Non-contact, Ultrasound-based Indentation Method for Measuring Elastic Properties of Biological Tissues Using Harmonic Motion Imaging (HMI)

Abstract

Noninvasive measurement of mechanical properties of biological tissues in vivo could play a significant role in improving the current understanding of tissue biomechanics. In this study, we propose a method for measuring elastic properties non-invasively by using internal indentation as generated by Harmonic Motion Imaging (HMI). In HMI, an oscillating acoustic radiation force is produced by a focused ultrasound transducer at the focal region, and the resulting displacements are estimated by tracking RF signals acquired by an imaging transducer. In this study, the focal spot region was modeled as a rigid cylindrical piston that exerts an oscillatory, uniform internal force to the underlying tissue. The HMI elastic modulus E_HMI was defined as the ratio of the applied force to the axial strain measured by 1D ultrasound imaging. The accuracy and the precision of the E_HMI estimate were assessed both numerically and experimentally in polyacrylamide tissue-mimicking phantoms. Initial feasibility of this method in soft tissues was also shown in canine liver specimens in vitro. Very good correlation and agreement was found between the actual Young’s modulus and the HMI modulus in the numerical study (r²>0.99, relative error <10%) and on polyacrylamide gels (r²=0.95, relative error <24%). The average HMI modulus on five liver samples was found to E_HMI=2.62±0.41 kPa, compared to E_MechTesting=4.2±2.58 kPa measured by rheometry. This study has demonstrated for the first time the initial feasibility of a non-invasive, model-independent method to estimate local elastic properties of biological tissues at a submillimeter scale using an internal indentation-like approach. Ongoing studies include in vitro experiments in a larger number of samples and feasibility testing in in vivo models as well as pathological human specimens.

Introduction

Measuring material properties with noninvasive, nondestructive methods has always been a challenging field in materials science. This is particularly true for biological tissues, where conventional mechanical testing methods are external and limited to either superficial tissues or to in vitro testing (Fung 1967, Sakuma et al 2003, Haslach Jr. 2005, Holzapfel et al 2000) and hence, do not allow to measure mechanical properties of major organs in vivo.

Elasticity Imaging is a research field that has been continuously expanding since its initial development more than two decades ago. The initial objective of such methods was to combine the principle of palpation with medical imaging modalities. Research in this field has developed beyond the sole initial objective of diagnosis by mechanical excitation, and a large number of methods have been proposed that offer more complex applications. In particular, there has been an increasing interest in the use of elasticity imaging methods for quantifying several aspects of biomechanics, as imaging offers the valuable advantage of measuring mechanical properties in vivo in a noninvasive manner. Examples of the use of elasticity imaging methods for biomechanical applications include: Magnetic Resonance Elastography (MRE) for the determination of brain and liver tissue viscoelastic properties in vivo (Klatt et al 2007, Green et al 2008, Atay et al 2008), evaluation of differences in elastic properties between brain tissue white and grey matter (Pattison et al 2010) and comparison between in vivo and post-mortem brain properties (Vappou et al 2008); MRE for the measurement of skeletal muscle contraction and muscle stiffness changes with aging (Jenkyn et al 2003, Domire et al 2009); Transient Elastography to assess skeletal muscle stiffness during contraction (Gennisson et al 2005); Dynamic ultrasound elastography for the rheological characterization of blood clots (Schmitt et al 2011, Viola et al 2004); Shear Waving Elasticity Imaging (SWEI) and Supersonic Shear Imaging (SSI) for the measurement of shear wave dispersion and characterization of soft tissue viscoelasticity (Sarvazyan et al 1998, Deffieux et al 2009) ; Acoustic Radiation Force Impulse (ARFI) Imaging has been shown capable of evaluating stiffness of in vivo liver (Palmeri et al 2008) and ex vivo colon cancer specimens based on tissue relaxation characteristics following acoustic radiation force excitation (Zhai et al 2008); Shearwave Dispersion Ultrasound Vibrometry (SDUV) for investigation of in vivo viscoelastic properties in porcine liver (Chen et al 2009, 2004). The reader is referred to review papers in the field of elasticity imaging for more details about this fast-growing field (Vappou 2012, Mariappan et al 2010). This paper introduces another new ultrasound-based method for the measurement of tissues elasticity, based on the principle of indentation.

