Identification of 3D heterogeneous modulus distribution
with the virtual fields method
Stéphane Avril, Fabrice Pierron
Mechanical Engineering and Manufacturing Research Group
Ecole Nationale Supérieure d’Arts et Métiers
Rue Saint Dominique, BP508, 51006 Châlons en Champagne, FRANCE
Jonathan M. Huntley
Wolfson School of Mechanical and Manufacturing Engineering
Loughborough University
Loughborough, Leicestershire, LE11 3TU, UK
Derek D. Steele
Departments of Radiology and Biomedical Engineering
University of Michigan Medical Center
3315 Kresge III, Ann Arbor, MI 48109-0553, USA
ABSTRACT
This paper presents an extension of the virtual fields method to the identification of heterogeneous stiffness properties. 3D
bulk full-field displacement data are used, provided by Magnetic Resonance Imaging (MRI). Two main issues are
addressed: 1. the identification of the stiffness ratio between two different media in a heterogeneous solid; 2. the
identification of stiffness heterogeneities buried in a heterogeneous solid. The approach is based on the equilibrium
equations. It is tested on experimental full-field data obtained on a phantom with the stimulated echo MRI technique. The
application deals with the characterization of a stiff spherical inclusion buried within a lower modulus material. Results
concerning the ratio between the modulus of the inclusion and the modulus of the surrounding material are in close
agreement with the reference. Results concerning the calculated modulus distribution through the whole investigated
volume are also quite promising. The inclusion location and the ratio of the average moduli are in agreement with the
reference. However, the resulting modulus distribution is highly variable. This is explained by the fact that the approach
relies on a second order differentiation of the data, which tends to amplify noise.
Introduction
In medicine, the accurate characterization of stiffness properties inside the human body is essential in order to detect
tumors and diagnose disease [1]. This entails addressing a challenging issue within the field of solid mechanics: the non
intrusive identification of mechanical properties inside a deformable body. This issue can be addressed in many different
ways, all involving the definition of a so-called inverse problem [2]. Many engineering applications of such inverse
problems have been addressed by combining external loadings and external measurements: displacements induced by an
applied static or dynamic loading are measured over the external surface of the solid and linked to the internal properties.
In medicine, this technique has been applied for many years by physicians when they try to detect hard tissues by external
palpations.
Even if techniques combining external loadings and external measurements have been successful, it is now possible to
combine external loading and internal (bulk) three-dimensional (3D) full-field measurements. In order to avoid any
confusion, it is worth pointing out what we mean by full-field measurement. In its most general form, the three components
of the displacement vector are measured at the nodes of a 3D lattice evenly meshing the whole volume of interest. In
medical sciences, thanks to ultrasound [3] and magnetic resonance imaging (MRI) [4,5], it is possible to measure
displacement fields in many human tissues.
Efficient and powerful approaches have to be developed for processing this type of full-field data with the final objective of
constitutive parameter identification. A lot of work has already been achieved in medical sciences, where the full-field
measurements have been available for more than a decade. The measurement and imaging of tissue elasticity has even
become an independent branch, widely called “elastography” [1]. The approaches used to process the full-field data
depend on the way the solid is deformed. Two different ways exist:
1. static experiments: the tissue is compressed quasi-statically,
2. dynamic experiments: a time harmonic excitation made on the boundary creates a time harmonic shear wave in the
tissue.
Many applications use dynamic experiments. The advantage over static experiments is that a direct relationship between
the measured strains and the unknown moduli can be derived, provided that only shear waves are present in the tissues.
Such a direct relationship is not available in static experiments because the strains and stresses are heterogeneous across
the whole the volume. Thus, an inverse problem has to be solved [6]. Nevertheless, the use of static experiments remains
of great interest since the results of dynamic experiments can be subjected to artifacts induced by the visco-elasticity of the
tissues.
The inversion to obtain stiffness distributions using quasi-static experiments has been solved in two ways in the literature.
