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A novel porous mechanical framework for modelling the interaction between coronary perfusion and myocardial mechanics

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A novel porous mechanical framework for modelling the interaction betweencoronary perfusion and myocardial mechanics
A.N. Cookson
a,1
, J. Lee
a,1
, C. Michler
a,1
, R. Chabiniok
a
, E. Hyde
b
, D.A. Nordsletten
a
, M. Sinclair
a
,M. Siebes
c
, N.P. Smith
a,
n
a
Imaging Sciences & Biomedical Engineering Division, St Thomas’ Hospital, King’s College London, SE1 7EH, UK
b
Department of Computer Science, Oxford OX1 3QD, UK
c
Biomedical Engineering and Physics, Academic Medical Center, University of Amsterdam, 1105 AZ, The Netherlands
a r t i c l e i n f o
Article history:
Accepted 4 October 2011
Keywords:
Incompressible poroelastic mediaMulti-compartmentCoronary perfusionMyocardial mechanicsFinite element method
a b s t r a c t
The strong coupling between the ﬂow in coronary vessels and the mechanical deformation of themyocardial tissue is a central feature of cardiac physiology and must therefore be accounted for bymodels of coronary perfusion. Currently available geometrically explicit vascular models fail to capturethis interaction satisfactorily, are numerically intractable for whole organ simulations, and are difﬁcultto parameterise in human contexts. To address these issues, in this study, a ﬁnite element formulationof an incompressible, poroelastic model of myocardial perfusion is presented. Using high-resolutionex vivo imaging data of the coronary tree, the permeability tensors of the porous medium were mappedonto a mesh of the corresponding left ventricular geometry. The resultant tensor ﬁeld characterises notonly the distinct perfusion regions that are observed in experimental data, but also the wide range of vascular length scales present in the coronary tree, through a multi-compartment porous model. Finitedeformation mechanics are solved using a macroscopic constitutive law that deﬁnes the couplingbetween the ﬂuid and solid phases of the porous medium. Results are presented for the perfusion of theleft ventricle under passive inﬂation that show wall-stiffening associated with perfusion, and that showthe signiﬁcance of a non-hierarchical multi-compartment model within a particular perfusion territory.
&
2011 Elsevier Ltd. All rights reserved.
1. Introduction
The perfusion of the heart is inherently coupled to its mechan-ical state. This coupling is perhaps most clearly evidenced in thelarge epicardial coronary vessels within which ﬂow is impededand even reversed during contraction. However, this epicardialphenomenon is fundamentally produced by compression of vessels within the myocardium which, in turn, increases theirresistance to ﬂow resulting in a reduced/reversed proximalpressure gradient. The interplay between the dynamics of vesselcompression with resistance and pressure gradients has moti-vated the development of a number of modelling frameworks.Early examples include the vascular waterfall model ﬁrst pro-posed by Downey and Kirk (1975), which was further developedwithin the seminal intramyocardial pump (Spaan et al., 1981),variable elastance (Krams et al., 1990) and coronary models (seeWesterhof et al., 2006 for a detailed review).Advances in modelling techniques and high performancecomputing have enabled the coupling between coronary ﬂowand myocardial deformation to be simulated within spatiallydiscrete frameworks (Huo et al., 2009; Smith, 2004; Smith and
Kassab, 2001). These approaches, combined with recent develop-ments in high resolution imaging, now provide frameworks(Kaneko et al., 2011; van den Wijngaard et al., 2011; Lee et al.,
2007) within which to analyse coronary ﬂow on an explicitcomputational representation of vascular networks.However, despite the progress of the ﬁeld and the insightswhich have already been gained, detailed coronary anatomicalinformation is unlikely to be available in clinical contexts whereperfusion is typically assessed by observing the passage of animaging contrast agent. Furthermore, the majority of existingmodels do not account for the effect of coronary blood pressureon myocardial tissue models, despite its inﬂuence being likely tobe physiologically signiﬁcant (Dijkman et al., 1998).Recent work has addressed this issue through the developmentof poro-elastic mechanical models of myocardial tissue (Huygheet al., 1992; May-Newman and McCulloch, 1998; Chapelle et al.,
2010), which have progressively incorporated additional complex-ities associated with representing perfused myocardium. Themodel of Huyghe et al. (1992) assumed quasi-static ﬂuid ﬂow
Contents lists available at SciVerse ScienceDirectjournal homepage: www.elsevier.com/locate/jbiomechwww.JBiomech.com
Journal of Biomechanics
0021-9290/$-see front matter
&
2011 Elsevier Ltd. All rights reserved.doi:10.1016/j.jbiomech.2011.11.026
n
Corresponding author.
