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An open source pipeline for design of experiments for

hyperelastic models of the skin with applications to

keloids

D. Sutulaa, A. Elounega, M. Sensalea,b, F. Choulyc, J. Chamberta,

A. Lejeunea, D. Barolid, P. Hauseuxb, S. Bordasb,∗∗, E. Jacqueta,∗

aUniv. Bourgogne Franche-Comt´e, FEMTO-ST Institute, Department of Applied

Mechanics, Besan¸con, France.

bUniversity of Luxembourg, Institute of Computational Engineering, Luxembourg.

Department of Computer Science and Information Engineering, Asia University,

Taichung, Taiwan. Department of Medical Research, China Medical University Hospital,

China Medical University, Taichung, Taiwan.

cUniv. Bourgogne Franche-Comt´e, Institut de Math´ematiques de Bourgogne, Dijon,

France.

dRWTH, AICES, Aachen, Germany

Abstract

The aim of this work is to characterize the mechanical parameters govern-

ing the in-plane behavior of human skin and, in particular, of a keloid-scar.

We consider 2D hyperelastic bi-material model of a keloid and the surround-

ing healthy skin. The problem of ﬁnding the optimal model parameters that

minimize the misﬁt between the model observations and the in vivo exper-

imental measurements is solved using our in-house developed inverse solver

that is based on the FEniCS ﬁnite element computational platform. The

paper focuses on the model parameter sensitivity quantiﬁcation with respect

to the experimental measurements, such as the displacement ﬁeld and re-

action force measurements. The developed tools quantify the signiﬁcance

of diﬀerent measurements on diﬀerent model parameters and, in turn, give

insight into a given model’s ability to capture experimental measurements.

Finally, an a priori estimate for the model parameter sensitivity is proposed

that is independent of the actual measurements and that is deﬁned in the

∗Corresponding author : emmanuelle.jacquet@univ-fcomte.fr

∗∗Corresponding author : stephane.bordas@alum.northwestern.edu

Preprint submitted to Journal of the Mechanical Behavior of Biomedical Materials

whole computational domain. This estimate is primarily useful for the de-

sign of experiments, speciﬁcally, in localizing the optimal displacement ﬁeld

measurement sites for the maximum impact on model parameter inference.

Key words: soft tissue, inverse problem, optimization, model sensitivity,

FEniCS.

1. Introduction

Biomechanical modelling can provide valuable insight into the response

of biological systems to mechanical loading (Hatze,1974). The study of

biomechanics ranges from the inner workings of cells (Niklas,1992) to the

mechanical properties of bones (Cowin,2001) and soft tissues (Fung,1993),

and to the development and movement of limbs. In the context of human

living tissues, the attention over the past few decades has been mainly on the

biomechanical characterization of such biological materials as brain tissues

(Van Dommelen et al.,2010;Rashid et al.,2014), tendons (Screen,2008),

meniscal tissue (Barrera et al.,2018), and skin (in vivo,in vitro and ex vivo)

(Maiti et al.,2016;Annaidh et al.,2012;Ottenio et al.,2015;Pan et al.,

2019).

In this paper, we focus on the patient-speciﬁc biomechanical characteri-

zation of human skin and, in particular, a keloid scar (Abdel-Fattah,1976;

Robles and Berg,2007). Keloids are non-cancerous tumors that grow con-

tinuously on the skin for several reasons. The evolution of keloids is related

to genetic (Halim et al.,2012), biological (Kim et al.,2000), and biomechan-

ical factors (Ogawa,2008;Ogawa et al.,2012). In this work we care about

the latter kind of inﬂuence; for example, the state of stress inside a keloid

and in the surrounding skin is known to play a signiﬁcant role in the growth

of a keloid (Akaishi et al.,2008;Ogawa et al.,2012;Nagasao et al.,2013).

However, in order to be able to characterize the mechanical behavior of a

skin-keloid tissue it is ﬁrst necessary to identify a suitable material model

and to determine the patient-speciﬁc model parameters. The model parame-

ter inference is centered around the idea of minimizing the error between the

model observations and the experimental measurements (Meijer et al.,1999).

There are many phenomenological hyperelastic models that are designed to

capture various non-linear behaviors of the skin undergoing large deforma-

tions (Evans and Holt,2009;Wex et al.,2015;Bahreman et al.,2015). We

cite for instance such models as Mooney-Rivlin (1948), Ogden (1972), Yeoh

(1993) or the simplest one, the Neo-Hookean model (Pence and Gou,2015).

2

Time-dependence, such as a time-varying Poisson’s ratio, can be modeled by

using a fractional viscoelastic constitutive model (Alotta et al.,2017,2018).

There are many types of experiments for measuring the mechanical re-

sponse of skin under static or even dynamic loading. The most common type

of test is the uniaxial traction test (Wan Abas and Barbenel,1982;Finlay,

1970;Meijer et al.,1999;Boyer et al.,2013). A biaxial test can be performed

for characterizing anisotropic material behavior (Wan Abas,1994). For small

and curved regions other types of tests may be more appropriate such as suc-

tion (M¨uller et al.,2018;Laiacona et al.,2019), bulge (Tonge et al.,2013) or

indentation (Pailler-Mattei et al.,2008) tests. Other non-invasive tests for

measuring the dynamic response of skin include using ultrasound (Diridol-

lou et al.,1998;Zhang and Greenleaf,2007), ballistometry (Fthenakis et al.,

1991) and other dynamic mechanisms (Finlay,1970).