The first is a direct method which was developed by Skovodora et al. [5]. The principle of this method is to cancel the curl
of the equilibrium equations. This yields a hyperbolic partial differential equation (PDE) with the shear modulus distribution
as the unknown. The strain field and its first order derivatives are involved in the coefficient functions of this PDE. This
approach presents some drawbacks, the main one being that it is required to differentiate at least two times the noisecorrupted displacement fields, yielding artifacts in the results [5].
The second method of inversion is referred to as an indirect approach in the literature because studies dealing with this
approach are based on model updating [6]. They involve the minimization of a residual norm defined over the volume of
measurements. This norm quantifies the difference between the measured displacement field and the theoretical
displacement field numerically generated by the predicted stiffness distribution. This approach tends to be computationally
expensive since the forward problem has to be solved repeatedly. Even if the use of an adjoint approach can significantly
reduce the computation time [2], it remains a computationally-expensive method for processing 3D full-field
measurements. Thus, it is mainly restricted to 2D applications.
In view of all the previous disadvantages, an alternative approach is proposed in this paper. The idea is to replace model
updating by the virtual fields method [7]. Until now, the virtual fields method (VFM) has been an approach dedicated to
engineering applications dealing with the identification of constitutive parameters of thin plates from 2D surface full-field
kinematic measurements. Since the development of the virtual fields method, the choice of the virtual fields has been
optimized [8], making possible the “direct” identification of stiffness parameters from full-field measurements. As no
iterative solution of the forward problem is required, this approach seems well suited for processing large amounts of data,
and hence to 3D bulk displacement fields.
This paper presents the first extension of the VFM to 3D bulk full-field displacement data. Only static experiments will be
addressed here. The approach is based on the approximation of displacement fields using a suitable functions basis,
similar to the so-called “equilibrium gap method” [9]. The virtual fields are also expanded using the same basis functions
[11].
Two issues are addressed independently:
1. identify the stiffness ratio between two different media in a heterogeneous solid by using 3D bulk full-field displacement
data.
2. to identify heterogeneities in the stiffness distribution inside a solid by using 3D bulk full-field displacement data.
Both issues are solved for statically undetermined tests, meaning that no assumption is made concerning the stress
distribution in the solid. First, the approach used to independently reach both objectives is described. Then, it is applied to
a practical example dealing with the characterization of a stiff spherical inclusion buried in a solid. This example has been
chosen because it is of great interest in biomedical applications for detecting cancerous tumors. The experimental data
were obtained on a phantom using the stimulated echo MRI technique [5].
Principle
Let us consider a parallelepiped included in the volume of the solid and across which displacement field measurements
are available, i.e. the domain scanned by the measurement system. The study will focus on the distribution of Young's
modulus across this parallelepiped. For solving the inverse problem, the volume has to be discretized. The parallelepiped
is partitioned into N × N × N brick-shaped finite elements.
Once the mesh has been generated, the displacement fields across the whole parallelepiped are approximated using the
so-called Galerkin approximation [11]. It involves replacing the displacement fields by the following series:
u ( x, y , z ) =
∑ a φ ( x, y , z )
i i
(1)
i
where the ai's denote the nodal degrees of freedom (DoF) and the φi's denote basis functions.
The basis functions φi(x,y,z) (where x, y and z are the usual cartesian coordinates) are equal to one at precisely one node of
the mesh, and precisely zero at all other nodes. Such basis functions are built up in a piece-wise manner over the brick-
shaped finite elements by combining trilinear shape functions which are only defined within a particular element. The ai's
are then identified by linear regression so as to make the finite element approximation match in a least squares sense the
measured displacement values.
The Young's modulus is assumed to be constant inside each brick-shaped finite element. Hence, the inverse problem is
reduced to the determination of a vector, denoted {E}, whose components Ek are the unknown Young's moduli of all the
finite elements.
The stiffness matrix of a single brick finite element ek, denoted [Ak], can be written:
∫
[ Ak ] = Ek [∇φ ][C ][∇φ ]dV
(2)
ek
where [∇φ ] is a matrix featuring the gradients of the basis functions. [C] is a matrix that only depends on Poisson's ratio.