Indentation is a widely used material testing method that allows to measure the Load-Displacement curve. Analytical solutions were proposed by Sneddon (Sneddon 1965) in the case of penetration of axisymetric indentors in an elastic half space. Based on these relationships, several methods have been proposed to derive the Young’s modulus of the tested material. For example, as proposed by Oliver and Pharr (Oliver and Pharr 1992), the Young’s modulus can be estimated using a complex iterative procedure and calibration process. While indentation is commonly used for materials such as polymers and metallic samples, the applications of such experimental approaches are limited in vivo, due to the external nature of indentation. Instead, we hereby propose a non-invasive internal indentation method that uses the acoustic radiation force resulting from focused ultrasound and the simultaneous measurement of the resulting displacement, using therefore the basic principle of the Harmonic Motion Imaging (HMI) method (Maleke et al 2006). A stiffness index named as the HMI modulus (E_HMI) is estimated using the applied acoustic radiation force divided by the resulting deformations measured using RF signal processing. The HMI modulus is compared to the actual Young’s modulus on both tissue-mimicking phantoms and in vitro biological specimens.

Harmonic Motion Imaging (HMI) is an ultrasound-based method that uses an amplitude modulated (AM) signal in order to generate an oscillating radiation force at the focal region using a focused ultrasound (FUS) transducer. The original objective of HMI was to constitute an all-in-one tool for tumor detection, characterization, targeting and monitoring of the thermal ablation. Within the small focal region (typically few mm³), force density oscillates at a single frequency within typically the 10–200 Hz range. At the same time, an imaging transducer that is concentric and confocal with the FUS transducer is simultaneously operated to acquire ultrasound radiofrequency (RF) data during the FUS excitation. Displacements resulting from the oscillating acoustic radiation force are estimated by using cross-correlation techniques on consecutively acquired ultrasound RF signals during force application. This paper investigates a new role for HMI, namely, a mechanical testing tool, where a known dynamic force density is applied and the resulting displacement is measured. The objectives of this paper are thus to (1) introduce a stiffness index, namely, the HMI modulus E_HMI, based on the applied force and on the measured strains, (2) assess the level of agreement between E_HMI and the true Young’s modulus E with finite-element simulations, (3) assess the quality and the experimental feasibility of the proposed method on phantoms, by comparing E_HMI to the Young’s modulus measured by mechanical testing, and (4) demonstrate feasibility of the proposed method on ex vivo biological tissues. The general objective of the proposed method is to provide an HMI-based mechanical testing setup that is particularly relevant for biomechanics as it is non-contact, non-invasive, and it can be used in vivo.

Materials and Methods

Calculation of the HMI Elastic Modulus

The method is based on the localized, internal acoustic radiation force, instead of an external mechanical indenter, as the source of stress. In HMI, a focused ultrasound (FUS) transducer is used to generate an oscillating force at the focal region. The force density can be measured experimentally as part of the transducer calibration, and its spatial profile can be therefore known. The resulting displacement is measured using the consecutive RF signals acquired during the application of the oscillatory force. The general experimental setup is summarized in figure 1(a).

Figure 1.

(a) General HMI setup, (b) actual distribution of the volumic acoustic radiation force field, and (c) equivalent volumic force modeled as uniform within the focal region.

In the proposed method, the focal region is modeled as a cylindrical region inside which the volumetric acoustic radiation force is uniform. This is an approximation that will be further addressed in this manuscript since the actual radiation force has a well-defined spatial distribution (Fig. 1 b,c). This cylindrical region is therefore assumed to act as an oscillating rigid piston.