In this work, we rely on in vivo uniaxial extension experiments and ap-

ply an inverse modeling approach to infer the material model parameters

for a supposed material model. We consider 2D models for the in-plane be-

havior of the skin assuming the Neo-Hookean type of material. The ﬁrst

model assumes just the healthy skin (without the keloid) whereas the second

model considers the bi-material of healthy skin and a keloid. Subsequently,

the sensitivity of the inferred model parameters is assessed with respect to

the measurements such as the displacement ﬁeld and reaction force measure-

ments. In summary, this paper makes the following contributions.

1. Introduction of a numerical process based on an open-source framework

to identify the mechanical parameters of a heterogeneous soft tissue.

2. Model parameter sensitivity quantiﬁcation with respect to displace-

ment ﬁeld and reaction force measurements. The sensitivities reveal

how signiﬁcant the diﬀerent measurements are on diﬀerent model pa-

rameters. They can also be used in troubleshooting a given model,

such as discovering why certain errors may be large between the model

observations and experimental measurements.

3. A prior estimate of the model parameter sensitivities with respect to the

displacement ﬁeld measurements. The estimate is deﬁned in the whole

computational domain and it is independent of the actual displacement

ﬁeld measurements. The estimate is useful primarily in that it reveals

the most important displacements measurement sites, which can be

leveraged in the design of experiments.

3

2. Problem statement

2.1. In vivo experiments

The in vivo experiments are carried out using a custom-made extensome-

ter (Jacquet et al.,2017a,b), as shown in Fig. 1b. The small device is ﬁxed on

a region of interest, such as around a keloid scar Fig. 1a, and a displacement-

controlled uni-axial displacement test is performed. The reaction force is

measured by a strain gauge whereas the surface displacement ﬁeld is cap-

tured using 2D digital image correlation (DIC) (Avril et al.,2008;Boyer

et al.,2013). A series of measurements of the reaction force and of the cor-

responding displacement ﬁeld are obtained.1As the pad displacements are

measured with a suﬃciently high accuracy – it is reported in (Jacquet et al.,

2017b) that the extensometer is equipped with a linear variable displace-

ment transducer whose positional accuracy is approximately 38 (µm) – the

pad displacement measurements are introduced directly as Dirichlet bound-

ary conditions in the hyperelastic problem. Unfortunately, we were unable

to quantify the errors in the displacement ﬁeld measurements obtained via

the DIC analysis including any errors due to the out of plane motion, which

is unavoidable in practice (Sutton et al.,2008). We did, however, identify

a small number of spurious displacements that we subsequently eliminated

from consideration.

2.2. Hyperelastic problem

Consider a domain Ω ∈R2of a hyperelastic medium. The domain bound-

ary ∂Ω is composed of the Dirichlet boundary ΩD(where displacements are

known) and the Neumann boundary ΩN(where tractions are known) such

that ∂Ω = ∂ΩD∪∂ΩNand ∂ΩD∩∂ΩN=∅. Considering a pure displace-

ment load uDprescribed on ∂ΩD, the potential energy of the hyperelastic

solid consists of just the strain energy of deformation that reads

W(u,m) = ZΩ

ψ(u,m) dx, (1)

where uis the displacement ﬁeld, ψ(u,m) is the strain energy density of

a chosen material model and mare the model’s parameters. The variational

1The present work uses experimental data that was previously published in Chambert

et al. (2019).

4

(a) Keloid scar located on the left-upper arm (X=

15 mm,Y= 47 mm).

(b) Extensometer. The speckle pattern is used for

capturing the displacements.

Figure 1: Experimental investigation into the mechanical response of human skin (Cham-

bert et al.,2019).

(or weak) form of static equilibrium is obtained by diﬀerentiating W(u,m)

with respect to uand requiring all the admissible variations to vanish:

F(u,m;v) = ∂uW(u,m)[v]=0 ∀v∈ V0,(2)

where ∂u(·)[v] denotes a partial derivative with respect to the argument

u∈ V in the direction of v∈ V0. Here, Vand V0are suitably chosen function

spaces such that the weak form (2) is well deﬁned, u∈ V satisﬁes the Dirichlet

boundary conditions and v∈ V0vanishes on the Dirichlet boundary.

2.3. Inverse problem

The cost functional for the surface displacement ﬁeld misﬁt is deﬁned as

the sum of the local costs over the measurement steps k= 1, . . . , Nmsr

Ju=

Nmsr

X

k=1

J(k)

u=

Nmsr

X

k=1

J(k)

uu(k)(m),u(k)

msr=

Nmsr

X

k=1

1

2ZΓuu(k)(m)−u(k)

msr2dx,

(3)

where u(k)(m) is the model displacement ﬁeld under the boundary condi-

tions of step k,u(k)

msr is the measured (experimentally obtained) displacement

ﬁeld, and Γuis the displacement ﬁeld measurement surface.

5

The functional (3) is generally insuﬃcient to capture all the information

needed to infer all the model parameters. In particular, the model param-

eters that carry units of pressure (N/mm2) can not be determined without

measurements of a related quantity. Therefore, an additional cost functional

is introduced that is a measure of the pad reaction force misﬁt

Jf=

Nmsr

X

k=1

J(k)

f=

Nmsr

X

k=1

J(k)

ff(k)(m), f (k)

msr=

Nmsr

X

k=1

1

2f(k)(m)−f(k)

msr2(4)

The model reaction force f(k)is obtained by integrating the pad traction

Ton the pad surface Γf

f(k)(m) = fu(k)(m),m=ZΓf

Tu(k)(m),m·Npad ds, (5)

where Npad is the pad displacement direction and f(k)

msr(m) is the experi-

mentally measured reaction force. Finally, the displacement and the reaction

force misﬁt costs can be combined in a weighted average sense to obtain the

total cost of the model

J(m) = Ju(m) + wJf(m) (6)

where w > 0 is the weight that also serves to rescale Jf(since its units

are diﬀerent from of Ju).2

The inverse problem may be stated as follows. Find the model parameters

m?such that the cost J(m) is minimized (or stationary) with respect to m

and the weak form of static equilibrium F(u(k),m;v) = 0 is satisﬁed at each

step k= 1,2, . . . , Nmsr. In summary, the inverse problem reads

m?= arg (DmJ(m) = 0) (7)

subject to: F(u(k),m;v) = 0

∀v∈ V0, k = 1,2, . . . , Nmsr (8)

where DmJis the total derivative of J(m) with respect to m.