Indeed, for an isotropic material, Young's modulus may be factored out of the Hooke matrix yielding an alternate form of
Hooke's law:
ν
ν
⎤
⎡
0
0
0 ⎥
⎢ 1
1 −ν 1 −ν
⎥
⎢ ν
ν
⎢
1
0
0
0 ⎥ ⎧ε xx ⎫
⎧σ xx ⎫
1 −ν
⎥⎪ ⎪
⎢1 − ν
⎪
⎪
ν
⎥ ε yy
⎢ ν
⎪σ yy ⎪
1
0
0
0 ⎥ ⎪⎪ ⎪⎪
⎢
⎪⎪σ zz ⎪⎪
ε
1 −ν
⎥ ⎪⎨ zz ⎪⎬
⎢1 − ν 1 − ν
(3)
⎨σ ⎬ = E
1
2
ν
−
ε
(1 + ν )(1 − 2ν ) ⎢ 0
0
0
0
0 ⎥ ⎪ xy ⎪
⎪ xy ⎪
⎥ ⎪ε ⎪
⎢
2(1 + ν )
⎪σ xz ⎪
⎥ ⎪ xz ⎪
⎢
1 − 2ν
⎪
⎪
0
0
0
0
0
⎥ ⎪ε yz ⎪
⎢
σ
⎩⎪ yz ⎭⎪
2(1 + ν )
⎥⎩ ⎭
⎢
1 − 2ν ⎥
⎢
0
0
0
0
⎢ 0
2(1 + ν ) ⎥⎦
⎣
The global stiffness matrix of the whole parallelepiped, denoted [A], is constructed by assembling all the elementary
stiffness matrices:
[ A] =
∑ E [A ]
k
(4)
k
k
Let us now write the equilibrium equations of the whole parallelepiped. Knowing the vector of nodal forces, denoted {b},
the following equation must be verified by the Ek:
[ A]{a} =
∑ E [ A ]{a} = {b}
k
(5)
k
k
where the {a} vector of nodal DoFs is the one derived from the least squares regression of experimentally measured
displacements.
Nodal forces in a finite element may be separated into internal body forces and external forces. Here, the body forces are
neglected. Therefore, in Eq. (5), all the nodal forces cancel out except at the nodes located over the boundary of the
parallelepiped where the reaction forces are unknown. Thus, vector {b} can be split up in two sub-vectors, one containing
all the known components, which are all zeros, and one containing the unknown components, denoted {F}. Those
unknown nodal forces will be removed from the equations [11]. This is one of the great advantages of the proposed
approach because no assumption is made concerning the boundary conditions.
The idea is to use the same procedure as the one used in the finite element method for eliminating Dirichlet's DoFs from
the equations [11]. It consists of separating the components of vectors and matrices corresponding to the unknown
components of {b} from the other components corresponding to the known components of {b}. Thus, all vectors are split
up into two sub-vectors, and all matrices are split up into four sub-matrices. Accordingly, the following equation must now
be satisfied by Ek:
[ K ]{U } + [ L]{V } =
∑ E [K ]{U }+ ∑ E [ L ]{V } = {0}
k
k
k
k
k
(6)
k
Actually, it is assumed that the modulus is known (or a given relative value is imposed) over n brick elements, with n >0.
Thus, the N 3-n modulus values that remain must satisfy:
N 3 −n
∑
k =1
Ek {[ K k ]{U } + [ Lk ]{V }} = −
n
(
∑ E {[K ]{U } + [L ]{V }} ⇔ [M ]{E} = {B}
l
l
l
(7)
k =1
where the Ĕl are the values of the known or imposed moduli.
Let us first assume that the structure of the investigated solid is known. For the sake of simplicity, let us consider the case
where the investigated solid is composed of only two different materials. Thus, only two different values are possible for
the modulus in each finite element. A value of one is prescribed for the finite elements which are known to be located in
the first material, called the reference material. An unknown value, denoted Er, is prescribed for the finite elements which
are known to be located in the other material, called the unknown material. Er therefore represents the ratio between the
modulus of both materials.
