Bilayer Stiffness Identification of Soft Tissues by Suction

Abstract

Background

In vivo mechanical characterisation of biological soft tissue is challenging, even under moderate quasi-static loading. Clinical application of suction-based methods is hindered by usual assumptions of tissues homogeneity and/or time-consuming acquisitions/postprocessing.

Objective

Provide practical and unexpensive suction-based mechanical characterisation of soft tissues considered as bilayered structures. Inverse identification of the bilayers’ Young’s moduli should be performed in almost real-time.

Methods

An original suction system is proposed based on volume measurements. Cyclic partial vacuum is applied under small deformation using suction cups of aperture diameters ranging from 4 to 30 mm. An inverse methodology provides both bilayer elastic stiffnesses, and optionally the upper layer thickness, based on the interpolation of an off-line finite element database. The setup is validated on silicone bilayer phantoms, then tested in vivo on the abdomen skin of one healthy volunteer.

Results

On bilayer silicone phantoms, Young’s moduli identified by suction or uniaxial tension presented a relative difference lower than 10 % (upper layer thickness of 3 mm). Preliminary tests on in vivo abdomen tissue provided skin and underlying adipose tissue Young’s Moduli at 54 kPa and 4.8 kPa respectively. Inverse identification process was performed in less than one minute.

Conclusions

This approach is promising to evaluate elastic moduli in vivo at small strain of bilayered tissues.

Appendices

Appendix A: Real time Evaluation of the Simulated Apparent Stiffness

The apparent stiffness $B_{i\, SIM}(\beta , \theta )$ is the slope of the pressure-shape curve at shape $S=0.1$ (equation (6), main paper body) when aspirating a bilayer phantom. This simulated stiffness is evaluated many times to find iteratively the minimum of the cost function $\Phi _{Param}$ (equation (8), main paper body) or to evaluate the identifiability of the material parameters (Appendix B). The apparent stiffness $B_{i\, SIM}(\beta , \theta )$ depends mainly on the different combination of four parameters, which are the aperture diameter $D_i$, the upper layer Young’s modulus $E_{R1}$ and its thickness $L_{R1}$, and the lower layer Young’s modulus $E_{R0}$ (Fig. 3).

If the simulated apparent stiffness $B_{i\, SIM}(\beta , \theta )$ was evaluated using, for example, a FE model implemented and updated for each calculation point, the time required to solve a single inverse identification would be phenomenal. Therefore, this appendix describes how the simulated apparent stiffness $B_{i\, SIM}(\beta , \theta )$ was evaluated in real time. The idea is mainly to define and interpolate precalculated abacuses as discussed in [57].

Four main steps are required:

• 1. Reducing, if possible, the number of parameters required for the database ("Database Definition"),

• 2. Defining a FE model for the suction experiment and creating the database in the required parameter range ("FE Model"),

• 3. Interpolate the database for any parameters $D_i$, $L_{R_1}$, $E_{R1}$ and $E_{R0}$ ("Database Interpolation"),

• 4. Validate the proposed method ("Validation").

A.1 Database definition

The four main parameters $D_i$, $L_{R_1}$, $E_{R1}$ and $E_{R0}$ can be combined to reduce the required dimension of the FE database from 4 to 2.

Scale Effect :

assuming the lower layer thickness is infinite (in practice, the total thickness of the layer is much larger than the aperture diameter $D_i$), the upper layer relative contribution to the shape $S_{tissue}$ is governed only by the depth ratio $\zeta =\frac{D_i}{L_{R_1}}$ between the aperture diameter $D_i$ and upper layer thickness $L_{R_1}$ [12]; redundant depth ratio $\zeta$ provides redundant information in the FE database.

Material Stiffness Contrast :

 considering a material stiffness contrast ratio $\eta =\frac{E_{R1}}{E_{R0}}$, the apparent stiffness $B_{i\, SIM}(\beta , \theta )$ can be seen as proportional to the bottom layer stiffness $E_{R0}$ (equation (10)).

