Computational assessment of blood flow heterogeneity in dialysis patients’ cardiac ventricles

Dialysis prolongs life but augments cardiovascular mortality. Imaging data suggests that dialysis increases myocardial blood flow (BF) heterogeneity, but its causes remain poorly understood. A biophysical model of human coronary vasculature was used to explain the imaging observations, and highlight causes of coronary BF heterogeneity. Post-dialysis CT images from patients under control, pharmacological stress (adenosine), therapy (cooled dialysate), and adenosine and cooled dialysate conditions were obtained. The data presented disparate phenotypes. To dissect vascular mechanisms, a 3D human coronary vasculature model was implemented. Simulations were performed to investigate the effects of altered aortic pressure and blood vessel diameters on myocardial BF heterogeneity which was quantified using relative dispersion, fractal dimension, and transmural BF distribution. Imaging showed that stress and therapy potentially increased mean and total BF, while reducing heterogeneity. BF histograms of one patient showed multi-modality. Using the model, it was found that total coronary BF increased as coronary perfusion pressure (CPP) was increased. BF heterogeneity was differentially affected by large or small vessel blocking. BF heterogeneity was found to be inversely related to small blood vessel diameters. Simulation of large artery stenosis indicates that BF became heterogeneous (increase relative dispersion) and gave multi-modal histograms. The total transmural BF as well as transmural BF heterogeneity reduced due to large artery stenosis, generating large patches of very low BF regions downstream. Blocking of arteries at various orders showed that blocking larger arteries results in multi-modal BF histograms and large patches of low BF, whereas smaller artery blocking results in augmented relative dispersion and fractal dimension. Transmural heterogeneity was also affected. Finally, the effects of augmented aortic pressure in the presence of blood vessel blocking shows differential effects on BF heterogeneity as well as transmural BF. Improved aortic blood pressure may lead to improved BF. Stress and therapy may be effective if they dilate small vessels. A potential cause for the observed complex BF distributions (multi-modal BF histograms) may indicate existing large vessel stenosis. The intuitive BF heterogeneity methods used can be readily used in clinical studies. Further development of the model and methods will permit personalised assessment of patient BF status.


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Dialysis is a life prolonging treatment but significantly reduces quality of life due to its 103 deleterious side effects on the heart. This mathematical modelling study explores 104 some of the coronary vasculature based causes of myocardial blood flow (BF) 105 heterogeneity. properties of vasculature has been studied to permit patho-physiological 133 investigations. The topology of coronary arterial trees has been quantified by Kassab 134 et al. (Kassab et al., 1997a;Kassab et al., 1997b;Kassab et al., 1993) using silicon 135 elastometer casts. In the absence of biophysical morphometry data, theoretical 136 vasculature topologies can also be generated (Keelan et al., 2016). The topology 137 data obtained from large animal hearts can be scaled to the human heart using 138 clinical angiograms (Dodge et al., 1992). To permit generating a 3D geometry from 139 the topology, the bifurcation properties of the network have been quantified. In 140 accordance with Murray's law (Murray, 1926a, b), the relationship between artery 141 segment lengths, diameters, and bifurcation angles and planes has been described 142 by Zamir and others (Zamir and Phipps, 1988;Zamir et al., 1983). Using the 143 topology and Murray's law, algorithms that distributed the topology as uniformly as 144 possible in 3D space were developed. The algorithm, which may overall be called 145 "space filling algorithm", is based on the principles of self-avoidance and boundary  (Mittal et al., 2005), and Smith et al. (Smith et al., 2000)), and have either 151 generated the complete or partial epicardial vasculature networks. Whereas vascular 152 resistance is regulated by geometry alone, the BF and pressure at each location in 153 the network also depends on the properties of fluid flowing through the network. 154 Blood is a biphasic fluid and alterations of its viscosity, dependent on haematocrit, 155 have been quantified by Pries et al. (Pries et al., 1996).The intricate problem of 156 vasculature involves optimising relative arterial diameters, bifurcation angles, 157 providing of BF to potentially empty regions, and being as widely distributed in the

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The effects of structural defects on haemodynamic distribution have been studied by 167 Yang and Wang (Yang and Wang, 2013). The availability of mathematical-168 computational tools such as the 3D coronary vasculature models has encouraged 169 the investigation of specific disease conditions in the heart (Zhang et al., 2014).   (32.5°C), after which they were scanned with and without adenosine stress.