Under the assumption of linear elasticity, the balance of forces on the z-axis acting on this cylindrical region can be written as:

$$\rho \frac{{\partial }^2{u}_z}{\partial t^2}V={\iiint }_V{f}_{V}dV-F_{\mathit{shear}}-F_{\mathit{top}}-F_{\mathit{bottom}}$$ (1)

F_shear, F_top and F_bottom correspond to the vertical forces acting on the lateral, superior and inferior surfaces of the cylinder, respectively (cf. Figure 2). V is the volume of the cylinder, u_z is the vertical displacement of the cylinder, ρ is the density of the tissue and f_v is the local, volumic acoustic radiation force. The inertia term on the left side of equation is considered negligible (1), as its order of magnitude is approximately 0.1% of the one of the other terms within the typical HMI frequency range (10–50 Hz). Therefore, equation (1) can be rewritten as:

$${\iiint }_V{f}_{V}dV=F_{\mathit{shear}}+F_{\mathit{top}}+F_{\mathit{bottom}},$$ (2)

which can be simply deemed as a static balance of forces, where the applied acoustic radiation force equals the sum of elastic forces related to the deformation of the tissue around the piston :

$$\{\begin{array}{c}{F}_{\mathit{top}}=F_{\mathit{bottom}}=\pi R^2\ast E\frac{\partial u_z}{dz}\\ F_{\mathit{shear}}=2\pi Rh\ast G\frac{\partial u_z}{dr}\end{array}$$ (3)

where R, h denote the radius and the height of piston, respectively, and E, G denote the Young’s and shear modulus of the tissue, respectively. Under the assumption of quasi-incompressibility (E=3G), equation (2) becomes:

$${\iiint }_V{f}_{V}dV=E\ast (2\pi R^2\ast \frac{\partial u_z}{dz}+\frac{1}{3}2\pi Rh\ast \frac{\partial u_z}{dr}).$$ (4)

Figure 2.

(a) Illustration of the focal region as a cylindrical piston exerting a vertical force, and corresponding deformations of the surrounding tissue, (b) illustration of the forces acting on this cylinder.

Eq. 4 can be rewritten with tensorial strain components:

$${\iiint }_V{f}_{V}dV=2E\pi R\ast (R\ast {\epsilon }_{zz}+\frac{2}{3}h\ast {\epsilon }_{rz}),$$ (5)

which assumes that the displacement occurs along the z-axis only with ${\epsilon }_{rz}=\frac{1}{2}\frac{\partial u_z}{dr}$.

Another major assumption in this study is that the compressive and shear strains are equal. As it will be demonstrated in the numerical study, they are indeed very close in a cylindrical geometry as the one used here. Consequently, eq. 5 can be written as:

$${\iiint }_V{f}_{V}dV=2E\pi R{\epsilon }_{zz}\ast (R+\frac{2}{3}h).$$ (6)

Finally, we define our HMI modulus as:

$$E_{\mathit{HMI}}=\frac{{\iiint }_V{f}_{V}dV}{2\pi R{\epsilon }_{zz}\ast (R+\frac{2}{3}h)}$$ (7)

The HMI elastic modulus method requires knowledge of the acoustic radiation force. In this study, we used the experimental pressure profile obtained using a needle hydrophone in water, as described elsewhere (Maleke et al 2010). The acoustic radiation force f_exp was then calculated as:

$$f_{exp}=\frac{2\alpha I}{c_s}$$ (8)

Where I is the acoustic intensity, c_s the speed of sound (c_s = 1530 m/s), and α the acoustic absorption of the medium, which was assumed to be equal to the acoustic attenuation. The attenuation of the tissue ahead of the focus was taken into account in order to obtain realistic estimates of the acoustic intensity at the focus. The corresponding 3-D acoustic radiation force distribution was used as the input in both simulations and experiments throughout the study.

Numerical Study

Finite-element simulations were performed using a commercial software package (Comsol Multiphysics 4.3). This numerical study had principally three objectives, namely (1) to validate the underlying theory summarized by equation 7, and to evaluate differences between E_HMI and the actual Young’s modulus numerically; (2) To evaluate the estimation error resulting from the assumption of a uniform cylindrical piston, by comparing results obtained using the actual experimental force field; (3) To assess the estimation error resulting from assumption of a uniform medium, by adding heterogeneities of different sizes.

In all studies shown, axisymmetric models were used, and the geometry of the medium was chosen as cylindrical (50 mm in height by 50 mm in diameter). All problems were solved as static, the input force being equal as the maximum of the oscillatory force. The focal region was set at the center of the axis of revolution with mesh refinement around this focal region.

