Finite Element Based Optimization of Human Fingertip Optical Elastography

Abstract

Dynamic elastography methods attempt to quantitatively map soft tissue viscoelastic properties. Application to the fingertip, relevant to medical diagnostics and to improving tactile interfaces, is a novel and challenging application, given the small target size. In this feasibility study, an annular actuator placed on the surface of the fingertip and driven harmonically at multiple frequencies sequentially creates geometrically focused surface (GFS) waves. These surface wave propagation patterns are measured using scanning laser Doppler vibrometry. Reconstruction (the inverse problem) is performed in order to estimate fingertip soft tissue viscoelastic properties. The study identifies limitations of an analytical approach and introduces an optimization approach that utilizes a finite element (FE) model. Measurement at multiple frequencies reveals limitations of an assumption of homogeneity of material properties. Identified shear viscoelastic properties increase significantly as frequency increases and the depth of penetration of the surface wave is reduced, indicating that the fingertip is significantly stiffer near its surface.

1. Introduction

1.1. Medical Motivation.

Viscoelastic properties of human skin are affected by disease and injury; measurement of these properties may be used as a diagnostic aid for detection and monitoring of conditions that affect the epidermis and dermis. For example, Raynaud's phenomena and scleroderma have been shown to increase shear elasticity and shear viscosity [1,2] as well as affect the thickness of skin [3]. However, skin viscoelastic properties are also affected by ambient temperature, humidity, moisture content [4,5], thickness, age [6–8], sex, and the direction of applied stress [9]. These confounders have limited the utility of elastography measurements.

Early measurement methods relied mainly on excised strips of skin and tensometer equipment [6]. These tests were accurate and could be performed alongside histological analysis to correlate the stress–strain relationship with changes in molecular structure. However, this was too invasive to be used as a diagnostic tool for skin disorders. Advances in more sensitive measuring equipment led to in vivo measurements of shear strain hysteresis curves [4]. Up to this point, the majority of the studies were based on static methods until [5] studied the dynamic viscoelastic properties of skin, over a frequency range of near zero to 1000 Hz, using propagating surface waves. It was found that the viscoelastic properties were dispersive (frequency-dependent). A similar study also found comparable results; using surface wave propagation, a direct measurement of the complex Young's modulus of excised rabbit skin was calculated [10]. Even though it has been determined that skin is viscoelastic, static methods, such as the suction method or indentation method, have proven reliable and practical for clinical use [9,11–13]; however, they cannot measure the frequency dependence or viscous behavior, which is potentially an additional biomarker.

Focusing on the dynamic response of skin, elastography techniques have been developed that use surface waves as the source for elastography imaging. Elastography uses the principle that the wavelength and attenuation of a propagating mechanical wave is dependent on the mechanical properties. Using this principle, several imaging modalities have been used to estimate shear viscoelastic properties. Optical coherence elastography has been used to measure skin viscoelasticity [14,15] and excised animal tissue [16]. Optical coherence elastography has the advantages of micrometer scale resolution, and three-dimensional (3D) imaging capabilities, with limited surface penetration. Optical elastography based on laser Doppler vibrometry has the advantage of high signal-to-noise ratio (SNR) due to the sensitivity of LASER probes, but it cannot penetrate the surface. Also, most optical elastography methods for skin measure waves that are propagating radially away from a vibratory point source which suffer significantly from attenuation due to geometric dispersion thus limiting the bandwidth of measurement.

In a recent paper by Kearney et al. [17], optical measurement of geometrically focused surface (GFS) waves was introduced that allows for wider bandwidth measurements. An analytical solution for the case of a radiating annular disk surface source was fit to experimentally measured GFS waves, enabling an estimate of the frequency-dependent complex-valued surface wavenumber, which can then be related to the dynamic complex-valued shear modulus. Several viscoelastic models were then fit to the dynamic shear modulus dispersion curve measured on the forearm of healthy volunteers. Viscoelastic models were evaluated based on their overall quality of fit and variability.

In the present paper, the analytical solution has been identified as an inaccurate model for GFS waves. A number of previous dynamic elastography studies have used numerical optimization approaches, some involving finite element (FE) approximations of the experimental system, to identify multiple and/or complex shear viscoelastic parameters simultaneously [18,19]. In the present paper, a new approach is proposed using a FE model of the finger. Material properties of the model are optimized by curve fitting the FE results to the experimental wave profiles measured using scanning laser Doppler vibrometer (SLDV). Further improvements to the method were studied such as the introduction of skin layers to the FE model.

