Improved identifiability of myocardial material parameters by an energy-based cost function

Article · February 2017with85 Reads
DOI: 10.1007/s10237-016-0865-3
Abstract
Myocardial stiffness is a valuable clinical biomarker for the monitoring and stratification of heart failure (HF). Cardiac finite element models provide a biomechanical framework for the assessment of stiffness through the determination of the myocardial constitutive model parameters. The reported parameter intercorrelations in popular constitutive relations, however, obstruct the unique estimation of material parameters and limit the reliable translation of this stiffness metric to clinical practice. Focusing on the role of the cost function (CF) in parameter identifiability, we investigate the performance of a set of geometric indices (based on displacements, strains, cavity volume, wall thickness and apicobasal dimension of the ventricle) and a novel CF derived from energy conservation. Our results, with a commonly used transversely isotropic material model (proposed by Guccione et al.), demonstrate that a single geometry-based CF is unable to uniquely constrain the parameter space. The energy-based CF, conversely, isolates one of the parameters and in conjunction with one of the geometric metrics provides a unique estimation of the parameter set. This gives rise to a new methodology for estimating myocardial material parameters based on the combination of deformation and energetics analysis. The accuracy of the pipeline is demonstrated in silico, and its robustness in vivo, in a total of 8 clinical data sets (7 HF and one control). The mean identified parameters of the Guccione material law were [Formula: see text] and [Formula: see text] ([Formula: see text], [Formula: see text], [Formula: see text]) for the HF cases and [Formula: see text] and [Formula: see text] ([Formula: see text], [Formula: see text], [Formula: see text]) for the healthy case.
Biomech Model Mechanobiol
DOI 10.1007/s10237-016-0865-3
ORIGINAL PAPER
Improved identifiability of myocardial material parameters
by an energy-based cost function
Anastasia Nasopoulou1·Anoop Shetty2·Jack Lee1·David Nordsletten1·
C. Aldo Rinaldi2·Pablo Lamata1·Steven Niederer1
Received: 2 July 2016 / Accepted: 9 December 2016
© The Author(s) 2017. This article is published with open access at Springerlink.com
Abstract Myocardial stiffness is a valuable clinical biom-
arker for the monitoring and stratification of heart failure
(HF). Cardiac finite element models provide a biomechanical
framework for the assessment of stiffness through the deter-
mination of the myocardial constitutive model parameters.
The reported parameter intercorrelations in popular consti-
tutive relations, however, obstruct the unique estimation of
material parameters and limit the reliable translation of this
stiffness metric to clinical practice. Focusing on the role of
the cost function (CF) in parameter identifiability, we inves-
tigate the performance of a set of geometric indices (based
on displacements, strains, cavity volume, wall thickness and
apicobasal dimension of the ventricle) and a novel CF derived
from energy conservation. Our results, with a commonly used
transversely isotropic material model (proposed by Guccione
et al.), demonstrate that a single geometry-based CF is unable
to uniquely constrain the parameter space. The energy-based
CF, conversely, isolates one of the parameters and in con-
junction with one of the geometric metrics provides a unique
estimation of the parameter set. This gives rise to a new
methodology for estimating myocardial material parameters
based on the combination of deformation and energetics anal-
ysis. The accuracy of the pipeline is demonstrated in silico,
Pablo Lamata and Steven Niederer acknowledge shared senior
authorship. Anastasia Nasopoulou is the single first author.
BPablo Lamata
pablo.lamata@kcl.ac.uk
BSteven Niederer
steven.niederer@kcl.ac.uk
1Department of Biomedical Engineering, Division of Imaging
Sciences and Biomedical Engineering, King’s College
London, London, UK
2Cardiovascular Department, Guy’s and St. Thomas’ NHS
Foundation Trust, London, UK
and its robustness in vivo, in a total of 8 clinical data sets
(7 HF and one control). The mean identified parameters of
the Guccione material law were C1=3000 ±1700 Pa and
α=45 ±25 (bf=25 ±14, bft =11 ±6, bt=9±5)
for the HF cases and C1=1700 Pa and α=15 (bf=8,
bft =4, bt=3) for the healthy case.
Keywords Parameter estimation ·Myocardium ·Patient-
specific modelling ·Passive constitutive equations
1 Introduction
Left ventricular (LV) stiffness is proposed as a diagnostic
indicator of cardiac function in heart failure (HF) patients
(Westermann et al. 2008). Ventricular stiffness has been
predominantly assessed in clinical practice through pressure–
volume (p–V) analysis (Bermejo et al. 2013;Burkhoff et al.
2005;Zile et al. 2004). However, this approach is unable
to distinguish between the anatomical and material contribu-
tions to LV stiffness. Specifically, an increment in ventricular
size due to myocardial hypertrophy or an increase in colla-
gen content with fibrosis may both lead to an equivalently
stiffer LV behaviour using this methodology. Differentiat-
ing between these two components, anatomical and material,
may improve the identification of HF aetiology in patients.
The development of biophysical models (Chen et al. 2016;
Crozier et al. 2016;Krishnamurthy et al. 2013;Lee et al.
2015;Nordsletten et al. 2011;Plank et al. 2009)forthe
simulation of cardiac mechanics allows the distinct rep-
resentation of the geometric and material components of
stiffness. Using these models, the assessment of myocardial
stiffness is posed as an inverse problem, where the material
parameters are determined from known mechanical loads and
deformations. Recent research in this field has focused on
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A. Nasopoulou et al.
developing tractable pipelines for linking model parameters
to data (Augenstein et al. 2005;Wang et al. 2009), evaluating
the available material models (Criscione et al. 2002;Schmid
et al. 2009) or improving the optimization strategies (Bala-
ban et al. 2016;Moireau and Chapelle 2011;Moireau et al.
2008,2009;Nair et al. 2007) and has led to the estimation of
material parameters from clinical data sets (Asner et al. 2015;
Gao et al. 2015;Wang et al. 2013;Xi et al. 2013). An inherent
limitation in these current methods is the intercorrelation of
the material parameters in myocardial material laws (Augen-
stein et al. 2006;Gao et al. 2015;Remme et al. 2004), which
results in multiple parameter combinations corresponding to
equivalent solutions in the optimization process. The exis-
tence of multiple solutions for the inverse problem limits the
interpretation of these parameters for characterizing patient
pathology and understanding changes in material properties
under conditions of HF.
In this paper we investigate the role of the cost func-
tion (CF) in parameter identifiability and develop a novel
energy-based CF that allows us to uniquely constrain the
myocardial material parameters. For our analysis we choose
a popular material model in cardiac mechanics, the trans-
versely isotropic constitutive equation proposed by Guccione
et al. (1991), which is reported to suffer from parameter cou-
pling (Augenstein et al. 2006;Xi et al. 2011). We examine
how a group of CFs based on geometric attributes, and the
energy-based CF, constrain the optimization in the search of
the parameters that best explain the clinical data of pressure
and deformation. After an evaluation on a synthetic data set,
a novel parameter estimation pipeline emerges based on the
combined use of the energy-based CF with one of the geo-
metric CFs, and is tested in 8 clinical cases demonstrating its
ability to identify unique material parameters from patient
data.
2 Methods
In this section we summarize the synthetic and clinical data
sets used (Sect. 2.1), the modelling framework (Sect. 2.2),
the evaluated CFs (Sect. 2.3) and the proposed parameter
estimation pipeline (Sect. 2.4). All data processing has been
performed in MATLAB, and the meshes and simulation out-
puts have been visualized with cmGui1(Christie et al. 2002).
2.1 Data sets
2.1.1 Synthetic
To provide a known ground truth for the material param-
eters a synthetic data set was employed (see top panel in
1www.cmiss.org/cmgui.
