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Abstract

The aim of this work is to characterize the mechanical parameters governing 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 surrounding healthy skin. The problem of finding the optimal model parameters that minimize the misfit between the model observations and the in vivo experimental 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 reaction 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 whole computational domain. This estimate is primarily useful for the design of experiments, specifically, in localizing the optimal displacement field measurement sites for the maximum impact on model parameter inference.
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 (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 = DNand DN=. Considering a pure displace-
ment load uDprescribed on D, the potential energy of the hyperelastic
solid consists of just the strain energy of deformation that reads
W(u,m) = Z
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)]TuJ(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)]1mF(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)]TuJ(k)(13)
DmJ(k)=mJ(k)+ [mF(k)]T[uF(k)]TuJ(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 vu(k)
msr in (18) the
solution to the model parameter sensitivity can be determined as
Du(k)
msr m=D2
mmJ1u(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 =PNdof1
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=M1Du(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
2Z
(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=Z
DmIuiDmJuidx+Z
(uiu
msri)D2
mImJuidx(29)
D2
mImJJ
uZ
DmIuiDmJuidx(30)
The second mixed derivative of J
uwith respect to mand u
msr reads
u
msr DmIJ
u[δu
msr] = Z
DmIuiδu
msridx(31)
As a first attempt, one may try to compute the sensitivity Du
msr mas
Du
msr m=D2
mmJ
u1u
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=ˆvTZ
Dmuiδu
msridx(33)
δˆm=ˆvI
ˆ
λZ
DmIuiφiJ dxδu
msrJ(34)
Subsequently, the model parameter sensitivity can be defined as
Du
msr ˆmJ=ˆvI
ˆ
λZ
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 =M1Du
msr ˆm, (36)
where M=RφTφdxis 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 (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(I1d2 ln J) + D1ln J2,(37)
where C1and D1are the material model parameters and
I1= tr C(38)
J= det F(39)
C=1
2(FTFI) (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=
(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. uumsr) 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. uumsr) 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 n2017-Y-06397, n2018-Y-04536, n2019-
Y-10541). P. Hauseux acknowledges the support of the Fonds National de la
Recherche Luxembourg FNR (grant nO17-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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... 25). Hence, to identify all the parameters with the latter 525 optimization function, the strategy to carry out is identifying the two parameters of healthy-skin first 526 (co-lateral test), and then identifying, in a second process, the two remaining keloid parameters [89]. We ...
... are deleted in the coarser mesh, and especially outside the keloid scar, where the nodal solutions are 489 spatially sensitive (because of significant transverse deformation gradient)[89]. Thus, we conclude490 that the proposed optimized coarser mesh is suitable for fast and accurate computations, knowing that 491 the displacements and reaction forces are assessed with low uncertainty levels: DIC < 120 m and 492 force < 10 mN. ...
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