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

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 −a−a134 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 −60◦at 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(eQ−1)(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/α (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 10−8mm error for |u|and 2 10−7

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 =

MRIF−1

n=RF

pn+pn+1

2(Vn+1−Vn). (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 C1−αparameter 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 −60◦to +60◦angle 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. 0171812−4−a12 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 10−9mm 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(I1−3)−1+af

2bfebf(I4f−1)2−1

+as

2bsebs(I4s−1)2−1+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(I1−3)−1+af

2bfebf(I4f−1)2−1,

(24)

Ψ=af

2bfγ

δeδbf(I1−3)−1+ebf(I4f−1)2−1.

(25)

From Eq. 25 the energy-based CF for the reduced H-O law

can be expressed as :

frH−O=

Wext1

Wext2

−

af

2bfγ

δeδbf(I11−3)−1+ebf(I4f1−1)2−1

af

2bfγ

δeδbf(I12−3)−1+ebf(I4f2−1)2−1

(26)

frH−O=

Wext1

Wext2

−

γ

δeδbf(I11−3)−1+ebf(I4f1−1)2−1

γ

δeδbf(I12−3)−1+ebf(I4f2−1)2−1(27)

and by ﬁxing the two parameter ratios γ,δ-in analogy to our

application for the Guccione law frH−Ois 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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- 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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