Transmural Variation and Anisotropy of Microvascular Flow

Conductivity in the Rat Myocardium

AMY F. SMITH,1 REBECCA J. SHIPLEY,2 JACK LEE,3 GREGORY B. SANDS,4 IAN J. LEGRICE,4

and NICOLAS P. SMITH3,5

1Oxford Centre for Collaborative Applied Mathematics, Mathematical Institute, University of Oxford, Woodstock Road,Oxford OX2 6GG, UK; 2Department of Mechanical Engineering, University College London, Torrington Place, London WC1E7JE, UK; 3Department of Biomedical Engineering, St. Thomas’ Hospital, King’s College London, London SE1 7EH, UK;

4Department of Physiology, Faculty of Medical and Health Sciences, The University of Auckland, Private Bag 92019,Auckland 1142, New Zealand; and 5Faculty of Engineering, The University of Auckland, Private Bag 92019, Auckland 1142,

New Zealand

(Received 3 February 2014; accepted 3 May 2014)

Associate Editor Dan Elson oversaw the review of this article.

Abstract—Transmural variations in the relationship betweenstructural and fluid transport properties of myocardialcapillary networks are determined via continuum modelingapproaches using recent three-dimensional (3D) data on themicrovascular structure. Specifically, the permeability tensor,which quantifies the inverse of the blood flow resistivity ofthe capillary network, is computed by volume-averaging flowsolutions in synthetic networks with geometrical and topo-logical properties derived from an anatomically-detailedmicrovascular data set extracted from the rat myocardium.Results show that the permeability is approximately ten timeshigher in the principal direction of capillary alignment (the‘‘longitudinal’’ direction) than perpendicular to this direc-tion, reflecting the strong anisotropy of the microvascularnetwork. Additionally, a 30% increase in capillary diameterfrom subepicardium to subendocardium is shown to translateto a 130% transmural rise in permeability in the longitudinalcapillary direction. This result supports the hypothesis thatperfusion is preferentially facilitated during diastole in thesubendocardial microvasculature to compensate for theseverely-reduced systolic perfusion in the subendocardium.

Keywords—Myocardial blood flow, Microvascular networks,

Transmural functional differences, Homogenization, Darcy

flow, Flow conductivity.

INTRODUCTION

During systole, myocardial contractions and highleft-ventricular pressure produced by cardiac pumpfunction induce significant compression of the suben-

docardial capillaries. This increased flow resistivityreduces perfusion8 which, combined with a highermetabolic demand, leads to a higher risk of ischemia inthe subendocardium than in subepicardial tissue.9

However, the exact link between transmural variationsin microvascular structure and flow have not yet beenresolved experimentally at the capillary scale.31 In thispaper, we seek to bypass these experimental limitationsby exploiting theoretical techniques to predict thetransmural gradient in tissue-scale flow properties di-rectly from microvascular network geometry. This isachieved by simulating flow in synthetic networks withgeometrical properties derived from 3D anatomicaldata extracted from the rat myocardium.

Detailed information on the 3D spatial arrangementof the coronary microvasculature has recently beenobtained via ex vivo vascular casting, high-resolutionimaging and automated post-processing technolo-gies.12,17 However, since these methods are currentlyonly possible for small tissue sections, many vessels(more than 104 in the data set of Lee et al.17) are bi-sected at the sample borders. Thus meaningful bloodflow simulations within these microvascular networksrequire prior knowledge of suitable boundary condi-tions (BCs). The definition of these boundary values ishindered by the limits of in vivo imaging and mea-surement techniques, with capillary flow measurementsrestricted to the endocardial surface15 and pressuresrecorded only in vessels >100 lm in diameter.4

The strong influence of capillary BCs on network-scale flow properties was recently demonstratedby Lorthois et al.,18 who conducted Poiseuille flow

Address correspondence to Nicolas P. Smith, Faculty of Engi-

neering, The University of Auckland, Private Bag 92019, Auck-

land 1142, New Zealand. Electronic mail: np.smith@auckland.ac.nz

Annals of Biomedical Engineering (� 2014)

DOI: 10.1007/s10439-014-1028-2

� 2014 Biomedical Engineering Society

simulations in a reconstructed 3D cerebral microvas-cular network. Enforcing zero-flux BCs at boundarycapillaries, effectively assuming the network wasindependent of neighboring arteriolar supply regions,minimized the total blood flow entering the network.Uniform pressure capillary BCs, chosen to satisfy zeronet flux out of the capillaries and thus requiring thatnetwork outflow was balanced by inflow from neigh-boring regions, maximized the total inflow (0.33 vs.2.36 mL min21 g21 with a discharge hematocrit ofHD = 0.4 at the main arterioles). Hence the linkbetween microvascular network geometry and effectivefluid transport continues to be difficult to elucidate.

Continuum-based mathematical modeling approacheswhich are independent of the choice of BCs seek to ad-dress this issue. Applying this approach, Shipley andChapman25 employed mathematical averaging tech-niques to derive amacroscopicDarcy flowmodel, whichensures computational tractability whilst capturing thekey capillary-scale fluid dynamics within the micro-scopic network geometry. In this paper, we define per-meability as the inverse of flow resistivity through themicrovascular network (analogous to electrical con-ductivity), rather than referring to the leakage of fluidthrough capillary walls. Under this definition, the per-meability tensor is a tissue-scale metric which links thepressure gradient and the averaged fluid velocity. Thepermeability tensor is calculated by volume-averagingthe flow solution to a periodic boundary value problemdefined on sub-units which represent the microvascularnetwork.

The formulation of this mathematical averagingmethod requires these sub-units to be spatially-peri-odic. Although microvascular networks in livingorganisms are unlikely to have a precisely self-repeat-ing structure, we hypothesize that capillary networks inthe myocardium are approximately periodic. This isjustified by observations of significant spatial organi-zation in these networks, which are composed of long,parallel (‘‘longitudinal’’) capillaries connected byshorter anastomoses.1,13 In this study, the theoreticalmodeling framework developed by Shipley andChapman25 is employed to compute a physiologically-meaningful permeability tensor for the 3D microvas-cular data set of Lee et al.17 This is achieved byparameterizing synthetic networks (which are spatiallyperiodic to comply with the modeling framework) withgeometrical statistics from the discrete data.

