Multi-Scale Parameterisation of a Myocardial Perfusion Model Using Whole-Organ Arterial Networks

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Annals of Biomedical EngineeringThe Journal of the BiomedicalEngineering Society ISSN 0090-6964Volume 42Number 4 Ann Biomed Eng (2014) 42:797-811DOI 10.1007/s10439-013-0951-y

Multi-Scale Parameterisation of aMyocardial Perfusion Model Using Whole-Organ Arterial Networks

Eoin R. Hyde, Andrew N. Cookson,Jack Lee, Christian Michler, AyushGoyal, Taha Sochi, Radomir Chabiniok,Matthew Sinclair, et al.

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Multi-Scale Parameterisation of a Myocardial Perfusion Model Using

Whole-Organ Arterial Networks

EOIN R. HYDE,1 ANDREW N. COOKSON,2 JACK LEE,2 CHRISTIAN MICHLER,2 AYUSH GOYAL,2 TAHA SOCHI,2

RADOMIR CHABINIOK,2 MATTHEW SINCLAIR,2 DAVID A. NORDSLETTEN,2 JOS SPAAN,3 JEROEN P. H. M. VAN

DEN WIJNGAARD,3 MARIA SIEBES,3 and NICOLAS P. SMITH2

1Department of Computer Science, University of Oxford, Oxford OX1 3QD, UK; 2Imaging Sciences & Biomedical EngineeringDivision, St Thomas’ Hospital, King’s College London, London SE1 7EH, UK; and 3Department of Biomedical Engineering

and Physics, Academic Medical Centre, University of Amsterdam, Amsterdam, The Netherlands

(Received 14 July 2013; accepted 20 November 2013; published online 3 December 2013)

Associate Editor Kerry Hourigan oversaw the review of this article.

Abstract—A method to extract myocardial coronary perme-abilities appropriate to parameterise a continuum porousperfusion model using the underlying anatomical vascularnetwork is developed. Canine and porcine whole-heartdiscrete arterial models were extracted from high-resolutioncryomicrotome vessel image stacks. Five parameterisationmethods were considered that are primarily distinguished bythe level of anatomical data used in the definition of thepermeability and pressure-coupling fields. Continuum multi-compartment porous perfusion model pressure resultsderived using these parameterisation methods were com-pared quantitatively via a root-mean-square metric to thePoiseuille pressure solved on the discrete arterial vasculature.The use of anatomical detail to parameterise the porousmedium significantly improved the continuum pressureresults. The majority of this improvement was attributed tothe use of anatomically-derived pressure-coupling fields. Itwas found that the best results were most reliably obtainedby using porosity-scaled isotropic permeabilities and ana-tomically-derived pressure-coupling fields. This paper pre-sents the first continuum perfusion model where allparameters were derived from the underlying anatomicalvascular network.

Keywords—Cryomicrotome, Anatomical vascular model,

Multi-compartment Darcy, Perfusion, Permeability.

INTRODUCTION

Coronary blood flow is a key determinant of cardiacfunction, and thus it is desirable to model perfusionacross all coronary vascular scales. There exist severaltypes of coronary perfusion model, which vary in the

level of anatomical vascular detail used. The majorityof these models solve for the blood flow and pressurevariables within the explicit vascular domain itself, vialumped parameter1,3 and/or reduced Navier–Stokesmethods.37,39 These discrete model types are particu-larly useful for investigating vascular wall properties41

and the rheological properties of blood.25 They arelimited, however, in terms of the number of vessels thatthe governing equations can be tractably solved on,and often have a large number of boundary conditions(BCs) which are difficult to determine—particularly ifone was to model the spatially heterogeneous flowknown to occur in the capillary bed.26

Complementary to the discrete model types, porousperfusion models have the ability to determine thepressure and flow in a continuous manner across the3D computational tissue domain.4,8,29 Darcy’s Law istypically assumed for the purposes of modelling fluidtransport within the porous domain, whereby flow isdefined to be the product of the pressure gradient andthe permeability of the medium, with flow proceedingfrom areas of high pressure to low pressure.2 Theapplication of multiple Darcy domains also providesfor an effective scale-separation of flow and materialcharacteristics (e.g., vessel wall stiffness, dynamic vis-cosity), which vary significantly with vessel diameteracross the multi-scale coronary network.26 Further-more, dynamic poroelastic perfusion models haveshown great promise, notably for reproducing knownphysiological features such as tissue swelling4 andincreased stiffness due to rising left ventricularpressure.8 Their ability to incorporate fluid-solidinteractions in a tractable manner is a key advantageover their discrete model counterparts, as such

Address correspondence to Eoin R. Hyde, Department of

Computer Science, University of Oxford, Oxford OX1 3QD, UK.

Electronic mail: [email protected], [email protected]

Annals of Biomedical Engineering, Vol. 42, No. 4, April 2014 (� 2013) pp. 797–811

DOI: 10.1007/s10439-013-0951-y

0090-6964/14/0400-0797/0 � 2013 Biomedical Engineering Society

797

Author's personal copy

mechanical ‘‘cross-talk’’ plays an important role incardiac function.44 Moreover, it should be able tocompute the model solution at a fraction of the cost ofa more traditional 1D/3D fully-coupled model giventhe large number of discrete vessels in question.

Due to a lack of experimental data, however, theseporoelastic perfusion results were obtained using asimplistic parameterisation of the vascular resistancefor the fluid components. In previous studies, weinvestigated the issue of porous domain parameteri-sation from discrete vascular models using both asynthetic network18 and a single arterial branch.8

While these studies were restricted in terms of ana-tomical realism and the volume of tissue being per-fused, they provided valuable insight into thechallenges of anatomical parameterisation, and high-lighted the importance of high quality vascular datasources. The emergence of the imaging cryomicrotomenow provides such anatomical data for the estimationof vascular resistance. Unlike other vascular imagingtechniques, such as confocal or micro-CT imaging,24,34

cryomicrotome imaging can cater for the larger organs,such as the human or porcine heart, while preservingan imaging resolution of 50 lm.15,42 The resultingcryomicrotome vessel data stack is used to createaccurate, whole-organ, discrete coronary vascularmodels14 upon which physiological porous perfusionmodel parameterisation can be based.