2In our speciﬁc test cases, the choice of wwas not found to be very important because

the costs Juand Jfhappened to aﬀect diﬀerent model parameters. As such, wacted more

like a stabilization than a weight.

6

3. Solution method

The total derivative of J(m) with respect to mis expressed as

DmJ=

Nmsr

X

k=1

∂mJ(k)+ [Dmu(k)]T∂uJ(k)(9)

The derivative Dmu(k)can be computed by considering the weak form

of static equilibrium (2). Firstly, suppose the current solution state (u(k),

m) satisﬁes (2). Now consider an arbitrary perturbation δmand the new

solution state (u(k)+δu,m+δm), where δu=Dmu(k)[δm]. We can ﬁnd

Dmu(k)such that the new state satisﬁes (2) for any admissible δmas follows

F(u(k),m;v) = F(u(k)+δu,m+δm;v)=0 ∀v∈ V0(10)

∂uF(u(k),m;v)[δmu(k)] + ∂mF(u(k),m;v)[δm] = 0 ∀v∈ V0(11)

Dmu(k)=−[∂uF(k)]−1∂mF(k),(12)

where [∂uF(k)]−1denotes the inverse of the tangent-stiﬀness ∂uF(k). How-

ever, the explicit computation of Dmu(k)can actually be avoided if we are

only concerned with determining DmJ(k):

DmJ(k)=∂mJ(k)+ [Dmu(k)]T∂uJ(k)(13)

DmJ(k)=∂mJ(k)+ [∂mF(k)]T−[∂uF(k)]−T∂uJ(k)(14)

DmJ(k)=∂mJ(k)+ [∂mF(k)]Tz(k)(15)

where z(k)is an intermediate solution vector (sometimes called the adjoint

variable) which is determined by solving the problem implied in (14)

[∂uF(k)]Tz(k)=−∂uJ(k)(16)

In review, computing DmJ(k)with the explicitly determined Dmu(k)re-

quires as many linear solves as there are model parameters; on the other

hand, computing DmJ(k)directly always requires a single linear solve (since

∂uJ(k)is a vector). Therefore, the second approach is preferred if Dmu(k)is

not explicitly needed. By extension, the second order derivatives D2

mmJ(k)

7

can also be determined by leveraging the second approach. In this case, how-

ever, although computing the derivative D2

mmu(k)can be avoided, computing

Dmu(k)can not be. Finally, the Newton-Rhapson’s algorithm can be used

to update the model parameters m[n]at the n’th iteration step as shown:

∆m[n]=−D2

mmJ[n]−1DmJ[n](17)

In case a model parameter is required to be of a certain sign, the simple

act of limiting the model parameter change is usually an adequate solution.

4. Model parameters sensitivity

The inverse problem of model parameter identiﬁcation is a deterministic

one. The solution is merely the best ﬁt solution to the data that leads to the

stationarity of the cost (6). Consequently, there are no additional outputs

from the solution process such as a measure of the uncertainty in the inferred

model parameters due to the uncertainties in the measurements. Therefore,

it is useful to estimate the sensitivity of the model parameters with respect

to the displacement ﬁeld and pad reaction force measurements.

4.1. Deterministic evaluation

The model parameter sensitivities with respect u(k)

msr and, similarly, with

respect to f(k)

msr can be computed by solving a problem of this form

δvDmJ=D2

mmJ[δvm] + ∂vDmJ[δv] = 0 (18)

where vdenotes a generic measurement and δvits directional change.

Equation (18) says that the model cost should maintain its slope (e.g. of

zero) by causing the model parameters to change accordingly based on a

change in a measurement. For instance, substituting v≡u(k)

msr in (18) the

solution to the model parameter sensitivity can be determined as

Du(k)

msr m=−D2

mmJ−1∂u(k)

msr DmJ,(19)

where ∂u(k)

msr DmJtakes the form a rectangular matrix that has Nmrows

and Ndof number of columns. Solely for the purpose of computing (19) we

suppose that umsr is discretized exactly like the displacement ﬁeld u, i.e.

umsr =PNdof −1

J=0 φJumsrJwhere Jis the DOF index, φis the Ndim -by-Ndof

8

matrix of the ﬁnite element nodal basis functions, and umsr is the vector

of DOFs. With this in mind, the entity Du(k)

msr mIJ refers to the I’th model

parameter’s J’th sensitivity value with respect to the degree of freedom umsrJ.

In the case of the reaction force measurements, the J’th index in Df(k)

msr mIJ

generally refers to the reaction force component. In the extensometer exper-

iment, however, the force is measured in just one direction; hence, Df(k)

msr mIJ

is eﬀectively a vector where index Jis unnecessary.

Equation (19) computes the model parameter sensitivities locally with

respect to a measurement and boundary conditions at step k. In order to es-

timate the total change in the model parameters, all the local changes would

need to be added up. This computation, however, is not straightforward

because the measurement perturbations can be arbitrary. The model param-

eter sensitivity can be assessed more simply by considering the measurement

variability in a statistical sense.