Accordingly, Eq. 7 can be modified to:
Er
∑{[ K ]{U } + [L ]{V }} = − ∑{[K ]{U }+ [L ]{V }}
k
k
l
unknown
l
(8)
reference
This overdetermined system of equations is solved by multiplying it with a suitable column vector, denoted {U*}. This
column vector features the nodal DoFs of a piecewise virtual displacement field. Finally, the following solution is obtained:
{U } ∑{[K ]{U }+ [L ]{V }}
=−
{U } ∑{[K ]{U } + [ L ]{V }}
*t
Er
*t
k
k
unknown
l
(9)
l
reference
Algorithms have been developed in 2D for providing optimal virtual fields [8,11]. However, they require the inversion of
large matrices which is computationally expensive when 3D data are processed. Therefore, an intuitive guess for vector
{U*} is the simplest solution here.
When the structure is composed of several different materials, or when the structure of the investigated solid is unknown,
the system of equations in Eq. 7 must be solved directly without any simplifications, such as the one in Eq. 8. The system
of equations in Eq. 7 is a rectangular system of equations because the number of unknown modulus values is lower than
the number of equations, as three independent equations are derived from the equilibrium written at each node. This
overdetermined system is solved by multiplying it with a suitable set of column vectors. Here also, these column vectors
feature the nodal DoFs of piecewise virtual displacement fields. A good choice from a numerical perspective is to use
vectors that can be written in the form:
{U }= [α ]{[ K ]{U } + [L ]{V }}
*
(10)
where [α] is a positive definite matrix and k varies between 1 and N -n. Indeed, such a choice for the column vectors
provides a system of equations that can be written:
k
k
3
[[M ] [α ][M ]]{E} = [M ] [α ]{B}
t
t
(11)
where [M] and [B] have been defined in Eq. 7. [[A] [α] [A]] is a large matrix that has the interesting property of being
symmetric positive definite. As [A] is also sparse if [α] is well chosen, iteratively solving Eq. 11 by the conjugate gradient
method is now possible. This provides a significant computational acceleration, allowing, for example, one million modulus
values to be calculated in less than 10 minutes. Therefore, the volume of interest can be refined in order to obtain an
acceptable spatial resolution for the modulus variation across the volume of the investigated solid. The solution of Eq. 11
can be interpreted as the weighted least-squares solution of Eq. 7. It consists of minimizing the following cost function:
1
ϕ ( E ) = {[ M ]{E} − {B}}t [α ] [ M ]{E} − {B}t
(12)
2
where [α] is a matrix used to minimize the effect of noise in the data [2]. A larger weight is given to nodes and components
where the confidence on the data is high and a lower weight where the confidence is small. The evaluation of the
confidence level depends on the data themselves and it is different from one application to another. An example will be
given in the next section. As [α] is a diagonal matrix, and hence sparse, the conjugate gradient method may be used to
solve Eq. 11.
t
-1
{
}
Application to experimental data
The approach described above was tested first using simulated data provided by a finite element analysis (FEA) package.
As expected, the modulus distribution input to the FEA package was reconstructed with only very small errors due to
numerical approximations. However, this result in itself does not guarantee the success of the approach when
experimental data are used because the spatial resolution of data sampling and the measurement uncertainties can affect
the procedure in a significant way. For this reason, we applied this method to the experimental data presented by
Steele et al. [5].
Figure 1. Dimensions (in mm) and geometry of an eighth of the tested specimen.
Figure 2. Experimental displacement components across cross section x = 0
containing the inclusion. (a): ux; (b): uy; (c): uz.
The experiments were performed on a phantom containing a stiff inclusion. The specimen has a 80×64×154 mm3
parallelepipedic shape and the inclusion is a 25 mm diameter sphere (Figure 1}). Semicosil 921 silicone gel (Wacker
Silicones Corporation, Adrian, MI) was used to construct the phantom qualitatively simulating the mechanical properties of
soft human tissues. Different ratios of the two components were used for the inclusion and for the surrounding material.