Fig. 1). An in silico model of the LV diastolic mechanics
was created from the passive inflation of a truncated confo-
cal prolate spheroidal of typical human cardiac dimensions
representing the myocardial domain (Evangelista et al. 2011;
Ho 2009;Humphrey 2002) to an end-diastolic pressure of
1.5 kPa (Humphrey 2002). A mesh of 320 (4 transmural,
16 circumferential, 4 longitudinal and 16 in the apical cap)
hexahedral elements and 9685 nodes was used for the pas-
sive inflation simulation (details on the interpolation schemes
and solver used are provided in Sect. 2.2.4). The pressure
was applied over 30 equal pressure increments of 0.05 kPa,
keeping the nodes of the ‘basal’ plane fixed in all directions.
The resulting 31 meshes (undeformed mesh and 30 deformed
meshes from each pressure increment) and their correspond-
ing cavity pressure values from the simulation compose the
synthetic data set used for the in silico study.
2.1.2 Clinical
In this study 8 clinical data sets are utilized, obtained from 7
Cardiac Resynchronization Therapy (CRT) patients (denoted
as PC1-PC7) and one healthy control (denoted as HC). The
clinical profile of the 8 cases is shown in Table 1. PC1-PC7
were obtained according to the clinical protocols followed
in St Thomas’ Hospital, London, and consist of LV cavity
pressure recordings and cardiac images covering the entire
cardiac cycle. The data collection conforms to the principles
of the Declaration of Helsinki and is guided by a local ethics
committee approved protocol with patient informed consent.
The healthy data set consists of pressure data and LV meshes
covering diastole and were described previously (Xi et al.
2013).
The cardiac images of the CRT patient data sets (PC1–
PC7) consist of 2-D short axis stacks of cine MRI with
SENSE encoding (1.19 ×1.19 ×8mm
3to 1.45 ×1.45 ×
10 mm3resolution), taken on a 1.5T—in six out of seven
cases—or 3T—in one case—Achieva Philips Medical Sys-
tems MRI scanner. Each MRI sequence had 25 to 35 frames
with a temporal resolution between 23 and 32 msec. The
LV domain was manually segmented in itksnap 2from the
end-diastolic frame. Images were processed using a non-
rigid registration (Shi et al. 2013) which enables a spatially
and temporally continuous description of the cardiac motion.
Mesh personalization was performed on the segmented LV, as
described previously (Lamata et al. 2014). A set of deform-
ing finite element (FE) meshes, consisting of 12 to 16 (4
circumferential, 3 to 4 longitudinal and 1 transmural) cubic
hexahedral elements and 436 to 580 nodes, were created
for each patient by warping the personalized end-diastolic
anatomical mesh using the motion field corresponding to
each frame of the cine MRI. As a result, correspondence
2www.itksnap.org.
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A novel cost function for unique parameter estimation...
Fig. 1 Synthetic and clinical data sets used to evaluate CFs. aSynthetic
data set was created by applying 30 equal pressure increments to an ide-
alized finite element (FE) model to generate 30 deformed geometries
‘frames’, bthe clinical data set combines an averaged pressure trace
with FE meshes that capture the deformation calculated from registra-
tion of the cine MRI frames. Combined, these provide a displacement
and pressure measurement for each MRI frame recorded over the car-
diac cycle. In our analysis only the diastolic frames, where a passive
inflation approximation is relevant, are utilized (marked as diastolic
window of interest)
of material points between frames is obtained from the cine
MRI images through the image registration and mesh per-
sonalization processes.
The LV cavity pressure transient was recorded during a
catheterization procedure before the beginning of the CRT
pacing protocols and separately from the MRI scans. For
each patient an average pressure trace was calculated over
5–13 beats and then synchronized to the cavity volume trace
estimated from the personalized FE meshes. The pressure–
volume synchronization was based on the assumption that
the inflection point in the pressure wave is approximately
aligned with the R wave (acquisition time of the first frame
of each MRI sequence) and finding the temporal offset that
maximized the p–V loop area and was less than 5% of the R-
R interval. The pressure transient was subsequently offset to
ensure a zero pressure at the MRI phase that corresponded to
the approximated reference configuration for the finite elas-
ticity analysis, described below. The schematic of the steps
followed for the processing of the clinical data sets is shown
in the lower panel of Fig. 1.
2.2 Mechanical model
LV diastolic passive inflation is simulated using large defor-
mation mechanics assuming that deformation is driven
principally by the LV pressure, the myocardium has homo-
geneous material properties, is incompressible, inertia or
viscoelastic effects are negligible, that the LV is in a sta-
ble relaxed state in late diastole and that the right ventricle
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A. Nasopoulou et al.
Table 1 Summary of the patient cases (PC1-PC7) and healthy data set
(HC) used
Case Age Sex EF (%) ESV (ml) EDV (ml) EDP (kPa)
PC1 61 M 13.5 266 307 2.59
PC2 61 M 6.2 348 371 1.21
PC3 70 M 19.5 174 216 4.44
PC4 76 F 32.3 86 127 1.84
PC5 57 F 19.3 214 265 2.99
PC6 65 M 29.7 122 173 1.17
PC7 39 M 19.7 176 219 2.98
HC 36 M aa134 1.89
aData not available
The abbreviations used are as follows: EF ejection fraction, ESV end-
systolic volume, EDV end-diastolic volume all corresponding to the
LV
Volumes were estimated from the personalized meshes following the
p–V synchronization
(RV), atria, pericardium and other neighbouring structures
have secondary roles.
2.2.1 Cardiac microstructure
The myocardial microarchitecture requires the continuous
description of the local fibre ( f), sheet (s) and sheet nor-
mal (n) directions. In the model, local tissue microstructure
was described by assuming a linearly varying preferential
myocyte orientation from 60at the epicardium to 60
at the endocardium based on the findings by Streeter et al.
(1969).
2.2.2 Material description
The myocardium is modelled as a hyperelastic incompress-
ible transversely isotropic material with the constitutive
relation introduced by Guccione et al. (1991). The math-
ematical description of the Guccione law is given in Eqs. 1
and 2. The parameters bf,bt,bft assign different mechanical
responses to the tissue along the fibre ( f) direction, across the
transverse planes (t) and in the fibre-transverse shear planes
(ft), respectively. The f,s,nindices in the strain components
express projections of the Green–Lagrange strain tensor (E)
along the fibre ( f), sheet (s) and sheet normal (n) directions.
Ψ=1
2C1(eQ1)(1)
Q=bfEff2+bft(2Efs2+2Efn2)
+bt(Ess2+Enn 2+2Esn2)(2)
After the reformulation proposed by Xi et al. (2013),
the strain energy density (Ψ) is expressed as a function of
the scaling (C1) and bulk exponential (α) parameters, along
which the primary coupling occurs (Eqs. 1,3).
Q=α[rfEff2+rft(2Efs2+2Efn2)
+rt(Ess2+Enn 2+2Esn2)](3)
α=bf+bft +bt(4)
rf=bf/α , rft =bft/α , rt=bt(5)
rf+rft +rt=1(6)
The rf,rft,rtparameters are referred to as anisotropy
ratios and range between 0 and 1, with the rfratio obtain-
ing usually the highest value in order to represent a stiffer
behaviour along the fibre direction. Throughout our analysis
the anisotropy ratios were kept constant at rf=0.55,rft =
0.25,rt=0.2 while focusing on fitting the coupled C1and
αparameters.
2.2.3 Reference configuration
The reference configuration represents an idealistic stress-
and strain-free geometry for the myocardium which is never
reached within the cardiac cycle. For simplicity the LV geom-
etry associated with the MRI frame corresponding to the
minimum pressure was chosen as an approximation of the
reference geometry.
2.2.4 Mechanical simulations and boundary conditions
The evaluation of the geometry-based CFs involves the
performance of mechanical simulations where the LV was
passively inflated to end-diastolic pressure applied on the
endocardial surface of the LV mesh. The motion of the basal
plane nodes was prescribed from the data, which for the case
of the synthetic data set translates to maintaining a fully fixed
basal plane. All boundary conditions (BCs) were applied in
30 equal increments. Figure 2schematically shows where
BCs are applied and how they are determined from the clin-
ical data.