Thus far, however, there does not exist a provenmethod to accomplish a realistic parameterisation ofporous perfusion models using anatomical models, norhas a systematic comparison of available techniquesbeen untertaken. This paper aims to address these is-sues. It describes a method to parameterise Darcyporous domains that incorporates realistic discretearterial data, resulting in an improved Darcy pressuresolution. Several Darcy permeability parameterisationmethods are evaluated quantitatively via a comparisonof continuum and spatially-averaged discrete modelpressures, from which the best of the tested methods isclearly identified. To our knowledge, this is the firstorgan-scale porous perfusion model to be directly pa-rameterised using detailed anatomical data.

MATERIALS AND METHODS

The steps outlined below in detail include: (i) theexperimental treatment of the heart, and the creationof the vascular and myocardium models, (ii) the hier-archical partition of the vascular model and parame-terisation of the Darcy perfusion model via a variety ofmethods, and (iii) the comparison of the parameteri-sation methods to a spatially-averaged Poiseuillepressure solution, solved for on the explicit vascular

model. For a schematic illustrating the parameterisa-tion process, see Appendix A in the SupplementaryMaterials.

Vascular Model Processing

Vascular Model Creation

Experimental procedures were undertaken in strictaccordance with the Institutional Animal Care and UseCommittee (IACUC) of the University of Amsterdam.Cryomicrotome imaging datawas acquired according tostandard procedures developed at the Department ofBiomedicalEngineering andPhysics,AcademicMedicalCenter, University of Amsterdam. While the full detailscan be found in the original literature for the experi-mental study,15,42 for reasons of conciseness the processis now only briefly elucidated. The excised heart wasperfused with adenosine-loaded (100 lg/L) phosphate-buffered saline, followed by the fluorescent vascularcasting agent [Batson #17 casting resin (PolysciencesEurope, Eppelheim, Germany)] at a pressure of90 mmHg. After the cast had set, the heart was sus-pended in 5% carboxymethylcellulose solution (AlfaAesar, Karlsruhe, Germany) and frozen. It was thentransferred to the cryomicrotome and serially sliced andimaged using fluorescent optical surface imaging in arepetitive fashion until the entire organ had been pro-cessed. Two data sets were processed in this manner,with an isotropic voxel resolution of 64 lm and 50 lmfor the porcine and canine models respectively.

The resulting data stack was enhanced using ourpreviously reported procedure.14 Specifically, a multi-scale vesselness filter13 was applied to the data stack,which was then binarised using a combination ofSauvola local adaptive and Otsu global threshold-ing.31,35 Non-vessel structures were removed and con-nected component analysis conducted to extract thefully-connected vascular network.14 This structure wasskeletonised via distance-ordered homotopic thin-ning33 and subsequently pruned to remove any spuri-ous branches. The vessel radii were evaluated at theskeleton voxels using a model-based radius estimationtechnique.14 As a simplifying assumption, cylindricalvessel segments were fitted to this data to provide amathematical model of the anatomical vasculature (seeFig. 1 for a schematic of the process and Appendix Bin the Supplementary Materials for information on theextent of vessel tapering). These vascular models arerepresentative of the coronary diastolic state.

A corresponding finite element left ventricle (LV)myocardial mesh was required to provide the compu-tational domain for solving the Darcy perfusionmodel, as the LV is the primary site of myocardialinfarction. To construct this mesh, the ‘‘white-light’’

HYDE et al.798


cryomicrotome data stack was used as the basis fromwhich a binary mask of the tissue domain was manu-ally segmented. Rather than segment the full cryomi-crotome data stack, which consists of thousands ofimages, the data stack was downsampled uniformly toa manageable size (50 images). These images weremanually segmented using the open-source imageanalysis tool, ‘‘itk-SNAP’’ (http://www.itksnap.org/).The resulting binary mask was used as input to ouronline binary-cubic Hermite finite element mesh gen-erator,23 which automatically fitted a cubic Hermitetemplate to the binary mask. Finally, a tetrahedralmesh of our desired resolution was constructed usingthe exterior surfaces of the Hermite mesh as input intothe meshing software NetGen (http://www.hpfem.jku.at/netgen/). Specifically, the canine (porcine) mesh had15,162 (25, 548) tetrahedra with an average edge lengthof 4.09 (2.82) mm. For a figure depicting the variousstages of this process, see Appendix C in the Supple-mentary Materials.

Vascular Model Reduction

The whole-organ vascular model features atrial andpapillary muscle vessels, which lie outside the LV tissuemodel. Such vessels, and more specifically their distal

terminal nodes, represent a serious challenge with re-gards to unknown BCs. They would also constitute aconfounding factor in our efforts to construct a faircomparison between the discrete and continuummodels. Thus, to reduce these possible ambiguities, thewhole-organ vascular model was reduced to the vesselsinvolved in perfusing the LV tissue model only.

This was done by constructing the path of leastresistance (PLR) from each terminal node to the rootnode, i.e., the most proximal node. Dijkstra’s algorithmwas employed for this purpose, using the Poiseuilleresistance as the cost function for each vessel.11 Theportion of each such PLR consisting of the first vesselinternal to the tissue model encountered from the ter-minal node through to the root node was added to thetotal reduced vascular model. To avoid erroneouslyremoving collateral vessels lying between any twoPLRs,a final post-processing step added any vessels that laycompletely within the LV tissue model (see Appendix Din the Supplementary Materials for a schematic of thisprocess). This reduced model is henceforth denoted asthe left ventricular model (LVM), and this vascularmodel is suitable for the characterisationof theLV tissuemodel only. The vessels removed in this process are notlocal to the tissue model, and as such would not affectthe Darcy permeability fields even if they were not

FIGURE 1. The top(bottom) row features short(long)-axis projections from the porcine vascular model creation process. (a, b)Maximum intensity projections of the original cryomicrotome imaging data stack. Note the varying intensity and globular struc-tures where leakage of the casting agent occurred. (c, d) Projections of the enhanced and binarised vascular tree, post-vesselnessfiltering. (e, f) After skeletonisation and radius estimation, a coronary arterial network is created. The radius field is displayed on alinear colour scale from 0 to 1.8 mm (blue–red), with an alpha channel applied to all radii <0.2 mm to enable the visualisation of thelarger arteries.