Measurements using an appropriate force-deformation system showed that the inclusion was four times stiffer than the
background [5].
The phantom was placed between two acrylic plates in a pneumatically driven device. A vertical deformation of
approximately 6 mm, or about 7.5 % average strain, was applied. A stimulated echo MRI sequence using displacement
encoding gradient pulses was employed to measure the phantom's internal static displacements. The purpose of this
3
paper is not to present this technique which was detailed in [5]. Only a 50×50×42 mm field of view, covered by a grid of
116×142×37 voxels, was kept around the inclusion. The resulting phase maps in this 50×50×42 mm3 region of interest
were unwrapped using a temporal unwrapping approach [10]. Once unwrapped, the phase maps were multiplied by the
respective sensitivity factors in order to get the displacement maps (Figure 2). A 9×9×1 median filter kernel (only averaging
within the slices) was applied to the unwrapped phase maps for reducing the noise level and eliminating inconsistent
values.
The material investigated in this study is almost incompressible. Although our approach is applicable to all types of
material, if the Poisson ratio is very close to 0.5, some components of the Hooke matrix in Eq. 3 will tend towards infinity.
Accordingly, Eq. 3 can be modified into the following equation:
⎧σ xx ⎫ ⎧ p ⎫
⎡1 0 0 0 0 0⎤ ⎧ε xx ⎫
⎪
⎪ ⎪ ⎪
⎢
⎥ ⎪ε ⎪
⎪σ yy ⎪ ⎪ p ⎪
⎢0 1 0 0 0 0⎥ ⎪ yy ⎪
⎪⎪σ zz ⎪⎪ ⎪⎪ p ⎪⎪
⎢0 0 1 0 0 0⎥ ⎪⎪ε zz ⎪⎪
(13)
⎥⎨ ⎬
⎨σ ⎬ = ⎨ ⎬ + 2G ⎢
⎢0 0 0 1 0 0⎥ ⎪ε xy ⎪
⎪ xy ⎪ ⎪ 0 ⎪
⎢0 0 0 0 1 0⎥ ⎪ε xz ⎪
⎪σ xz ⎪ ⎪ 0 ⎪
⎢
⎥⎪ ⎪
⎪
⎪ ⎪ ⎪
⎣⎢0 0 0 0 0 1 ⎦⎥ ⎩⎪ε yz ⎭⎪
⎩⎪σ yz ⎭⎪ ⎩⎪ 0 ⎭⎪
where p is the hydrostatic pressure variation induced by the deformation of the solid and G is the shear modulus. The
hydrostatic pressure variation can be related to the relative volume variation according to:
dV
p =κ
= κdiv (ε )
(14)
V
where κ is the bulk modulus, which is almost uniform in human tissues, close to the bulk modulus of water [1]. As both κ
and div(ε) have negligible variations across the volume, the gradient of the hydrostatic pressure will be neglected in the
following. Furthermore, it can be shown that the hydrostatic pressure will not be involved in the equations. Indeed, let us
consider a virtual displacement field, denoted u*, which is null over the whole boundary of the solid (like all the virtual
displacement fields introduced previously). Integrating by parts, one can prove that p is not involved in the virtual work for
such a virtual field that is null at the boundary. As this is the case for the virtual fields considered before, the hydrostatic
pressure is not involved in equations derived previously. Thus, only the G-dependent part of the elementary stiffness
matrix is needed to construct the modified global stiffness matrix, denoted [K•].
Thus, the unknowns for the problem defined here will be the values of the shear modulus instead of the values of Young's
modulus. For an incompressible material, the two are related through E=3G. The vector of unknown shear moduli, denoted
{G}, will satisfy similar equations to Eq. 9 and Eq. 11:
- for the modulus ratio identification:
- for the shear modulus distribution:
Figure 3. (a) Identified modulus distribution (dimensionless values) across all the available cross sections. (b) Reference
modulus distribution (dimensionless values) across all the available cross sections.