The finite elasticity problem was solved within a multi-
field variational principle approach, with incompressibility
enforced through a Lagrange multiplier. Cubic and linear
Lagrange interpolation were chosen for the displacement
field and pressure, respectively (Hadjicharalambous et al.
2014). The mechanical simulations were performed in the
CHeart3nonlinear FE solver following a Galerkin FE
method (Lee et al. 2016).
2.3 Examined CFs and their evaluation
2.3.1 Methodology to assess CF performance
To assess the parameter identifiability provided by the geo-
metric and energy-based CFs, we computed the CF residual
3www.cheart.co.uk.
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A novel cost function for unique parameter estimation...
Fig. 2 Overview of the proposed parameter estimation pipeline. The
required input consists of LV meshes and corresponding pressure values
at the defined diastolic window of interest (covering the frames from
minimum pressure until before the beginning of contraction, see also
Fig. 1). The evaluation of the energy-based CF is entirely data based,
while the evaluation of the geometry-based CF requires the performance
of mechanical simulations with sweeps over C1and α. The combination
of the CFs ensures the unique estimation of these parameters
across the α-C1parameter space. We then visualize the
landscapes of the CF residuals, and locate the parameter sub-
spaces that could potentially be identified as solutions to the
inverse problem. This, for the case of the geometry-based
CFs, requires the performance of mechanical simulations
with parameter sweeps over C1and α. Conversely, the
energy-based CF relies only on data analysis, as highlighted
below.
2.3.2 Geometry-based CFs
We evaluate the ability of six geometrically based CFs to
uniquely constrain the passive material parameters either
independently or in combination. The geometry-based CFs
are based on popular CFs from the literature, comparing
displacement, strain and cavity volume (Asner et al. 2015;
Gao et al. 2015;Hadjicharalambous et al. 2015;Mojsejenko
et al. 2015;Xi et al. 2013), and extended to include widely
used clinical indices, such as wall thickening and apicobasal
deformation. The required data for all these CFs can be read-
ily provided from available imaging modalities. For the CF
evaluation we consider the time at end diastole (ED), where
the largest amount of deformation can be observed in the
diastolic window. Specifically, the examined geometry-based
CFs are:
L2displacement norm.TheL2displacement norm CF
is estimated by comparing the displacements between
the simulated displacement (usim) and the clinically mea-
sured or synthetic data (udat):
|u|=Ω(usim udat)·(usim udat )d
Ωd.(7)
L2strain norm.TheL2norm of the difference in Green–
Lagrange strains between simulated (Esim) and synthetic
or clinical data (Edat):
|E|=Ω(Esim Edat):(Esim Edat )d
Ωd.(8)
The L2 displacement and L2 strain norm CFs were
estimated using 4 Gauss points per element direction.
Increasing the Gauss points to 5 per element direction
led to a maximum 5 108mm error for |u|and 2 107
error for |E|, which is well within the expected mag-
nitude of error due to data noise.
Cavity Volume. The Cavity Volume CF (|V|) describes
the absolute difference between the LV cavity volumes in
the clinical or synthetic data (Vdat) and model simulations
(Vsim):
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A. Nasopoulou et al.
|V|=|Vsim Vdat|.(9)
Wall Thickness. The wall thickness metric |dWT |com-
pares the average wall thickness at the equatorial nodes
between the simulation (dsim
WT) and the data (ddat
WT):
|dWT|= nn
n1|dsim
WT ddat
WT|
nn
,(10)
where n1,...,nnare the node pairs at the equator used
for the wall thickness measurements.
Apicobasal distance. The endocardial |dABendo|(or
epicardial |dABepi|) apicobasal distance metrics esti-
mate the average difference between the distance of the
endocardial (or epicardial) basal nodes to the endocardial
(or epicardial) apical node at the data ddat
ABendo (or ddat
ABepi)
and simulation dsim
ABendo (or dsim
ABepi) meshes:
|dABendo|= mn
m1|dsim
ABendo ddat
ABendo|
mn
,(11)
|dABepi|= mn
m1|dsim
ABepi ddat
ABepi|
mn
,(12)
where m1,...,mnare the basal nodes whose distance
from the apex is calculated.
2.3.3 Energy-based CF
Based on the modelling assumptions described in Sect. 2.2,
the energy conservation dictates the equality of the external
work Wext (the work performed by the external loads acting
on the tissue) to the internal energy Wint (the work of the
internal stresses and strains), giving:
Wext =Wint.(13)
The internal energy for hyperelastic materials can be
expressed via the strain energy function Ψ, which with the
chosen constitutive law (Eqs. 1and 3) yields the internal
energy expression in Eq. 14 as a function of the Green–
Lagrange strain tensor E.
Wint =V
ΨdV =V
1
2C1(eQ(E)1)dV (14)
The external work is estimated as the increase in LV cavity
volume (V) from the reference configuration (V0)toagiven
volume VDcaused by the LV pressure p.
Wext =VD
V0
pdV(15)
Defining two points in diastole with corresponding vol-
ume, Green–Lagrange strain, external work and internal
energy of V1,E1,Wext1 and Wint1 and V2,E2,Wext2 and
Wint2, respectively, we can write :
Wext1
Wext2
=Wint1
Wint2 .(16)
The ratios of the external work and internal energy at these
two points must be equal and the difference of these two ratios
should tend to zero. This provides the energy-based CF, f:
f=Wext1
Wext2
Wint1
Wint2 .(17)
Substituting in the definition of Wint (Eq. 14) and Wext
(Eq. 15) then gives :
f=V1
V0pdV
V2
V0pdV
V
1
2(eQ(E1)1)dV
V
1
2(eQ(E2)1)dV .(18)
We can see in Eq. 18 that the constant C1of the constitutive
Eq. 14 is cancelled out from the numerator and denominator
of the right part of Eq. 17. Note that here the strain field is
directly derived from the deformation field extracted from the
medical images, without any forward simulation involving a
choice of C1, giving a CF dependent only on the material
parameters in Q.
In implementing the energy CF we select two time points.
In the clinical study these correspond to diastolic frames
(DF) of the MRI sequence. To avoid potential artefacts from
slow decaying active tension we choose to use frames from
the MRI that corresponded to the last two frames of end dias-
tole (this choice is reviewed in Appendix 7). We define DF
2
as the end-diastolic frame, and DF
1as the frame prior to
DF
2(see also Fig. 2). For consistency, we also chose to use
the last two ‘frames’ in the analysis of the synthetic dataset.
These correspond to the last two increments of the simula-
tion used to generate it (DF
1corresponds to the solution after
inflation to 1.45 kPa and DF
2to 1.5 kPa ).
The external work for each time point is calculated by
integrating the product of the pressure and change in volume
between sequential MRI frames, giving:
Wext =
MRIF1
n=RF
pn+pn+1
2(Vn+1Vn). (19)
In Eq. 19,RF corresponds to the index of the cine
sequence that corresponds to the reference frame, MRIF
corresponds to the index of the MRI frame of interest (for
example the index for DF
1or DF
2), pnis the pressure at
MRI frame nand Vnis the LV cavity volume at MRI frame
n.
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A novel cost function for unique parameter estimation...
The internal energy Wint is calculated solely from the
Green–Lagrange strain field which is derived from the dis-
placement field between the geometries of the DF under
consideration and the MRI frame employed as the reference
frame. This tensor field can be calculated directly from the
image registration algorithm without any further requirement
for mechanical simulations of the LV model.
The energy-based CF is only dependent on the parameters
in Qin the Guccione law. Assuming constant anisotropy
ratios then allows the αparameter (Eq. 3) to be uniquely
inferred form the energy-based CF.
2.4 Proposed parameter estimation workflow
The proposed workflow relies on the combination of the
energy-based and the L2displacement norm (|u|)CFs,
inferring the unique C1αparameter set from the point
of intersection between the minimum residual contours of
the two CFs (Fig. 2). Thus, the steps are:
Step 1. Estimate αthrough minimization of the energy-
based CF from analysing the data.