Multi-Scale Parameterisation of a Myocardial Perfusion Model 799


http://www.itksnap.org/

http://www.hpfem.jku.at/netgen/

http://www.hpfem.jku.at/netgen/

removed. Note that the porcine model does not have aRight Coronary Artery (RCA) due to difficultiesencountered during the casting process. Thus, to effec-tively remove this portion of the computational domainfrom the simulation, zero pressure was assigned to finiteelement mesh nodes in this region. This reduction pro-cess yielded arterial networks with 42,752 and 27,416vessels for the canine and porcine LVM, respectively(Fig. 2). Of the vessels removed from the canine (por-cine) model, 26% (29%) were atrial vessels, theremainder being papillary muscle vessels. Some indica-tivemorphological distributions for the canine LVMarepresented in Fig. 3.

1D Model Component

A proximal portion of the LVM was designated asthe 1D model component, and its vessels were not used

to parameterise the Darcy permeability or pressure-coupling fields. Instead, the 1D model component de-fines the source fields driving the flux within the Darcymodel i.e., si in Eq. (1b). To this end, a Poiseuille flowsolution was solved on the entire LVM to produce avessel-defined volumetric flux field on the entire net-work. The flux exiting the terminal vessels of the 1Dmodel component was then incrementally added to theassociated element of the finite element tissue mesh,thereby defining the element-wise volumetric sourcefields.

A variable yet robust method of defining this 1Dmodel component was required such that its size can beeasily adjusted, according to the needs of the specificapplication or available data. The chosen method ofextraction was to fix a threshold radius value, denoteds, and then determine the set of all LVM nodes thatreside within the LV tissue model and have a radius

FIGURE 2. Processed vascular models after reduction to only those vessels perfusing the LV tissue model. The canine model(left), and the porcine model (right), consisting of 42,752 and 27,416 vessels, respectively.

FIGURE 3. Canine reduced vascular model morphological distributions. Displayed are the length (a) and radius (b) segmentaldata. Both variables were fitted to log-normal distributions, the Probability Density Functions (PDF) of which are illustrated by theblue lines. Coronary arterial lengths are known to be log-normally distributed as is the radii of the coronary microcirculation.20

Inset figures display the original variable histograms.

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r ‡ s, denoted S(s). The 1D model component wasthen defined to be the union of the vessels from eachPLR from each node of S(s) to the root node of theLVM (Fig. 4).

Multi-Compartment Static Darcy Perfusion Model

Model Equations

The multi-compartment static Darcy system is anextrapolation of the single compartment Darcy modelto N porous domains and has been applied to severalperfusion modelling problems.8,18,29,43 The Darcy sys-tem for compartment i 2 [1, N] is (Einstein summa-tion is not in use)

wi þ Ki � rpi ¼ 0 in X; ð1aÞ

r � wi ¼ si �XN

k¼1bi;kðpi � pkÞ in X; ð1bÞ

where subscripts i and k are compartment indices, K isa permeability tensor of the material, w and p denotethe Darcy velocity and pore pressure respectively, s is avolumetric source field with units of mm3 s21, and b isthe array of inter-compartment pressure-couplingcoefficients with units of mm3 Pa21 s21, and unit fluiddensity is assumed. Note that bi;k 2 Rþ0 and bi,k = bk,ifor i, k = 1, …, N in order to conserve fluid massacross the system [see Eq. (6)]. In this manner, volu-metric source terms allowed for inter-compartmentflow scaled by the relevant pressure differences,whereas intra-compartment flow represented the flowfor the set of vessels within a defined scale range (seeSection ‘‘Parameterisation of K and b’’). Equations(1a, 1b) were set on an open bounded domain X � Rnd

with spatial dimension nd and a piece-wise smooth

boundary @X; upon which zero flux BCs were en-forced. In the case of our static domain X; this implieda compatibility condition of

XN

i¼1

Z

X

si dv ¼ 0; ð2Þ

where sN is a volumetric sink field representing thevenous system, and is discussed in Section ‘‘BoundaryConditions.’’

Note that the permeability tensor field, K; is theconstant of proportionality linking the Darcy velocity,w; to the pressure gradient. K is a material property ofthe porous medium, and it has some key propertiesthat are related to the physical constraints of the sys-tem2: (i) K is symmetric, by the Onsager’s reciprocalrelation. In other words, this ensures that there is nobias in the material and a reversal in the pressuregradient should bring about a similar reversal in flow,and (ii) K is positive definite. This property ensuresthat the direction of flow is always in the same direc-tion as the pressure drop i.e., fluid flows from volumesof high pressure to lower pressure.

Continuum model simulations were solved numeri-cally with the finite element method, using linear La-grange pressure basis functions over tetrahedralelements and applying a reduced Darcy formulation.29

Compartmentalisation of Vascular Network

The capillary vessels are below the resolution of thedata and thus do not feature in the LVM. Due to thislack of explicit data, the capillary compartment wasparameterised using a simple isotropic permeabilityfield of 0.1 mm2 Pa21 s21.4 Note that the subsequentresults were found to be insensitive to this parameterwithin the range 0.01–1.0 mm2 Pa21 s21. Experimental

FIGURE 4. The porcine left ventricular vascular model is shown in each subplot, with the various proximal 1D componentsdisplayed in red, and defined by a radius threshold, s, of values (a) 0.2, (b) 0.25, and (c) 0.3 mm. The spatial distributions of theincoming volumetric flux fields for the continuum component, si, are dependent on the 1D model component size.



data suggests that N = 2,3 are reasonable values forthe number of Darcy compartments (see Section‘‘Limitations of Darcy Parameterisation via Anatomi-cal Networks’’ for discussion). Thus, in conjunctionwith the capillary compartment, the remaining N 2 1arterial Darcy compartments were anatomically pa-rameterised using the discrete vascular data.