3
The 50×50×42 mm volume of interest was meshed using 53×66×36 brick-shaped finite elements for the identification. The
elements located inside the inclusion have been detected using the magnitude of the same MR signal as the one used for
measuring the displacement fields. This information is used to complete the former equation where the “reference”
becomes the background material and the “unknown” becomes the inclusion. Accordingly, the ratio can be written:
The components of vector {U*} have been chosen as the DoFs of the following virtual displacement field:
0
⎧
⎫
⎪
⎪
u* ( x, y, z ) = ⎨( x − L / 2)( x + L / 2)( x − l / 2)( x + l / 2)( x − h / 2)( x + h / 2)⎬
(15)
⎪
⎪
0
⎩
⎭
Only the y component has nonzero values because it is the direction of loading, hence the direction along which the
sensitivity is the best. When applied to the experimental results with 53×66×36 brick elements and the previous virtual
field, one obtains: Ğ=4.2. This is in agreement with the reference value Ğ =4 and therefore appears very promising for
further applications of our approach.
3
The 50×50×42 mm volume of interest was again meshed using 53×66×36 brick-shaped finite elements. Here, the
modulus was assumed to be known only on the boundary of the volume of interest. The same assumption was used by
Steele et al. [5]. Thus, 51×64×34=110976 values are unknown inside the volume of interest. The modulus of the boundary
elements was set to unity to get a relative value of the moduli inside the specimen. 500 iterations of the conjugate gradient
method were used to solve the system of equations (2 minutes using a Pentium M processor, 1400 MHz). The [α] matrix
was chosen to give zero weight to the nodal forces in the x and z directions and uniform weight to the nodal forces in the y
direction.
Reconstruction results are presented in Figure 3. As there are 36 finite elements along the z direction, each plane passing
through one of the centroids of these finite elements represents a cross section where a 53×66 matrix of results is
available. All of the 53×66 result matrices across all 36 cross-sectional planes are plotted in the same picture, starting from
the x = -20.4 mm plane on the top left to the z = 20.4 mm plane on the bottom right in Figure 3. This plotting scheme allows
one to visualize the three-dimensional results.
It can be noticed in the results that the highest modulus values delimit a volume that is similar to the actual volume of the
inclusion. The reference region occupied by the inclusion in Figure 3 has been determined from the magnitude of the MR
signal. Moreover, the average value of the modulus identified in the surrounding material is 0.69 and the average value of
the modulus identified in the inclusion is 2.98. Thus the ratio between both modulus values is 4.3. This is in agreement with
the reference value of 4.
The main source of errors in Figure 3a seems to be the noise in the experimental data. The standard deviation of the
results is 1 in the region of the inclusion and 0.4 in the surrounding material. This means that the coefficient of variation
(ratio between the standard deviation and the average) is 33 % in the region of the inclusion and 57 % in the surrounding
material. The scatter in the reconstruction results is likely induced by the noise amplification of the ill-conditioned
equations. As explained previously, a small error in the coefficients of the matrix can induce large errors in the results [2].
The coefficients of the system are derived from the first derivatives of the displacement field. Thus, they contain significant
errors because the differentiation of noisy data amplifies the noise effects. Noise effects could be lowered by using a
regularization approach [2]. However, this was not implemented because the qualitative distribution identified here is
sufficient to determine the volume occupied by both different media. Once this volume is determined, one can switch to the
identification of the modulus ratio by averaging the results. The obtained ratio is much more stable due to the filtering effect
of the averaging process, and therefore, no more regularization is required.