A recent study by Zhang [20] has also introduced a dynamic elastography approach to quantify the stiffness of the skin on the fingertip using an optical approach, a type of laser displacement sensor. The phase gradient cross spectrum method is used to estimate the propagation speed of Rayleigh (surface) waves generated via a 100 Hz harmonic excitation. While using a related, but different means of generation and acquisition of surface wave motion on the fingertip, some of the same issues addressed in the present paper, with respect to identification of shear storage and loss moduli, choice of rheological model and other modeling assumptions, are relevant regardless of differences in means of surface wave generation and measurement.

1.2. Sensory Tactile Motivation.

Physical touch is one of the most basic ways humans interact with the world. Fingertips heavily impact the touch perception. Understanding the mechanical properties of the fingertip is essential to manipulating touch perception. For example, a rough surface can be made to feel smooth or rough because of vibrations at a certain frequency. The early work of Watanabe and Fukui showed that a surface that undergoes a normal displacement at low ultrasonic frequency sees its friction decreased when explored with a bare fingertip [21]. But the skin and subcutaneous tissues of the fingertip are far from rigid. As a consequence, the surface wave created by a harmonic excitation of 20 to 40 kHz has a wavelength on the order of 1 mm. The wavelength is approximately one order of magnitude lower than the diameter of a typical contact patch between the fingertip and a flat surface [21]. Further studies have shown that the increase in amplitude of vibrations reduces the true contact area between the fingertip and the surface as well as the interfacial friction [22]. Consequently, there is an interest in determining the viscoelastic properties of the skin on the fingertip in the range of frequencies spanning from tactile exploration (100 Hz) through the lower end of the ultrasonic frequencies (<100 kHz).

1.3. Objectives.

The objectives of this study were to design and use an optical-based dynamic elastography setup to quantitatively map the viscoelastic properties of the index finger tip of a volunteer using an FE-based optimization method. Dynamic elastography entails three technical challenges: (1) generation of vibratory waves of sufficient but safe amplitude to enable assessment of shear viscoelasticity in the region of interest; (2) noninvasive measurement of wave motion; and (3) reconstruction or inverse modeling to estimate shear viscoelastic properties based on the measured wave motion. The rest of the paper is arranged as follows: Sec. 2 introduces background theory on shear and surface wave motion and viscoelastic modeling options relevant to the third technical challenge. Section 3 introduces the novel experimental design (Sec. 3.1) that tackles the first two technical challenges and then Secs. 3.2–3.4 review the practical application of theory and numerical methods to address the third reconstruction challenge. Results are summarized in Sec. 4, followed by a discussion in Sec. 5 and conclusion in Sec. 6.

2. Theory

2.1. Analytical Approximation for Geometrically Focused Surface Waves.

The steady-state exact analytical solution for geometrically focused shear waves that are radially converging over a linear isotropic viscoelastic cylindrical volume can be found in Yasar et al. and is

$$u_z(r,t,k_s)=u_z\left(a\right)\frac{J_0\left(r{k}_s\right)}{J_0\left(a{k}_s\right)}{e}^{j\omega t}$$ (1)

where r is the radial distance from the central axis, k_s is the complex-valued shear wavenumber, a is the radius of the cylinder, ω is the frequency in radians/s, u_z is the displacement amplitude in the axial (z) direction, and $u_z(a)$ is the magnitude of the displacement amplitude in the axial (z) direction at r = a.

This can also be expressed as a frequency response function relating vertical (axial) motion $u_z(r)$ at a radial location r < a to the imposed vertical motion $u_z(a)$ at the outer boundary of the cylinder r = a as follows:

$$\text{FRF}=\frac{u_z\left(r\right)}{u_z\left(a\right)}=\frac{J_0\left(r{k}_s\right)}{J_0\left(a{k}_s\right)}$$ (2)

Previously, in Kearney et al. [17], by analogy to Yasar et al. [23] and in comparison to the theory for outward traveling surface waves driven by a disk of radius a derived in Royston et al. [24], the following frequency response function was proposed to relate steady-state vertical (normal) motion $u_z(r)$ at a radial location r < a to the imposed vertical motion $u_z(a)$ at r = a on a surface driven normally by an annular (ring-shaped) actuator of inner radius r = a

$$\text{FRF}=\frac{u_z\left(r\right)}{u_z\left(a\right)}=\frac{I_0\left(jr{k}_{\text{su}}\right)}{I_0\left(ja{k}_{\text{su}}\right)}$$ (3)

Here, k_su denotes the complex-valued Rayleigh (surface) wave number, whose real part correlates with surface wave phase speed and whose imaginary part is linked to surface wave attenuation due to viscous losses.