The distinctive anisotropy of the coronary capillar-ies, which is hypothesized to play a key role inmicrovascular function within the rhythmically-con-tracting myocardium in vivo,14,17 is incorporated intothis model through an anisotropic permeability tensor,which is calculated via a principal component analysis(PCA) on discrete sections of the network data. The

calculated principal values quantify the degree ofmicrostructural anisotropy. Capillaries in the data setare grouped into three types (longitudinal, or in-sheet/sheet-normal cross-connecting) according to theiralignment with the principal axes, and the distributionsof lengths and diameters for each capillary type areused as inputs for generating the synthetic networks.

Finally, calculation of the permeability tensor fordiscrete sections of the data enables quantification of thedegree of transmural gradient in tissue-scale flow prop-erties as a result of anatomical variation, which isimportant in the context of the increased risk of ischemiain the subendocardium. It is predicted that diastolicpermeability is higher in the subendocardium than thesubepicardium, which is hypothesized to compensate inpart for the strong reduction in subendocardial perfu-sion during systole as a result of increasedmicrovascularresistance produced by compression of the embeddedcoronary vasculature during cardiac contraction.

MATERIALS AND METHODS

Structural Properties of the Coronary Capillaries

The methods presented in this study, summarized inFig. 1, were implemented in MATLAB R2013a andapplied to the rat microvascular data set, which wasobtained ex vivo by perfusing the adenosine-dilatedvasculature of the rat myocardium with a corrosioncasting mixture at a perfusion pressure of 80 mmHg.Confocal laser scanning microscopy, high-precisionsectioning and automated image processing techniqueswere employed by Lee et al.17 to digitally reconstructa �2 mm3 section of the vasculature at sub-micronresolution (see Fig. 2). In this data set, the x, y and z-axes correspond to the circumferential, transmural(endocardium to epicardium) and apex-base direc-tions. The region below y = 600 lm was neglectedbecause of large regions absent of vessels, likely due tothe uneven surface typical of the endocardium.

Since a gold-standard reference data set for theradius-detection algorithm was unavailable and theextracted diameters were found to be significantlylarger than those reported previously in the rat,21,22,29

the approach of Secomb et al.24 was applied to thenetwork data. Specifically, a scaling factor was appliedto all diameters to match the mean of diametersreported for the rat myocardium (5.1 lm),21,22,29

obtaining a standard deviation (SD) of 2.1 lm. Afterthis scaling, the capillary volume fraction was 5.7%,similar to measurements in the dog myocardium.19

Firstly, themain arterioles and venules were excludedfrom subsequent analysis by employing a geometry-based vessel classificationmethod (see Smith26). In brief,this algorithm distinguishes branching trees from an

SMITH et al.

interconnected capillary mesh by stepping through thenetwork in a sequence that depends on both branchingorder and vessel diameter, and then identifying loopswithin the network. A PCA weighted by vessel lengthwas performed on the remaining capillaries to quantifythe anisotropy in capillary orientation and identify theaxes best describing this alignment.5 The Cartesiancomponents of the covariancematrixwere computed forthe set of vectors from the start node to the end node ofeach capillary segment (unbranched section of capil-lary). The normalized eigenvalues of this covariancematrix, sorted in order of descendingmagnitude, are theprincipal values {k1, k2, k3} which indicate the propor-tion of the variance in capillary orientation accountedfor by each axis. The corresponding eigenvectors of thecovariance matrix are the principal axes {e1, e2, e3},giving the principal directions of capillary alignment. Tocapture the gradually-varying capillary alignment, thedata was discretized into five sections transmurally, five

sections in the circumferential direction and two sectionsin the apex-base direction, to yield sub-blocks ofdimension 363 9 329 9 336 lm3. The principal axesand corresponding principal values were computed forthe capillaries within each section.

Capillaries were grouped into three types accordingto their spherical polar angles (u, h) with respect to thelocal principal axes (see Fig. 3). The regions occupiedby each capillary type were specified by critical anglesuc and hc: longitudinal capillaries (CL) satisfied |h| £ hcand |u| £ uc; cross-connecting capillaries were sub-categorized into in-sheet (CS) capillaries (|h| £ hc and|u|> uc) or sheet-normal (CN) capillaries (|h|> hc).The mean and SD of lengths and diameters for eachcapillary type were recorded, and the sensitivity ofthese metrics to uc and hc was investigated.

Calculation of the Permeability Tensor

Mathematical averaging (‘‘homogenization’’) tech-niques were employed to predict tissue-scale flowproperties following the approach of Shipley andChapman,25 who derived equations for effective fluidand drug transport in vascular networks. In the currentstudy it was assumed that vessels were healthy i.e.,non-leaky, so that there was no interstitial flow, and ano-slip, no-flux BC was applied for the blood velocityat capillary walls. Assuming well-separated capillary(micro) and tissue (macro) length-scales and startingwith viscous-dominated Stokes flow at the capillaryscale, an asymptotic expansion was performed for theblood velocity and pressure in terms of the ratio oflength scales. In brief, it was deduced that the flowsolution was given by a linear superposition of con-tributions proportional to the tissue-scale pressuregradient in each of the principal directions ej. Theweightings of these contributions were calculated bysolving a periodic boundary value problem in terms of

FIGURE 1. Workflow diagram of the methods applied in this paper.

FIGURE 2. The 3D rat coronary microvascular block of Leeet al.,17 colored by diameter (lm). This figure was producedusing CMGUI, a 3D visualization software package available todownload via http://www.cmiss.org/cmgui/.

Flow Conductivity in the Coronary Microcirculation

the capillary-scale variations in pressure, Pj, andvelocity, w

j, defined on periodic sub-units (‘‘micro-cells’’) of the capillary network.