4.2. Statistical quantiﬁcation

Assume the displacement ﬁeld measurements can be characterized by the

variance σ2

u(k)

msr

that is spatially and temporally uncorrelated. The I’th model

parameter variance σ2

mI|u(k)

msr

due to σ2

u(k)

msr

can be estimated for step kas follows

σ2

mI|u(k)

msr = Var Du(k)

msr mIJ δu(k)

msrJ(20)

σ2

mI|u(k)

msr =Du(k)

msr mIJ Cov δu(k)

msrJ K Du(k)

msr mIK (21)

σ2

mI|u(k)

msr =σ2

u(k)

msr X

JDu(k)

msr mIJ 2(22)

The model parameter variance with respect to the reaction force mea-

surement variance is obtained similarly

σ2

mI|f(k)

msr =σ2

f(k)

msr Df(k)

msr mI2(23)

In practice, it is more meaningful to report the sensitivity of a model

parameter in terms of the relative standard deviation

ˆσmI|u(k)

msr =σmI|u(k)

msr /|mI|(24)

ˆσmI|f(k)

msr =σmI|f(k)

msr /|mI|(25)

9

The statistical quantiﬁcation of the model parameter sensitivity is use-

ful because the model parameter dependence on the measurements can be

summarized simply. The results can give useful insight into how signiﬁcant

diﬀerent measurements are on the model parameters.

4.3. Smoothing projection

We would like to visualize the model parameter sensitivity with respect

to the displacement ﬁeld measurements umsr because the matrix Du(k)

msr mis

too abstract. To this end, each vector Du(k)

msr mIcan be transformed into a

continuous vector function. The transformation problem is posed as follows.

Suppose the sought sensitivity function is deﬁned using the ﬁnite element

basis functions as gI=PNdof −1

K=0 φKgIKwhere gIis the DOFs vector for the

I’th model parameter sensitivity. Then, for any admissible perturbation in

the displacement ﬁeld measurements δumsr =PNdof−1

J=0 φJδumsrJ, require the

model parameter change δmI=PNdof

J=0 δumsrJDu(k)

msr mIJ to be equivalently

computable by the inner product of the functions δumsr and gI:

δumsrJZΓu

φT

JφKgIKdx=δumsrJDu(k)

msr mIJ ∀δumsrJ(26)

The solution vector for the sensitivity function gIis determined as

gI=M−1Du(k)

msr mI(27)

where M=RΓuφTφdxis the ﬁnite element mass-like matrix. Note

that because the displacement ﬁeld measurements are deﬁned only on the

subdomain Γuthe sensitivity values outside this subdomain will be zero.

4.4. A prior estimate

The main drawback of (19) is that the model parameter sensitivity is

deﬁned only in the measurement subdomain. However, we would also like to

have some idea of the signiﬁcance of the displacement ﬁeld measurements in

the whole domain so that, for example, we could determine a more optimal

subdomain for gathering displacement ﬁeld measurements. Let us pretend

we have displacement ﬁeld measurements u∗

msr deﬁned on the whole domain

Ω. The fabricated cost J∗

udue to the displacement ﬁeld misﬁt is

J∗

u(m) = 1

2ZΩ

(u(m)−u∗

msr)2dx(28)

10

Let us suppose that the material model is very good such that the error

between the measured and the model displacements is negligible, i.e. u≈

u∗

msr. In this case, the second derivative of J∗

uwith respect to the material

model parameters msimpliﬁes to the following:

D2

mImJJ∗

u=ZΩ

DmIuiDmJuidx+ZΩ

(ui−u∗

msri)D2

mImJuidx(29)

D2

mImJJ∗

u≈ZΩ

DmIuiDmJuidx(30)

The second mixed derivative of J∗

uwith respect to mand u∗

msr reads

∂u∗

msr DmIJ∗

u[δu∗

msr] = −ZΩ

DmIuiδu∗

msridx(31)

As a ﬁrst attempt, one may try to compute the sensitivity Du∗

msr mas

Du∗

msr m=D2

mmJ∗

u−1−∂u∗

msr DmJ∗

u,(32)

where ∂u∗

msr DmJ∗

uis a rectangular matrix that has Nmrows and Ndof

number of columns corresponding to the DOFs of u∗

msr. Unfortunately, it

may not be generally possible to determine Du∗

msr mbecause D2

mmJ∗

umay

not have an inverse. Nonetheless, the sensitivity can always be measured in

an eigenvector direction. Suppose the non-zero eigenvalue and eigenvector

are ˆ

λand ˆ

vrespectively. Also suppose the displacement ﬁeld measurements

are discretized as u∗

msr =PNdof −1

J=0 φJu∗

msrJwhere Jis the DOF index, φis

matrix of ﬁnite element shape functions and u∗

msr is the vector of DOFs. The

model parameter change δˆmin the direction of ˆ

vcan be computed as follows

ˆvTD2

mmJ∗

uˆvδˆm=ˆvTZΩ

Dmuiδu∗

msridx(33)

δˆm=ˆvI

ˆ

λZΩ

DmIuiφiJ dxδu∗

msrJ(34)

Subsequently, the model parameter sensitivity can be deﬁned as

Du∗

msr ˆmJ=ˆvI

ˆ

λZΩ

DmIuiφiJ dx(35)

11

The vector Du∗

msr ˆmof discrete sensitivities can be converted into a vector-

valued function using the method described in Sec. 4.3. Suppose this function

is deﬁned as ˆg =PNdof

J=0 φJˆgJ. The DOFs vector ˆg can be computed as

ˆg =M−1Du∗

msr ˆm, (36)

where M=RΩφTφdxis the ﬁnite element mass matrix. The model

parameter sensitivity function ˆg, which is independent of the actual dis-

placement ﬁeld measurements and that is deﬁned on the whole domain, can

be used as an indicator for the most optimal displacement ﬁeld measurement

subdomain that has the most impact on the model parameters.