The discrete vessels were partitioned, or compart-mentalised, into groups corresponding to each of theN 2 1 anatomically parameterised Darcy compart-ments via the hierarchic parameter field f.18,43 Thisfield is based on the network morphology via a nor-malised distal vessel length metric i.e., f at a LVMnode is the sum of the lengths of the distal vessels fromthat node divided by the total network’s vessel length.The use of f enforces a hierarchic partition of thenetwork i.e., given two vessels, a and b, with meanf(a) ‡ f(b), then vessel a(b) is used to parameteriseDarcy compartment i(k) where i ‡ k. In order to definef, and overcome the issue of looping pathways, eachvessel was assigned to a unique proximal vessel bymeans of its greatest incoming flux from the Poiseuillemodel simulation. For N = 2, there is a trivial parti-tion vector of f = [1, 0] i.e., all LVM vessels not in the1D model are used to parameterise the first Darcycompartment. For the case of N = 3, the set of discretevessels to be spatially-averaged is divided equallybetween the two compartments (in terms of totalnumber of vessels), to the extent that the hierarchicparameter field will allow. For example, the porcinemodel partition vector for s = 0.3 mm isf = [1, 8.67 9 1025, 0]—determined via the bisectionmethod, and it yields an almost equal split in vesselsbetween the first and second Darcy compartments(13,497 and 13,498 vessels respectively). This vascularpartition method is suitable for this study where theparameterisation process itself is being investigated.Otherwise, the point of partition with respect to fshould be guided by the specific model application.

Parameterisation of K and b

A full description of how the anatomically-derivedKi and bi,k fields were determined is provided in Hydeet al.18 Note here that anatomically-derived parame-ters are in contrast to constant, or homogeneous, pa-rameterised fields. The goal is to parameterise theDarcy permeability and pressure-coupling fields foreach porous compartment, given a LVM partition. Forthis study, a total of five parameterisation methodswere considered for the perfusion simulations. Indecreasing order of their usage of detailed vascularinformation these were—(i) HvC—permeability is de-fined by the method of Huyghe and van Campen17 andanatomically-derived b fields were applied, (ii)

/iI—porosity-scaled isotropic permeability fields andanatomically-derived b fields were applied, (iii) I—iso-tropic permeability fields and anatomically-derived bfields were applied, (iv) /iIcb—porosity-scaled isotropicpermeability fields and constant/homogeneous b fieldswere applied, and (v) Icb—isotropic permeability fieldsand constant/homogeneous b fields were applied (i.e.,no vascular data was used).

The method of Huyghe and van Campen defines apermeability tensor field K via

Kij¼p

128volRVEðxÞdx0l

X

v2nsðxÞ

d44xi4xjl

; i; j¼ 1;2;3;

ð3Þ

where volRVEðxÞ is the volume of the representativevolume element (a.k.a RVE, or spatial-averagingwindow) applied for the spatial averaging that is withinX while centred at x; dx0 is an infinitesimal element oftheir hierarchic parameter field, nsðxÞ is the set ofvessels within the hierarchic parameter range dx0 andwithin RVEðxÞ; d is the vessel diameter, l is the vessellength, l is the dynamic viscosity, and 4xi is the dif-ference in spatial coordinate i between the vessel endpoints. The remaining field used in the permeabilityparameterisation is the porosity of the medium. Theporosity of compartment i, denoted /i, is defined to be

/iðxÞ ¼P

v2nsiðxÞ volv

volRVEðxÞ; ð4Þ

where volv is the volume of the vessel v. The overallporosity of the material is

/fðxÞ ¼XN

i¼1/iðxÞ: ð5Þ

The pressure-coupling fields between any two Darcycompartments are defined as

bi;kðxÞ ¼0 if piðxÞ � pkðxÞ ¼ 0;

Qi;kðxÞjpiðxÞ�pkðxÞj

otherwise;

(ð6Þ

where Qi,k is the spatially-averaged mass flux betweenvessel groups associated to compartments i and k, andpi is the spatially-averaged Poiseuille pressure given by

piðxÞ ¼P

v2nsiðxÞ Pv volvPv2nsiðxÞ volv

; ð7Þ

where Pv is the average nodal pressure for the vthvessel within nsiðxÞ: Note that this definition yields aspatially heterogeneous continuum model field that isrepresentative of the local Poiseuille pressure values.

As per Hyde et al.,18 each Darcy porous domainparameter field was individually scaled (via a global

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parameter sweep followed by a local Newton–Raphsonprocedure) in order to produce the minimal discrep-ancy between the total pressure fields from the con-tinuum and discrete models. For example, in the N = 2case, three scales are determined that multiply the twoDarcy permeability tensor fields and the b1,2 pressure-coupling field. This quantitative comparison was car-ried out via a root-mean-square (rms) metric. Firstly, atotal pressure field was defined by the cumulativeporosity-weighted compartment pressures i.e.,

a0 ¼XN�1

i¼1/i � ai; ð8Þ

where a is p or p for the spatially-averaged discrete orcontinuum pressures respectively. Secondly, theporosity-weighted spatially-averaged discrete pressurefield p0 was then normalised between 0 and 1, and thesame normalising transformation was applied to theporosity-weighted continuum pressure field, p0. Thediscrepancy was then defined to be

4 ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiPx2N ðp0ðxÞ � p0ðxÞÞ2

#N

s

; ð9Þ

where N is the set of FEM nodal coordinates withp0 ‡ 3 kPa. Note that a pressure comparison waschosen in preference to a velocity comparison in orderto avoid potential confounding factors such as an RVEwith non-zero Poiseuille flow having spatially-averagedzero flow.

All system parameters were determined at the tet-rahedral mesh nodes via the parameterisation pro-cesses outlined, using a spatial-averaging window ofsize 1.5 times the average finite element edge lengths(see Hyde et al.18 for a detailed study on model sensi-tivity to the RVE size).

Boundary Conditions

As stated previously, zero flux BCs were applied on@X; and the flow was driven by volumetric sourcefields, si. These source fields were derived from thePoiseuille flow through the LVM, which had pre-scribed pressure BCs of 12.5 kPa at the root node, and3 kPa at its terminal nodes. The si are also dependenton the 1D model component size as determined via s.Specifically, the flux through each terminal vessel ofthe 1D model component was taken to be the flux fromthe corresponding vessel of LVM, and these fluxeswere spatially mapped to the corresponding finite ele-ment from the LV myocardial mesh to form the sourcefields. To ensure consistency amongst 1D modelcomponents, for a given value of s the terminal fluxeswere linearly scaled such that their total was equal tothe total flux through the entire LVM. Finally, a

pressure-dependent sink term, compatible with thediscrete model BCs, was applied for sN of the form

sN ¼ �0:1ðpN � 3000Þ; ð10Þ

similar to Michler et al.29

Poiseuille Flow Model

A flow solution derived solely from the explicitnetwork was used as a common reference againstwhich the continuum solutions were compared. ThePoiseuille flow model was chosen as it is the leastcomplex flow model that supplies the pressure and flowfields throughout the vasculature, it is compatible withthe HvC parameterisation method (the derivation ofwhich assumes the Poiseuille conductance of flowwithin the vessels), and it is arguably the natural dis-crete analogue of Darcy flow through a rigid domain.