1) Evaluation of the differences between E_HMI and Young’s modulus

This section aimed at investigating the reliability of the underlying theory that allows the estimation of the elastic modulus as summarized by eq. (7). The numerical phantom was considered as uniform. Its Young’s modulus E was varied across a range representing typical stiffness of the soft tissue (0.5kPa, 1kPa, 2.5kPa, 5kPa, 10kPa, 25kPa, 50kPa, 100kPa). The dimensions of the cylindrical piston were chosen as the zone inside which the volumetric force is greater than 10% of the maximum force using experimental beam profiles, resulting in a region of 3mm in height and 1.5mm in diameter. The input force was implemented as uniform within this cylinder, and equal to the spatial average of the experimentally-measured force. For each case, the axial strain was averaged over a small rectangular region of interest (ROI) immediately below the focus (1.5mm in diameter, 1mm in height), in order to estimate the HMI modulus using equation (7).

2) Evaluation of the approximation of uniform radiation force density distribution

The proposed method to estimate the Young’s modulus, summarized by Eq. 7, assumes that the cylindrical piston is perfectly rigid, i.e., that the distribution of the volumic force inside this piston is perfectly uniform. In reality, the spatial distribution of the acoustic radiation force is non-uniform, despite the fact that virtually all of the energy is concentrated in the focal spot region. Here, we evaluate the error related to the assumption of uniformity of the force. The same simulations were performed as in the previous subsection under two different conditions: (1) using the true, experimentally-measured spatial force distribution and (2) using the equivalent rigid piston with a uniform force equal to the spatial average of the actual force, as required by the theory. It is important to emphasize that the total force, defined as the integration of the volumic force within the piston volume, is identical in the two cases. The Young’s modulus of the medium was chosen equal to E = 10 kPa.

3) Influence of Heterogeneity

Our method assumes that the tested tissue is uniform. Here, we evaluated the error resulting from the fact that the tissue around the focal region could be not perfectly uniform. A spherical inclusion with different mechanical properties was added around the focal region. The background stiffness was fixed to E_background = 5kPa while E_inclusion = 20kPa. The diameter of the inclusion was varied between 2 and 10 mm. E_HMI was calculated for each case.

Tissue mimicking phantom study

The 1-D HMI modulus was measured on six polyacrylamide (PA) phantoms (concentrations of 15%, 20%, 25%, 30%, 35% and 40%), and compared to the Young’s modulus measured by mechanical testing. For each gel, the same liquid mixture (before cross-linking) used for the construction of HMI phantoms was used to make samples for mechanical testing. For the HMI experiments, parallelepiped-shaped phantoms were prepared (approx. 10cm by 5cm by 5cm). Agar powder (4%) was added to each mixture for scattering. The HMI experimental protocol used here has been described in details previously (Maleke et al 2006). The amplitude-modulated (AM) frequency was 20Hz, resulting in an excitation frequency of f=40Hz. The HMI system was comprised of a PZT HIFU 4.5 MHz transducer (outer diameter 80 mm, inner diameter 16.5 mm, focal depth 9 cm) transducer (Riverside Research Institute, New York, NY) and a confocal 7.5 MHz single-element pulse-echo transducer (Olympus-NDT, Waltham, MA, U.S.) with a diameter of 15 mm and a focal length of 6 cm for simultaneous RF signals acquisition at a frame rate of 4 kHz. The focal region was approximately at 1.25cm depth from the surface of the gel. Displacements were measured using cross-correlation on raw ultrasonic RF signals. The size of the window for displacement estimation was equal to 1.8 mm, with an overlap of 90%. The compressive strain was calculated using a 1^st derivative Savitzky-Golay filter (kernel=7). Figure 3 illustrates how strain is calculated from displacement images. Absorption α was measured by transmission. It was found to be equal to α = 0.25 cm⁻¹ at f_center = 4.5 MHz.

Figure 3.

(a) Estimated displacement using 1D cross correlation, and (b) Estimated axial compressive strain. Red line indicates the region of investigation for compressive strain, i.e., end of acoustic focal zone. Note that the displacement profile is plotted in m-mode.