5. Implementation

The inverse solver and the sensitivity analysis routines are implemented

in Python based on the ﬁnite element computational platform of FEniCS

(Logg,2007;Logg and Wells,2010;Logg et al.,2012a). FEniCS is a col-

lection of open-source libraries that enable highly automated solutions of

diﬀerential equations. The FEniCS’ Python interface called DOLFIN (Kirby

and Logg,2006;Logg and Wells,2010;Logg et al.,2012c) provides an easy

means to deﬁne ﬁnite element variational forms (Alnæs et al.,2014;Alnæs,

2012;Alnæs et al.,2012) that can then be translated (Kirby and Logg,2006;

Logg et al.,2012b) into low-level C++ ﬁnite element assembly code (Alnæs

et al.,2009,2012). Consequently, all the expensive ﬁnite element related

computations can be handed over to the C++ runtime. Our implementation

also heavily relies on the Python’s popular scientiﬁc computing libraries,

namely: scipy (Virtanen et al.,2020), numpy (van der Walt et al.,2011),

matplotlib (Hunter,2007), IPython (P´erez and Granger,2007) and other

parts of the toolstack (Oliphant,2007;Millman and Aivazis,2011). Our im-

plementation, including the studied cases and other examples, is available at

https://github.com/danassutula/model_parameter_inference.git.

6. Application

We consider a 2D hyperelastic model and plane-stress conditions. The

eﬀective load carrying thickness of the skin (and keloid) is assumed to be one

millimeter. The computational domain is discretized using linear triangu-

lar ﬁnite elements in an unstructured mesh. The surface displacement ﬁeld

measurements, which are read-in in raw form as point-displacement values

12

on a regular grid, are projected at the nodes of the underlying ﬁnite element

mesh using a second degree least-squares meshless interpolation (Belytschko

et al.,1996;Duarte and Oden,1996;Nguyen et al.,2008).

6.1. Material model

Human skin is generally considered to be an anisotropic (Reihsner and

Menzel,1996;Meijer et al.,1999;Limbert,2017) incompressible (Weiss et al.,

1996;Nolan et al.,2014;Chagnon et al.,2015) material. In our case, how-

ever, ﬁtting an anisotropic model would likely lead to an unreliable solution

since all the measurements come from a uni-directional extension, i.e. there

is no information along other load axes (Reihsner and Menzel,1996;Groves

et al.,2013;Boyer et al.,2013). On the other hand, (near-)incompressible

material behavior is numerically burdensome since an additional incompress-

ibility constraint and a new variable for the hydro-static stress must be in-

volved (Lapeer et al.,2011;Nolan et al.,2014). As we consider a simple 2D

plane-stress model, it makes little sense to choose an incompressible model

over a compressible one given the added numerical complexity. Therefore, we

assume the material model for both the healthy skin and the keloid to be a

compressible hyperelastic Neo-Hookean material whose strain energy density

is deﬁned as shown

Ψ = C1(I1−d−2 ln J) + D1ln J2,(37)

where C1and D1are the material model parameters and

I1= tr C(38)

J= det F(39)

C=1

2(FTF−I) (40)

I1is the ﬁrst invariant of the right Cauchy-Green deformation tensor C,

Fis the deformation tensor whose components are Fij =δij +∂ui/∂xjwhere

i, j = 0, ..., d −1, and dis the geometric dimension (d= 2). For consistency

of (37) with linear-elasticity C1=µ/2 and D1=λ/2 where λis the Lam´e’s

ﬁrst parameter and µis the shear modulus (Lam´e’s second parameter). It

is useful to express µand λin terms of a pseudo Young’s modulus Eand

pseudo Poisson’s ratio νas shown

13

λ=Eν

(1 + ν) (1 −2ν)(41)

µ=E

2(1 + ν)(42)

The advantage is that the material parameters are separated by physics;

speciﬁcally, Ecarries units of pressure whereas νis unitless. Besides this,

the parameters are consistent with linear-elasticity.

6.2. Healthy skin model parameter identiﬁcation

The model parameter identiﬁcation is ﬁrst carried out for the healthy

skin. The extensometer devise is placed on the patient’s arm without the

keloid at an equivalent location as that of the keloid on the other arm. The

rectangular computational domain is illustrated in Fig. 2. The square holes

indicate the ﬁxed pads of the extensometer. The boundaries of the holes have

prescribed displacements: the left pad translates rigidly leftward relative to

the right pad, which is ﬁxed at zero displacement. The external boundaries

are unrestrained. Fig. 2highlights the displacement ﬁeld measurement sub-

domain; it is roughly in the center between the two pads. This subdomain

was chosen because the measurements could be acquired more reliably.

6.2.1. Measurements

The highlighted subdomain in Fig. 2shows the location of the displace-

ment ﬁeld measurements. An example measurement, as shown in Fig. 3,

consists of point-displacements on a regular grid. All displacements were

computed relative to the in-plane position of the right pad. This was pos-

sible since the in-plane rigid translations and rotations of the extensometer

had been tracked via landmarks assigned on the vertices of both pads. The

reaction force vs. pad displacement measurements are shown in Fig. 4. The

eﬀective force-displacement relationship is taken to be the solid curve which

is obtained by mean-ﬁltering the measurements. The force-displacement re-

lationship is deﬁned up to a maximum displacement of 8 (mm). The hyper-

elastic behavior is largely exhibited from 6 (mm) of displacement.

6.2.2. Results

By virtue of the simplicity of the Neo-Hookean material model it was

possible to ﬁt the model parameters at each measurement step (i.e. the

14

Figure 2: Computational domain of the healthy skin model. The highlighted grid of points

indicates the displacement ﬁeld measurements. The square holes represent the ﬁxed pads.