In brief, for each discrete vessel the flow rate, Q, iscalculated via

Q ¼ C4p; where C ¼ pr4

8ll: ð11Þ

In this equation, C is the Poiseuille conductance of thevessel, which is dependent on the vessel radius,r, length, l, and the dynamic viscosity of the fluid, l.The pressure difference between the vessel end-pointsis denoted by 4p: The dynamic viscosity was assumedto be constant throughout the vasculature, at a valueof 0.0035 Pa s. The transport of fluid mass at eachvascular junction was conserved via

X

i2T;j2SLijQj ¼ 0; ð12Þ

where T is the set of internal nodes, S is the set ofvessels, and the entries of the matrix L are defined to be

Lij ¼þ1; if for node i; segment j is incoming;�1; if for node i; segment j is outgoing;0; if node i is not part of segment j:

8<

:

ð13Þ

This linear system of equations was solved for thePoiseuille pressure at the LVM vascular nodes, andsegmental flux can be calculated using Eq. (11). Thespatially-averaged Poiseuille pressure field, p; was thendetermined via Eq. (7).

RESULTS

Comparison to Experimental Data

It is extremely difficult to get pressure and/or flowdata on the scale of the microvasculature due to the sizeconstraints of most measuring devices. However, it is



possible in some limited circumstances to obtain suchdata from epi- and sub-epicardial vessels. For example,the work of Chilian et al.7 reports experimental mea-surements of pressure in vessels less than 400 lm indiameter. In this study, the researcherswere interested indetermining the active sites of vasodilation due toendogenously-produced adenosine in the beating myo-cardium of anesthitised cats (n = 24). To investigate theredistribution of coronarymicrovascular resistance theysought to detail microvascular pressure under controland extra-adenosine conditions, via a complex system ofmicropipette movement control and high-frequencystroboscopic imaging.6,7 To induce the extra-adenosinecondition, dipyridamole was administered (0.4 mg kg21

min21, iv) which caused an approximately 40-fold in-crease in myocardial adenosine levels.

This pressure data is used to define Dirichlet pres-sure BCs on the distal terminal nodes of the porcinearterial network, and to compare the total networkpressures to the same data for a qualitative assessment.To this end, the arterial pressure data contained in Fig.3 of Chilian et al.7 was extracted using bespoke data-extraction software. As recommended by the authors,a sigmoidal function was chosen to produce a contin-uous fit to this discrete data. Specifically, the parame-ters of the following function were optimised withrespect to the experimental data:

Sðx;a;bÞ¼ 100 1� 1

1þ e�aððx=100ÞþbÞ

� �� ; x2 ½�400;0�:

ð14Þ

The optimised parameters were found to be a = 1.0057,b = 0.7407, and this fitwas calculated using a non-linearfitting function from a commercial software package(MATLAB R2012b, The MathWorks Inc., Natick,MA, 2000). The extracted experimental data and theoptimally fitted sigmoidal curve, S, are shown inFig. 5a. The root pressure was assigned a value of14.6 kPa (see Table 2 of Chilian et al.7), and theremaining terminal pressureswere assigned according toS. The resulting Poiseuille pressure field is visualised onthe network geometry in Fig. 5b. To compare the sim-ulated pressure from the network, all nodes with diam-eters in the range 0–0.4 mm (the experimental data didnot report pressure data for vesselswith diameter greaterthan 0.4 mm) were grouped in consecutive bins of width0.04 mm and their average pressure per bin was calcu-lated. The simulated pressures are directly compared tothe experimental data in Fig. 5a.

Continuum Porosity Field

It is important to consider the spatial distribution ofthe continuum porosity field for two reasons—(i) to

investigate if there is an observable physiological trendfor this field with respect to some spatial gradientwithin the heart, either transmural or apex-base, and(ii) to assess the quality of the vascular model, bycomparing our data against reported values in the lit-erature. For clarity of presentation, the results pre-sented in this section are for the porcine N = 2,s = 0.3 mm model as (i) the porcine heart is moresimilar to the human heart than the canine heart interms of vascular anatomy,9 and (ii) this selection of Nand s yields the largest number of vessels to be spa-tially-averaged across all porcine model scenarios.

The porosity was evaluated at the finite elementmesh nodes as per Eq. (5) (see Fig. 6a). Quantitativeanalysis was undertaken to investigate potential dif-ferences in porosity amongst the septum, the inferiorLV, and the lateral LV (the anterior LV was notconsidered due to the lack of an RCA model). Themyocardium was further discretised into equal trans-mural sections representing the epicardium, the mid-myocardium, and the endocardium. The calculatedregional mean arterial porosities (see Fig. 6b) areindicative of a physiological difference between septaland non-septal LV arterial morphology. Furthermore,the differences in mean porosity between the inferiorand septal regions, and the lateral and septal regionswere statistically significant (using a one-sample t test,p< 0.05). The average arterial porosity across all threeregions was 4.8%. No substantial apical-base porositygradient was found.

Effect of Anatomical Parameters

We consider 12 model scenarios, consisting of amodel variation of each combination of N = 2,3,s = 0.2, 0.25, 0.3 mm and the canine or porcine cor-onary model. The dominant factor in terms of reducingthe rms discrepancy is the presence of the anatomi-cally-derived b fields (Fig. 7). The effect of removinganatomical data from the b parameterisation (/I to/Icb) yielded an average increase of 9.66 ± 6.11 timesthe rms achieved when anatomical data were not usedto parameterise the K fields but anatomically-derived bfields were used (/I to I). This clearly demonstrates theimportance of anatomically-derived b fields. Overall,the parameterisation methods which did not apply theparameterised b fields did not sustain locally highpressure gradients, and this feature is the primarycause of the increase in rms. For an example of thisfeature, a typical subset of the canine results is dis-played in Fig. 8 and other relevant fields such as thecompartmental velocities, porosities and pressures canbe seen in Appendix E of the Supplementary Materials.

