adapted from [30].

**Figure 8.**Results of the simulations carried out: (

**a**) healthy, (

**b**) ischemia, and (

**c**) infarction. (

**b**,

**c**) indicate the stenosis position, where a reduced flow is imposed. For the ischemic case, the flow was decreased by a factor of 25 ($\alpha =25$), whereas for the infarction, it was decreased by a factor of 30 ($\alpha =30$). Panel (

**d**) shows the regions of interest (remote and damaged) for the evaluation of the signal intensity of the contrast agent.

**Figure 9.**Results of the grid independence test in terms of the total extravascular concentration as a function of time for meshes consisting of $100\times 100$, $200\times 200$, and $400\times 400$ nodes.

**Figure 10.**Profiles of (

**a**) blood flow (mL/min); (

**b**) length of the segments; and (

**c**) radius of the segments for the arterial trees provided by the CCO method.

**Figure 11.**Dynamics of the CA at the end (50 s) of the exams first pass (FP) and after the late enhancement (LE) at 600 s.

**Figure 12.**Contrast agent concentration for the (

**a**) infarction and (

**b**) ischemia scenarios at every 50 s until the final time of 450 s.

Parameter (Units) | Description | Healthy/Ischemic | Infarction |
---|---|---|---|

$P\left({\mathrm{s}}^{-1}\right)$ | endothelial permeability | 1.0 | 1.0 |

${k}_{e}$ (${\mathrm{s}}^{-1}$) | decay parameter | 0.007 | 0.002 |

${k}_{f}\left({\mathrm{s}}^{-1}\right)$ | decay parameter (fibrotic) | 0 | 0.001 |

${k}_{ef}\left({\mathrm{s}}^{-1}\right)$ | rate that contrast moves from the interstitium to the fibrosis | 0 | 0.01 |

$\varphi (-)$ | porosity | 0.10 | 0.10 |

$\lambda (-)$ | fraction of the extravascular domain that is occupied by the interstitial space | 0.25 | 1.0 |

${\lambda}_{f}(-)$ | fraction of fibrosis at the extravascular media | 0 | 0.5 |

$D\left({\mathrm{mm}}^{2}{\mathrm{s}}^{-1}\right)$ | diffusion coefficient | ${10}^{-2}$ | ${10}^{-2}$ |

$\sigma $ | variance of the CA infusion | 6.0 | 6.0 |

${t}_{peak}\left(\mathrm{s}\right)$ | Gaussian mean | 25 | 25 |

${v}_{out}(\mathrm{mm}\xb7{\mathrm{s}}^{-1})$ | velocity of the recirculatory system | 0.06 | 0.06 |

${D}_{out}\left({\mathrm{mm}}^{2}{\mathrm{s}}^{-1}\right)$ | diffusion coefficient of the recirculatory system | 0.05 | 0.05 |

$k\left({\mathrm{s}}^{-1}\right)$ | contrast agent clearing rate | 0.01 | 0.01 |

Disclaimer/Publisher’s Note: The statements, opinions and data contained in all publications are solely those of the individual author(s) and contributor(s) and not of MDPI and/or the editor(s). MDPI and/or the editor(s) disclaim responsibility for any injury to people or property resulting from any ideas, methods, instructions or products referred to in the content. |

© 2023 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Alves, J.R.; Berg, L.A.; Gaio, E.D.; Rocha, B.M.; de Queiroz, R.A.B.; dos Santos, R.W.
A Hybrid Model for Cardiac Perfusion: Coupling a Discrete Coronary Arterial Tree Model with a Continuous Porous-Media Flow Model of the Myocardium. *Entropy* **2023**, *25*, 1229.
https://doi.org/10.3390/e25081229

**AMA Style**

Alves JR, Berg LA, Gaio ED, Rocha BM, de Queiroz RAB, dos Santos RW.
A Hybrid Model for Cardiac Perfusion: Coupling a Discrete Coronary Arterial Tree Model with a Continuous Porous-Media Flow Model of the Myocardium. *Entropy*. 2023; 25(8):1229.
https://doi.org/10.3390/e25081229

**Chicago/Turabian Style**

Alves, João R., Lucas A. Berg, Evandro D. Gaio, Bernardo M. Rocha, Rafael A. B. de Queiroz, and Rodrigo W. dos Santos.
2023. "A Hybrid Model for Cardiac Perfusion: Coupling a Discrete Coronary Arterial Tree Model with a Continuous Porous-Media Flow Model of the Myocardium" *Entropy* 25, no. 8: 1229.
https://doi.org/10.3390/e25081229