The required FE database to compute the apparent stiffness $B_{i\, SIM}(\beta , \theta )$ can thus be reduced to evaluate a two-parameter function, $f_{sim}$, so that:

$$\begin{aligned} B_{i\, SIM}(\beta , \theta )= E_{R0}\, f_{sim}(\zeta , \eta ) \end{aligned}$$

(10)

where $f_{sim}$ is an adimensional function depending on the depth ratio $\zeta =\frac{D_i}{L_{R_1}}$ and on the layer stiffness contrast ratio $\eta =\frac{E_{R1}}{E_{R0}}$.

Note that equation (10) implies that the cost function $\Phi _{Param}$ (equation (8), main paper body) is linearly conditional on the parameter $E_{R0}$ [44]. It means that once $\zeta$ and $\eta$ are chosen, the parameter $E_{R0}$ minimising $\Phi _{Param}$ is simply obtained by solving a linear problem.

The range of both the ratio parameters $\zeta$ and $\eta$ were chosen to build the database, i.e. to estimate the function $f_{sim}(\zeta , \eta )$ :

• 1. The chosen range for the stiffness contrast ratio $\eta$ was from 1 to 120 to anticipate application to in-vivo cases.

• 2. Aspirating with an aperture of diameter $D_i$ extracts data mainly at a depth of one diameter [12]. Let us consider the case where the layer thickness is greater than $D_i$, i.e. for example, $L_{R_1}>3 D_i$. A small increase of the layer thickness should have negligible influence on the result in this case [12, 36, 40]. Therefore, a limit scale ratio $\zeta =\frac{D_i}{L_{R_1}}>\frac{1}{3}$ was chosen. Moreover, the smallest aperture diameter being of $D_{i}=4$ mm, it was decided that the identification of mechanical properties of layers thinner than 0.25 mm would be out of the identification range of this work. The largest aperture diameter being of $D_{i}=30$ mm, the maximum depth ratio $\zeta$ for such a thin layer is of 120. Therefore, the range of the scale ratio $\zeta$ required in this work was $[\frac{1}{3}, \, 120]$.

FE Model

Model Definition

An FE model was parameterized using a Matlab code to provide $f_{sim}(\zeta , \eta )$ for the chosen ranges of $\zeta$ and $\eta$:

$$\begin{aligned} f_{sim}(\zeta , \eta )=\frac{B_{i\, SIM\, db}}{E_{R0\, db}} \end{aligned}$$

(11)

where $B_{i\, SIM\, db}$ is the slope of the FE pressure-shape curve. To compute the database, an arbitrarily chosen lower layer stiffness of $E_{R0\, db}=4000$ Pa was used.

A static, implicit, axisymmetric model (ANSYS APDL) was defined to describe suction onto cylindrical phantoms. The model takes into account large displacements. A constant aperture diameter of $D_{i}=10$ mm was chosen (Fig. 13(a)); the depth ratio $\zeta =\frac{D_i}{L_{R_1}}$ was changed by modifying the layer thickness $L_{R_1}$. To allow the use of a unique mesh for all simulations in the database, a geometry of $M=20$ pre-meshed layers was defined (Fig. 13(b)). The ratio $\zeta =\frac{D_i}{L_{R_1}}$ was thus modified between simulations by attributing a Young modulus of $E_{R1\, db}$ to the first $[1, \, m]$ upper-layers and a Young’s modulus of $E_{R0\, db}$ to the other layers in $[m+1, \, M]$. The mesh used to compute the whole database was composed of 6 bilinear axisymmetrical elements (Q8, Plane183, ANSYS) in each layer thickness. A zoom-in of the mesh size is reported in Fig. 13(b).

Note that the parts of the 3D printed cups in contact with the tissue (wall thickness, fillet radius) were all proportional to the cup aperture $D_i$; the model cup geometry in contact with the phantom is representative of the reality for all cup sizes.

The boundary conditions are presented in Fig. 13(a). The vertical line AG is the axisymetric axis of the model; a single planar section of the model defines the whole model geometry. The top of the suction aperture (line CD) is clamped in all directions. A partial vacuum $-\Delta P_{tissue}$ is applied to the line AB. Contact elements were defined between the suction aperture and the line AB. With these boundary conditions, the whole tissue is free to move up or down relatively to the cup, depending on the applied pressure $-\Delta P_{tissue}$. These boundary conditions account for the fact that external loads applied on the cup were as small as possible during the experiments (Fig. 4, illustration on phantom A). No additional external loads were taken into account in the simulations. Furthermore, the dimensions of the phantom were large enough so that the application of a rigid casing outside the tissue phantom (Fig. 13(a)) had a negligible impact on the aspirated volume (numerically tested).