An analytical solution to this micro-cell problemwas found by assuming that capillary diameters weretypically much less than the length of the micro-cell(lubrication scalings) and by assuming that the flow ineach capillary was radially-symmetric. In 2D (which isjustified later) this led to the following expression forthe micro-cell flux qm

j (in lm5 s kg21), defined as wj

integrated over the cross-section of capillary m:

qjm ¼pD4

m

128lm

DPjm

Lmþ ej � em

� �em; j ¼ 1; 2; ð1Þ

where Dm, Lm, and lm are the diameter, length andviscosity respectively, DPj

m is the drop in micro-cellpressure Pj along the vessel length, and em is the unitvector directed down the center-line of capillary m.This is a modified Poiseuille law, with an extra forcingterm ej � em due to the tissue-scale pressure gradient. Toaccount for the non-Newtonian rheology at this scale,the in vivo viscosity law of Pries and Secomb23 wasused with a constant discharge hematocrit ofHD = 0.45 to give lm as a function of Dm. Conserva-tion of micro-cell flux at nodes and periodicity of mi-cro-cell flux and pressure at boundary nodes wereenforced. To obtain a unique solution, the volumeaverage of the micro-cell pressure was defined to bezero. Together these conditions led to two (for j = 1, 2)linear systems of equations in terms of the micro-cellpressure Pj at each node, which was solved usingstandard linear solvers.

The permeability tensor K, defined as the volume-average of micro-cell velocity wj, transmits capillary-scale flow properties within specific micro-cell net-works to the tissue-scale fluid transport equation(Darcy’s Law):

u ¼ �K � $p; ð2Þ

where u is the volume-averaged tissue-scale bloodvelocity and $p is the tissue-scale pressure gradient(note that u and $p are not calculated in this paper).The permeability tensor K = [K1 K2] was calculatedvia:

Kj ¼ K1j

K2j

� �¼ 1

Xj jXMm¼1

qjmLm; ð3Þ

where |X| is the micro-cell domain volume, M is thetotal number of vessel segments, and qm

j is obtained interms of the micro-cell pressure solution Pj via Eq. (1).The permeability components K1j and K2j represent thevolume-averaged flow in the e1 and e2 directionsrespectively due to a forcing in the ej-direction.

Construction of Synthetic Networks

To parameterize physiologically-based micro-cellswhich are constrained to be spatially periodic asrequired by the homogenization framework, the inbuiltstructural organization of the coronary capillaries wasexploited. Specifically, synthetic spatially-periodic mi-cro-cell networks were constructed by sampling fromthe morphological data (lengths and diameters) col-lected from the three capillary types. Within theseabstracted networks only longitudinal (CL) and in-sheet cross-connecting (CS) capillaries were incorpo-rated into the micro-cell networks, since a large pro-portion (quantified in ‘‘Results’’ section) of capillarieslay within a smoothly-varying two-dimensional mani-fold following the sheet architecture of cardiacmuscle,16 with longitudinal capillaries preferentiallyoriented in the direction of the muscle fibers.17 Thusthe synthetic networks were constructed to be two-dimensional (2D) with coordinates 0<X<Xmax and0<Y<Ymax, embedded in a 3D domain with thick-ness given by the mean length of CN capillaries (seeFig. 4).

Each individual segment was assumed to be cylin-drical with constant diameter along its length.19 Lon-gitudinal capillaries running the entire length of themicro-cell, parallel to the X-axis (referred to as ‘‘ele-ments’’) were linked by perpendicular CS segments.The placement of longitudinal elements was deter-mined, followed by the location of CS linking neigh-boring elements. Each section of longitudinal elementbetween successive nodes (segment junctions) was

FIGURE 3. Definition of spherical polar angles u and h withrespect to the principal axes {e1, e2, e3} for a sub-section ofthe microvascular block. Vessels are colored by diameter(lm).

SMITH et al.

defined to be a CL segment. A typical micro-cell net-work is shown in Fig. 4.

The separation of longitudinal elements is equiva-lent to the length of CS linking element i to elementi + 1, and determined by sampling from a log-normaldistribution with mean and SD given by CS lengthsextracted from the data set (see Algorithm 1 in Sup-plementary Material). Starting at Y = 0, this proce-dure was repeated to find the Y-coordinate Yi ofelement i until the height of the micro-cell Ymax wasreached, at which stage a final element was placed atYn = Ymax.

Following this step, the exact placement of cross-connections CS were determined (see Algorithm 2 inSupplementary Material). Starting at X0 = 0 on ele-ment 1, the X-coordinate of the jth CS, Xj, was given byXj21 plus sampled length Hj taken from a log-normaldistribution with mean m and standard deviation s, sothat this segment has start node at (Xj, Y1) and endnode at (Xj, Y2). This was repeated until X = Xmax

was reached, with the same procedure then iterated forCS starting on the elements at Y = Y2, Y = Y3, etc.until the element at Y = Yn was reached. The period-icity assumption requires that boundary segments alignat opposite sides of the micro-cell: CL segments run-ning in parallel across the length of the micro-cellautomatically satisfy this condition; however CS seg-ments starting on the final element at Y = Yn mustconnect to the first element at Y = Y1 (shown in yellowin Fig. 4). To avoid the placement of CS segmentsunrealistically close to each other a minimum separa-tion of 1.41 lm (the minimum segment length in thedata set) between nodes was enforced.