HMI modulus study on ex vivo specimens

Canine liver specimens (lobe = 2, location =5) were excised immediately upon sacrifice. All procedures were approved by the Institutional Animal Care and Use Committee of Columbia University. The tissue specimens were immersed in a degassed Phosphate Buffered Saline (PBS) bath upon excision and throughout the experimental procedure. Both the HMI imaging and experimental protocol used here has been described in details previously (Maleke and Konofagou 2008) where the AM frequency was 25 Hz, resulting in an excitation frequency of f=50Hz. The HIFU transducer had a center frequency of 4.75 MHz, an outer diameter and inner diameter of 80mm and 16.5 mm, respectively, and its focal depth was 9 cm (Riverside Research Institute, New York, NY). The transducer was positioned in such a way the end of axial focal zone (i.e., the region where the axial compressive strain can be estimated) was well within the liver specimen. The focal region was approximately at 1.5cm depth from the surface of the liver. The compressive strain was calculated using a 1^st derivative Savitzky-Golay filter (kernel=7). The size of the window for displacement estimation was equal to 1.05 mm, with an overlap of 95%. The absorption value for force calculation from ex vivo studies was chosen to be α = 0.75 cm⁻¹ at f_center = 4.5 MHz based on previous literature (Hou et al 2011).

Mechanical testing

Polyacrylamide Phantoms

Cylindrically-shaped samples were prepared for each gel (N=5–8 per gel, diameter = 12mm, thickness = 2.5 to 3.5 mm) used in the study. The samples were tested on an ARES rheometer (TA Instruments, NewCastle, DE, USA). Dynamic analysis was performed at frequencies between 1Hz and 10Hz at ε=0.5% strain. The value of the shear storage modulus G_Mech.Test. measured at f_AM=10Hz was selected as the reference value, and the Young’s modulus was obtained by E_Mech.Test. = 3 G_Mech.Test. under the assumption of quasi-incompressibility.

Liver samples

Samples were collected from the same livers as those tested by HMI, and were constantly kept in Phosphate Buffered Saline (PBS) (Fisher Scientific, Pittsburgh, PA) during ex vivo transportation and mechanical testing. To minimize the variations in tissue properties, all tests were completed within 12 hours post-mortem. Five cylindrical test specimens of diameter d =5.19±0.41mm and height h =4.22±0.80mm (N=3 and N=2 for liver 1 and 2, respectively) were prepared using a biopsy punch. The specimens remained unconfined between the top/bottom plates, pre-compressed (to induce enough grip between the tissue and grip surfaces) (ε=5%), and tested under oscillatory shear test (ε=1%, f=1Hz–10Hz) in order to measure the shear modulus (G_Mech.Test.). The value of the shear storage modulus G_Mech.Test. measured at f=10Hz was chosen as the reference value, and the Young’s modulus was obtained by E_Mech.Test. = 3 G_Mech.Test. under the assumption of quasi-incompressibility.

Results

Numerical Study

1) Evaluation of the differences between E_HMI and Young’s modulus

The HMI modulus E_HMI was calculated from equation (7) using the output of finite-element simulations. Excellent correlation was found between the actual Young’s modulus E and E_HMI (r²>0.99). E and E_HMI were found to be very close, as described in Table 1. The relationship between them was found to be: E_HMI=0.915*E-60 (Pa), illustrating that E_HMI estimates are slightly but systematically lower than E.

Table 1.

Results from simulations on uniform phantoms whose Young’s modulus is varied between 0.5 and 100kPa, and corresponding HMI modulus found by using equation 7.

Input Young’s modulus (kPa)	Axial strain εzz	Shear strain εrz	EHMI calculated from eq. (7), kPa	Relative error (%)
0.5	0.203	0.196	0.45	10
1	0.1	0.975	0.914	8.6
2.5	0.04	0.038	2.25	10
5	0.02	0.019	4.57	8.6
10	0.01	0.098	9.05	9.5
25	0.004	0.0039	22.7	9.2
50	0.002	0.0019	45.2	9.6
100	0.001	0.0096	91.3	8.7

2) Evaluation of the approximation of uniform radiation force density distribution

E_HMI values were compared in two cases, namely, (1) using the actual force density field derived from experimental measurements and (2) using the corresponding modeled force field uniform within the focal region. Figure 4 illustrates the strain profile around the focal region (zoomed) for both cases. The values of ε_zz in the uniform case differ from the ε_zz measured using the actual force density field, but this difference was deemed to be relatively limited (≈ 12% relative error).