The material of aperture and, optionally of the rigid casing, were modelled with an elastic Hookean model with steel mechanical properties. An incompressible Neo-Hookean model simulated the material behaviour of each tissue layer. The apparent stiffness $B_{i\, SIM\, db}$ was evaluated at shapes equal to 0.1; for such a small deformation state, the incompressibility of the material (Poisson coefficient $\nu \in [0.45\, \, 0.5[$) did not influence the results (numerically tested).

The friction coefficient between the tissue and the cup was chosen of $f=0.2$. During the experiment, this parameter was actually unknown and was affected by the ultrasound gel cord. The influence of the friction coefficient has been tested numerically (no friction to glued boundary conditions). Its effect was considered negligible (as also reported in [58]) when the upper layer is stiffer than the lower layer.

The model solution was computed for an initial small partial vacuum $-\Delta P_{tissue}$. The 2D displacement of line AB was converted by numerical integration into the simulated volume $V_{tissue}$ aspirated into the cup. This volume was normalised into shape $S_{tissue}$ (equation (4)), main paper body). The partial vacuum $-\Delta P_{tissue}$ was gradually and monotonically increased. The output result needed to include the shape $S_{tissue}=0.1$ to be validated (Fig. 14(a)). The results obtained around this reference shape were used to compute the sought slope $B_{i\, SIM db}$, which provided in turn the adimensional value $f_{sim}$ (equation (11), illustration in Fig. 14(b)) for $\eta =\frac{E_{R1}}{E_{R0}}=120$).

The FE database was calculated on stiffness ratios ranges: $\eta =\frac{E_{R1}}{E_{R0}}\in [1, \, 120]$ and $\zeta =\frac{D_i}{L_{R_1}} \in [\frac{1}{3},~133]$ (Fig. 15).

Mesh Convergence

To be trustworthy, the database results should be independent of the mesh used. To test this point, a specific curve of the simulation output $f_{sim}$ is presented for 6 meshes with different sizes. Mesh 1 is the coarsest mesh, with only 1 elements in each pre-meshed layer thickness ($6\,561$ elements in the tissue). The number of elements in each pre-meshed layer thickness is progressively increased up to 6 elements ($65\,918$ elements in the tissue). The thinnest mesh is noted Mesh 6 (Fig. 13(b)).

The case with the stiffness contrast ratio $\eta =\frac{E_{R1}}{E_{R0}}=120$ was considered to be the most demanding case, i.e. inducing stress concentrations that could most affect the results. The curve of interest $f_{sim}$ is presented in (Fig. 14(b)) for all 6 meshes. At first sight, all the results overlap. A closer inspection (zoom-in Fig. 14(b)) confirms that the curves obtained for all 6 meshes are slightly different. The convergence of this curve is illustrated in Fig. 14(c) for different depth ratios $\zeta =\frac{D_i}{L_{R_1}}$ and taking the output curve of Mesh 6 as reference to compute relative variations. Therefore, the variations between Mesh 1 and 6 are less than $2\%$ even if the total number of elements is multiplied by 10. Mesh 6 is considered converged and has been used to compute the entire database.

Database Interpolation

The database (Fig. 15) was analysed using the Principal Component Analysis (PCA) method (based on the well known Singular Value Decomposition method). For a detailed description of the model reduction using the PCA method, the reader is kindly referred to [59]. Only the 3 first eigenvectors and associated weighting functions were kept, representing more than $99.99\%$ of the database information:

$$\begin{aligned} f_{sim\, PCA}(\zeta , \eta )=f_{sim0}+\sum\limits_{p=1}^{3} \alpha _p(\eta )\, \, V_p(\zeta ) \end{aligned}$$

(12)

where $V_p(\zeta )$ are the three first PCA normalised eigen vectors and $\alpha _p(\eta )$ are the associated weighing functions. $f_{sim0}=0.7885$ is the FE output for a stiffness ratio $\eta =1$ subtracted from the database prior to PCA. The eigen vectors $V_p(\zeta )$ and their spline interpolation are presented in Fig. 16(a). The weighing functions $\alpha _p(\eta )$ are presented in Fig. 16(b) and (c). Note that the database is dominated by the first weighing function $\alpha _1(\eta )$ and associated first eigen vector $V_1(\zeta )$; the simulated value $f_{sim\, PCA}$ is mainly proportional to the first eigen vector $V_1(\zeta )$.

