Yunlong Huo

Intraspecific scaling laws of vascular trees

A fundamental physics-based derivation of intraspecific scaling laws of vascular trees has not been previously realized. Here, we provide such a theoretical derivation for the volume–diameter and flow–length scaling laws of intraspecific vascular trees. In conjunction with the minimum energy hypothesis, this formulation also results in diameter–length, flow–diameter and flow–volume scaling laws. The intraspecific scaling predicts the volume–diameter power relation with a theoretical exponent of 3, which is validated by the experimental measurements for the three major coronary arterial trees in swine (where a least-squares fit of these measurements has exponents of 2.96, 3 and 2.98 for the left anterior descending artery, left circumflex artery and right coronary artery trees, respectively). This scaling law as well as others agrees very well with the measured morphometric data of vascular trees in various other organs and species. This study is fundamental to the understanding of morphological and haemodynamic features in a biological vascular tree and has implications for vascular disease.

The Flow Field along the Entire Length of Mouse Aorta and Primary Branches

There is a spatial disposition to atherosclerosis along the aorta corresponding to regions of flow disturbances. The objective of the present study is to investigate the detailed distribution of hemodynamic parameters (wall shear stress (WSS), spatial gradient of wall shear stress (WSSG), and oscillatory shear index (OSI)) in the entire length of C57BL/6 mouse aorta with all primary branches (from ascending aorta to common iliac bifurcation). The detailed geometrical parameters (e.g., diameter and length of the vessels) were obtained from casts of entire aorta and primary branches of mice. The flow velocity was measured at the inlet of ascending aorta using Doppler flowprobe in mice. The outlet pressure boundary condition was estimated based on scaling law. The continuity and Navier–Stokes equations were solved using three-dimensional finite element method (FEM). The model prediction was tested by comparing the computed flow rate with the flow rate measured just before the common iliac bifurcation, and good agreement was found. It was also found that complex flow patterns occur at bifurcations between main trunk and branches. The major branches of terminal aorta, with the highest proportion of atherosclerosis, have the lowest WSS, and the relatively atherosclerotic-prone aortic arch has much more complex WSS distribution and higher OSI value than other sites. The low WSS coincides with the high OSI, which approximately obeys a power law relationship. Furthermore, the scaling law between flow and diameter holds in the entire aorta and primary branches of mice under pulsatile blood flow conditions. This model will eventually serve to elucidate the causal relation between hemodynamic patterns and atherogenesis in KO mice.

Effects of vessel compliance on flow pattern in porcine epicardial right coronary arterial tree

The compliance of the vessel wall affects hemodynamic parameters which may alter the permeability of the vessel wall. Based on experimental measurements, the present study established a finite element (FE) model in the proximal elastic vessel segments of epicardial right coronary arterial (RCA) tree obtained from computed tomography. The motion of elastic vessel wall was measured by an impedance catheter and the inlet boundary condition was measured by an ultrasound flow probe. The Galerkin FE method was used to solve the Navier-Stokes and Continuity equations, where the convective term in the Navier-Stokes equation was changed in the arbitrary Lagrangian-Eulerian (ALE) framework to incorporate the motion due to vessel compliance. Various hemodynamic parameters (e.g., wall shear stress-WSS, WSS spatial gradient-WSSG, oscillatory shear index-OSI) were analyzed in the model. The motion due to vessel compliance affects the time-averaged WSSG more strongly than WSS at bifurcations. The decrease of WSSG at flow divider in elastic bifurcations, as compared to rigid bifurcations, implies that the vessel compliance decreases the permeability of vessel wall and may be atheroprotective. The model can be used to predict coronary flow pattern in subject-specific anatomy as determined by noninvasive imaging.