Figure 4.

(a) Volumic force profile implemented in simulations from experimental pressure measurements, (b) Resulting axial strain using this profile and (c) Axial strain obtained using the equivalent uniform force profile (where the force value is the spatial average of (a)).

3) Dependence on heterogeneity

The inclusion size was varied between 2mm and 10mm in diameter (focal region height = 3mm). For an inclusion size of 6 mm or higher, the HMI modulus was found to be very similar to the HMI modulus found in the uniform case (<10% relative error). For inclusion sizes smaller than 6 mm, the HMI modulus was found to be significantly different (Figure 5). This difference was equal to 22% in the 4-mm inclusion size. These results are indicative of the fact that as long as a heterogeneity is larger than ~6mm, it will have very limited influence on the estimated E_HMI and we will estimate the E_HMI of this inclusion. If the heterogeneity is smaller than ~6mm, the calculated E_HMI is affected by both the elasticity of the inclusion and of the background.

Figure 5.

Effect of the size of the heterogeneity on the estimated HMI modulus. (a) Description of the geometry used for the simulation; (b) HMI modulus vs size of the inclusion. E_HMI in the uniform case is equal to 18.4kPa (corresponding to input E=20kPa)

Tissue-mimicking phantom study

Excellent correlation (r² = 0.95) was found between the Young’s modulus and HMI modulus. Figure 6 illustrates the HMI modulus versus the Young’s modulus for the 6 tested gels. The linear relationship between E and E_HMI was found as E_HMI=0.83*E+0.51 (kPa).

Figure 6.

HMI modulus vs Young’s modulus measured by rheometry on the six polyacrylamide phantoms (N=5–8 for each phantom). Vertical and horizontal errorbars denote standard deviations among the different samples tested by HMI and rheometry, respectively.

Ex vivo liver study

The estimated E_HMI and the Young’s Modulus measured on all liver specimens are listed on Table 2 and Table 3. The average value of the HMI modulus was found to 2.62±0.41 kPa whereas the measured Young’s Modulus from mechanical testing was 4.20±2.58 kPa.

Table 2.

Estimated E_HMI on liver samples

Specimen #	Axial Strain	EHMI (kPa)
1	0.011±0.0025	2.98
2	0.014±0.002	2.34
3	0.011±0.003	2.98
4	0.016±0.006	2.05
5	0.012±0.005	2.73
Average	0.0125±0.002	2.62±0.41

Table 3.

Young’s Modulus of ex vivo liver tissue measured using mechanical testing

Specimen #	Estimated E (kPa)
1	2.64
2	2.19
3	2.16
4	7.29
5	6.72
Average	4.2±2.58

Discussion

In this paper, a methodology for internal, non-contact indentation of soft tissues using HMI was proposed and investigated both theoretically and experimentally. Currently, there are no techniques available for a localized, noninvasive, internal, indentation-based assessment of mechanical properties for biological specimens. Based on the principle of HMI, our proposed method utilized the applied force and measured the resulting deformation. We hereby propose a stiffness modulus, namely, the HMI modulus (E_HMI). We initiated our study from numerical simulation, followed by a feasibility study using tissue-mimicking phantoms, and ex vivo canine liver samples. We compared our findings against those obtained by dynamic mechanical testing. The numerical study has established a very high correlation between E_HMI and E, and has shown that E_HMI could be related to E through E_HMI=0.915*E-60 (Pa). E_HMI underestimates therefore E, which is partially related to the assumption that compressive and axial strains are equal (ε_zz=ε_rz). As illustrated in Table 1, these two strains were found slightly different. Using both ε_zz and ε_rz in equation (5) allows to decrease the difference between E and E_HMI. However, measuring ε_rz would be particularly challenging as it would require ultrasound imaging with very high lateral resolution in order to properly estimate lateral derivatives of the displacement. Therefore, our method here proposed to measure only ε_zz, which can be achieved by using only a single-element ultrasound imaging transducer.