A connectivity matrix was created recording thestart and end nodes of each segment (defining the

direction of positive flow in subsequent simulations).The sampled lengths Hj do not translate exactly to themean and SD of CL segment lengths, because nodes onelement imark the junction with CS linking to elements(i 2 1) and also (i + 1). For this reason, Hj weresampled from a log-normal distribution with meanm = 2 9 38.0 lm and SD s = 2 9 29.0 lm. The net-work periodicity was accounted for when recordingboundary segment lengths: on each element the CL

boundary segments (shown in red in Fig. 4) arerecorded as one segment, and similarly for the CS

boundary segments (shown in yellow). Diameters wereassigned to each CL and CS segment by sampling froma log-normal distribution using the correspondingmean and SD for each capillary type. The micro-cellproblem was then solved (as detailed in ‘‘Calculationof the Permeability Tensor’’ section) on 1000 synthet-ically-generated networks. The sensitivity of the per-meability tensor K to micro-cell length Xmax and heightYmax and to the input geometrical properties of themicro-cell networks was quantified in order to high-light the important parameters in determining tissue-scale flow properties.

Transmural Functional Variations

Transmural variations in permeability K werecomputed by constructing 1000 micro-cell networksusing geometrical data collected over discrete x and zsections in terms of the y-coordinate of the mid-pointof each discrete slice (corresponding to the transmuraldistance).

RESULTS

Results of PCA

The discretized principal axes e1 and e2 exhibited asmooth variation across the microvascular block (seeFig. 5). The first principal axis e1 was oriented roughlyparallel to the epicardial surface, angled toward theapex at the endocardium and twisting to point moretoward the base near the epicardium. Meanwhile the e2direction pointed generally toward the apex but,moving from endocardium to epicardium, twistedfrom pointing toward the endocardium to lyingapproximately parallel to the epicardial surface. Theprincipal values (mean ± SD) were k1 = 0.74 ± 0.05,k2 = 0.18 ± 0.04 and k3 = 0.08 ± 0.02. The distribu-tion of angles u and h for capillaries in one discretesection of the block is shown in Fig. 6. Using criticalthreshold angles uc = 45� and hc = 41.8� to ensurethat each region used to group capillaries had equalvolume in parameter space, the resulting geometricalstatistics for each capillary type collected over the

FIGURE 4. Example of a stochastically-generated micro-cellnetwork. Here, V1, V2 and V3 indicate the first sampled verticallengths (see Algorithm 1 in Supplementary Material) while H1,H2 and H3 indicate the first three sampled horizontal lengthson the element at Y 5 Y1. Cross-connections CS are shown inturquoise (in the interior of the micro-cell) or yellow (crossingthe periodic boundary) while CL are colored dark blue (interiorsegments) or red (boundary segments).

Flow Conductivity in the Coronary Microcirculation

whole block are summarized in Table 1. These statis-tics were not significantly sensitive to changes in criti-cal angles uc and hc, and there was no clear cut-offbetween vessel types: a 30% increase in uc led to lessthan 1% change in the mean diameters of CL and CS

segments, while the mean CN length increased by 6.1%in response to a 30% decrease in hc. The assumption ofa 2D micro-cell composed of CL and CS only is sup-ported by the data: 91.5% of capillaries were orientedwithin 41.8� of e1–e2 plane. The distribution of

600 800 1000 1200 1400 1600 1800 2000 2200−140

−120

−100

−80

−60

−40

−20

0

20

40(a)

y, μm

Ang

le w

ith x

−ax

is in

x−

y pl

ane

EPIENDO

e1

e2

600 800 1000 1200 1400 1600 1800 2000 2200−150

−100

−50

0

50

100

y, μm

Ang

le w

ith x

−ax

is in

x−

z pl

ane

EPIENDO

e1

e2

(b)

FIGURE 5. Transmural variation in orientation of discretized principal axes e1 and e2 for the microvascular block of Lee et al.17;mean (solid line) 6 SD (dotted lines) in the angles that each principal axis makes with the x-axis when projected onto (a) the x–yplane and (b) the x–z plane. Values are averaged over x- and z-sections for each transmural slice.

SMITH et al.

diameters was similar for all capillary types, while themean length of the longitudinal (CL) segments(38.0 lm) was 53% longer than that of the CS seg-ments (and 76% longer than the mean CN length). Thedistributions of lengths and diameters for all capillarytypes were approximately log-normal. The mean andSD in lengths and diameters of CL and CS capillarieswere used as inputs into the micro-cell network con-struction procedure, and the micro-cell domain thick-ness was prescribed by the mean length of CN segments(21.6 lm).

Permeability of Synthetic Networks

Appropriate micro-cell length Xmax and height Ymax

were determined by computing the permeability of1000 stochastically-generated networks for a range ofmicro-cell sizes. A large micro-cell size would challengethe assumption of length-scale separation, while asmall micro-cell size increases the sensitivity of theextracted permeability values to the stochastically-generated network structure. The mean of the K11

permeability component was more sensitive to micro-cell length than height, and increased sharply for mi-cro-cell lengths <200 lm (see Fig. 7). A micro-cell of

size 450 9 300 9 21.6 lm3 or greater was needed toobtain mean K11 and K22 values within 10% of themean values obtained with a 1000 9 700 9 21.6 lm3

micro-cell. Subsequent simulations were performed ona micro-cell of size 500 9 300 9 21.6 lm3, for whichK11 = (3.3 ± 0.8) 9 1023 mm3 s kg21 and K22 =

(3.1 ± 1.0) 9 1024 mm3 s kg21 (K11 was 10.8 9 K22

on average), reflecting the anisotropic structure of theunderlying network. The off-diagonal permeabilitycomponents K12 and K21 were equal and close to zero((0.0013 ± 9.0) 9 1025 mm3 s kg21).

Due to the constraints imposed during the con-struction process, the distribution of segment lengthsin the stochastically-generated synthetic networks didnot exactly match those of the original data set (seeTable 2). The mean length of CL segments in the mi-cro-cell networks was 5.4% higher than in the data set(with a 5.9% higher SD), while the mean CS length was4.8% lower than in the data set (with 4.7% lower SD).These length errors are predicted to lead to a 6.4%overestimate in the mean K11 (6.3% overestimate in theSD of K11) combined with a 10.0% underestimate inthe mean of K22 (11.0% underestimate in SD).