Step 2. Perform mechanical simulations in order to opti-
mize C1from the |u|CF.
The L2displacement norm is chosen as the geometric CF
(Sect. 2.3.2)fortheC1parameter estimation due to its robust-
ness (see Sect. 3.2.1) and the comprehensive data model
deformation comparison it provides compared to simpler
metrics. Note that the optimization of C1can be achieved by
setting αto the value obtained in step 1 and sweeping over
C1, resulting in 1D searches of the C1value that minimizes
|u|. However, C1estimation through 1D optimization may
not always be possible, as for certain C1,αcombinations the
nonlinear mechanical solver may not converge (Land et al.
2015b). To overcome this, simulations with 2D sweeps over
both C1and αcan be performed in order to allow an exponen-
tial fit of the parameter combinations yielding the minimum
|u|residual (for a justification on the choice of the expo-
nential fit, see Xi et al. (2013)). C1can then be uniquely
estimated from the intersection of this curve with the flat line
corresponding to the αsolution from step 1 (see Fig. 5cfor
an example). The latter approach was followed in our study
for parameter estimation from both synthetic and clinical
datasets.
3 Results
3.1 Parameter estimation in the synthetic data set
To determine if geometric CFs or the energy-based CF can
constrain passive stiffness parameters, in the absence of data
noise and under conditions of absolute model fidelity to the
data, we evaluate the CF performance on a synthetic data
set with baseline Guccione constitutive law parameters set
to α=30, C1=1000 Pa, rf=0.55, rft =0.25, rt=0.2
(Eqs. 1,3).
3.1.1 Identifiability of the geometry-based CFs
The reported coupling between the C1-αparameters (Xi
et al. 2013) is confirmed for the L2displacement norm
(Fig. 4a) and extended for the remaining geometry-based
CFs (Fig. 4b–f). Fitting an inverse exponential function to the
parameters with the minimum residual for each CF reveals
that the CF minimization contours are highly coincident
(Fig. 6). This shows that for the in silico case the geometry-
based CFs, independently or in combination, are unable to
uniquely constrain the parameters of the Guccione law.
3.1.2 Identifiability of the energy-based CF
The landscape of the energy-based CF residual in the C1-α
parameter space is shown in Fig. 3. Due to its formulation
the energy-based CF is independent of the C1parameter, as
is evident by the fact that its minimization contour is parallel
to the C1axis and the minimum occurs for a unique value
of α. Combining the energy-based CF with the L2displace-
ment norm the ground truth C1,αparameters of the synthetic
dataset were recovered (Fig. 3).
3.2 Parameter estimation in the clinical data sets
Following the in silico analysis we investigated the CF per-
formance in 8 clinical cases.
3.2.1 Evaluating Geometric CFs on Clinical Data
The energy-based CF must be paired with a geometric CF
to constrain both the C1and αparameters. To determine the
geometric CF to pair with the energy-based CF we evaluated
the six proposed geometric CFs on the 8 clinical data sets.
The identifiable parameter combinations for each CF for each
clinical data set are presented in Fig. 5as summary plots of
the exponential fits to the CF residual minimization param-
eter contours. This figure confirms that the C1-αparameter
coupling exists in vivo for all the geometric CFs. However,
the minimization contours are not always coincident in the
clinical setting, with some of the CF producing discordant
parameter solutions as in cases PC2 and PC7 in Fig. 5.
The L2norm of displacements was selected as the geomet-
ric CF to pair with the energy CF, as it is based on a thorough
global comparison of the agreement of the deformation field
between model and data and consistently accorded well with
the majority of the other geometric CFs across cases.
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A. Nasopoulou et al.
0 2500 5000
0
150
300
C1(Pa)
α
0
3.65
7.31
0 2500 5000
0
150
300
C1(Pa)
α
0
0.15
0.3
0 2500 5000
0
150
300
C1(Pa)
α
0
49
98
0 2500 5000
0
150
300
C1(Pa)
α
0
1.15
2.31
0 2500 5000
0
150
300
C1(Pa)
α
0
5.98
11.96
0 2500 5000
0
150
300
C1(Pa)
α
0
5.36
10.73
(a) (b) (c)
(d) (e) (f)
Fig. 3 Plots of landscapes of the examined geometry-based CF resid-
uals over the C1and αparameter space for the synthetic data set: a|u|
CF (residual in mm). b|E|CF. c|V|CF (residual in ml). d|dWT|
CF (residual in mm). e|dABendo|CF (residual in mm). f|dABepi |
CF (residual in mm). The parameter grid used is shown as the empty
black circles and is in the range of 200–5000 Pa for C1and 5–300 for
α. The parameter combinations yielding the minimum CF residual are
plotted in blue. The white patches in the plots a–f signify parameter
combinations that resulted in simulations that could not solve with the
defined loading paradigm
3.2.2 Identifiability of the energy-based CF
The energy-based CF was estimated for the 8 clinical data
sets. Its independence on C1is verified in clinical data, as the
CF minimizing parameter combinations form a horizontal
line parallel to the C1axis (see Fig. 5c, where the fitted line
to the minimum residual contour is overlain on top of the
exponential fits to the geometry-based CF minimums).
3.2.3 Estimated Parameters from the proposed pipeline
Following the proposed pipeline (Sect. 2.4), the passive mate-
rial parameters for the 8 clinical data sets were determined
from the intersection of the fits to the minimum residuals
of the energy and L2displacement norm CFs (Fig. 5). The
identified parameters are shown in Table 2along with the L2
displacement norm residual for each case.
A certain level of variability is evident in the estimated
parameters. In HF patient cases, C1ranges from 820 Pa for
case PC2 to 5300 Pa for PC1, and αfrom 5 for PC4 to 66
for PC3 and PC7. The healthy volunteer data set yielded
1700 Pa for C1and 15 for α. To provide context for the
fitted parameters in the parameter estimation results of this
study, previous estimates of the Guccione parameters fitted
to human data are presented in Table 3.
Fig. 4 Lines of minimal residual for all the geometry-based CFs in the
synthetic dataset, after an exponential fitting (each line is an exponential
fitting to the points of minimum residual, see Fig. 5c for an example).
The result shows that the lines of minimal residual are nearly identical
for all geometry-based CFs in silico
3.3 Comparison with Previous Methods
We test if the proposed energy CF method for unique param-
eter estimation predicts significantly different local stresses
as compared to one of the previous approaches, specifically
where C1was fixed to 2000Pa (Xi et al. 2013). The difference
in stresses developed at the ED frame using the two methods
is presented in Table 4. The large discrepancies observed in
123
A novel cost function for unique parameter estimation...
0 2500 5000
0
150
300
C1(Pa)
α
0
3.65
7.31
0 2500 5000
0
40
80
C1(Pa)
α
0
0.01
0.02
(a) (b) (c)
Fig. 5 Proposed CF combination for unique parameter estimation
emerging from the in silico analysis. aThe landscape of the |u|CF
(residual in mm), which is the chosen geometric CF (see Sect. 2.4 for
more details). bLandscape of the energy-based CF. A straight line pat-
tern emerges that demonstrates the independence of the CF on C1and
its ability to uniquely identify α.cUnique parameter estimation with
the combined use of the energy-based CF for αparameter identification
and the L2displacement norm CF. A superimposed black curve is fitted
to the parameter combinations corresponding to the minimum |u|CF
residual values. The horizontal black line corresponds to the minimum
energy-based CF residual. The identified parameters (indicated by the
black circle at the intersection of the energy-based and geometry-based
CF minimization lines) coincide with the ground truth values (α=30,
C1=1000 Pa). (Minimum and maximum CF residuals are shown in
blue and red, respectively.)