Database interpolation results using the PCA is presented as black continuous curves in Fig. 15. Although each point of Fig. 15 required to solve a FE model for different partial vacuums $-\Delta P_{tissue}$, the interpolation of the whole database requires only the interpolation of the eigen vectors and weighing function in equation (12). Also note that any other interpolation scheme could have been chosen to interpolate the FE database.

Validation

To validate the apparent stiffness $B_{i\, SIM}(\beta , \theta )$ predicted by the PCA interpolation (equations (12) and (10)), additional tests were performed. Seven FE models were created with overmeshed models (200 elements in diameter $D_i$) and implementing the exact parameters $D_i$, $L_{R_1}$, $E_{R1}$ and $E_{R0}$. The other parameters of the model were kept similar to the ones used to compute the whole database.

The input parameters and the associated apparent stiffness results by direct FE simulation or PCA interpolation ($B_{i\, SIM\, FE}$ and $B_{i\, SIM\, PCA}$, respectively) are reported in Table 8. Note that both the dimension ratio $\zeta$ and the stiffness ratio $\eta$ were chosen so as not to be directly represented in the database (Figs. 15 and 16). For all tests performed, the relative error between the PCA and the direct FE model is less than $1\%$, which is considered to be fully satisfactory.

Table 8 Comparison between the apparent stiffness computed by interpolating the PCA analysis ($B_{i\, SIM\, PCA}$) or with a FE model ($B_{i\, SIM\, FE}$) implementing the exact parameters $D_i$, $L_{R_1}$, $E_{R1}$ and $E_{R0}$. The data input and output for the FE models are highlighted in light grey. The data input and output for the PCA estimation are highlighted in darker grey (equation (12)). For illustration, the particular interpolated points for $f_{sim\, PCA}$ are plotted in (Fig. 15) using the markers reported in first column

Fig. 13

Axisymmetric FE model. Subplot (a) Geometry, boundary conditions, and main dimensions. The nodes of the CD line are completely clamped. Line AG nodes cannot move horizontally and are free in the vertical direction to account for the axisymmetric conditions. A partial vacuum homogeneous pressure $-\Delta P_{tissue}$ is applied to the AB line and is represented by the green area and arrows. Contact elements are defined between the line AB and the suction aperture. Note that with the defined boundaries conditions, the suction cup is fixed and the tissue can freely move up and down into the suction aperture under partial vacuum $-\Delta P_{tissue}$. This set of boundary conditions ensures that load between tissue and suction aperture is only due to the cup internal pressure; no external normal or shear loads are added to the model. Subplot (b) Local mesh zoom in: pre-meshed layers are defined at different depths ($L_{R_1}=\{0.075, \, 0.3, \, 0.67, \, 1.2, \, 1.8, \, ...\}$) to use the same converged mesh for all calculations in the database (six Q8 element minimum in each layer thickness, noted Mesh 6). The mechanical property of the material $E_1$ is applied to the elements of the upper pre-meshed layers (illustration of the layer thickness $L_{R_1}=1.2$ presented as a darker gray, i.e. a ratio $\zeta =8.3, m=4$)

Fig. 14

Mesh convergence demonstration for a specific database curve $f_{sim}$. Subplot (a) Pressure-shape curves obtained for the thinnest mesh (Mesh 6), for a stiffness ratio $\eta =\frac{E_{R1}}{E_{R0}}=120$ and different values of the depth ratio $\zeta =\frac{D_i}{L_{R_1}}$. Subplot (b) Simulation output curve $f_{sim}$ for a stiffness ratio $\eta =\frac{E_{R1}}{E_{R0}}=120$. The output curves for 6 different meshes (from coarse to thin) overlap in this plot. Local zoom-in for a depth ratio $\zeta =\frac{D_i}{L_{R_1}}=3.7$ illustrates convergence with mesh refinement. Subplot (c) Relative variations of $f_{sim}$ for meshes 1 to 6 (total number of elements in the model multiplied by 10) using the results of Mesh 6 as reference