The permeability tensor was highly sensitive tochanges in segment diameters: see Table 3. A 10% in-crease in the mean CL diameter led to 52.4% increase inmean K11 (combined with 46.0% increase in SD) and30.6% increase inmeanK22 (25.5% increase in SD). Thisstrong effect on K22 occurs because flow forced in the e2direction (i.e., when j = 2 in Eq. (1)) is diverted alongCL

as well as CS due to the off-set cross-connections, and isthus strongly affected by diameters of both capillarytypes. A 10% rise in the meanCS diameter had a similareffect on K22 (22.0% increase in the mean and 25.7%increase in SD), while less than 10% change was foundfor the mean or SD inK11. Changing the SD ofCL orCS

diameters by 10% led to less than 10% change in eitherthe mean or SD of K11 and K22, demonstrating that thevariance in diameters was much less important than themean. Increasing the input mean horizontal length (m)by 10% resulted in an 9.4% decrease in K22 (see Ta-ble 4), while a 10% rise in the mean CS length led to an9.9% drop in the mean K11.

Across all results the permeability was consistentlymuch less sensitive to changes in segment length thanto changes in the mean diameter. This is to be expectedsince the permeability is proportional to the fourthpower of diameters, but inversely proportional tocapillary length (see Eq. (1)). This relationship com-bined with the effect of a non-Newtonian viscositydirectly explains the impact of CL diameter changes tothe mean K11; however, K22 was affected by diametersof both CL and CS segments whilst also experiencing amore subtle dependence on the anisotropic networktopology.

0 20 40 60 800

20

40

60

80

φ

θ

CL

CS

CN

FIGURE 6. A scatter plot of the (absolute values of) sphericalpolar angles u and h for each capillary relative to the principalaxes. Capillaries are labeled as longitudinal (CL), in-sheet (CS)or sheet-normal (CN) according to their u and h values.

TABLE 1. Geometrical properties (mean 6 SD) of capillarytypes CL, CS and CN, collected over discrete sections of the

microvascular block of Lee et al.17

CL CS CN

N 76,508 29,414 9839

VF 71.6 21.8 6.6

L 38.0 ± 29.0 24.9 ± 15.7 21.6 ± 12.3

D 5.0 ± 1.9 5.2 ± 1.9 5.3 ± 2.0

N is the number of capillaries, VF is the percentage of the total

capillary volume occupied by each type, L is the length (lm) and D

is the diameter (lm).

Flow Conductivity in the Coronary Microcirculation

Transmural Variations

The principal values did not exhibit a correlationwithtransmural distance, indicating that the level of struc-

tural anisotropy does not vary from the subendocardi-um (ENDO) to the subepicardium (EPI). The capillarydensity (number of capillaries per mm3) was 100.2%higher in ENDO than EPI on average over all capillarytypes, thoughmost significantly the density of capillarieswas 131.5% higher in the mid-myocardium in compar-ison to the EPI (see Fig. 8a). Figure 8b) shows the sig-nificant increase in mean diameter from EPI to ENDO(a 30.1% rise averaged over the three capillary types),confirming the findings of May-Newman et al.19 MeanCL lengths increased slightly with transmural depth

100 200 300 400 500 600 700 800 900 10000

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80

Micro−cell Length, μm

Per

cent

age

erro

r in

mea

n K

11

100 μm300 μm500 μm700 μm

Micro−cell Height

FIGURE 7. Percentage error in mean K11, relative to the mean K11 for a micro-cell of dimensions 1000 3 700 3 21.6 lm3. Simu-lations in this study were performed for micro-cells of 500 3 300 3 21.6 lm3, for which the percentage error was 6.4%.

TABLE 2. Mean and SD of length and diameter of CL and CS

segments, averaged over 1000 stochastically-generated mi-cro-cell networks.

Capillary type CL CS

Length (lm) 40.0 ± 30.7 23.7 ± 14.9

Diameter (lm) 5.0 ± 4.2 5.2 ± 4.3

TABLE 3. Sensitivity of permeability components K11 andK22 to 610% changes in the mean and SD of diameter for CL

and CS segments with N 5 105.

% change in diameter

% change in permeability

K11 K22

Mean SD Mean SD

CL Mean 210 234.5 233.3 224.6 222.0

+10 +52.4 +46.0 +30.6 +25.5

SD 210 +5.2 +1.7 +4.2 +2.5

+10 25.1 21.9 24.0 21.9

CS Mean 210 25.0 24.7 219.7 222.1

+10 +4.5 +4.4 +22.0 +25.7

SD 210 +0.8 +0.5 +3.9 +3.3

+10 21.1 21.0 23.7 22.4

TABLE 4. Sensitivity of permeability components K11 andK22 to 610% changes in the mean and SD of length for CL and

CS segments with N 5 105.

% change in length

% change in permeability

K11 K22

Mean SD Mean SD

CL Mean 210 21.8 23.9 +10.1 +12.3

+10 +1.6 +3.4 29.4 210.0

SD 210 +0.1 20.6 +5.9 20.8

+10 20.3 +0.1 25.2 +0.3

CS Mean 210 +11.9 +12.4 26.4 27.6

+10 29.9 29.9 +5.7 +6.3

SD 210 20.7 25.7 20.6 23.0

+10 +0.3 +4.9 +0.2 +4.1

SMITH et al.

(a 10.6% rise from 36.2 to 40.0 lm); the length of CS

segments was slightly lower in the mid-myocardium(24.2 lm) than peripheral regions (26.1 lm in the EPI);themeanCN length increased33.3%fromEPI toENDO.Using these results as inputs to the micro-cell networksrevealed the strong dependence of the permeability ontransmural location (see Fig. 8c). The mean of K11

increased 130% from EPI (2.0 9 1023 mm3 s kg21) toENDO (4.6 9 1023 mm3 s kg21); similarly, the mean ofK22 ranged from 2.3 9 1024 mm3 s kg21 up to4.9 9 1024 mm3 s kg21 (an increase of 117%).