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To assist the dynamic contrast enhanced CT scanning, a contrast agent (lopamidol) 208 was administered. The heart rate was also reduced using a beta-blocker that 209 permitted a longer diastolic phase in the left ventricle. The details of the imaging 210 protocol and image processing that computed the BF maps are given in 211 Supplementary Methods Section S1.     (Strahler, 1957). Within this numbering system the largest  provide BF transmurally (Kaimovitz et al., 2005). As a computationally manageable 251 approximation that permitted simulation of whole heart BF, the arterial trees were 252 generated for SN 6 to SN 11, where SN 6 was identified based on its diameter and 253 the number of bifurcations that would be needed to reach the capillary level (SN 0). 254 First, arterial elements, defined as a series of connected segments of the same SN, 255 were generated stochastically using the segment to element ratios in the  The total tree lengths were bounded to avoid non-physiologically short or long trees 265 (Kaimovitz et al., 2005). The total tree length of RCA was limited to between 120 mm 266 and 192 mm, and those of LAD and LCX were both limited to between 100 mm and 267 160 mm.

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Using the segment lengths (L) and radius (r) information assigned during topology The downstream resistance at any bifurcation node was then computed recursively.

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Elements that include segments of SN 8 to SN 11 constitute epicardial vessels point for the self-avoidance and boundary-avoidance algorithms as described by 296 Beard and Bassingthwaighte (Beard and Bassingthwaighte, 2000). The methods are 297 described briefly for completeness.

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To implement the self-avoidance algorithm, a vascular supply vector specific to an 300 arterial node, v s , was computed as follows: taken to be 2 throughout the arterial trees (Beard and Bassingthwaighte, 2000).

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In addition to self-avoidance, boundary avoidance was implemented to constrain the  numbers of values in each bin were counted and a BF histogram was constructed.

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The means and standard deviations of the BF histograms were computed. (2 ) ( (2 )) ( Simulation experiments and data analysis were performed using a local cluster. In The values of FD were found to fit an exponential decay curve exactly (Figure 2, 444 Biii). From the fitted curve, it was found that the asymptotic value was 1.14. The 445 difference between our control BF map and the asymptotic value was 7%, and 446 therefore the error was deemed to be negligible.

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Further, the transmural BF heterogeneity was quantified (Figure 3). BF within layers   Figure S4). Severe stenosis 479 also reduced the total perfusion by approximately 15% (Supplementary Figure S4). 480 However, increased BF heterogeneity is reflected in the increase of RD as the terminals were blocked. The PDF was seen to be uni-modal. Blocking root segments 500 of SN 7 or higher sub-trees also increased relative dispersion, but also gave rise to 501 bi-modal PDFs with peaks at low and high values (Figure 6, B). The amplitude of the 502 lower relative flow peak was greater at SN 10 blocking as compared to SN 7 503 blocking. The total BF when each SN order was blocked is shown in Figure 6, C.

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The main conclusions of this study are: instances was generated whose average haemodynamic properties are presented.

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The presented models BF heterogeneity (FD = 1.14) is in agreement with FD of 1. In this study, multi-modality in BF histograms due to large vessel severe stenosis 575 was observed in the model, which agrees with the imaging data. However, single 576 segment stenosis, or stenosis of high order segments, may be accompanied by an 577 auto-regulatory response. Although overall BF became multi-modal, the FD 578 (computed using the same methods as well as binning) was observed to be reduced.

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In contrast, the modelling study by Meier et al. (Meier et al., 2004) which included 580 autoregulation showed that FD remained unchanged under stenosis. Remarkably, 581 the inclusion of autoregulation in the above study maintained uni-modality of BF 582 histograms, while increasing RD when low aortic pressure was applied.

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The present model was constructed using a limited topology (SN 6 to SN 11). It was 585 also optimised using relatively straightforward space filling and boundary avoidance  Generation of geometry is based on space filling, whereas optimisation of energy 656 expenditure or some measure will give improved distribution in the future. In the 657 future, we aim to develop a 4D XCT phantom that will permit patient specific BF 658 assessment along with other parameters (Fung et al., 2011).

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The lack of autoregulation is a significant simplification in our study. The inclusion of In this study, imaging data acquired from patients was analysed. Whereas it is 673 probably true that dialysis causes vascular dysfunction by affecting the micro-674 vasculature, this study's imaging observation combined with the modelling results, 675 indicates that large vessel dysfunction may significantly affect patient's coronary 676 perfusion.

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In the process of quantifying BF heterogeneity, we have now developed algorithms 679 that compute simple yet informative measures of BF heterogeneity. Such a tool will 680 provide rapid assessment of whether imaging data reflect the effectiveness of 681 therapy.

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An important development during this study was the implementation of a method to