E-mail address:
nicolas.smith@kcl.ac.uk (N.P. Smith).
1
Joint ﬁrst authors, listed alphabetically. Journal of Biomechanics 45 (2012) 850–855
and a quasi-linear viscoelastic constitutive law. The approachtaken by May-Newman and McCulloch (1998) was to assume anidealised representation of the vascular embedding within thetissue, and perform a homogenisation procedure to obtain a non-linear constitutive relation for the solid mechanical behaviour.More recently, the model presented by Chapelle et al. (2010)included unsteady ﬂuid ﬂow, and a non-linear hyperelastic mate-rial, which was assumed to be nearly incompressible.While each of these studies has made a signiﬁcant contribu-tion to the development and application of poro-mechanicalframeworks for capturing coronary perfusion, a number of limita-tions persist. Speciﬁcally all of these models were applied toidealised, axisymmetric representations of the left ventricle, andall three employed a single ﬂuid phase/porous compartment torepresent the coronary vasculature. This approach limits theability of the model to directly parameterise permeability tensorsof the porous medium from vascular data and thus exploitinformation derived from detailed coronary anatomies.In this paper we aim to address these issues through thedevelopment and application of a multi-compartment coupledporous-mechanical model of the cardiac left ventricle, providing amodelling framework for the future integration of both detailedstructural and clinical imaging data.
2. Mathematical formulation of a multi-compartmentporoelastic model
2.1. Multi-compartment porous medium
Consider a simple saturated porous medium consisting of aﬂuid and a solid phase. Its total mass can be deﬁned as
m
total
¼
Z
O
ð
r
f
f
f
þ
r
s
f
s
Þ
d
O
,
ð
1
Þ
where
f
and
r
refer to the volume fractions occupied by, anddensity of, each of the phases. Note that the integral bound is thetotal domain
O
, such that
f
f
þ
f
s
¼
1
:
ð
2
Þ
Note however that unless the solid is rigid, the bulk porousmedium will be compressible, regardless of the compressibility of the individual constituents, as the pore space can undergo volumechanges due to net ﬂuid inﬂow.In the vascularised tissue such as the cardiac tissue, the vesselﬂow direction displays signiﬁcant heterogeneity even in closespatial proximity, and the vessels themselves span multiple scales(radius
m
m
2
mm, length
m
m
2
cm). Due to these reasons it is moreuseful to treat each of the functional groups of the vascular tree asits own pore space, or compartment, which is connected to othercompartments.Thus the single ﬂuid phase generalised to
N
compartments, allof which occupy a fraction of the total volume such that
X
N i
f
f
,
i
þ
f
s
¼
1
:
ð
3
Þ
This formulation offers two advantages. Firstly, it preserves thehierarchy of ﬂow between nearby vessels of disparate lengthscales, which a single-compartment formulation would otherwisesmear out. Secondly, it allows the parameters such as perme-ability to be represented distinctly for each compartment, thusallowing greater accuracy in material characterization.A similar concept was previously introduced by Vankan et al.(1997) in a 2D model of perfusion in rat calf muscle, in which theﬂow between compartments was restricted to a strictly hierarch-ical manner. In the framework presented below, the cross-compartment connections between all levels, which are evidentfrom studies of vascular morphology (Kassab et al., 1993), arecaptured using a generalization of the previously applied doubleporosity network concept (Coussy, 2004).
2.2. Multi-compartment poroelastic ﬂow equations
The evolution equations for both the ﬂuid and solid phases aredescribed here, formulated in a more naturally suited Lagrangianform. The equations governing ﬂow through deformable porousmedia are given by Coussy (2004). Following standard conven-tions, reference and deformed coordinates are denoted by
X
and
x
, respectively,
y
¼
x
X
is the displacement of the skeleton, and
F
¼
@
x
@
X
and
J
¼
det
F
ð
4
Þ
denote the deformation gradient tensor and its associated Jaco-bian, respectively.The equations of Darcy ﬂow for compartment
i
in a system of
N
compartments, where
i
,
k
¼
1
,
. . .