DISCUSSION

The PCA conducted in this paper confirmed that the3D structure of the coronary capillary network ishighly anisotropic. Data extracted from this analysiswas used to compute the network permeability tensorvia flow solutions in microvascular sub-units (‘‘micro-cells’’). In this way the link between this structuralanisotropy and tissue-scale fluid transport propertieswas quantified, showing that the permeability in thedirection of longitudinal capillaries was 10.89 largerthan in the direction of cross-connecting capillaries.The distribution of capillary diameters was a crucialdeterminant of tissue-scale perfusion, with changes tothe mean diameter of longitudinal segments having thestrongest effect on the permeability tensor.

Previously, Wieringa et al.,34 Beard and Bassi-ngthwaighte,2 and Goldman and Popel7 studied similarspatially-periodic synthetic networks composed oflong, parallel capillaries with cross-connections; how-ever, capillaries were assigned uniform diameters. Thesynthetic networks developed here had large SD incapillary diameters, and although a 10% change to thediameter SD had little effect, if the SD of diameterswere set to zero (i.e., imposing uniform diameters in CL

and CS segments), the mean K11 and K22 would be 60.6and 95.8% higher respectively. Non-zero diametervariance reduces the permeability due to local pressuregradients which force flow to deviate along the cross-connections rather than following the shortest pathacross the micro-cell, hence increasing the total net-work resistance. Note that an approximate estimate ofK can be obtained via a simple calculation using thecapillary density and mean capillary diameter andlength, which predicts similar values but with alarger transmural range (K11 = (0.9–6.1) 9 1023 mm3

s kg21, K22 = (2.9–16.2) 9 1024 mm3 s kg21) mostlikely because this does not account for the non-zerodiameter SD which is a feature of the micro-cell net-works.

One limitation to this model is that a persistence indiameters between neighboringCL capillaries in the data

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FIGURE 8. Transmural variation in (a) capillary density (cap-illaries per mm3) for CL, CS and CN segments; (b) mean diameter(percentage change relative to subepicardial values) for CL, CS

and CN segments; and (c) permeability components K11 and K22

(percentage change relative to subepicardial values). The y-coordinate corresponds to the mid-point of the discrete sectionof capillaries in the data set used to compute the principal axes.

Flow Conductivity in the Coronary Microcirculation

set was not replicated in the micro-cell networks. Theratio of diameters at bifurcations was 1.05 ± 0.36 in thedata, compared with 1.14 ± 0.63 in the micro-cells. Toinvestigate the effect of this error, new micro-cells weregenerated inwhich the firstCL segment diameter on eachlongitudinal element was sampled as before, but sub-sequent CL segment diameters were a specified ratio(sampled from a distribution obtained from the data) ofthe previous neighboring segment’s diameter. OutputCL diameters were 5.0 ± 3.0 lm (with SD significantlyhigher than 1.9 expected from the data), while K11 was(4.46 ± 1.73) 9 1023 mm3 s kg21. For comparison,micro-cells were generated without diameter correlationbut with CL diameters sampled from the distributionobtained above, yielding a larger diameter ratio of1.41 ± 1.41 and K11 = (2.41 ± 0.73) 9 1023 mm3

s kg21. The mean K11 was 54.0% of the value obtainedwith diameter correlation, indicating that by neglectingdiameter correlation in the micro-cell networks we mayhave been underestimatingK11 (although this differencemay be exaggerated by the high diameter SD).However,incorporating this feature in a manner which is alsoconsistentwith the expected SDof diameters appears farfrom straightforward but would be an interesting andpotentially important extension to this work.

The simple, 2D micro-cell construct assumed in thispaper with cross-connections at 90� to the longitudinalelements agrees with the ‘‘H’’-type of branching, whichwas the most common capillary branching typeobserved by Kassab and Fung.13 A clear extension tothe model which is beyond the scope of this paperwould be to construct 3D micro-cell networks includ-ing CN segments. Since CN had similar geometricalproperties to CS but approximately one third thecapillary density, it is predicted that the permeability inthe sheet-normal direction (K33) could be approxi-mately 1024 mm3 s kg21, which while small couldcontribute to tissue-scale flow if there were a largepressure gradient in this direction.

Due to experimental limitations, there is very littlequantitative information about tissue-level pressuregradients within the myocardium in any of the coor-dinate directions. A non-negligible transmural pressuredifference (corresponding roughly to the sheet-normaldirection) in vessels of diameter >100 lm wasreported by Chilian,4 though this may be due in part tothe spatial location of terminal arterioles supplying thecapillary network. A key advantage of the approachpresented here is that K is de-coupled from tissue-scalepressure gradients (which are implicit in coarse-scaleexperimental measurements of myocardial blood per-fusion) and so could be used in poroelastic models todescribe the potential flow capacity of the capillarynetwork.

In this study, variations in the geometrical proper-ties of the capillary network were related to transmuralfunctional differences. The mean capillary diameterwas �30% larger in the subendocardium than thesubepicardium, contributing to a significant transmu-ral increase in the computed permeability (�130% inthe longitudinal direction). The method presented hereprovides a novel technique for combining sophisticatedmodeling techniques with microstructural statistics topredict flow properties from anatomical data. Thepermeability tensor in itself provides fundamentalinformation about (the inverse of) network flow resis-tivity. In addition, these permeability results could beused to parameterize tissue-scale models of volume-averaged flow in the myocardium.3,10,11,27

Since the data used in this paper was obtainedex vivo with relaxed cardiac muscle and fully-dilatedvessels, the reconstructed vasculature is most repre-sentative of the diastolic phase of the cardiac cycle.The results of this study support the hypothesis thatsubendocardial network permeability is significantlyhigher than subepicardial permeability during diastole,most likely in order to compensate (to some extent) forthe reduced systolic perfusion, and the associatedhigher risk of ischemia, in the subendocardium.30 Thepermeability was shown here to be strongly dependenton capillary diameters, and so would be expected toreduce significantly in the subendocardium in responseto increased microvascular constriction as a result ofsystolic myocardial contraction. The methods devel-oped in this paper could be used in future to quantifythe microvascular contribution to systolic flowimpediment in the context of active myocardial con-traction.27

Although capillaries cannot actively dilate or con-strict, flow across the capillary network in the livingmyocardium depends strongly on driving pressures ofupstream terminal arterioles, which respond to vas-oregulatory signals. However in microvascular disease,coronary flow reserve is impaired and arterioles arealmost fully dilated at rest.20 In this context, K repre-sents an important metric of passive flow conductivityonce the vasoactive response is exhausted. Thisframework provides a tool which could in future beused to quantify the effects of structural changes in-duced by microvascular diseases such as diabetes,27

which are important for understanding whole-organcoronary flow in the clinic.