Table 2 Parameter estimation results from the application of the pro-
posed pipeline to the clinical data sets
Case αsol C1(Pa) |u|(mm)
PC1 61 5300 0.95
PC2 61 820 1.96
PC3 66 1960 2.56
PC4 5 4780 5.61
PC5 24 3140 b
PC6 29 1460 1.91
PC7 66 3300 1.18
HC 15 1700 3.35
bIn case PC5 there is no available residual as the forward simulation
with the identified parameters did not converge.
The αparameter is estimated by the energy-based CF and the C1param-
eter from the |u|CF
Table 3 Estimated C1,αparameters for human myocardium from pre-
vious studies
Case C1(Pa) α|u|(mm)
Human
Healthya2000e43 1.78
Patient 1a2000e105 1.58
Patient 2a2000e95 1.39
Healthyb3600 ±1200e38 –
HTb,c12000 ±2600e38 –
NI-HFb,d11800 ±3400e38 –
aXi et al. (2013). bWang et al. (2013). cPatients with hypertrophic LV.
dPatients with non-ischaemic HF with reduced EF. eIn these studies
the scaling constant is defined as half the C1parameter in Eq. 1,and
therefore, values reported here have been doubled for consistency in the
results
Table 4 Differences in stress calculated with parameters estimated by
proposed method (A) and a previous one (Xi et al. 2013), where C1was
fixed at 2000 Pa (B)
Case Mean ˆ
SA
ff ˆ
SB
ff
(Pa)
Standard
deviation (Pa)
αB
Synth 31.7 102.8 17
PC1 461.8 47103.7 142
PC2 577767.5 34820092.3 25
PC3 3127.3 52912.1 65
PC4 7.3 284.9 10
PC5 0.5 82.2 35
PC6 32.6 114.1 22
PC7 18.7 598 92
HC 45.7 972.7 13
Stress values are the deviatoric second Piola–Kirchhoff stress in the
fibre direction ( ˆ
Sff) computed at end diastole. Estimated αfrom fixing
C1(αB) are also reported for completeness
some cases, notably PC2, can be explained by the existence
of large local strains that amplify differences due to the expo-
nential term in the strain energy function. In addition, Fig. 7
illustrates the difference in stress–strain curves correspond-
ing to a 1-D fibre stretch for the parameter pairs estimated
with both methods.
4 Discussion
We have shown that unique identification of myocardial
material parameters is possible with a suitable choice of
the CF. To our knowledge, this is the first time that the
123
A. Nasopoulou et al.
two coupled parameters in the Guccione model have been
uniquely constrained by clinical data; previously this issue
was addressed by fixing part of the parameter set (Asner et al.
2015;Hadjicharalambous et al. 2015;Wang et al. 2013;Xi
et al. 2013).
4.1 Identifiability by an energy-based CF
The core methodological contribution of this work is the
proposal of a CF that removes the parameter coupling. The
energy-based CF identifies αdue to its independence to the
C1parameter. Its accuracy was tested in silico, where it esti-
mates the correct αparameter and combined with the L2
displacement norm provides accurate estimates of the ground
truth parameter values. Results in 8 real clinical data sets with
the complete pipeline demonstrate that the CF is robust to the
inherent noise in clinical data and finite model fidelity.
The novel energy-based CF is also a data driven met-
ric. Only the data of deformation (strain and cavity volume)
and pressure are required to compute it, without the need
of computational simulations or data assimilation pipelines.
This has three main benefits. Firstly, the data derived defor-
mation field employed in the CF is unaffected by the C1-α
coupling that arises from the simulation. Secondly, the com-
putationally expensive search over the full parameter space
involved in current data assimilation schemes has one dimen-
sion of the parameter space reduced since the αparameter is
fixed. Thirdly, the reduction of methodological complexity
to obtain the αparameter opens the possibility for a quicker
and easier clinical adoption.
It is important to note that the energy-based CF raises
the demands on data quality and quantity, since it requires
strain data of the entire myocardium at two time points during
diastole and the pressure–volume information covering the
filling phase of the cycle. The importance of data quality on
parameter estimation is demonstrated in a sensitivity study,
provided in Appendix 6. In the absence of high fidelity strain
data it is possible to recast the energy CF in terms of a pressure
CF. This allows unique parameter estimates from pressure
and volume transient data alone. The efficacy of this approach
is presented in Appendix 8.
In our calculations the external work is estimated using a
pressure–volume approach (see Eq. 19) which is fully accu-
rate for the case of a deformation field consistent to the
passive inflation assumption we have adopted. However, the
image driven Dirichlet boundary conditions applied on the
basal plane in the clinical data sets contribute to external
work. This contribution is quantified as a mean 5% of the
elastic energy in the clinical cases based on forward simula-
tions with the identified parameters.
The efficiency of the energy-based CF was demonstrated
for the Guccione material law, but can be extended to other
exponential constitutive relations for reducing the mate-
rial parameter redundancy by one, such as the Holzapfel–
Ogden law (Holzapfel and Ogden 2009) as demonstrated in
Appendix 9 or the pole–zero (Nash and Hunter 2000) among
others.
4.2 Geometry-based CFs
Geometry-based CFs, and their combination, were not able
to identify unique myocardial material parameters, agreeing
with previous reports (Augenstein et al. 2006;Xi et al. 2011).
We investigated if a combination of geometrical CFs could
improve parameter identifiability. The C1-αparameters
would then be identified by an intersection of the minimiza-
tion contour of two or more CFs. Nevertheless, in the in silico
data set the minimization contours are almost identical for
the different CFs, suggesting the low complementary value
of the CFs (Fig. 4). On the contrary, results with real data
report a large variability in the agreement between the dif-
ferent CFs in half of the cases (see the offsets between lines
that identify the coupling in Fig. 6), suggesting that this strat-
egy does not lead to a unique set of parameters in practice.
The disagreement between CFs with real data, and not with
simulated data, is interpreted as a reflection of the mismatch
between model and real data, caused by a combination of
lack of model fidelity and data quality.
Note that while the geometric cost functions are based
on a single frame, in contrast to the two frames used in the
energy CF, the addition of an adjacent frame is not expected
to improve the identifiability of parameters from geomet-
ric cost functions when working with clinical data due to
the presence of noise that is sufficiently large to obscure the
global minimum, as reported in Xi et al. (2013).
4.3 Parameter estimation workflow
The proposed parameter estimation pipeline lead to a unique
estimation of the Guccione material parameters in the 8 clin-
ical data sets analysed in this study, and with |u|residuals
(Table 2) comparable to previously reported errors (Table 3).
It is important to note that the set of kinematic BCs in this
study was lighter (only constraining the base, and not also
the apex as in Xi et al. (2013)), thus making the task of repro-
ducing the clinical observation more challenging.
In the proposed methodology the unique C1and αparam-
eters, where the coupling occurs (Xi et al. 2013), are found
while fixing two of the less intercorrelated ratios rf,rt,rft
(the third is bound by Eq. 6). Once C1and αare found, the
ratios can be uniquely found (as reported in Xi et al. (2013)).
The impact of a wrong initial choice of rf,rt,rft on the
estimation of C1and αparameters was evaluated in the sen-
sitivity study (Appendix 6) and was found to be relatively
low.
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A novel cost function for unique parameter estimation...
0 5000 10000
0
150
300
C1(Pa)
α
0 2500 5000
0
150
300
C1(Pa)
α
0 5000 10000
0
150
300
C1(Pa)
α
0 5000 10000
0
30
60
C1(Pa)
α
0 2500 5000
0
130
260
C1(Pa)
α
02500 5000
0
90
180
C1(Pa)
α
0 2500 5000
0
150
300
C1(Pa)
α
0 5000 10000
0
40
80
C1(Pa)
α
(a) (b) (c)
(d) (e) (f)
(g) (h) (i)
Fig. 6 Lines of minimal residual from the geometry-based CF superimposed to the solution for the αparameter resulting from the energy-based
CF (flat line) in each of the 8 cases studied: aPC1, bPC2, cPC3, dPC4, ePC5, fPC6, gPC7, hHC. iLegend
An important remark in the methodology is the existence
of challenges associated with the convergence of nonlinear
mechanics solvers in incompressible applications (Land et al.