The robustness of EHMI was shown to be independent of the heterogeneity of the medium as long as the heterogeneity remained more than approximately 6 mm in size. This threshold value is valid for the focal size used in this study, i.e., a cylinder of 3mm in height by 1.5mm in diameter. Additional simulations were performed by varying the size of the focus, and preliminary results indicate that the threshold value is approximately equal to twice the height of the focal region for this cylindrical geometry (see Appendix). However, the exact relationship between this threshold and the focal size needs to be further investigated. To summarize, E_HMI can be computed locally and independently of the elasticity distribution around the focal point, as long as the typical heterogeneity size is larger than approximately twice the height of the focal region. Such independence makes the method particularly robust.

Experimentally, high correlation was found between E_HMI and E on polyacrylamide phantoms, and they could be related linearly through E_HMI=0.83*E+511 (Pa). The standard deviation of the measurements was significant, especially in the most rigid phantoms (30%, 35% and 40% concentration) because the estimated strain values and hence the strain signal to noise ratio were lower than that of the softer phantoms. In order to overcome this uncertainty and enhance the accuracy of the method, the input force can be increased so that the SNR increases as well. However, in this phantom study, the same input force was used for all the phantoms in order to allow direct comparison in the same conditions.

Preliminary HMI modulus studies using ex vivo specimens were performed on liver in order to demonstrate the feasibility of the method on soft tissue. E_HMI and E_MechTesting were found to be significantly different (2.62±0.41 kPa and 4.2±2.58 kPa, respectively). However, experiments were performed on a very limited number of samples (N=10 in total), and this conclusion should be considered with caution, especially since the inter-sample variability was found to be particularly high in mechanical testing. This can be explained by the difficulty in performing reproducible experiments on excised tissue samples: obtaining uniform, regular cylindrically-shaped samples and ensuring same conditions of tissue/plate adherence becomes particularly challenging under the same boundary conditions. The differences between our method and other conventional mechanical testing should be more thoroughly investigated on a much larger number of samples. Despite these differences that require further investigation, these preliminary in vitro studies demonstrate the feasibility of measuring the elasticity of biological tissues with the proposed methodology.

In this study, we demonstrated that this method was capable of providing quantitative measurements of the Young’s modulus using a principle similar to indentation. This is unique in so far as our method allows to perform such indentation non-invasively, directly inside the tissue, as opposed to superficial, conventional indentation methods. Compared to quantitative shear wave elasticity imaging methods, including those based on the HMI principle (Vappou et al 2009), it allows to estimate elasticity directly at the focal region at a submillimetric scale; shear wave methods allow to map elasticity within the whole image, but their resolution is substantially poorer at usual imaging frequencies; they are also strongly related to the resolution of the inverse problem which may be a particularly complex issue in several tissues. The proposed method does not require an ultrasound imaging system: a focused ultrasound transducer and a single-element imaging transducer are sufficient to estimate the HMI modulus.

Several methods use the acoustic radiation force for estimating mechanical responses of biological tissues. For example, Acoustic Radiation Force Impulse Imaging (Nightingale et al 2002) and conventional HMI (Maleke et al 2006) consist in measuring the tissue displacement resulting from an internal excitation. However, the displacement alone is only a qualitative indicator of tissue elasticity, because it is highly sensitive to boundary conditions and heterogeneity of the medium. Displacement can therefore particularly be useful for certain clinical diagnostic purposes, but it remains limited for biomechanical applications that aim at quantitatively measuring tissue elasticity.