These permeability calculations can be validatedusing experimental quantifications of myocardialblood flow (MBF). The perfusion (lm3 s21) into a gi-ven region of the myocardium is obtained by inte-grating the Darcy velocity over a cross-sectional area A(lm2):

SMITH et al.

Perfusion ¼ �ZA

K � $pð Þ � n dA; ð4Þ

where n is the unit vector normal to A. If n is parallel tothe principal axis e1 and the pressure gradient is givenby a uniform pressure drop4 of Dp = 15 mmHg overan arteriolar–venular path length28 of l = 310 lm,then

Perfusion ¼ K11 �Dpl� 0:133� 10�3 � A; ð5Þ

where the factor 0.133 9 1023 converts from mmHg tokg lm21 s22, and K11 is in lm3 s kg21. Alternatively,

Perfusion ¼MBF

60� q� A� l; ð6Þ

where the factor of 60 converts from min to s, andq = 1.053 g mL21 is the myocardial density.32

Substituting Eqs. (6) into (5) yielded an estimatedMBF in the range 2.4–5.5 mL min21 g21 dependingon transmural depth. Using magnetic resonanceimaging via a spin-labeling technique, Waller et al.33

measured a MBF of 3.5 mL min21 g21 in the rat,which is within this predicted range.

This study has revealed the transmural variation inthe orientation of the myocardial capillary network,which can be characterized by capillaries aligned withthe longitudinal, sheet or sheet-normal directions.Combining mathematical averaging techniques withdetailed anatomical data enabled the extraction of apermeability tensor, which is a physiologically relevantmetric for characterizing myocardial perfusion. Thispermeability tensor was shown to be highly sensitive tocapillary diameters, which exhibit a strong dependenceon transmural location. Future research will involvecomparing a continuum-based Darcy flow modelparameterized by these permeability values to a flowsolution within the discrete vascular network estimatedvia the optimization procedure of Fry et al.,6 whichconstrains capillary shear stress and pressure to withina physiological range.

ELECTRONIC SUPPLEMENTARY MATERIAL

The online version of this article (doi:10.1007/s10439-014-1028-2) contains supplementary material,which is available to authorized users.

ACKNOWLEDGMENTS

The authors acknowledge support from the VirtualPhysiological Rat Project (NIH1 P50 GM094503-1),the EPSRC (Engineering and Physical Sciences

Research Council) under grant numbers EP/F043929/1and EP/G007527/2, and Award No. KUK-C1-013-04made by King Abdullah University of Science andTechnology (KAUST). The authors would also like tothank Prof. Timothy W. Secomb (University of Ari-zona) for helpful scientific discussions.

REFERENCES

1Bassingthwaighte, J. B., T. Yipintsoi, and R. B. Harvey.Microvasculature of the dog left ventricular myocardium.Microvasc. Res. 7:229–249, 1974.2Beard, D. A., and J. B. Bassingthwaighte. Advection anddiffusion of substances in biological tissues with complexvascular networks. Ann. Biomed. Eng. 28:253–268, 2000.3Chapelle, D., J.-F. Gerbeau, J. Sainte-Marie, and I. E.Vignon-Clementel. A poroelastic model valid in largestrains with applications to perfusion in cardiac modeling.Comput. Mech. 46:91–101, 2010.4Chilian, W. M., S. M. Layne, E. C. Klausner, C. L. Eas-tham, and M. L. Marcus. Redistribution of coronarymicrovascular resistance produced by dipyridamole. Am. J.Physiol. Heart. Circ. Physiol. 256:H383–H390, 1989.5Cookson, A. N., J. Lee, C. Michler, R. Chabiniok, E.Hyde, D. A. Nordsletten, M. Sinclair, M. Siebes, and N. P.Smith. A novel porous mechanical framework for model-ling the interaction between coronary perfusion and myo-cardial mechanics. J. Biomech. 45:850–855, 2012.6Fry, B. C., J. Lee, N. P. Smith, and T. W. Secomb. Esti-mation of blood flow rates in large microvascular net-works. Microcirculation 19:530–538, 2012.7Goldman, D., and A. S. Popel. A computational study ofthe effect of capillary network anastomoses and tortuosityon oxygen transport. J. Theor. Biol. 206:181–194, 2000.8Goto, M., A. E. Flynn, J. W. Doucette, C. M. Jansen, M.M. Stork, D. L. Coggins, D. D. Muehrcke, W. K. Husseini,and J. I. Hoffman. Cardiac contraction affects deep myo-cardial vessels predominantly. Am. J. Physiol. Heart. Circ.Physiol. 261:H1417–H1429, 1991.9Hoffman, J. I. E. Transmural myocardial perfusion. Prog.Cardiovasc. Dis. 29:429–464, 1987.

10Hyde, E. R., R. Chabiniok, D. A. Nordsletten, and N. P.Smith. Parameterisation of multi-scale continuum perfu-sion models from discrete vascular networks. Med. Biol.Eng. Comput. 51:557–570, 2013.

11Hyde, E. R., A. N. Cookson, J. Lee, C. Michler, A. Goyal,T. Sochi, R. Chabiniok, M. Sinclair, D. Nordsletten, J.Spaan, J. P. van den Wijngaard, M. Siebes, and N. P.Smith. Multi-scale parameterisation of a myocardial per-fusion model using whole-organ arterial networks. Ann.Biomed. Eng. 42:797–811, 2014.