By substituting $I_n(x)=i^{-n}{J}_n(ix)$, where n = 0, into Eq. (3), it can be rewritten in terms of Bessel function of the first kind. Since Bessel functions of the first kind and zeroth-order are even functions, further simplifications can be made to yield

$$\text{FRF}=\frac{u_z\left(r\right)}{u_z\left(a\right)}=\frac{J_0(-r{k}_{\text{su}})}{J_0(-a{k}_{\text{su}})}=\frac{J_0\left(r{k}_{\text{su}}\right)}{J_0\left(a{k}_{\text{su}}\right)}$$ (4)

To assess the validity of this analytical analogy, finite element analysis (FEA) simulations were conducted and are presented in this paper.

Soft biological tissue can be considered as nearly incompressible since the bulk modulus is similar to that of water and is as much as six orders of magnitude larger than the tissue's shear and Young's moduli. This results in Poisson's ratio ν approaching, but not reaching, 0.5. Using ν = 0.49995, we have the following approximation from Graff [25] relating shear and surface wavenumbers:

$$\frac{k_{\text{su}}}{k_s}=\frac{0.87+1.12\nu }{1+\nu }\approx 0.95$$ (5)

This suggests then that if one can estimate the complex surface wavenumber by fitting Eq. (4) to measured steady-state geometrically focused surface wave motion, then by Eq. (5) one can estimate the complex shear wave number. This in turn can be used to estimate the complex shear modulus $\mu ={\mu }_R+j{\mu }_I$, where μ_R and μ_I denote shear storage and loss moduli, respectively, as

$$k_s=\omega \sqrt{\frac{\rho }{\mu }}$$ (6)

where ρ is the tissue density.

2.2. Viscoelastic Models.

Once the complex shear modulus at multiple frequencies has been estimated, different linear viscoelastic models can be fit to it. The value of identifying an appropriate rheological (viscoelastic) model by fitting the complex shear moduli obtained at multiple frequencies is that it enables one to predict moduli at other frequencies not measured in the experiment. Additionally, it potentially provides insight into the structure of the material. Candidate rheological models considered in this study are introduced in Fig. 1. The shear storage modulus, μ_R, and the loss modulus, μ_I for these viscoelastic models can be seen in Table 1.

Fig. 1.

Shear viscoelastic models used for study

Table 1.

Shear storage and loss moduli for selected viscoelastic models

Viscoelastic model	Storage modulus μR	Loss modulus μI
Maxwell	$\frac{{\omega }^2{\eta }^2{\mu }_1}{{\mu }_1^2+{\omega }^2{\eta }^2}$	$\frac{\omega \eta {\mu }_1^2}{{\mu }_1^2+{\omega }^2{\eta }^2}$
Voigt	μ 0	$\omega \eta$
Standard linear solid (SLS)	$\frac{{\mu }_0{\mu }_1^2+{\omega }^2{\eta }^2({\mu }_0+{\mu }_1)}{{\mu }_1^2+{\omega }^2{\eta }^2}$	$\frac{\omega \eta {\mu }_1^2}{{\mu }_1^2+{\omega }^2{\eta }^2}$
Spring-pot	${\mu }_{\alpha }{\omega }^{\alpha }\hspace{0.17em}\text{cos}(\frac{\pi }{2}\alpha )$	${\mu }_{\alpha }{\omega }^{\alpha }\hspace{0.17em}\text{sin}(\frac{\pi }{2}\alpha )$
Fractional Voigt	${\mu }_0+{\mu }_{\alpha }{\omega }^{\alpha }\hspace{0.17em}\text{cos}(\frac{\pi }{2}\alpha )$	${\mu }_{\alpha }{\omega }^{\alpha }\hspace{0.17em}\text{sin}(\frac{\pi }{2}\alpha )$

3. Methods

3.1. Experimental Setup.

The annular actuator imparts a ring of dynamic displacement around the fingertip that generates radially converging surface waves. As the waves converge toward the center, they add constructively to counteract attenuation due to viscosity. This allows motion to be measured over a broader frequency range.