Despite its potential and the promising opportunities our method offers in biomechanics, it is important to emphasize that this method relies on several assumptions and very fine tuning of several parameters. From equation (7), we can see that the HMI modulus estimation relies first on an accurate knowledge of the applied force. The force profile can be easily obtained by a calibration process as used in this study. However, the question of the magnitude of this force is a more complex issue. As illustrated in equation (8), the value of the force is related to the knowledge of the local tissue absorption. The uncertainty of our HMI modulus estimate will be simply proportional to any uncertainty in this value of absorption. Whereas such an acoustic parameter has been characterized on a large variety of tissues by different authors in the literature, there will always be some degree of uncertainty related to a particular given tissue. Knowing the absorption at the focal region is not the only issue: estimating correctly the pressure at the focus supposes that we know how much energy has been lost in the tissue before reaching the focal region, which raises the question of in situ actual pressure versus pressure measurements obtained in a water bath. In our ex vivo experiments, the focal region was at a depth of 1.5cm. We have taken into account 1.5cm of absorption in tissue when compared to the values that were calibrated in a water bath. However, this may be more complex when applied in vivo, where ultrasound beams propagate through several layers of different tissues. Accurate measurement of the attenuation is necessary in order to obtain an accurate estimation of acoustic intensity at the focus. Our group has recently shown that HMI can be used to evaluate HMI displacement attenuation, which can be used to approximate acoustic attenuation in soft tissues (Chen et al 2014). Once we assume that we know the applied force, other experimental parameters have to be finely tuned as well. The HMI modulus is calculated from the experimentally-measured axial strain. The estimation of this strain is related to both the quality of the displacement estimates, the axial resolution and the kernel of the Savitzky-Golay filter. With our experimental setup, we have determined that a window of ~1 mm with 90 to 95% overlap for the displacement estimation yielded reproducible and robust results. However, increasing this window beyond a certain degree would introduce averaging effect across the depth and therefore result in underestimation of the strain. On the other hand, decreasing this window below 1 mm may lead to displacement estimation with low SNR quality. Therefore, a tradeoff (or optimized window) has to be established based on the respective beam profile of the focused ultrasound transducer and on ultrasound acquisition parameters. Furthermore, assuming a uniform cylindrical piston can also lead to substantial limitations, because the focusing quality of the transducer must be very high for such an assumption to remain valid. In other words, any environmental conditions that may affect the quality of focusing, such as secondary lobes, must be minimized as much as possible.

Finally, it is important to re-emphasize again on the importance of all of the aforementioned limitations: Although this method is able to provide quantitative elastic parameters of biological tissues, thorough calibration and tuning procedures are required for every acquisition system and environment conditions. The presence of other uncontrolled parameters such as tissue absorption results in intrinsic uncertainty of the modulus estimate. Nevertheless, after this thorough calibration process, the proposed method can be served as a new internal indentation-based approach for measuring tissue elasticity quantitatively. In addition to its value as a biomechanical testing tool, it might also be used as a HIFU therapy monitoring tool through information on tissue stiffness during the HIFU treatment.

Conclusion

In this work, a noninvasive, noncontact, model-independent method for the measurement of tissues elasticity is presented based on the principle of indentation by applying the principle of HMI. The method is capable of estimating the Young’s Modulus from estimated focal strain values and knowledge of the applied focal force density. The method has been validated theoretically with Finite Element Analysis (FEA), and feasibility studies have been completed experimentally using both phantoms and liver specimens. Very good agreement and correlation were found between the Young’s modulus and our HMI modulus in the numerical studies, and feasibility was demonstrated ex vivo on liver samples. Ongoing work is being carried out in investigating the relevance of E_HMI as a monitoring parameter for HIFU, and investigating feasibilities on in vivo tumor model mice as well as ex vivo human pathological specimen.

Appendix: Effect of the size of the heterogeneity on E_HMI for different focus sizes

Figure 5(b) illustrates the effect of the size of the heterogeneity on the estimated HMI modulus. It is found that for inclusions larger than 6mm in diameter, E_HMI is very close to the equivalent E_HMI found in a uniform medium (relative error < 5%). For this original case, the focal size was a cylinder of 3 mm in height by 1.5 mm in diameter. The objective of this additional, preliminary study, is to assess how this threshold (equal to 6mm in the original case) depends on the size of the focus. Similar simulations as those described in P12 and P13 (Influence of heterogeneity) were performed. Two additional configurations were tested with scaling factors of 1.5 and 2, which correspond to focal sizes of 4.5 mm* 2.25 mm and 6 mm * 3 mm, respectively.

Figure A.1.

Effect of inclusion size on HMI modulus for scaling factor of 1.5 (top row) and 2 (bottom row)

The threshold values (relative error <5%) were found to 7mm and 11mm for the 1.5 and 2 scaling factor, respectively. This preliminary study confirms that this threshold is in the order of magnitude of twice the length of the focal size. Ongoing work includes a more thorough investigation on the actual resolution of the proposed method, by considering different shapes and sizes of focal regions.