2015b). The lack of convergence can often obstruct the calcu-
lation of the geometry CF residual for the whole C1range of
interest under the known α. Knowing that the set of coupled
parameters that lead to very similar minimum costs draw a
line in the α-C1log-scale space (Xi et al. 2013) allows the
problem to be reformulated as finding the parameters of this
exponential line.
4.4 Model assumptions
There are a series of model assumptions which, although not
affecting the contribution of the proposed pipeline to param-
eter identifiability, may have an impact on the parameters
found.
One important factor determining diastolic filling is the
residual active tension (AT), which is known to be present
in diastole (Bermejo et al. 2013;Xi et al. 2013). In this
study the parameters are estimated from late diastolic instants
and the inclusion of an earlier frame suggested the presence
of remaining AT as detailed in Appendix 7. End-diastolic
events, where the AT can be assumed to be limited and
its contribution to the work estimation negligible, are the
most suitable observations. Following this approach, any
remaining AT at end diastole leads to an apparent increased
myocardial stiffness (Asner et al. 2015), specifically in the
fibre direction (Xi et al. 2013).
The most important element in the proposed methodology
was revealed to be the choice of the reference frame (see
Appendix 6), in concordance with previous studies (Xi et al.
2013). The reference geometry directly affects the observed
myocardial stiffness as it dictates the measured strain under
a given cavity pressure. In this study the LV geometry at the
lowest pressure frame was chosen to describe this geometry
following a popular approach (Asner et al. 2015;Gao et al.
2015;Land et al. 2012;Nikou et al. 2016;Wang et al. 2009)
123
A. Nasopoulou et al.
in order to simplify the workflow. Inclusion of a reference
frame estimation (Krishnamurthy et al. 2013) can possibly
enhance the parameter estimation in future applications.
Assumptions are also made regarding the definition of the
myocardial microstructure and material. Conforming to the
majority of FE studies in the field of cardiac mechanics, the
myocardium is assumed to be incompressible although cap-
illary and coronary flow are known to locally violate this
assumption (Ashikaga et al. 2008;Yin et al. 1996). This
assumption affects both the simulated deformation fields
from the mechanics solver, as well as the novel energy-based
CF, where deformation due to cardiac perfusion (increment
of volume during diastole) is assimilated to contribute to
the tissue strain energy. Also in the definition of myocardial
microstructure, the inclusion of a more realistic fibre field
might improve the accuracy in the estimation of the projected
strain components (Eq. 3) and thus that of the energy-based
CF (Eq. 18). Nevertheless, results in the sensitivity study
(Appendix 6) suggest that the impact of this assumption is
very small, in accordance to previous studies (Land et al.
2015a).
A central assumption in the modelling approach followed
here is myocardial material homogeneity, which by reduc-
ing model complexity facilitates the parameter estimation
procedure. However, this currently restricts the application
of the proposed method to disease where this assumption is
valid. Although rendering our method suitable for cardiac
disease with localized stiffness alterations -such as myocar-
dial infarction- is within our future plans, our workflow is
readily suitable for applications on disease, such as dilated
cardiomyopathy, diffuse fibrosis or heart failure with nor-
mal ejection fraction (HFnEF), where tissue properties are
expected to be more homogeneous.
One last set of assumptions are needed to define the BCs
of the model. First, a homogeneous pressure load is assumed
to act on the endocardial boundary during ventricular fill-
ing. This is a reasonable simplification based on the reported
cavity pressure variations in the literature (de Vecchi et al.
2014), and is commonly taken for computational efficiency
(Gao et al. 2015;Hadjicharalambous et al. 2015;Mojsejenko
et al. 2015;Nikou et al. 2016;Wang et al. 2013). However
the impact of the RV, atria and pericardium on achieving
more physiological deformations is known (Belenkie et al.
2001;Tyberg and Smith 1990;Williams and Frenneaux
2006), and thus the use of more advanced mechanical mod-
els (Fritz et al. 2014) is anticipated to improve model fidelity
and therefore parameter estimation accuracy. We would also
expect that more realistic BCs will enable the model to
better reproduce the recorded myocardial deformation, thus
leading to smaller residual |u|—and this should be espe-
cially beneficial in 3 of our cases (PC3, PC4 & HC, see
Table 2).
4.5 Estimated parameters in vivo
Few studies have reported passive material properties for
multiple patients. The identified parameters for the 8 cases
in this study (Table 2) are within the reported range in the
literature (Table 3). In our results, C1falls within a range of
800–5300 Pa for human HF patients, as opposed to higher
values obtained when αwas held fixed. The αvalues fitted
here span from 5 to 66 and lie within the literature range,
while the higher values corresponding to HF patients are sig-
nificantly lower to the ones previously reported when fixing
C1at 2000 Pa (Table 3).
In our results we found that in four out of seven patient
data sets (PC1, PC3, PC5, PC7) both identified parameters
were higher than those of the healthy data set, in two cases
(PC2, PC6) only the αparameter was increased while in
one case (PC4) C1was increased instead. The sample of
7 HF subjects already reveals functional differences by the
linear and exponential components of the material law, and
we can speculate that this may have links with the aetiology
of the disease. While the general trend is for the HF patient
data sets processed to have higher stiffness than the healthy
case in agreement with previous clinical studies (Wang et al.
2013), further cases need to be investigated to confirm this
observation.
4.6 Cardiac Mechanics Application
Computational models are increasingly used in clinical appli-
cations. Inferring material properties from these models
offers three potential applications with each benefiting from
uniquely constrained parameters. Firstly, material param-
eters may provide a more sensitive descriptor of patient
pathology (Lamata et al. 2016). To test the utility of this
application first requires a method for inferring constitutive
parameters from clinical data. Assuming that these param-
eters reflect some underlying material property, and hence
reflect the patients pathology, then the inferred parame-
ter values should be unique and independent of the fitting
method.
Secondly, the use of biophysical models, that are con-
strained by physical laws, increases the capacity of models
to predict outside of the data used to constrain them. This is
particularly important for patient-specific models that could
be used to predict response to treatments from pre proce-
dure data. The use of non-unique parameters will make these
predictions dependent on the parameter inference method,
increasing the uncertainty in the model predictions.
Finally, models can be used for predicting values of inter-
est that can not easily be measured, including material stress,
regional work and local mechanical efficiency. We have
shown that an alternative approach for a unique estimation
123
A novel cost function for unique parameter estimation...
0 0.6 1.2
1
1607
3212
λf
σf(kPa)
A: α=61 C1=5300
B: α=142 C1=2000
0 0.6 1.2
0
27
53
λf
σf(kPa)
A: α=61 C1=820
B: α=25 C1=2000
0 0.6 1.2
0
79
157
λf
σf(kPa)
A: α=66 C1=1960
B: α=65 C1=2000
0 0.6 1.2
0
2.5
5
λf
σf(kPa)
A: α=5 C1=4780
B: α=10 C1=2000
0 0.6 1.2
0
18
35
λf
σf(kPa)
A: α=24 C1=3140
B: α=35 C1=2000
0 0.6 1.2
0
9
18
λf
σf(kPa)
A: α=29 C1=1460
B: α=22 C1=2000
0 0.6 1.2
0
235
470
λf
σf(kPa)
A: α=66 C1=3300
B: α=92 C1=2000
0 0.6 1.2
0
3.5
7
λf
σf(kPa)
A: α=15 C1=1700
B: α=13 C1=2000
0 0.6 1.2
0
7
13
λf
σf(kPa)
A: α=30 C1=1000
B: α=17 C1=2000
(a) (b) (c)
(d) (e) (f)
(g) (h) (i)
Fig. 7 Cauchy stress vs stretch curves for an idealized 1-D extension
along the fibre direction (up to 120% stretch) of an incompressible cube
with the Guccione law using the material parameters estimated with the
proposed methodology (A: energy-based & |u|CFs) and a previous
one where one of the parameters was assigned a certain value (B: C1
fixed at 2000 Pa & |u|CF as in Xi et al. (2013))
of parameters, by fixing C1, leads to significant discrepancy
in model predictions (see Fig. 7).