With regards to K parameterisation, the porosity-scaled isotropic method (/I) is demonstrably the most

HYDE et al.804


reliable in terms of producing the minimum rms dis-crepancy, obtaining the minimum in 11/12 of themodel scenarios (see Section ‘‘Limitations of DarcyParameterisation via Anatomical Networks’’ for dis-cussion of this finding, and Appendix F in the Sup-plementary Materials for histograms detailingproperties of the parameterised K fields).

Model Sensitivity to s and N

We wish to gauge the sensitivity of the perfusionmodel results to the remaining two parameters ofinterest—specifically s which determines the size of the1D model component, and the number of Darcycompartments, N. Considering first the /I parame-terisation method results (Fig. 7), the average increase

in rms going from s = 0.2 mm to s = 0.3 mm was0.49 ± 4.53%. Such a relatively small change in rms isindicative of a high level of robustness to the modelresults with respect to s, at least across the tested rangeof 0.2–0.3 mm. The addition of an extra Darcy com-partment also yielded a reduction in rms, with theminimum rms decreasing by 21% and 2% for the ca-nine and porcine models respectively.

To provide further quantitative detail, Table 2shows the determined parameter field scales and keycompartment-averaged statistics for the HvC parame-terisation method. Several consistent findings can beobserved from this data: (i) increasing s causes an in-crease in the average non-zero nodal value for s1, as thesame fluid influx is entering the porous domain viafewer finite elements; (ii) the scale on K1 decreases with

Vessel Diameter ( m)

Pre

ssu

re(m

gH

g)

Comparison to Experimental Data

Fitted CurveExp. DataSim. Data

(a) (b)

FIGURE 5. (a) Sigmoidal curve (green line) fitted to the data (black circles) reported in Chilian et al.7 Negative diameters corre-spond to the arterial system, as per the original experimental report. The simulated pressures averaged by radii bins are alsoshown (blue dots). (b) Poiseuille flow pressure solution on the porcine anatomical network using the experimental pressureboundary conditions derived from the sigmoidal curve in (a).

TABLE 1. Effect of various actions on different coronary vascular ranges.

Action Diameter range A Diameter range B Animal References

Dipyridamole >170 lm

67% decrease in CVR

170–150 lm*

92% decrease in CVR

Cat 7

Nitroglycerin 300–100 lm

90% relaxation

100–60 lm

No effect

Pig 36

Coronary arterial occlusion >150 lm

11.9% constriction

<150–10 lm

10.4% dilation

Dog 10

Critical stenosis >150 lm

6.9% constriction

<150–10 lm

15% dilation

Dog 10

Coronary perfusion pressure (60 mmHg) >100 lm

No effect

<100–10 lm

17% dilation

Dog 19

Coronary perfusion pressure (40 mmHg) >100 lm

13% constriction

<100–10 lm

11% dilation

Dog 19

*Venous as opposed to arterial vessel diameter.



increasing s as there are now an increased number oflarge conduit vessels included in the compartment 1vessel group from which the permeability is derived;

and (iii) the scales the compartment 2 parameter fieldsare less sensitive to s changes than the compartment 1scales, as expected.

FIGURE 6. Overview of the porosity field for the spatially-averaged porcine coronary arterial tree. Three short-axis slices aredisplayed at the base, the equator, and the apex (a). Three distinct regions were considered for quantitative analysis—the septum,the inferior LV, and the lateral LV. The myocardium was further discretised into equal transmural sections representing theepicardium, the mid-myocardium, and the endocardium. The observed regional mean porosities (b) are indicative of a physio-logical difference between septal and the remaining LV arterial morphology.

TABLE 2. Tables displaying various quantitative results from the continuum model simulations using the HvC parameterisationmethod (see Fig. 7). Each animal model has a total of six simulation results to report on, corresponding to variations in the numberof porous compartments used and the size of the 1D model component. The optimised variables displayed in the tables are: ki, thescale on Ki i.e., the anatomically-derived permeability tensor field for compartment i; and bi, the scale on bi,i+1 i.e., the anatomi-cally-derived inter-compartment, or pressure-coupling, field between compartments i and i + 1. The derived data displayed in thetables are: bi ;j ; the average value for the bi,j field; jKi j; the average value for the determinant of the Ki field; and s1; the averagenon-zero value for the compartment 1 source field.

Canine model

N2 3

s 0.2 0.25 0.3 0.2 0.25 0.3

k1 1.81e+00 1.21e+00 1.07e+00 1.11e+01 7.08e+00 6.44e+00

k2 – – – 1.00e+00 1.00e+00 1.00e+00

b1 1.04e203 1.05e203 1.05e203 3.78e201 3.87e201 3.90e201

b2 – – – 8.02e201 8.02e201 8.02e201

b1;2 4.49e202 4.62e202 4.87e202 1.23e201 1.23e201 1.22e201

b2;3 – – – 1.44e204 1.37e204 1.34e204

jK1j 1.14e207 1.14e207 1.14e207 1.14e207 1.14e207 1.14e207

jK2j – – – 1.54e215 1.54e215 1.60e215

s1 4.97e+00 7.63e+00 4.98e+01 4.97e+00 7.63e+00 4.98e+01

Porcine model

N2 3

s 0.2 0.25 0.3 0.2 0.25 0.3

k1 4.98e+00 4.18e+00 2.88e+00 2.50e+01 1.90e+01 1.18e+01

k2 – – – 2.33e+00 1.76e+00 1.10e+00

b1 2.63e203 2.64e203 2.69e203 2.67e201 2.91e201 3.17e201

b2 – – – 3.26e201 4.37e201 7.26e201

b1;2 1.12e+00 1.04e+00 1.03e+00 2.79e+00 2.82e+00 2.91e+00

b2;3 – – – 2.63e202 2.56e202 2.51e202

jK1j 1.13e207 1.13e207 1.13e207 1.13e207 1.13e207 1.13e207

jK2j – – – 9.73e212 1.05e211 1.20e211

s1 2.18e+01 3.20e+01 4.98e+01 2.18e+01 3.20e+01 4.98e+01

HYDE et al.806


DISCUSSION

In this study, a method has been proposed for theincorporation of whole-organ high-resolution ana-tomical vascular networks within a continuum perfu-sion model via spatial-averaging techniques. Thismethod accounts for the morphology and orientationof vessels across multiple scales of the vascular model,and it provides for a natural scale separation of flowand anatomical characteristics. It has been shown thatthe inclusion of vascular data significantly improvesthe continuum perfusion results in comparison to asimplistically parameterised model. The results forboth the canine and porcine models were definitivewith regards to this matter, as shown in Fig. 7 and thequalitative images of Fig. 8. In particular, the use ofanatomically-derived b fields is crucial for the forma-tion of pronounced pressure gradients. The parame-terisation method consisting of the porosity-scaledisotropic K fields and the anatomical b fields per-formed the best with respect to the minimum rmsdiscrepancy, producing the minimum rms discrepancy