,
N
denote the compartmentindices, are
F
M
i
r
f
¼
J
K
i
F
T
r
X
p
i
,
ð
5a
Þ
d
s
m
i
dt
þ
r
X
M
i
¼
X
N k
¼
1
J
b
i
,
k
ð
p
i
p
k
Þþ
r
f
q
i
,
ð
5b
Þ
variable
M
i
is the Lagrangian Darcy ﬂow vector, related to theEulerian relative ﬂow vector of ﬂuid mass by
M
i
¼
J
F
1
w
i
ð
6a
Þ
with
w
i
¼
r
f
f
f
,
i
ð
V
f i
V
s
Þ
,
ð
6b
Þ
Nomenclature
b
coupling coefﬁcient between ﬂuid compartments
F
deformation gradient tensor
K
permeability tensor of porous medium
M
Lagrangian Darcy ﬂow vector
S
second Piola–Kirchhoff stress tensor
w
i
Eulerian relative ﬂow vector
X
reference coordinates
x
deformed coordinates
l
Lagrange multiplier that enforces volume constraint
V
f
velocity of ﬂuid
V
s
velocity of solid
m
f
dynamic viscosity of the ﬂuid
f
f
porosity of ﬂuid phase
f
s
porosity of solid phase
r
f
density of ﬂuid phase
J
determinant of
F
m
ﬂuid mass increase
p
pressure
q
i
volumetric source term
A.N. Cookson et al. / Journal of Biomechanics 45 (2012) 850–855
851
where
V
f i
and
V
s
denote the velocity of the ﬂuid and of theskeleton particle, respectively.The ﬂuid density is
r
f
, pressure is
p
i
,
K
i
is the permeabilitytensor of the porous medium, and
q
i
is a volumetric source term.The pressure is determined using a constitutive law, the details of which are given in Eqs. (10)–(13).In the equation of mass conservation, Eq. (5b), the operator
d
s
m
i
=
dt
is the particle derivative with respect to the skeleton,such that
d
s
m
i
dt
¼
@
m
i
@
t
þð
r
x
m
i
Þ
V
s
,
ð
7
Þ
where
m
i
is the ﬂuid mass increase of compartment
i
, deﬁned perunit volume of the reference conﬁguration. The coefﬁcient
b
i
,
k
describes the ﬂow coupling between compartments and isdeﬁned as (Coussy, 2004)
b
i
,
k
¼
n
i
,
k
r
f
m
f
,
ð
8
Þ
where
n
i
,
k
is a dimensionless coefﬁcient, which describes thepermeability between two compartments and
m
f
is the dynamicviscosity of the ﬂuid. It is assumed that the communicationbetween compartments occurs solely as a mass exchange propor-tional to the pressure difference. Momentum transfer associatedwith this mass exchange is assumed to be negligible. To attain aconservative exchange between compartments
i
and
k
it followsthat
b
i
,
k
must be symmetric, i.e. the mass drained from (fed into)compartment
i
equals the mass fed into (drained from) compart-ment
k
, and
b
i
,
i
¼
0.
2.3. Finite-deformation elasticity
We model the kinematics of the skeleton by ﬁnite-deformationelasticity to allow for large strains. The skeleton kinematics aregoverned by the linear momentum balance
r
X
ð
FS
Þ¼
0
,
ð
9a
Þ
where
S
is the second Piola–Kirchhoff stress tensor. Eq. (9a) issolved subject to the volume constraint
J
¼
1
þ
P
N i
m
i
r
f
,
ð
9b
Þ
which is enforced using a Lagrange multiplier, denoted by
l
,where
m
i
is the ﬂuid mass increase determined from Eq. (5b).Incompressibility of the solid matrix is implied by Eq. (9b), that is,any volume increase (decrease) can occur only by pore dilatation(contraction) due to an increase (decrease) of pore ﬂuid mass.The constitutive law is given by Eq. (10), which is an isotropicexponential-form material law, modiﬁed to account for the effectof ﬂuid mass increase on the strain energy through the use of the
m
i
=
r
f
terms. Compared to the standard ﬁnite elasticity constitu-tive laws this law additionally governs the pore pressure devel-opment, as well as the skeletal stress, and characterises thecoupling between solid and ﬂuid media. For multi-compartmentmodels,
Q
i j
is allowed to differ for each compartment, to capturethe differing vessel compliances that exist.
C
s
¼
a
exp
D
1
I
1
1
þ
X
N i
Q
i
1
m
i
r
f
!
3
!"
þ
D
2
I
2
1
þ
X
N i
Q
i
2
m
i
r
f
!
3
!
:
þ
D
3
ð
J
1
Þ
2
þ
X
N i
Q
i
3
m
i
r
f
!
2
0@1A1A
1
35
,
ð
10
Þ
where
I ¯
1
and
I ¯
2
denote the modiﬁed invariants of the rightCauchy–Green deformation tensor
C
¼
F
T
F
, deﬁned as
I ¯
1
¼
J
2/3
I
1
and
I ¯
2
¼
J
4/3
I
2
. The terms
a
,
D
1
,
D
2
,
D
3
,
Q
i
1
,
Q
i
2
and
Q
i
3
are material parameters.The constitutive law is then subjected to the constraintEq. (9b) to form
C
scons
¼
C
s
þ
l
J
1
X
N i
m
i
r
f
!