A three-dimensional model of an actuator was created using Autodesk Inventor Professional. The model was printed using titanium alloy via Materialise (Leuven, Belgium) and is shown in Fig. 2. It contains a 10 mm ring for actuating and a hole for line of sight necessary for SLDV measurements. The actuator is connected to a piezo-ceramic stack from Physik Instrumente GmbH & Co. (Karlsruhe, Germany) 7 × 7 × 36 mm which provides 30 μm displacement at 100 V. The other end of the piezo-ceramic stack is connected to the ground via a tabletop breadboard. The study volunteer inserts their finger into the actuator and gently presses up against the ring. Their wrist rests on an elevated surface to keep the arm level and comfortable.

Fig. 2.

Titanium alloy actuator (left) with arrows showing direction of motion and line of sight required by SLDV (right) showing finger in actuator

A sinusoidal 3 V peak-to-peak (V_PP) waveform is created using an internal function generator on the SLDV computer. The waveform is sent to an amplifier (Yamaha Power Amplifier P3500S, Buena Park, CA) to bring amplitude to 40 V_PP. This is applied to the piezo ceramic stack along with a 20 V DC bias (BK Precision 1672, Yorba Linda, CA) to prevent damage to the actuator caused by applying a negative charge.

The SLDV (Polytec PSV-400, Irving, CA) is setup to measure a single line across the surface of the fingertip along an internal radius of the actuator. About 60 equispaced points are used along the radius. Additionally, a single point vibrometer (Polytec PDV-100) is aimed at the surface of the actuator to be used as the reference. Frequencies are measured from 1 kHz to 3 kHz in steps of 500 Hz. Complex displacement measurements (real and imaginary that can also be represented as amplitude and phase) were taken and corrected for any phase delay from the reference signal. A three-point moving average filter was applied to reduce noise. Additionally, due to line of sight constraints, the SLDV could not measure points directly at the contact point of the actuator and finger. Therefore, a complex valued displacement is extrapolated as the maximum amplitude and zero phase at the radius of the actuator.

3.2. Postprocessing Using Analytical Approximation.

Displacement measurements were processed using matlab. A custom code was created using the Global Optimization Toolbox as used by Yasar [23]. Additional parameters were used as in Kearney et al. [17] to account for variations and incorporated into Eq. (3) to form the objective function

$${\text{FRF}}_{\text{obj}}=X_1{e}^{j{X}_2}\frac{I_0\left(j{X}_3\right(r-X_4\left)\right)}{I_0\left(ja{X}_3\right)}+X_5$$ (7)

where X₁ is the amplitude, X₂ is the phase offset, X₃ is the complex surface wavenumber k_su, X₄ is the symmetry shift, and X₅ is the zero offset. The data are fit using a least square approach. Two hundred initial guesses are used for k_su over a range from 10 to 2000. The optimization is run for ten iterations. Using more trials did not lead to different or better results. Once all guesses are evaluated, the least square fit (r²) is used to determine k_su with highest correlation. The resulting k_su is converted to a complex shear modulus using Eqs. (5) and (6).

3.3. Postprocessing Using Finite Element Based Optimization.

A finite element model is created in comsol 5.3 Multiphysics in 3D space using the Solid Mechanics Module and Optimization Module in the Frequency Domain. A computer aided design model of the fingertip, created from segmented slices of magnetic resonance imaging images taken using a Bruker Preclinical 7T magnetic resonance imaging (Bruker Instruments, Billerica, MA), is imported into the model (Fig. 3). A circular ring is defined on the surface using the same dimensions as the experimental setup. The bottom of the fingertip is defined as a fixed surface. The base of the model that would connect with the rest of the finger is treated as a low reflecting boundary condition. A low reflecting boundary is a boundary condition used in comsol to minimize reflections from the edge by using material data from adjacent domain in order to create a perfect impedance match for compression and shear waves. It is primarily used to reduce the computational domain to a practical size while ensuring accurate simulation results [26]. Rather than simulating an actuator, the model treats the area of contact, in this case a circle on the surface, to be a prescribed displacement in the z direction, orthogonal to the surface of the fingertip. The ring is excited at a given displacement and frequency. The amplitude of displacement is selected based on measurements of the actuator during the experimental procedure. The materials are considered viscoelastic, isotropic, nearly incompressible (Poisson's ratio 0.49995) which allow us to approximate bulk modulus (K) as $K=(2\mu (1+\nu )/3(1-2\nu ))$. The mesh is automatically generated through comsol's built in physics-controlled mesh (122,699 tetrahedral elements, maximum element size 0.733 mm, minimum element size 0.0314 mm). The second finest mesh setting was selected. Selecting the finest mesh possible increased computation time significantly and had a negligible impact on the results.