in 11/12 model scenarios. Of course one cannot excludethe possibility that some future parameterisationmethod not included in this study could produce a K/bfield combination that results in a lower rms error.Even in this hypothetical scenario, we now have a testsuite of methods and a systematic approach for eval-uating potential parameterisation methods. We believethat this study represents a substantial step towards ananatomically parameterised porous perfusion model.However, the quantitative results also point to key is-sues associated with using experimentally-derivedvascular models to parameterise the Darcy component.

Limitations of Darcy Parameterisation via AnatomicalNetworks

The creation of an anatomical vascular model fromthe experimental data is a difficult and complex pro-cess.14,15 The occurrence of an error or artefact duringthis process could lead to the failure to extract allvessels within the theoretical range established by theimaging modality. If an extraction failure occurs at a

FIGURE 7. Array of plots comparing the various parameterisation methods used. The rms error, 4; compares the continuum tothe discrete porosity-weighted pressure fields. The total number of Darcy compartments applied is N, and the 1D model compo-nent size is determined by the variable s. There is a marked decrease in 4 when anatomically-derived pressure-coupling fields areused (methods HvC;/I and I) as opposed to the methods that assume homogeneous pressure-coupling fields (methods /Icb andIcb).



proximal portion of the network (or indeed if anunsuitably small s value is applied) then a region ofartificially high volumetric source and low permeabil-ity would result. This lack of local permeability impliesthat the mass influx could only dissipate through‘‘vertical’’ transport (i.e., mass transport to anotherDarcy compartment), potentially causing an unphysi-ological spike in pressure. This phenomenon would beexaggerated in the N = 2 scenarios as the only othercompartment is then the capillary compartment, whichhas a prescribed distributed pressure sink term therebyconstraining the pressure field for this compartment.Thus all efforts should be made to perfect the vascularmodel creation process.

More importantly, the HvC method failed to pro-duce the minimum rms discrepancy in all but onemodel scenario (see Fig. 7 for the canine N = 2,s = 0.3 mm case). This is contrary to the observationsin Hyde et al.18 where this method consistently yieldedthe best results. The principal difference between thatstudy and this present work is the form of the vascularmodel. Unlike a synthetic network constructed in arepeatable and robust manner with determinable net-

work statistics, the anatomically-derived networks areless regular. A statistical analysis of the HvC perme-ability field determinants for the porcine case N = 2,s = 0.3 mm (i.e., the case with the worst performanceof the HvC method) revealed that there were 10.8%mild outliers (defined to be >Q3 + 1.5 9 IQR whereQ3 is the 75th percentile and IQR is the inter-quartilerange of the determinant field respectively) and 5.5%extreme outliers (defined to be >Q3 + 3 9 IQR).These extreme outliers were all associated with rela-tively small length vessels. While the cause of some ofthese small length vessels was traced to the skeletoni-sation of the imaging data stack, some also appearedto be genuine features of the vasculature, observable inthe original data images. As an illustrative example,imagine the scenario of two adjacent bifurcations.Now allow the vessel connecting these bifurcations todecrease in length. As the two bifurcations tend to-wards a single trifurcation, the connecting vessel lengthtends towards zero, and thus by Eq. (11) the Poiseuilleconductance, C, tends to infinity. In this fashion, wecan appreciate how a minor change to the network interms of the discrete model can have a tremendous

FIGURE 8. Qualitative results for the canine model scenario of two anatomically parameterised Darcy compartments and the 1Dmodel created using s 5 0.25 mm. (a) The porosity-weighted spatially-averaged Poiseuille pressure field. (b) The Darcy solutionwith the minimum rms error of the methods tested was given by the porosity-scaled isotropic permeability with anatomi-cally-derived pressure-coupling fields method (i.e., method /I). (c) The Darcy solution with the largest rms error of the methodstested was given by the most basic method, which used no discrete vascular data at all (i.e., method Icb). To further demonstratethe improvements in the model parameterisation by applying the discrete data, the bottom row shows surface plots of the absolutedifferences. (d) is the difference between (a) and (b), whereas (e) is the difference between (a) and (c).

HYDE et al.808


effect on the spatially-averaged permeability whenapplying the HvC method. Our results suggest that amodification of the HvC method is required if ana-tomical vascular data is to be used.

The discrete flow solution is highly sensitive to thenetwork radii, due to the formulation of Poiseuilleconductance, C. Any error in the estimated vessel radii,however, will impact both the discrete and continuumpressure solutions in a similar manner, and thus theresults of this study with respect to the parameterisa-tion process are valid. Moreover, there is strong evi-dence as to the efficacy of the vascular extraction andradii estimation techniques applied, such as our earlierin silico and phantom-based studies.14 Further evi-dence is provided by the comparison to availableexperimental data in Section ‘‘Comparison to Experi-mental Data.’’ Caution should be exercised whencomparing unequal network preparations, however, asthe experimental pressures were measured in vivo incats whereas our simulated results were computed on acryomicrotome-derived, and hence static, porcine net-work. In addition, the transmural porosity datadetermined in this study is within a reasonable physi-ological range (Fig. 6). Previous works have reportedthe total myocardial blood volume to be 12% of thetissue volume,21 with 30% of this blood volumeresiding in the arterial system.22 Thus our estimate ofarterial porosity of 4.8% supports the anatomicallyaccurate nature of the vascular models used in thisstudy. The observed transmural porosity gradient forthe LV inferior and lateral regions is expected con-sidering the known transmural gradient in flow andconductance.12,45 These porosity results are alsoindicative of a clear physiological difference betweenthe vasculature in the septum and that of the LV freewall, as has been previously noted.30