Figure 3: Displacement ﬁeld measurements at the ﬁnal measurement step for the healthy

skin model parameter identiﬁcation problem.

inverse problem (7)-(8) was solvable). The optimal model parameter values

for each measurement step are shown in Fig. 5. The results show that the

15

Figure 4: Pad reaction force vs. displacement measurements. The solid red curve, which

was obtained by mean-ﬁltering the data, represents the eﬀective force-displacement load

curve that was used in the healthy skin model parameter identiﬁcation problem.

model parameter νvaries with each measurement. The inferred value steadily

decreases from νt=7 = 0.286 to νt=25 = 0.157 with increasing load. On the

other hand, the parameter Estays relatively constant at around E= 0.028

(N/mm2). The model parameter dependence on measurements suggests that

the choice of the model is not a very good one because the model parameter ν

changes steadily with increasing load magnitude. Tab. 1shows the obtained

model parameter values when the model was ﬁtted for all measurement steps.

Table 1: Inferred Neo-Hookean material parameters for the healthy skin model

E0.028 N/mm2

ν0.195 [-]

The model parameter evolution with diﬀerent ﬁnite element mesh sizes

is shown in Fig.6. Although the convergence of the model parameters is not

obvious at the ﬁnest ﬁnite element discretization (about 104k linear triangle

elements), the values change suﬃciently little for practical purposes.

The ﬁtted Neo-Hookean model was able to match the measured reaction

force well as shown by the results in Fig. 7. On the other hand, the dis-

16

Figure 5: Healthy skin Neo-Hookean material parameters ﬁtted at each measurement step.

Figure 6: Healthy skin Neo-Hookean material model parameter value vs. mesh size.

placement ﬁeld misﬁt turned out to be large. The error between the model

displacement ﬁeld and the measured displacement ﬁeld is shown in Fig. 8.

17

The displacement misﬁt error is largest at the start of the load when the

deformations are relatively small and decreases steadily with larger deforma-

tions (albeit the smallest error is still very large). The reason that the error

is inversely correlated with the deformation magnitude can be attributed to

the formulation of the displacement cost functional (3); speciﬁcally, the cost

is biased towards larger deformations. For example, Fig. 9shows that the

model cost increases monotonically. An example displacement misﬁt error

ﬁeld at the ﬁnal measurement step is shown in Fig. 10. The error vector ﬁeld

is principally in the direction of extension. Unfortunately, the Poisson’s-like

model parameter νhas very little to do with this type of deformation (it

mainly aﬀects the lateral deformation). Insofar as the total cost functional

(6) is concerned, its derivatives are plotted in Fig. 11. The derivatives are

zero on average over the measurement steps, which attests to the fact that

the cost is minimized. Nevertheless, ﬁtting the parameter νis demonstra-

bly hard. The parameter νis depressed with respect to roughly the ﬁrst

half of the measurements and elevated with respect to the remaining half of

the measurements. Even so, the parameter νhas little eﬀect on the cost as

implied by Fig. 10 since the error is principally in the direction of extension.

Figure 7: Model reaction force-displacement curve vs. measurements.

The model parameter sensitivities (refer to Sec. 4.1 for details) revealed

18

Figure 8: Error between the model and the measured displacements.

Figure 9: Cost of the healthy skin material model using the optimal material parameters.

that the parameter Ewas sensitive to the reaction force measurements and

insensitive to the displacement ﬁeld measurements. The opposite was ob-

19

Figure 10: Displacement ﬁeld misﬁt (i.e. u−umsr) for the ﬁnal measurement step. (This

plot is deﬁned over the subdomain highlighted in Fig. 2.)

Figure 11: Cost derivatives with respect to model parameters of the healthy skin.

served regarding the sensitivity of the model parameter ν; it was found to be

sensitive to the displacement ﬁeld measurements and insensitive to the re-

action force measurements. The sensitivities were quantiﬁed in terms of the

relative standard deviation in the model parameters as caused by a unit stan-

20

dard deviation in the displacement ﬁeld measurements (at the mesh nodes)

and a unit standard deviation in the reaction force measurements; the re-

sults are shown in Fig. 12 and Fig. 13 respectively. More details about the

sensitivity of parameter νwith respect to the displacement ﬁeld measure-

ments are revealed in Fig. 14. The vector ﬁeld conveys the information that

perturbing the displacement ﬁeld measurements in the direction of this vec-

tor ﬁeld will result in a positive change in the parameter ν. Note that ν

is principally sensitive to the vertical component of the displacement ﬁeld

measurements, which is almost perpendicular to the displacement ﬁeld error,

as seen in Fig. 10.

An estimate of the model parameter νsensitivity in the whole domain

(refer to Sec. 4.4) is shown in Fig. 15. The reason that the intensity of the

ﬁeld is almost an order of magnitude smaller than that of the ﬁeld shown

in Fig. 14 is because of the diﬀerence in the domain size. Speciﬁcally, the

average intensity of the sensitivity ﬁeld for a bigger domain should be smaller.

Figure 12: Healthy skin model parameter relative standard deviation assuming one stan-

dard deviation in the displacement ﬁeld measurements at the mesh nodes.

6.3. Keloid scar model parameter identiﬁcation

The model parameter identiﬁcation for the keloid scar is performed con-

sidering the bi-material domain shown in Fig. 16. The Neo-Hookean material

21

Figure 13: Healthy skin model parameter relative standard deviation assuming one stan-

dard deviation in the reaction force measurements.

Figure 14: Model parameter νsensitivity with respect to the displacement ﬁeld measure-

ments at the ﬁnal measurement step. (This plot is deﬁned over the subdomain highlighted

in Fig. 2.)

model (refer to Sec. 6.1 for details) is assumed for each material subdomain.