Fig. 3.

Finite element solid model of finger. Blue (dark gray in grayscale) circle denotes actuation.

Although the wave motion can be in multiple directions, the SLDV only measures motion normal to the surface of the fingertip, and measurements were taken along a radial profile within the annular region of excitation. Therefore, we only extract the displacement in the z direction from the FE simulation along this same radial profile for comparison. Displacement measurements from the SLDV are imported into comsol as experimental data points. A function is created from the points using interpolation to create an experimental profile to account for any gaps in measurements. The model is run through the optimization module where the shear storage and the loss modulus are varied. The resulting displacement is then compared to the experimental profile using

$${\int }_{-a}^a{(v_{\hspace{0.17em}\text{exp}\hspace{0.17em}}(r)-v_{\text{optim}}(r))}^{2}dr*{10}^{15}$$ (8)

where v_exp and v_optim are displacement in the z direction of the experimental and FE-based optimization data, respectively. The scaling of 10¹⁵ at the end is to improve the sensitivity of the error function as per recommendation through comsol support. The optimization uses the Nelder–Mead method. The optimization tolerance is set to 0.0001 for the error function. The maximum iterations are set to 1000 although optimization completes before the maximum iteration limit is reached due to the tolerance limit.

3.4. Postprocessing Using Viscoelastic Models.

Once μ_R and μ_I are obtained using either the analytical approximation or FE-based Optimization method detailed earlier, best fit viscoelastic models are estimated. Using another custom matlab code utilizing the Global Optimization Toolbox, the storage and the loss modulus simultaneously are fit to each viscoelastic model in Table 1 using the objective function ${({\mu }_{\hspace{0.17em}\text{exp}\hspace{0.17em}}-{\mu }_{\text{optim}})}^2$. The parameters yielding the best correlation are then selected to represent each viscoelastic model.

4. Results

4.1. Evaluation of Analytical Approximation and Finite Element Based Optimization Using Synthetic Data.

To assess the validity of assumptions in the analytical approximation and FE-based optimization methods introduced in Secs. 3.2 and 3.2, FE models with material properties comparable to soft biological tissue were generated and tested. Two cases are studied to see how the equations fit to semi-infinite half-space and semi-infinite cylinder models. To reduce calculation time and memory usage, a two-dimensional axisymmetric model was used to create a cylinder of 1 m radius and 5 m length to simulate the infinite half-space. Low reflecting boundaries were used to minimize boundary effects, such as reflections and mode conversion. The material is given a springpot model to be similar to materials previously studied by our group (Yasar et al.), where ${\mu }_{\alpha }$ is 4571 $\text{Pa}\cdot {\mathrm{s}}^{\alpha }$ and α is 0.34. The mesh was automatically generated by comsol and the finest mesh was selected (30,965 triangular elements, 239 elements distributed along surface from r = 0 to 4 mm, and remaining elements sized between 0.1 mm and 50 mm). The model is actuated for multiple frequencies by an 8 mm ring on the surface.

Then, to simulate a cylinder driven harmonically with a displacement at its outer radial boundary in the axial direction, the model is modified to be only 4 mm in radius and the simulation is repeated with the entire outer wall of the cylinder being actuated. Again, the mesh was automatically generated by comsol and the finest mesh setting was selected (20,550 triangular elements, 239 elements distributed along surface from r = 0 to 4 mm, and remaining elements sized between 0.1 mm and 50 mm). The resulting synthetic data of displacements seen in Fig. 4 are run through the analytical curve fit procedure (Sec. 3.2) yielding results seen in Fig. 5 along with the actual values of storage and loss moduli; identified viscoelastic model values can be seen in Table 2.

Fig. 4.

Wave propagation at 1 kHz for (left) ring actuator on surface and (right) cylindrical wall actuation

Fig. 5.