Fig. 15

The FE normalized results $f_{sim}$ (equation (11)) in the database are represented as coloured point markers versus depth ratio $\zeta =\frac{D_i}{L_{R_1}}$. The stiffness ratios range is $\eta =\frac{E_{R1}}{E_{R0}} \in [1, \, 120]$. Interpolation of the PCA eigen vectors and weighing functions enables interpolation of the database (equation (12)), as presented with the black curves joining the point markers. Integer values of depths ratio $\frac{D_i}{L_{R_1}}$ are visually represented under the abscissa axis. Illustrations of particular interpolated points $f_{sim\, PCA}$ (equation (12)) used to compute the values $B_{i\, SIM\, PCA}$ in Table 8 are also reported as specific markers. Consult Table 8 for corresponding legend

Fig. 16

PCA three first eigen vectors and weighing functions representing the FE database (equation (11)). Illustrations of particular interpolated points on the eigen and weighing functions to compute $f_{sim\, PCA}$ (equation (12)) and $B_{i\, SIM\, PCA}$ (equation (10)) in Table 8 are also reported in this figure as specific markers. Consult Table 8 for corresponding legend. Subplot (a) Three first normalised eigen vectors and associated interpolation with splines. Subplot (b) Pondering functions $\alpha _1$ and spline interpolation. Subplot (c) Pondering functions $\alpha _2$ and $\alpha _3$ and spline interpolation

Appendix B: Parameters’ Identifiability and Experimental Variance

As mentioned in the main body of the paper, choosing weights $w_i^2$ representative of the experimental variance $\sigma _{i}^2$ is important if the parameter identifiability is directly inferred from the cost function $\Phi _{Param}$ (equation (8), main paper body). This appendix develops the mathematical approach chosen to evaluate the parameter identifiability and the variance estimation derived from the residual vector $u_{ij}$.

Parameters’ Identifiability

The parameter identifiability under heteroscedastic variance is usually computed using different variance-covariance estimators [42, 43]. In this work, a classic variance-covariance matrix $\widehat{V}_{WLS}$ is used [43]:

$$\begin{aligned} \widehat{V}_{WLS}=\bigg [F^T\big (\widehat{\beta } \big )\, F\big (\widehat{\beta } \big ) \bigg ]^{-1} \end{aligned}$$

(13)

where $F\big (\widehat{\beta }\big )$ is the $N_m\times P$ Jacobian matrix of the function $w_i\, Ln\big (B_{i\, SIM}(\beta , \theta )\big )$ (equation (8), main paper body) evaluated at $\beta =\widehat{\beta }$ . The variance-covariance matrix $\widehat{V}_{WLS}$ is of dimension $P\times P$ and is a linear approximation of the inverse of the Hessian matrix of $\Phi _{Param}$. Its graphical representation is an hyperelipsoid of dimension P known as Indifference Regions (IR). In this work, IR with a confidence level of $95\%$ will be plotted.

With this approximation, the Confidence Interval (CI) for parameter $\widehat{\beta }$ is computed as [43]:

$$\begin{aligned} \beta _{p\, CI}= \widehat{\beta }_p \pm z_{\alpha /2} \sqrt{{diag(\widehat{V}_{WLS})}_p} \end{aligned}$$

(14)

where $\widehat{\beta }_p$ is the pth element of $\widehat{\beta }$ and $z_{\alpha /2}$ is the cumulative distribution of a normally centered distribution function for a confidence level $\alpha$.

Note that the particular residual error vector $e_{ij}=w_i u_{ij}$, which is the residual value for a specific noise copy $\epsilon _{ij}$, is not taken into account to compute the variance-covariance matrix $\widehat{V}_{WLS}$ (equation (13)). The variances and associated weights $w_i$, taken into account while computing the Jacobian matrix F of $w_i\, Ln\big (B_{i\, SIM}(\beta , \theta )\big )$, must be properly estimated so that the calculated CIs are meaningful.