The model parameter values previously determined for the healthy skin were

22

Figure 15: Model parameter νestimated sensitivity with respect to displacement ﬁeld

measurements.

re-used for the healthy skin subdomain, namely Eskin = 0.028 (N/mm2) and

νskin = 0.195. Therefore, only the keloid model parameters needed to be

determined, speciﬁcally: Ekeloid and νkeloid.

Figure 16: Bi-material skin-keloid model. The displacement ﬁeld measurements are high-

lighted by the rectangular grid of points. The square holes represent the ﬁxed pads.

23

6.3.1. Measurements

The highlighted subdomain in Fig. 16 shows the location of the displace-

ment ﬁeld measurement points. This subdomain was chosen because the dis-

placement ﬁeld measurements could be acquired more reliably. An example

measurement of the point-displacements evaluated at the ﬁnal measurement

step is shown in Fig. 17. The measurements contained some spurious data

points; consequently, these data were omitted from consideration. The mea-

sured reaction force vs. pad displacement relationship is shown in Fig. 18.

The force is deﬁned up to a displacement of roughly 4 (mm). The hy-

perelastic behavior is mainly observed from 2 (mm) of pad displacement.

Since a Neo-Hookean material model can reproduce only a weakly nonlinear

force-displacement relationship, the model parameter ﬁtting was limited to

a maximum pad displacement of 2 (mm).

Figure 17: Displacement ﬁeld measurements at the ﬁnal measurement step for the keloid

model parameter identiﬁcation problem. (The point vacancies in the regular grid indicate

spurious data that had to be discarded.)

6.3.2. Results

The model parameter evolution with mesh reﬁnement is shown in Fig. 19.

The ﬁnest resolution mesh consisted of 105k linear triangle elements. The

convergence in the Young’s modulus-like parameter Ekeloid is fairly obvious.

On the other hand, the Poisson’s-like model parameter νkeloid fails to con-

24

Figure 18: Pad reaction force vs. displacement measurements. The solid curve, which was

obtained by mean-ﬁltering the discrete data, represents the eﬀective force-displacement

load curve for the keloid model parameter identiﬁcation problem.

verge. The presumed bi-material model parameters are given in Tab. 2.

Fig. 20 shows that the bi-material model can match the measured force-

displacement curve reasonably well. However, the displacement error, as

shown by Fig. 21, is large – up to 14% for the last measurement step. Fig. 22

shows the distribution of this error in the measurement subdomain. The

concentration of error likely arises due to the 3D shape of the keloid (above

and within the skin) whose eﬀects can not be accounted for by our simple 2D

model. Another source of error is believed to be the 2D DIC technique that

was used. In this case, a single camera is not so well suited for capturing

the displacements on a curved surface of the skin. The measurements could

have been more accurate if a stereoscopic camera system was used instead.

Table 2: Inferred Neo-Hookean material parameters for the skin-keloid bi-material model

Ekeloid 0.077 N/mm2

νkeloid 0.35 [-]

Eskin 0.028 N/mm2

νskin 0.195 [-]

25

Figure 19: Keloid Neo-Hookean material model parameter values vs. mesh size.

Figure 20: Bi-material model reaction force-displacement curve vs. measurements.

The keloid material model parameter sensitivities with respect to the

displacement ﬁeld measurements are given in Fig 23. The sensitivities with

26

Figure 21: Error between the model and the measured displacements.

Figure 22: Displacement ﬁeld misﬁt (i.e. u−umsr) for the ﬁnal measurement step. (This

plot is deﬁned over the subdomain highlighted in Fig. 16.)

respect to the reaction force measurements are shown in Fig. 24. The param-

eter νkeloid is mainly aﬀected by the displacement ﬁeld measurement whereas

the reaction force measurements have a signiﬁcant eﬀect on both parameters

(although the eﬀect is much greater on parameter Ekeloid than on νkeloid).

Fig 25 shows the sensitivity ﬁeld for parameter νkeloid with respect to the dis-

placement ﬁeld measurements in the measurement subdomain. An estimate

of the sensitivity in the whole domain is shown in Fig. 26.

27

Figure 23: Keloid model parameter relative standard deviation assuming one standard

deviation in the displacement ﬁeld measurements at the mesh nodes.

Figure 24: Keloid model parameter relative standard deviation assuming one standard

deviation in the reaction force measurements.

28

Figure 25: Model parameter νkeloid sensitivity with respect to the displacement ﬁeld mea-

surements at the ﬁnal measurement step. Note that the parameter is principally sensitive

to the vertical component of the displacement ﬁeld measurements. (This plot is deﬁned

over the subdomain highlighted in Fig. 2.)

Figure 26: Model parameter νkeloid estimated sensitivity with respect to displacement

measurements.

7. Discussion and perspective

The idea behind the inverse method is quite simple: the aim is to mini-

mize a cost functional that is a measure of the misﬁt between the model and

the experiment. In practice, the inverse solution can be challenging since

its success hinges on the experimental measurements being suﬃciently rich

to, in a sense, activate every model parameter. Although human skin is

usually regarded as an incompressible, anisotropic and visco-elastic material,

we considered in this study a much simpler model, namely, a compressible

29

isotropic Neo-Hookean model. This decision was in part due to the already

very strict 2D plane-stress modeling assumption and because the experimen-

tal measurements consisted only of uni-axial extensions, which lacked the

information for ﬁtting an anisotropic model. The present results indicate

that the Neo-Hookean model, while capable of reproducing the almost-linear

branch of the force-displacement relationship (e.g. Fig. 7and 20), has sig-

niﬁcant displacement ﬁeld misﬁt errors (e.g. Fig. 8and 21). Although a

more complex model may very well reduce these errors, there is the possi-

bility of overﬁtting the 2D model. A similar work (Boyer et al.,2013) to

ours assessed the mechanical properties of skin albeit on a forearm. In their

case, a 2D orthotropic linear elastic plane-stress (also 1 mm thick) model

was assumed. The results from 10 young female patients were the following.