5 Conclusions
A novel and clinically tractable pipeline for passive myocar-
dial material estimation is proposed, which manages to
decouple the material constitutive law parameters and guar-
antee reliable material estimation. This is an important step
towards the use of myocardial stiffness as a reliable tool for
the understanding of cardiac pathophysiology and the devel-
opment of biomechanically relevant biomarkers. This work
highlights the central role of CFs in the identifiability of mate-
rial parameters.
Acknowledgements This work was supported by the British Heart
Foundation (PG/13/37/30280), the UK Engineering and Physical Sci-
ences Research Council (EPSRC) (EP/M012492/1), the King’s College
London Wellcome Trust and EPSRC Medical Engineering Centre and
the Department of Health via the National Institute for Health Research
(NIHR) comprehensive Biomedical Research Centre award to Guy’s &
St Thomas’ NHS Foundation Trust in partnership with King’s College
London and King’s College Hospital NHS Foundation Trust. PL holds
a Sir Henry Dale Fellowship funded jointly by the Wellcome Trust and
the Royal Society (Grant No. 099973/Z/12/Z). SN has received support
from Boston Scientific and St Jude Medical. CAR receives research
funding and Honoraria from St Jude Medical, Medtronic, and Boston
Scientific.
Conflict of interest The authors declare that they have no other con-
flicts of interest.
Open Access This article is distributed under the terms of the Creative
Commons Attribution 4.0 International License (http://creativecomm
ons.org/licenses/by/4.0/), which permits unrestricted use, distribution,
and reproduction in any medium, provided you give appropriate credit
to the original author(s) and the source, provide a link to the Creative
Commons license, and indicate if changes were made.
123
A. Nasopoulou et al.
6 Sensitivity analysis
This section tests the performance of the novel energy-
based CF in the presence of data quality issues or arbitrary
modelling assumptions that are necessarily made in clinical
models. For this purpose we use the in silico workbench with
access to ground truth parameter values, and then perform a
sensitivity analysis. We focus on the impact in the estimation
of the αparameter (ground truth is 30), since it is provided
by the novel energy-based CF.
6.1 Methods
6.1.1 Sensitivity to pressure offset errors
Artefacts in the pressure data were studied introducing an
offset, which can be caused by calibration errors and are
hypothesized to be the main source of material parameter
errors (see the large impact reported in Xi et al. (2014)). The
effect of offset pressure values is investigated by shifting
all pressure data points by +/10% of the mean pressure ¯p
(corresponding to a 75 Pa increase/decrease in the pressure
values).
6.1.2 Sensitivity to strain data quality
The quality of the strain field in the imaging data can affect the
accuracy of the estimated α, as the energy-based CF relies on
the integration of strains in two diastolic timepoints (Eq. 18).
To assess the anticipated error in parameter estimation from
strain data noise, we inserted white noise to the strain tensor
at the gauss points used for the integration of strain energy
density function in Eq. 14. The noise was independently
applied to each strain component in fibre coordinates, and
corresponded to a standard deviation of 5, 10 and 20% of
the mean strain ¯
Eij, which varied from 0.15 to 0.10 in the
employed diastolic frames (corresponding to increments 29
and 30 of the original simulation).
6.1.3 Sensitivity to modelling assumptions
Two modelling assumptions are challenged, the choice of the
reference configuration and the mathematical description of
the myocardial microarchitecture (fibre field).
The reference frame, which represents the geometrical
configuration at which the myocardium is unloaded and
the developing stresses and strains are zero, is difficult to
calculate. In the clinical application of the parameter esti-
mation pipeline, the reference frame is estimated by taking
an observed geometry at the beginning of diastole, similar to
previous approaches (Hadjicharalambous et al. 2015;Mojse-
jenko et al. 2015;Wang et al. 2013). To estimate the effect of
an inaccurate reference configuration on the parameter solu-
Table 5 Sensitivity analysis results with respect to the identified α
parameter
Data/model modification αsol
Pressure
+10% ¯poffset 22
10% ¯poffset 40
Enoise
STD 5% ¯
Eij 28
STD 10% ¯
Eij 26
STD 20% ¯
Eij 37
Ref. Fr.
126
222
53
Fibres
/+ 90 o29
rf-rft-rt
0.85-0.1-0.05 29
0.34-0.33-0.33 28
The ground truth value for αis 30
tion, later frames (frames 1 to 5, where frame 0 is the original
reference frame) were used as reference configurations in the
mechanical model.
The cardiac microarchitecture is approximated by rule-
based mathematical models (Bayer et al. 2012;Nielsen et al.
1991) with 60to +60angle variation. To assess the effect
of this choice of fibre microarchitecture, a set of extreme
vertical fibres was selected (fibre angles varying from 90
to +90).
6.1.4 Effect of fixed parameter values
Throughout the parameter estimation procedure fixed values
for the exponential stiffness parameter ratios were assumed.
To evaluate the effect of this arbitrary choice, two different
stiffness ratio combinations were used: one pertaining to a
highly anisotropic material (rf=0.85, rft =0.10, rt=
0.05) and one to an isotropic material (rf=0.34, rft =
0.33, rt=0.33).
6.2 Results and discussion
Results (see Table 5) demonstrate a satisfactory robustness
of the approach to possible sources of disagreement between
model and data. The largest effect on parameter estimation
accuracy derives from an unsuitable choice of reference con-
figuration (choice of frame 5, αsol =3). The offset in pressure
recordings has the second largest impact on the parame-
ter estimation, with smaller parameters found with larger
pressure values and vice-versa. The anticipated effect of the
123
A novel cost function for unique parameter estimation...
Table 6 Results of the estimation of αwith an energy cost function extended to consider the last 3 frames of diastolic filling (α3fr.), instead of the
2 of the original formulation (referred to here as α2fr.)
Case Synth PC1 PC2 PC3 PC4 PC5 PC6 PC7 HC
α3fr. 30 78 79 52 1 a41 77 17
α3fr. α2fr. 01718124a12 11 2
aPC5 had data inconsistent with passive inflation, see text for further details
quality of the strain field on the estimated parameter values
is also confirmed. The microstructural representation in the
model seemed to have a negligible effect on the identified
values. The stiffness anisotropy ratios did not bias the esti-
mation of α(despite the pronounced anisotropy vs isotropy
scenarios tested) but had a larger effect on the estimation of
C1, which was 2100 Pa for the highly anisotropic assumption
and 620 Pa for the isotropic one. In both cases the resulting
|u|residual of the forward simulations with the identified
parameters was increased from 3 109mm for the ground
truth parameters to 1.15 and 1.07 mm, respectively.
7 The effect of using additional frames in the
energy-based CF on α
In our analysis the two last frames of diastole were used as
they provide larger strain signals with lowest probability for
presence of residual AT. Alternatively, additional late dias-
tolic frames can be included in the analysis, leading to a
modified functional using the ratios of the paired combina-
tions of the last 3 diastolic frames f3DF (Eq. 20):
f3DF =
Wext1
Wext2
Wint1
Wint2
+Wext2
Wext3
Wint2
Wint3
+Wext1
Wext3
Wint1
Wint3
(20)
Results in Table 6show that the change in αis generally
small (up to a 25% except for a 40% in PC6). In most of
the cases the additional earlier frame led to a larger value in
α, suggesting the presence of residual AT. The data of the
additional earlier diastolic frame in PC5 indicated a decrease
in strain energy, despite pressure increasing, which is incom-
patible with underlying model assumptions and results in a
negative αvalue.