Input Noise Variance Evaluation

In this work, the variances $\sigma _i^2$ of the noise copies $\epsilon _{ij}$ (equation (7), main paper body) for each aperture diameter $D_i$ were evaluated in two different ways.

Given equation (7) (main paper body), the classic way is to compare the experimental values $Ln\big (B_{ij\, EXP}\big )_k$ obtained on the phantom k, aperture diameter $D_i$ and cycle j, with the averaged value $\overline{Ln\big ({B}_{ij\, EXP}\big )}_k$ over the number of cycles $J_{ki}$ measured on the phantom k and with aperture diameter $D_i$, so that:

$$\begin{aligned} \sigma ^{2}_{i\, Classic} = \frac{1}{(N_{ki}-K)} \sum\limits_{k=1}^{K} \sum\limits_{j}^{J_{ki}} \bigg ( Ln\big (B_{ij\, EXP}\big )_k - \overline{Ln\big ({B}_{ij\, EXP}\big )}_k \bigg )^2 \end{aligned}$$

(15)

where K is the number of phantoms, and $J_{ki}$ is the total number of cycles for the phantom k and aperture diameter $D_i$. Thus, the parameter $N_{ki}= \sum _{k=1}^{K} J_{ki}$ is the number of tests that one has at hand for aperture diameter $D_i$.

The unbiased variance $\sigma ^{2}_{i\, Classic}$ is an exact evaluation under the hypothesis that the model perfectly fits the data and that the random disturbance $\epsilon _{ij}$ is of zero mean: in equation (15), the average value $\overline{Ln\big ({B}_{ij\, EXP}\big )}_k$ plays the role of a model that ’perfectly’ fits the data.

In the cases where these hypotheses are not perfectly met, the classic variance underestimates the actual variance. Another variance estimation, also known as the Almost Unbiased Estimator (AUE), has been implemented based on [60]:

$$\begin{aligned} \sigma ^{2}_{i\, AUE} = \frac{1}{N_{ki}} \sum _{k=1}^{K} \sum _{j}^{J_{ki}} \frac{u^2_{ij\, k} }{(1- \widehat{h_{ij}}_k)} \end{aligned}$$

(16)

where $u_{ij\, k}$ is the residual error vector obtained on phantom k, aperture diameter $D_i$ and cycle j after fitting a model on all phantom k experimental data (one cost function $\phi _{param}$ per phantom k, (equation (8), main paper body). The leverages $\widehat{h_{ij}}_k$ are the diagonal values of the ’hat’ matrix $H_k$ of dimensions $J_{ki} \times J_{ki}$ defined for the kth non-linear model on the phantom k. The hat matrix $H_k$ defined for non-linear models on phantom k writes [43]:

$$\begin{aligned} H_k= F_k\big (\widehat{\beta } \big ) \bigg [F_k^T\big (\widehat{\beta } \big ) F_k\big (\widehat{\beta } \big ) \bigg ]^{-1} F_k^T\big (\widehat{\beta } \big ) \end{aligned}$$

(17)

where $F_k\big (\widehat{\beta }\big )$ is the $J_{ki}\times P$ Jacobian matrix of $w_i\, Ln\big (B_{i\, SIM\, k}(\beta , \theta )\big )$ evaluated at $\beta =\widehat{\beta }$ on the phantom k.

In this contribution, the AUE variance was computed iteratively. The starting weights were chosen so that $w_i^2=1$ to define the function $\Phi _{Param}$ in equation (8), (main paper body). The residual error vector $u_{ij\, k}$ minimizing $\Phi _{Param}$ (equation (9), main paper body) was then computed and injected in equation (16) to provide a variance estimation $\sigma ^{2}_{i\, AUE}$. This estimation has then been used to compute new weights ($w_i^2=1/\sigma ^{2}_{i\, AUE}$) and a new iteration was performed. Iterations were performed until the convergence of $\sigma ^{2}_{i\,AUE}$ (few iterations in practice).