Comparison with other approaches
Let us compare our approach to other ones. The distribution of the strain component in the loading direction εyy may be
used as an indicator for the modulus distribution when the stress distribution is almost homogeneous [6]. This could be the
case here because a uniform compression has been applied to the solid and the stress component in the direction of
loading should be nearly homogeneous. However, looking at the displacement fields (Figure 2), it can be seen that friction
forces prevent the material from sliding at the top and at the bottom, close to the contact with the loading plates. Therefore,
the stress field will not be homogeneous here. Local gradients in the stress fields can also be expected close to the
boundary of the inclusion due to the steep change of modulus. However, because it is sometimes assumed in
elastography [6], let us assume that the stress is homogeneous here. Therefore, the ratio σyy/εyy should represent the
distribution of the Young modulus across the volume. This approach for determining the modulus distribution is called the
“strain imaging” approach. As σyy is unknown here, the ratio εyy/εyy (where εyy is the average value of εyy) has been computed
across the whole volume and plotted for all the cross sections along the z direction (similarly as in Figure 3). The results
are shown in Figure 4. They are less noisy than the ones of our approach because they are obtained by a mere first order
differentiation. However, different errors occur, that prevent the strain imaging approach from being useful either in a
quantitative way or in a qualitative way for the example addressed here. Quantitatively, the modulus of the inclusion is
significantly underestimated (average of 2.5 instead of 4). Qualitatively, the location of the inclusion is well found but its
boundary is blurred in the plane z=0. Moreover, other heterogeneities are erroneously detected at the top of the volume of
interest. These result from the stress heterogeneities induced by the existence of friction forces at these locations. Due to
this double failure, it can be concluded that the example processed here can only be addressed satisfactorily using an
inverse method like our approach or like the one developed by Skovodora et al. [5].
Skovodora’s approach is similar to ours in principle, but instead of the nodal forces, the curl of the nodal forces is cancelled
in their approach. To remove the hydrostatic pressure from the equilibrium equations, we have assumed that its gradient is
negligible and that using particular virtual fields which are null on the boundary cancel its contribution. Computing the curl
of the nodal forces is another method for ridding the hydrostatic pressure from the equilibrium equations. Eventually, one
obtains a hyperbolic PDE with the shear modulus values as unknowns. But the derivation of these differential equations is
based on a third-order differentiation of the data instead of the second-order one in our case. This third-order differentiation
brings even more instability. The instability is tackled by the authors via numerical integration of the equations. This yields
a system of equations similar to ours, but the coefficients of their matrix involve the second order derivatives of the data
whereas ours involve only the first order derivatives.
The poor stability of the equations solved by Steele et al. [5] may explain the artifacts encountered in their results. Our
results show that even when there is one order less in the differentiation, results are still very noisy (Figure 3). Then, only
regularization approaches based on relevant a priori information can improve the stability. The stability which is gained in
our approach compared to Skovodora's one relies on the assumption that the gradient of the hydrostatic pressure can be
neglected. According to the success of our approach, it seems that this assumption is justified. Due to the relative
uniformity of the bulk modulus in human tissues and due to their incompressibility, this assumption may also be justified in
most applications dealing with human tissue deformations. Anyway, in other applications where the material would not be
incompressible (typical material engineering applications for instance where the VFM is applicable), this assumption is not
needed. Instead of neglecting the hydrostatic pressure, it would be included in the equations. The original stiffness matrix,
denoted [Kk] in Eq. 3, would be kept instead of using modified stiffness matrix [Kk•].
Figure 4. Identified modulus distribution obtained by the “strain imaging” approach (dimensionless values)
Conclusion
In this paper, the first extension of the virtual fields method to 3D bulk full-field measurements has been presented. It has
been tested on displacement fields measured by stimulated echo MRI. Two main issues have been addressed:
1. the identification of the stiffness ratio between two different tissues.
2. the identification of the stiffness distribution inside heterogeneous tissues.
Experiments were carried out on a silicone gel phantom containing a stiff spherical inclusion. Using the virtual fields
method, the ratio between the modulus of the inclusion and the modulus of the surrounding material is in close agreement
with the reference value. Furthermore, the inclusion location is correctly determined. However, the range of modulus
values is very scattered. This is not surprising because the approach is based on a second order differentiation of the data,
which amplifies the noise.
In the future, this approach could be extended to dynamic analysis. This would permit faster loading-unloading cycles as
the quasi-static requirement would no longer be needed. A drastic reduction of the data acquisition time (which was 6h for
the data processed here [5]) could then be envisaged.
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