A Scaling Law of Vascular Volume

Vascular volume is of fundamental significance to the function of the cardiovascular system. An accurate prediction of blood volume in patients is physiologically and clinically significant. This study proposes what we believe is a novel volume scaling relation of the form: V(c)=K(v)D(s)(2/3)L(c), where V(c) and L(c) are cumulative vessel volume and length, respectively, in the tree, and D(s) is the diameter of the vessel segment. The scaling relation is validated in vascular trees of various organs including the heart, lung, mesentery, muscle, and eye of different species. Based on the minimum energy hypothesis and volume scaling relation, four structure-function scaling relations are predicted, including the diameter-length, volume-length, flow-diameter, and volume-diameter relations, with exponent values of 3/7, 1(2/7), 2(1/3), and 3, respectively. These four relations are validated in the various vascular trees, which further confirm the volume scaling relation. This scaling relation may serve as a control reference to estimate the blood volume in various organs and species. The deviation from the scaling relation may indicate hypovolemia or hypervolemia and aid diagnosis.

A validated predictive model of coronary fractional flow reserve

Myocardial fractional flow reserve (FFR), an important index of coronary stenosis, is measured by a pressure sensor guidewire. The determination of FFR, only based on the dimensions (lumen diameters and length) of stenosis and hyperaemic coronary flow with no other ad hoc parameters, is currently not possible. We propose an analytical model derived from conservation of energy, which considers various energy losses along the length of a stenosis, i.e. convective and diffusive energy losses as well as energy loss due to sudden constriction and expansion in lumen area. In vitro (constrictions were created in isolated arteries using symmetric and asymmetric tubes as well as an inflatable occluder cuff) and in vivo (constrictions were induced in coronary arteries of eight swine by an occluder cuff) experiments were used to validate the proposed analytical model. The proposed model agreed well with the experimental measurements. A least-squares fit showed a linear relation as (Δp or FFR)experiment = a(Δp or FFR)theory + b, where a and b were 1.08 and −1.15 mmHg (r2 = 0.99) for in vitro Δp, 0.96 and 1.79 mmHg (r2 = 0.75) for in vivo Δp, and 0.85 and 0.1 (r2 = 0.7) for FFR. Flow pulsatility and stenosis shape (e.g. eccentricity, exit angle divergence, etc.) had a negligible effect on myocardial FFR, while the entrance effect in a coronary stenosis was found to contribute significantly to the pressure drop. We present a physics-based experimentally validated analytical model of coronary stenosis, which allows prediction of FFR based on stenosis dimensions and hyperaemic coronary flow with no empirical parameters.

Which diameter and angle rule provides optimal flow patterns in a coronary bifurcation

The branching angle and diameter ratio in epicardial coronary artery bifurcations are two important determinants of atherogenesis. Murrays cubed diameter law and bifurcation angle have been assumed to yield optimal flows through a bifurcation. In contrast, we have recently shown a 7/3 diameter law (HK diameter model), based on minimum energy hypothesis in an entire tree structure. Here, we derive a bifurcation angle rule corresponding to the HK diameter model and critically evaluate the streamline flow through HK and Murray-type bifurcations. The bifurcations from coronary casts were found to obey the HK diameter model and angle rule much more than Murrays model. A finite element model was used to investigate flow patterns for coronary artery bifurcations of various types. The inlet velocity and pressure boundary conditions were measured by ComboWire. Y-bifurcation of Murray type decreased wall shear stress-WSS (10%-40%) and created an increased oscillatory shear index-OSI in atherosclerosis-prone regions as compared with HK-type bifurcations. The HK-type bifurcations were found to have more optimal flow patterns (i.e., higher WSS and lower OSI) than Murray-type bifurcations which have been traditionally believed to be optimized. This study has implications for changes in bifurcation angles and diameters in percutaneous coronary intervention.

Optimal diameter of diseased bifurcation segment: a practical rule for percutaneous coronary intervention.