8 An alternative approach for unique parameter
estimation
In this work a novel CF was proposed that can isolate α,
allowing for C1to be uniquely determined from previous geo-
metrical CF. The proposed energy CF relies on strain fields
derived directly from the images, and thus on the availability
of high quality data and robust image registration tools (Shi
et al. 2012). A similar approach can be followed by exploiting
Table 7 Parameter estimation results from the pressure-based CF
(pCF), compared to the energy-based CF (eCF)
Case αpC F C1pC F αpC F αeC F C1pC F C1eC F
Synth 30 1000 0 0
PC1 18 21630 43 16330
PC2 6 8335 55 7515
PC3 >120a<885a>54a<1075a
PC4 50 240 45 4540
PC5 60 1070 36 2070
PC6 130 275 101 1185
PC7 >400a<210a>334a<3090a
HC 50 310 45 1390
Note that C1here is obtained in both cases by the L2displacement norm
aResults in PC3 and PC7 are bounded since the passive inflation sim-
ulation for larger αvalues did not converge
the fact that the cavity pressure that inflates the myocardium
to a given volume scales linearly with the parameter C1.This
leads to the definition of an alternative pressure based CF,
which relies on the good quality of the pressure data and
their synchronization to the volume trace:
fp=
pdat1
pdat2
psim1
psim2
(21)
where pdat1,pdat2 are the measured cavity pressures at
two timepoints in diastole and psim1,psim2 are the esti
at the two last diastolic frames (DF2, DF1 are again the
end-diastolic and pre-end-diastolic frames from the MRI
sequence, respectively) and psim DF
1,psim DF
2are the esti-
mated cavity pressures at the respective timepoints, which are
required to inflate the reference mesh to the measured cavity
volumes. For consistency with the energy CF analysis, these
diastolic timepoints are defined as frames DF1and DF2cor-
responding to the pre-end-diastolic and end-diastolic frames
of the MRI sequence (Sect. 2.3.3). As psim scales with C1,
the ratio psim1
psim2 is independent of C1and therefore fpdepends
only on α(for fixed anisotropy ratios). Note that the pressure
CF can only be evaluated through forward simulations, while
in contrast the energy CF can be evaluated directly from the
clinical data.
To evaluate the pressure form of the cost function we find
the αparameter for the 8 cases of this study. For this purpose,
123
A. Nasopoulou et al.
parameter sweeps over αfor an arbitrary fixed C1at 2000 Pa
were performed. For each αthe myocardium was inflated to
an elevated cavity pressure (3 times the end-diastolic pres-
sure) in 90 increments. The values psim1 and psim2 were
found by linear interpolation between the two simulation
increments with volumes that bound the cavity volume from
the data. Following estimation of αfrom Eq. 21,C1was
estimated from the already fitted exponential curves to the
minimum L2displacements norm CF residual (as explained
in Sect. 2.4) and the results are presented in Table 7.
Despite the fact that both the energy CF and the pressure
CF do find the correct unique set of parameters in an in-silico
test, parameters show large differences depending on the CF
when working with real data (see Table 7). This is a similar
finding as when working with the collection of geometrical
CFs, and reflects the incongruity between data and model
accentuated by the different origins and types of data errors.
The assumption that these patients, who share a common
pathology, have similar material properties would suggest
that for the available data the energy-based CF provides a
more robust estimate of model parameters. Nevertheless,
there is no ground truth methodology available for the evalua-
tion of the performance of the different CFs. Further research
in data acquisition and analysis, on model fidelity, and on
alternative in-vivo strategies may improve the agreement
between the different CFs and lead to the true material param-
eters of the beating myocardium.
9 Application of the energy-based CF to the
reduced Holzapfel–Ogden material model
The method can be extended to a variety of exponential
parameter laws where scaling parameters are used in one
or more terms. An example that suits this description is the
Holzapfel–Ogden (H-O) law (Holzapfel and Ogden 2009):
Ψ=a
2beb(I13)1+af
2bfebf(I4f1)21
+as
2bsebs(I4s1)21+afs
2bfs ebfsI2
8fs 1(22)
where:
I1=tr(C)I4f=f0·(Cf
0)
I4s=s0·(Cs0)I8fs =f0·(Cs0)(23)
and f0,s0the basis vectors aligned with the local fibre
and sheet directions, respectively. Following the approach
by Hadjicharalambous et al. (2015) we will use the reduced
version of this law where the parameters associated with the
sheet and fibre–sheet response are set to zero (as,afs =0).
This provides an equivalence in the number of unknown
parameters between the Guccione and reduced H–O (both
have four parameters). Aiming to address the main direction
of coupling in the reduced H–O, which is expected to occur
between the scaling parameters and those inside the expo-
nential function (pairs: a-b,af-bf), we use the parameter
ratios γ=a
afand δ=b
bfto reformulate the reduced H–O
relationship as:
Ψ=γaf
2δbfeδbf(I13)1+af
2bfebf(I4f1)21,
(24)
Ψ=af
2bfγ
δeδbf(I13)1+ebf(I4f1)21.
(25)
From Eq. 25 the energy-based CF for the reduced H-O law
can be expressed as :
frHO=
Wext1
Wext2
af
2bfγ
δeδbf(I113)1+ebf(I4f11)21
af
2bfγ
δeδbf(I123)1+ebf(I4f21)21
(26)
frHO=
Wext1
Wext2
γ
δeδbf(I113)1+ebf(I4f11)21
γ
δeδbf(I123)1+ebf(I4f21)21(27)
and by fixing the two parameter ratios γ,δ-in analogy to our
application for the Guccione law frHOis only a function of
bfif I1,I4fare estimated directly from the data deformation
field. In Eq. 26,27 Wext1,I11 and I4f1and Wex t2,I12 and
I4f2denote the external work, I1and I4finvariants at two
timepoints which correspond to diastolic frames DF
1and
DF
2respectively. Following estimation of bf, the scaling
parameter af-in analogy to the formulation for the Guccione
law can be estimated from |u|.
To demonstrate the ability of this CF to isolate bfwe
have performed an in silico analysis, where the synthetic LV
presented in Sect. 2.1.1 with the same cardiac microstructure
(Sect. 2.2.2) was assigned a material behaviour described by
Eq. 25 with af=10,000 Pa, γ=2.5, bf=5, δ=1asin
Hadjicharalambous et al. (2015), and was passively inflated
to 1500 Pa in 15 increments. In analogy to the in silico test
for the Guccione law, the ‘frames’ corresponding to the last
two increments of the simulation were chosen as DF
1and
DF
2and the energy-based CF presented a unique minimum
at bf=5, thus identifying the ground truth value.
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123
    • 2.2) with the identified parameters to simulations with parameter sets that lie on the 'principal' coupling line [6] demonstrates the good performace of the identified parameters in terms of deformation prediction (Table 1). On the contrary, not addressing the param eter coupling problem can lead to an error in the predicted developing stresses within the range of cardiac deformation (Fig. 3) [4]. The sensitivity analysis showed that the biggest source of error in parameters was the noise in the deformation field, which is an anticipated result given the exponential nature of the material law and the formulation of the proposed CF (Eq.
    [Show abstract] [Hide abstract] ABSTRACT: Myocardial stiffness is a useful diagnostic and prognostic biomarker, but only accessible through indirect surrogates. Computational 3D cardiac models, through the process of personalization, can estimate the material parameters of the ventricles, allowing the estimation of stiffness and potentially improving clinical decisions. The availability of detailed 3D cardiac imaging data, which are not routinely available for the conventional cardiologist, is nevertheless required to constrain these models and extract a unique set of parameters. In this work we propose a strategy to provide the same ability to identify the material parameters, but from 2D observations that are obtainable in the clinic (echocardiography). The solution combines the adaptation of an energy-based cost function, and the estimation of the out of plane deformation based on an incompressibility assumption. In-silico results, with an analysis of the sensitivity to errors in the deformation, fibre orientation, and pressure data, demonstrate the feasibility of the approach.
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