The minor value of the Young’s modulus was E1= 0.040 ±0.007 (N/mm2),

the major value was E2= 0.146 ±0.069 (N/mm2) and the Poisson’s ratio

was ν= 0.062 ±0.061. Although these results can not be compared directly

to ours, they are at least indicative of our ﬁndings: Eskin = 0.028 (N/mm2)

and νskin = 0.195.

Accuracy of experimental measurements is of crucial importance for model

parameter identiﬁcation. Unfortunately, acquiring accurate measurements in

vivo is a signiﬁcant challenge and, consequently, measurement uncertainties

are unavoidable. Therefore, when possible, it is important to quantify the

model parameter uncertainties due to measurement uncertainties as part of

an inverse solution. Addressing model uncertainties in biomechanics in gen-

eral is of fundamental importance because of the severe variability between

patients. This variability creates signiﬁcant challenges for modeling and iden-

tifying which model parameters are the most important and which statistical

distributions are best-suited to represent those parameters. The amount of

cohort data is usually limited, although modern experimental methods enable

the acquisition of multi-modal data which can enrich biological tissue models

(Mascheroni and Schreﬂer,2018). Once baseline information on a patient co-

hort is available, Bayesian updating can be used to develop a patient-speciﬁc

understanding and to update the prior cohort-based knowledge (Hawkins-

Daarud et al.,2013;Rappel et al.,2018,2019,2020). Subsequently, once

parameter distributions are available, Monte Carlo simulations can be used

to predict the impact of such parameters on quantities of interest to clinicians.

Recent advances in this area show that, as expected, the relative importance

of those biomechanical parameters depends on the set of boundary condi-

tions applied upon the biomechanical system (Hauseux et al.,2018). Such

30

approaches are also notoriously slow, which has led to the inception of accel-

erated methods such as derivative-based Monte Carlo (Hauseux et al.,2017).

The next step in stochastic modeling of biological tissue is the design of ex-

periments. Because experiments are costly and often invasive, it is critical to

decrease the number of tests required to achieve a given level of understand-

ing of the biomechanical system under consideration. The logical next steps

for this work will be to develop a stochastic inverse method aimed at the

identiﬁcation of the sensitivity of parameters on boundary conditions and

on other external factors, such as in (Barbone and Gokhale,2004;Risholm

et al.,2011;Hale et al.,2016). It is clear to the authors that the above rep-

resents a very large amount of work which is being largely undertaken by the

community at large and requires the conjunct eﬀorts from experimental to

computational, through theoretical and statistical researchers. By working

jointly, these three communities can make signiﬁcant progress in the design

of new experimental devices, new in silico experiments, new phantom ma-

terials, and in devising digital twins to better understand human biological

tissues (Bordas et al.,2018;Ley and Bordas,2018).

8. Summary

The paper focused on the problem of hyperelastic model parameter identi-

ﬁcation and model parameter sensitivity analysis for a keloid scar and healthy

skin. The patient speciﬁc experimental measurements were acquired in vivo

using a custom-made uni-axial extensometer. A two-parameter Neo-Hookean

material model was assumed for both the keloid and the healthy skin. The

model parameter identiﬁcation was carried out in two separate analysis. The

model parameters were ﬁrst identiﬁed for the healthy skin. Then, using the

healthy skin parameters in the skin-keloid bi-material model, the keloid ma-

terial parameters were determined. The ﬁtted models could reproduce the

measured reaction forces well. On the other hand, the displacement ﬁeld mis-

ﬁt errors were signiﬁcant. The healthy skin model displacements were more

than 65% oﬀ from measurements, whereas the skin-keloid bi-material model

displacements were up to 14% oﬀ. It is understood that the unquantiﬁed er-

rors the displacement ﬁeld measurements from the DIC analysis play a part

in the current results. On the other hand, the sensitivity analysis could also

explain the large errors. Speciﬁcally, the parameter governing the model’s

kinematic ﬁeld turned out to be insensitive to displacement measurements

in the direction of the observed error. In essence, the model’s displacement

31

misﬁt error would be large no matter the value of the model parameter. Fi-

nally, we proposed a simple way to estimate the model parameter sensitivity

with respect to the surface displacement ﬁeld measurements in the whole

domain. The estimate depends only on the boundary conditions and on the

assumed model parameter values and is otherwise independent of the actual

surface displacement ﬁeld measurements. This information can aid in the de-

sign of experiments, speciﬁcally in localizing the optimal displacement ﬁeld

measurement sites for the maximum impact on model parameters.

Acknowledgments

The authors acknowledge the ﬁnancial support of Region de Bourgogne

Franche-Comt´e, France (grants n◦2017-Y-06397, n◦2018-Y-04536, n◦2019-

Y-10541). P. Hauseux acknowledges the support of the Fonds National de la

Recherche Luxembourg FNR (grant n◦O17-QCCAAS-11758809). S. Bordas

thanks the FEMTO-ST Institute, Department of Applied Mechanics, and

the Laboratoire de Math´ematiques de Besan¸con for their fellowship opportu-

nities, and also acknowledges the discussions with Dr. Olga Barrera, Oxford

University. D. Baroli acknowledges the support of FEDER funding innova-

tion and DFG Clusters of Excellence: Internet of Production. M. Sensale

thanks the University of Luxembourg, Institute of Computational Engineer-

ing for the visiting fellowship opportunity.

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