12Kaneko, N., R. Matsuda, M. Toda, and K. Shimamoto.Three-dimensional reconstruction of the human capillarynetwork and the intramyocardial micronecrosis. Am. J.Physiol. Heart. Circ. Physiol. 300:H754–H761, 2011.

13Kassab, G. S., and Y. C. B. Fung. Topology and dimen-sions of pig coronary capillary network. Am. J. Physiol.Heart. Circ. Physiol. 267:H319–H325, 1994.

14Kassab, G. S., K. N. Le, and Y. C. B. Fung. A hemody-namic analysis of coronary capillary blood flow based onanatomic and distensibility data. Am. J. Physiol. Heart.Circ. Physiol. 277:H2158–H2166, 1999.

Flow Conductivity in the Coronary Microcirculation

15Kiyooka, T., O. Hiramatsu, F. Shigeto, H. Nakamoto, H.Tachibana, T. Yada, Y. Ogasawara, M. Kajiya, T. Mor-imoto, Y. Morizane, S. Mohri, J. Shimizu, T. Ohe, and F.Kajiya. Direct observation of epicardial coronary capillaryhemodynamics during reactive hyperemia and duringadenosine administration by intravital video microscopy.Am. J. Physiol. Heart. Circ. Physiol. 288:1437–1443, 2005.

16LeGrice, I. J., B. H. Smaill, L. Z. Chai, S. G. Edgar, J. B.Gavin, and P. J. Hunter. Laminar structure of the heart:ventricular myocyte arrangement and connective tissuearchitecture in the dog. Am. J. Physiol. Heart. Circ. Phys-iol. 269:H571–H582, 1995.

17Lee, J., S. Niederer, D. Nordsletten, I. LeGrice, B. Smaill,D. Kay, and N. Smith. Coupling contraction, excitation,ventricular and coronary blood flow across scale andphysics in the heart. Philos. Trans. R. Soc. A 367:2311–2331, 2009.

18Lorthois, S., F. Cassot, and F. Lauwers. Simulation studyof brain blood flow regulation by intra-cortical arterioles inan anatomically accurate large human vascular network:Part I: methodology and baseline flow. NeuroImage54:1031–1042, 2011.

19May-Newman, K., O. Mathieu-Costello, J. H. Omens, K.Klumb, and A. D. McCulloch. Transmural distribution ofcapillary morphology as a function of coronary perfusionpressure in the resting canine heart. Microvasc. Res.50:381–396, 1995.

20McDonagh, P., and J. Y. Hokama. Microvascular perfu-sion and transport in the diabetic heart. Microcirculation.7:163–181, 2000.

21Poole, D. C., S. Batra, O. Mathieu-Costello, and K.Rakusan. Capillary geometrical changes with fiber short-ening in rat myocardium. Circ. Res. 70:697–706, 1992.

22Potter, R., and A. Groom. Capillary diameter and geom-etry in cardiac and skeletal muscle studied by means ofcorrosion casts. Microvasc. Res. 25:68–84, 1983.

23Pries, A. R., and T. W. Secomb. Microvascular bloodviscosity in vivo and the endothelial surface layer. Am. J.Physiol. Heart. Circ. Physiol. 289:H2657–H2664, 2005.

24Secomb, T., R. Hsu, N. Beamer, and B. Coull. Theoreticalsimulation of oxygen transport to brain by networks of

microvessels: effects of oxygen supply and demand on tis-sue hypoxia. Microcirculation. 7:237–247, 2000.

25Shipley, R. J., and S. J. Chapman. Multiscale modelling offluid and drug transport in vascular tumours. Bull. Math.Biol. 72:1464–1491, 2010.

26Smith, A. F. Multi-Scale Modelling of Blood Flow in theCoronary Microcirculation. DPhil Thesis, University ofOxford, 2013.

27Smith, N. P., and G. S. Kassab. Analysis of coronary bloodflow interaction with myocardial mechanics based onanatomical models. Philos. Trans. R. Soc. A 359:1251–1262, 2001.

28Toborg, M. The microcirculatory bed in the myocardiumof the rat and the cat. Z. Zellforsch. 123:369–394, 1972.

29Tomanek, R. J., J. C. Searls, and P. A. Lachenbruch.Quantitative changes in the capillary bed during develop-ing, peak, and stabilized cardiac hypertrophy in the spon-taneously hypertensive rat. Circ. Res. 51:295–304, 1982.

30van de Hoef, T. P., F. Nolte, M. C Rolandi., J. J. Piek, J. P.van den Wijngaard, J. A. Spaan and M. Siebes. Coronarypressure-flow relations as basis for the understanding ofcoronary physiology. J. Mol. Cell. Cardiol. 52:786–793,2012.

31van den Wijngaard, J. P. J. C. Schwarz, P. van Horssen, M.G. van Lier, J. G. Dobbe, J. A. Spaan, and M. Siebes. 3Dimaging of vascular networks for biophysical modeling ofperfusion distribution within the heart. J. Biomech. 46:229–239, 2013.

32Vinnakota, K. C., and J. B. Bassingthwaighte. Myocardialdensity and composition: a basis for calculating intracel-lular metabolite concentrations. Am. J. Physiol. Heart.Circ. Physiol. 286:1742–1749, 2004.

33Waller, C., E. Kahler, K. H. Hiller, K. Hu, M. Nahrendorf,S. Voll, A. Haase, G. Ertl, and W. R. Bauer. Myocardialperfusion and intracapillary blood volume in rats at restand with coronary dilatation: MR imaging in vivo with useof a spin-labeling technique. Radiology 215:189–197, 2000.

34Wieringa, P. A., J. A. E. Spaan, H. G. Stassen, and J. D.Laird. Heterogeneous flow distribution in a three dimen-sional network simulation of the myocardial microcircula-tion—a hypothesis. Microcirculation 2:195–216, 1982.

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