Identifying shear storage and loss moduli using analytical approximation: (left) ring actuator on surface; (right) cylindrical wall actuation. Key: actual storage (blue) and loss (green) moduli; X and O best fits to storage and loss moduli, respectively, using analytical approximation. Resulting best fit rheological (springpot) model based on X and O best fits (color online).

Table 2.

Viscoelastic models estimated using analytical (Sec. 3.2) and FE-based optimization (Sec. 3.3)

Model	μ 0	α	R 2
Actual	4571 Pa· ${\mathrm{s}}^{\alpha }$	0.34	—
Analytically optimized ring	475 Pa· ${\mathrm{s}}^{\alpha }$	0.59	0.015
Analytically optimized cylinder	4628 Pa· ${\mathrm{s}}^{\alpha }$	0.33	0.99
FE optimized ring	4571 Pa· ${\mathrm{s}}^{\alpha }$	0.34	1.00
FE optimized cylinder	4570 Pa· ${\mathrm{s}}^{\alpha }$	0.34	1.00

In place of the analytical approximation, the FE-based optimization method was applied to the same synthetic data to reconstruct material properties as well. To reduce bias, starting points for the optimization method were varied from the known material properties. Both solutions for the ring and cylindrical excitation can be found in Fig. 5. Additional reconstructed values can be seen in Table 2.

4.2. Finite Element Based Optimization to Identify Viscoelastic Properties Based on Experimental Data.

Experimentally measured displacements from SLDV were postprocessed in comsol using the method detailed in Sec. 3.3. The procedure was repeated multiple times with varied initial guesses to avoid reaching a local extremum. In the rare case that a limit was reached, the limits were expanded and the initial guess was based on where the previous optimization left off. Figure 6 shows an example of the optimization output. Table 3 shows the derived storage and loss modulus at each frequency. 3 kHz was rejected due to a failure to converge to an appropriate solution.

Fig. 6.

FEA optimization fit at 1 kHz

Table 3.

Storage and loss moduli from an FE-based optimization

Frequency	Storage modulus μR (kPa)	Loss modulus μI (kPa)	Error $({10}^{15})$
1000	92.8	3.06	0.011
1500	121	6.43	0.045
2000	131	12.3	0.201
2500	173	39.1	0.0092

Values from Table 3 were used for fitting viscoelastic (rheological) models. Table 4 shows the derived values for the viscoelastic models along with corresponding R² values to show correlation. The best fit was the SLS model, while the fractional Voigt and spring-pot models were comparable. Their corresponding plots can be seen in Fig. 7.

Table 4.

Viscoelastic models based on FE optimization

Model	First parameter	Second parameter	Third parameter	R 2
Fractional Voigt	μ0 = 96.2 kPa	${\mu }_{\alpha }$ = 1.45 kPa· ${\mathrm{s}}^{\alpha }$	α = 0.35	0.89
Maxwell	μ1 = 131 kPa	η = 122 Pa·s	—	0.85
SLS	μ0 = 114 kPa	μ1 = 51.7 kPa	η = 3.17 Pa·s	0.91
Spring-pot	${\mu }_{\alpha }$ = 51.2 kPa· ${\mathrm{s}}^{\alpha }$	α = 0.10	—	0.89
Voigt	μ0 = 130 kPa	η = 1.59 Pa·s	—	0.88

Fig. 7.

Best fits of different viscoelastic models to FE-based estimate of storage and loss moduli based on experimental measurements: (left) storage and (right) loss moduli

4.3. Layered Materials.

From Tables 3 and 4, as well as Fig. 7, it is clear that the simple viscoelastic models considered here that are based on an assumption of homogeneity and isotropy are a poor fit and cannot capture the frequency dependence of the apparent shear storage and loss moduli of the fingertip based on experimental measurements from 1 to 2.5 kHz. In fact, over this frequency range, there is a near doubling of the shear storage modulus and an increase in shear loss modulus by more than a factor of ten. Such significant increases defy conventional models of viscoelasticity and highlight the limitation of assuming homogeneity of material properties. To investigate this, we return to the numerical FE simulation study of Sec. 4.1, but modify the halfspace model by tripling the complex shear modulus for a 1 mm thick layer of the material at the surface. Using the output of the FE simulation of this layered half-space, the FE optimization routine, which assumes homogeneity, produced estimates of shear storage and loss moduli shown in Table 5. These estimates are closer to the measurements shown in Fig. 7 than the estimates obtained from a model assuming homogeneous properties. Note, the complex shear modulus of both the top 1 mm layer and the underlying layer is modeled with a spring-pot model with α = 0.34. So, from 1 to 2.5 kHz, both of their storage and loss moduli should increase by a factor of ${2.5}^{0.34}=1.37$. Yet, by neglecting the layering effect and assuming homogeneity, the optimization routine predicts increases in shear storage and loss moduli by factors of 1.52 and 6.62. Moderate overprediction of shear storage modulus and significant overprediction of shear loss modulus is consistent with trends in values calculated based on the experimental fingertip measurements, which is evident in Fig. 7.