:
ð
11
Þ
The second Piola–Kirchhoff stress tensor
p
is then deﬁned as (seeBonet and Wood, 2008, Chapter 5.5.1)
p
¼
@
C
s
@
E
þ
l
J
C
1
,
ð
12
Þ
while the relationship between the compartmental ﬂuid pressure,
p
i
, and the constitutive law is given by
p
i
¼
@
C
s
@
ð
J
f
f
,
i
Þ
l
:
ð
13
Þ
3. Computational methods
3.1. Partitioned solution strategy of the coupled equations
A partitioned scheme is chosen to solve the coupled solid–ﬂuidequations, with ﬁxed point sub-iterations between solid and ﬂuidsolution steps used to obtain a converged solution at eachtime step.It is possible to eliminate
M
i
, reducing the number of equa-tions to three, by substituting
M
i
deﬁned in Eq. (5a) into Eq. (5b)to yield
d
s
m
i
dt
¼
r
X
ð
r
f
J
F
1
K
i
F
T
r
X
p
i
Þþ
r
f
q
i
þ
X
N k
¼
1
J
b
i
,
k
ð
p
i
p
k
Þ
:
ð
14
Þ
Although the ﬂow vector
M
i
is a quantity of interest, it is notnecessary for the coupling with the solid, which occurs throughmass increase,
m
i
and Lagrange multiplier,
l
. Therefore, duringthe sub-iterations between solid and ﬂuid solution steps, a savingin computational cost can be obtained by solving only Eq. (14),rather than both Eqs. (5a) and (5b). Once the converged solutionis reached, the ﬂuid velocity is then determined using Eq. (5a).
3.2. Finite element formulation
The discretization of the governing equations (9) and (14) isperformed using a standard Galerkin ﬁnite-element discretizationin space with Lagrange basis functions. In particular, displace-ment
y
is discretized using a quadratic basis, while
l
,
p
i
,
M
i
andmass increase
m
i
are represented by a linear basis. A backwardEuler method is used to perform the time integration of Eq. (14).
3.3. Model parameterisation 3.3.1. Poroelastic constitutive law
High resolution cryomicrotome imaging data of myocardialvasculature strongly suggests that the myocardium is partitionedinto regional perfusion zones, and that these regions are perfusedby distinct arterial subnetworks (Spaan et al., 2005). To capturethis within the model every such region is associated to its uniquesubnetwork using the morphology to determine regional bound-aries of zero normal ﬂux, and to derive a region-speciﬁc perme-ability tensor. This regional partitioning for a selection of sixsubnetworks that supply the left ventricle (LV) is shown inFigs. 1 and 2. Such inclusion of accurate anatomical data is an
A.N. Cookson et al. / Journal of Biomechanics 45 (2012) 850–855
852
essential step towards reproducing physiologically based perfu-sion of the myocardium. The division of the vascular tree intothree compartments is illustrated in Fig. 3.A Principal Components Analysis (PCA) technique wasemployed in order to base this spatially averaged parameter ﬁeldderivation on the vascular data. Speciﬁcally, the regional subnet-work was discretized into a number of smaller segments. The unitorientation vector of each segment from compartment
i
consti-tutes a single datum in the PCA, which was then weighted by itssegment conductance. This segment property was chosen assum-ing parabolic ﬂow through an idealised cylindrical segment, withthe product of the conductance and the applied pressure gradientyielding the ﬂux through the segment. Once constructed, thegenerated data set for that regional subnetwork was then sub- jected to PCA, yielding a positive, semi-deﬁnite permeabilitytensor,
K
i
. The proportion of ﬂuid volume to total materialvolume within a region deﬁnes the regional porosity.The constitutive parameters were obtained through manualtuning, which yielded the following values
a
¼
1.0,
D
1
¼
2
:
0,
D
2
¼
0
:
2,
D
3
¼
2
:
0 and
Q
i
1
¼
1
:
0,
Q
i
2
¼
0
:
5 and
Q
i
3
¼
1
:
0.