Our choice of N was motivated by the availableexperimental data with respect to changes in CoronaryVascular Resistance (CVR) distribution after variouschemical or physiological actions. The data presentedin Table 1 is not an exhaustive list, but it does indicatethat two arterial compartments are sufficient to modelthe key vasoactive agents and their effect on CVR.These two compartments could be classified as largeconduit arteries and small conduit arteries and arteri-oles, with a vascular scale boundary of approximately150lm in diameter, but this can be adjusted accordingto the available experimental data for a specific modelapplication. Regardless of the amount of functionaldata available, one must also respect the assumptionsunderlying Darcy’s Law, thereby effectively placing alimit on the number of Darcy compartments whichcould be applied using an anatomical parameterisationapproach as the number of observable discrete vesselsis currently restricted by the imaging resolution.

Alternative approaches that are found in the literaturefor large scale vessels or scales with unknown topologyare to (i) assume an isotropic constant value of per-meability,4,16 (ii) assume spatially-periodic arterial andvenous network systems in order to satisfy the math-ematical requirements of the model being applied,5,38

and (iii) the construction of discrete networks.27 All ofthese approaches allow for novel insight and are wor-thy contributions to biological engineering, yet it isknown that their approximations are not fully repre-sentative of the physiology of coronary networks.40 Inthis paper, we chose to be faithful to the underlyingnetwork topology. It is possible that we are alreadyseeing the effects of these assumptions vis-a-vis vascu-lar density, as the increase in N led to a much greaterdecrease in rms for the canine model than the porcinemodel (21 vs. 2%). This disparity could be due to thecanine vascular model having 56% more vessels rela-tive to the porcine model, due to the absence of theRCA in the latter model.

With respect to the dynamic viscosity, we chose aconstant value throughout, regardless of the vesselscale. We acknowledge that useful variable expressionsfor l of the form l(r) have been experimentallydetermined from the study of flow of red blood cellsuspensions through glass capillaries and mesenterynetworks.32 However, it was believed that the use ofsuch an expression would not significantly alter theresults of our present study given the vascular scale inquestion, as the effect of phase separation is mostprevalent in vessels of diameter less than 40 lm.32 Avarying form of the dynamic viscosity could be in-cluded within the parameterisation process, in con-junction with a higher resolution vascular modelincorporating small arterioles.

As a general limitation of the model, our study wasfocused on the static perfusion of the circulatory sys-tem only, and it did not feature any autoregulatory ormechanical components. The autoregulatory aspect inparticular poses a significant challenge for the futuremodelling of cardiac function.

Clinical Implications

In a clinical setting, the 1D model component size isdetermined by the available data. Thus, with respect topatient-specific modelling, it was important to dem-onstrate that our parameterisation process is insensi-tive with respect to reasonable 1D model componentsizes. Of particular interest are the results for thes = 0.3 mm model scenarios, as this 1D model com-ponent size is representative of clinically measurableepicardial vessels. The results of this study indicate thatour process is robust with respect to changes in s, andthat s = 0.3 mm produces a satisfactory rms level in



comparison to larger 1D model components. Thisfurther suggests that epicardial vascular modelsderived via angiography data may be adequate for ourperfusion modelling needs. Moreover, the possibility ofdefining patient-specific permeability fields via clini-cally acquired porosity data has already been noted,18

and the findings of this study with regards to the /Ipermeability parameterisation adds further credence tothis idea.

It is important to note that this study has not fo-cused on the functional role played by collateral ves-sels, and a ‘‘healthy’’ LV has been assumedthroughout. While collaterals may play a more minorrole in terms of porcine or human hypoperfusion, thisis not the case for the canine heart, where significantcollateral flow has been measured.28,42 However, thismodelling framework can be extended to include theaction of a coronary stenosis via the appropriateapplication of additional flux BCs on internal perfu-sion boundary finite elements. In this manner, onecould restrict intra-compartment flow between perfu-sion regions, or allow flow across a regional boundaryin the presence of collateral vessels.

Future Directions

This study has successfully demonstrated amethod toparameterise both canine and porcine LV porous per-fusion models. It would be valuable to investigate whe-ther there are consistent intra-species trends observablein the parameterised fields, and to characterise theuncertainty of the estimated parameter fields across acohort of same-species experimentally-derived vascularmodels. Not only would this be worthwhile from a basicphysiological point-of-view, but it might also indicatewhether or not a statistical atlas of the parameterisedfields could be constructed for use in perfusion modelswhen the underlying network is not known.

Assuming a static model of perfusion is reasonablegiven the source of the data and the hypothesis beingconsidered in this paper. However, exploiting theknowledge gained from this study, we aim to incor-porate the anatomically-derived parameters within anexisting dynamic hyperporoelastic model of perfusion,8

including any necessary modifications to the parameterfields due to tissue deformation. Poroelasticity allowsfor the strong coupling between the myocardialdeformation and the coronary flow in a computation-ally tractable manner, which is widely-acknowledgedto be of fundamental physiological importance tocardiac functionality.39,44 We believe that embeddingthe anatomically-derived Darcy parameters within aporoelastic perfusion modelling framework wouldrepresent a significant step forward towards a clini-cally-useful dynamic porous perfusion model.

ELECTRONIC SUPPLEMENTARY MATERIAL

The online version of this article (doi:10.1007/s10439-013-0951-y) contains supplementary material,which is available to authorized users.

ACKNOWLEDGMENTS

The authors would like to acknowledge fundingfrom the Engineering and Physical Sciences ResearchCouncil (EP/G007527/2) European Community’sSeventh Framework Program FP7-ICT Grant No.224495: euHeart, the Wellcome Trust Medical Engi-neering Centre at King’s College London, the NationalUniversity of Ireland (ERH), the Netherlands HeartFoundation Grant 2006B226 (JAS and MS). JvdWwas supported by a Veni grant from the NetherlandsOrganization for Scientific Research (NWO 91611171).

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Multi-Scale Parameterisation of a Myocardial Perfusion Model Using Whole-Organ Arterial Networks

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