AIMS The percutaneous repair of a diseased segment should consider the dimensions of other segments of a bifurcation in order to ensure the optimality of flow through the bifurcation. The question is, if the diameters of two segments of a bifurcation are known, can an optimal diameter of the third diseased segment be determined such that the bifurcation has an optimal geometry for flow transport? Various models (i.e., Murray, Finet, area-preservation and HK models) that express a diameter relationship of the three segments of a bifurcation have been proposed to answer the question. METHODS AND RESULTS In this study the four models were compared with experimental measurements on epicardial coronary bifurcations of patients and swine. The HK model is found to be in agreement with morphometric measurements of all bifurcation types and is based on the minimum energy hypothesis while Murray and area-preservation models are in agreement with experimental measurements for bifurcations with daughter diameter ratio (i.e., small daughter diameter/large daughter diameter) ≤0.25 and Finet model is in agreement for bifurcations with daughter diameter ratio ≥0.75. CONCLUSIONS The HK model provides a comprehensive rule for the percutaneous reconstruction of the diameters of diseased vessels and has a physical basis.

Capillary Perfusion and Wall Shear Stress Are Restored in the Coronary Circulation of Hypertrophic Right Ventricle

It has been shown that right ventricle (RV) hypertrophy involves significant compensatory vascular growth and remodeling. The objective of the present study was to determine the functional implications of the vascular growth and remodeling through a full flow analysis of arterial tree down to first capillary segments. A computer reconstruction of RV branches including the proximal right coronary artery to the posterior descending artery was established based on measured morphometric data in arrested, vasodilated porcine heart. The flows were computed throughout the reconstructed trees based on conservation of mass and momentum and appropriate pressure boundary conditions. It was found that the flow rate was significantly increased in large epicardial coronary arteries in hypertrophic as compared with control hearts but normalized in the intramyocardial coronary arteries and smaller vessels in RV hypertrophy primarily because of the significant increase in number of arterioles. Furthermore, the wall shear stress was restored to nearly homeostatic levels throughout most of the vasculature after 5 weeks of RV hypertrophy. The compensatory remodeling in RV hypertrophy functionally restores the perfusion at the arteriolar and capillary level and wall shear stress in most of larger vessels. This is the first full analysis of coronary arterial tree, with millions of vessels, in cardiac hypertrophy that reveals the compensatory adaptation of structure to function.

Biophysical Model of the Spatial Heterogeneity of Myocardial Flow

The blood flow in the myocardium has significant spatial heterogeneity. The objective of this study was to develop a biophysical model based on detailed anatomical data to determine the heterogeneity of regional myocardial flow during diastole. The model predictions were compared with experimental measurements in a diastolic porcine heart in the absence of vessel tone using nonradioactive fluorescent microsphere measurements. The results from the model and experimental measurements showed good agreement. The relative flow dispersion in the arrested, vasodilated heart was found to be 44% and 48% numerically and experimentally, respectively. Furthermore, the flow dispersion was found to have fractal characteristics with fractal dimensions (D) of 1.25 and 1.27 predicted by the model and validated by the experiments, respectively. This validated three-dimensional model of normal diastolic heart will play an important role in elucidating the spatial heterogeneity of coronary blood flow, and serve as a foundation for understanding the interplay between cardiac mechanics and coronary hemodynamics.

The Scaling of Blood Flow Resistance: From a Single Vessel to the Entire Distal Tree

Although the flow resistance of a single vessel segment is easy to compute, the equivalent resistance of a network of vessel segments or the entire vasculature of an organ is difficult to determine in an analytic form. Here, we propose what we believe is a novel resistance scaling law for a vascular tree (i.e., the resistance of a vessel segment scales with the equivalent resistance of the corresponding distal tree). The formulation can be written as (R(s)/R(c)) proportional, variant(L(s)/L(c)) (where R(s) and L(s) are the resistance and length of a vessel segment, respectively, and R(c) and L(c) are the equivalent resistance and total length of the corresponding distal tree, respectively), which was validated for the coronary vascular systems of the heart. The scaling law was also shown to apply to the vascular systems of the lung, mesentery, muscle, eye, and so on. The novel resistance scaling law, coupled with the 3/4-power scaling law for metabolic rates, can predict several structure-function relations of vascular trees, albeit with a different exponent. In particular, the self-similar nature of the scaling law may serve as a diagnostic tool with the help of noninvasive imaging modalities.