4. Results
In order to test the accuracy of the continuum porous model inrepresenting the discrete blood velocity, a Poiseuille ﬂow compu-tation was performed on the discrete vessel network. Thisdiscrete solution was then volume averaged to allow comparisonwith the continuum model. The pressure in the Darcy model wasobtained by a volume-weighted average of all the compartmentalpressures. The results of this comparison are shown in Fig. 4, andreveal a good agreement in the magnitude and spatial variation of pressure.As discussed in the Introduction and outlined in Section 2 themulti-compartment formulation allows for the possibility of connections between non-neighbouring compartments, ratherthan enforcing a strict hierarchy of ﬂow, whereby ﬂuid must pass
Fig. 1.
The vascular tree for vessels with a radius in the range 0.031–1.9 mm,which was reconstructed from cryomicrotome imaging data. This is embeddedwithin the surface representation of the left ventricle geometry, with the vesselscoloured by radius (
left
). The main subtrees of the vasculature are then identiﬁedfrom this tree, and the tissue associated with each subnetwork determined by adiscrete distance metric (
right
).
Fig. 2.
Various cross-sections of the left ventricle (LV) model showing the resultsof the regional partition for a chosen set of six subnetworks that perfuse the LV.Each colour is a distinct, contiguous region of tissue associated with a particularsubnetwork. The central ﬁgure indicates the position of the cross-sections relativeto the LV model, the left-hand column is a long-axis cross-section while the right-hand column shows short-axis cross-sections.
Fig. 3.
The full vascular tree (
upper left
), vessels belonging to compartment one (
upper right
), vessels belonging to compartment two (
lower left
), and vessels belonging tocompartment three (
lower right
).
A.N. Cookson et al. / Journal of Biomechanics 45 (2012) 850–855
853
through compartment two if it is to ﬂow from compartment oneto three. The provision for this was motivated by the properties of the vascular tree data. In Fig. 4a comparison of the two models ispresented, which clearly illustrates the higher pressure valuesthat occur in the strictly hierarchical model. This is caused by thelower overall permeability of the hierarchical system, than in thecase where the non-hierarchical connections are included.To study the ﬂuid–solid coupling behaviours of the model wesimulated passive inﬂation of a full left ventricle (Lamata, 2011),with a diastolic cavity pressure ramp. In order to demonstrate thewall stiffening behaviour caused by ﬂuid mass increase, a ﬂuidsource, corresponding to a linear ramp up to an 8% overall wallvolume increase, has been simultaneously applied with the cavityinﬂation in one of the simulations. To isolate the mechanism of the garden-hose effect, spatially uniform source strengths andpermeabilities were employed, thereby avoiding the potentiallyconfounding effects of detailed parameterisation.Fig. 5 shows that as the ﬂuid content of the tissue increases,the LV progressively stiffens. Estimations of instantaneous com-pliance show a settling trend to a relatively constant differencebetween the perfused and unperfused cases, consistent with thatreported in May-Newman et al. (1994). However, the initialtrends in compliances show that at very low states of perfusion,the tissue stiffening caused by the mass inﬂow can be overcomeby the deformation that the ﬂuid mass itself induces. With thecurrent isotropic constitutive law, this strain occurs largely in thelongitudinal direction, as opposed to the radial direction asreported in the experimental study, highlighting the need forfurther work required in cardiac poroelastic constitutive laws.
5. Conclusions
The presented results show that the proposed modellingframework of a multi-compartment poroelastic medium has thepotential to serve for the investigation of coronary perfusionphenomena. In particular, the multi-compartment porous med-ium was shown to provide a good approximation to a discretemodel of the coronary tree. In addition, the model reproduced theexpected wall stiffening due to perfusion increasing the ﬂuidmass content in the wall.One of the difﬁculties found in these initial modelling studieswas the application of realistic ﬂuid sources and boundaryconditions on the ﬂuid model. The choice of distributed sourcesover an input velocity condition was justiﬁed by the morpholo-gical characteristics of the vascular network, in which the pro-gressive branching to smaller vessel segments can be regarded asentering a ﬂuid compartment over a distributed volume. Simi-larly, the sinks correspond to smaller vessel segments which arefurther distributed in space and carry the ﬂow to the nextcompartment. Nevertheless, further work is underway to movebeyond the simple speciﬁed volume rate conditions, by coupling
Fig. 4.
The discrete network embedded within its distinct perfusion territory on which a comparison of a Poiseuille ﬂow model and a continuum model based on Darcy’slaw is made (
upper left
). Volume-averaged pressure, plotted on ﬁve axial slices, from the Poiseuille ﬂow model (
upper right
) is matched well by the static Darcy model(
lower right
). Comparison with the strictly hierarchical model (
lower left
) shows that removing the connections between non-neighbouring compartments results in anover-estimation of pressure.
A.N. Cookson et al. / Journal of Biomechanics 45 (2012) 850–855
854

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