Table 5.

Storage and loss moduli from an FE-based optimization of layered materials

Frequency	Storage modulus μR (kPa)	Loss modulus μI (kPa)	Error $({10}^{15})$
1000	121	4.45	0.477
1500	144	7.09	0.133
2000	163	17.5	0.060
2500	184	29.4	0.047

5. Discussion

The FE study in Sec. 4.1 suggests that the analytical approximation for geometrically focused surface waves caused by an annular actuator on the surface is not exact. It was also noticed that the surface wave response within the annular region was affected by the width (radial extent) of the annular actuator, which is not a factor in the analytical approximation. Nonetheless, this simple approximation, as it is, which took only a few seconds to obtain can be used to provide an initial estimate of shear viscoelasticity, which can serve as a starting point for numerical optimization using the FE-based approach. Figure 8 and Table 2 show that the FE-based optimization more accurately approximates the material properties in both annular and cylindrical actuation cases. Speed and accuracy of optimization using FEA for this complex optimization problem with multiple local optima improves when the starting point is within a factor of the optimal solution.

Fig. 8.

Identifying shear storage and loss moduli using FE-based optimization method: (left) ring actuator on surface; (right) cylindrical wall actuation. Key: actual storage (blue) and loss (green) moduli (shear storage modulus is large than loss modulus); X and O best fits to storage and loss moduli, respectively, using FE-based optimization method. Resulting best fit rheological (springpot) model based on X and O best fits (exact match and overlays the actual values) (color online).

The FE-based optimization to estimate the complex shear modulus of the fingertip, with geometry derived from ultrahigh field magnetic resonance imaging, could be improved. By viewing an animation of the wave motion, it appears that the boundary conditions have a non-negligible influence on the propagation. Since the fingertip FE model is truncated, it would be better to extend the geometry of the fingertip to include the rest of the finger. There is a nonzero displacement near the truncation boundary, which in turn results in reflection of wave motion that would not occur in a an untruncated model. Additionally, the entire fingertip was treated as having the same homogeneous material properties. This is not accurate as the actual composition of the finger includes bone, muscle, fatty tissue, and layers of skin. Each of these would have their own material properties. The numerical study in Sec. 4.3 highlighted how increased shear viscoelasticity near the surface is likely the reason for the significant increase in shear moduli with respect to frequency based on the FE optimization strategy that assumes homogeneity.

Although the feasibility study in this paper covers the methods of material property reconstruction, the derived values should not be considered as normal values. Multiple samples must be taken to test for repeatability. It has been seen that the air humidity as well as the moisture content of the fingertip can alter the wave propagation, and therefore, the material properties. Future work could involve measuring both humidity and moisture content and testing on the same volunteer on different days as well as multiple volunteers on the same day.

6. Conclusion

Dynamic elastography of the fingertip was attempted in order to identify its viscoelastic properties relevant to both medical diagnostics and to the improvement of communication technology that utilizes haptic (tactile) interaction with a vibrating surface. A novel approach using geometrically focused surface wave generation, measured by scanning laser Doppler vibrometry, provided experimental data from 1 to 3 kHz. A two-stage reconstruction strategy was ultimately developed, which utilized an initial estimate of viscoelastic properties based on a simple analytical approximation that was then input into a numerical optimization routine that included a subject geometry-specific finite element model. Improved fits to experiment will require consideration of heterogeneity, for example, explicitly modeling the skin surface with a different (larger) complex shear modulus relative to the underlying tissue, and possibly anisotropy in modeling the material properties in the fingertip.

Funding Data

• National Institute of Biomedical Imaging and Bioengineering (Grant No. EB012142).

• National Science Foundation (Grant No. 1302517).