2.1.

Tissue Deformation in Compression Elastography

A compressive load applied to a sample is described in terms of stress, $\sigma =F/A$, where $F$ is the applied force and $A$ is the cross-sectional area over which the force is applied. The resulting bulk deformation of the sample along the axis of compression is quantified by strain, $\epsilon =\mathrm{\Delta }l/l_0$, where $\mathrm{\Delta }l$ is the change in length and $l_0$ is the original length of the sample. The axial compression of a sample is generally accompanied by some lateral expansion, to conserve volume. This shape change is characterized by Poisson’s ratio, $\upsilon$, which is equal to 0.5 for a completely incompressible material and commonly assumed to be in the range $\sim 0.49$ to 0.5 for most soft tissues.²⁸

If we make the assumption that tissue deforms in a linearly elastic manner, which has been reported to be an accurate approximation for strains $<0.1$ in tissues such as breast and prostate,²⁹ stress and strain are related through a three-dimensional (3-D) set of elastic constants. In compression elastography, the applied load can be approximated as uniaxial, i.e., occurring along one axis of the sample. Under the further assumption that the sample is isotropic, i.e., its response to stress is direction independent, the uniaxial stress and strain are related through one elastic constant, defined as the Young’s modulus, $E=\sigma /\epsilon$. The Young’s modulus is commonly used to characterize the elasticity of tissues and, for soft tissues, has been reported to range from hundreds of Pascals (Pa) for healthy liver tissue,³⁰ to tens and hundreds of kilopascals for healthy and malignant breast tissues,²⁹ to a few megapascals for arterial wall tissues.³¹

The objective of compression OCE is to form a map of Young’s modulus within the sample—a so-called elastogram. The equation for Young’s modulus given above describes a “bulk” response of a sample, but a map of Young’s modulus requires knowledge of the “local” stresses and strains throughout the sample. Local strain is obtained by calculating the spatial derivative of the measured displacement, i.e., the change in displacement per unit length of the sample. Local stress, on the other hand, cannot be directly measured at depth within the sample. As a result, the elastogram in compression OCE is typically a map of strain, which gives a relative measure of Young’s modulus, under the assumption that stress is uniformly distributed throughout the sample. However, in practice, stress concentrations arise at feature boundaries within heterogeneous samples³² and at the sample surface where friction is present. Thus, strain elastograms are subject to mechanical artifacts that limit their accuracy in representing the true elasticity distribution in a sample.

2.2.

Quantification of Elastogram Accuracy

A measure of elastogram accuracy (fidelity to the true elasticity distribution) is the contrast transfer efficiency (CTE), which was defined by Ponnekanti et al.³² in ultrasound elastography as the ratio of strain contrast observed between features in an elastogram, $C_{\mathrm{o}}$, to true elasticity contrast of the features, $C_t$: $\mathrm{CTE}=C_{\mathrm{o}}/C_t$. We employ this definition of CTE to quantify elastogram accuracy in this article and address the first question in the introduction: “How accurately do OCE elastograms represent the elasticity distribution in a sample?”

In compression elastography, the CTE of strain elastograms depends on the validity of the assumption of uniform distribution of stress. This assumption holds for the trivial case of a mechanically homogeneous sample undergoing uniaxial compression, which results in a uniform strain field but provides no elasticity information without a measurement of the applied stress. We consider here the more relevant case of a mechanically heterogeneous sample, in which the aim is to differentiate features based on mechanical contrast within an image. In particular, we consider two heterogeneous sample geometries: bilayer (soft on stiff) and stiff inclusions embedded in a soft matrix. Analytical expressions for the CTE of strain elastograms have been derived for each of these geometries, assuming an infinite medium of isotropic, linear elastic material undergoing uniaxial compression, in which each material in the structure has equal Poisson’s ratio. For a layered structure, by considering an equivalent system of springs, it can be shown that the stress is uniformly distributed among the layers, such that the strain contrast between layers is inversely proportional to the Young’s modulus contrast between layers.³³ In other words, a strain elastogram of a layered geometry is expected to have $\mathrm{CTE}=1$.

Kallel et al.³⁴ used an analytical solution to the elasticity equations to derive expressions for the CTE of strain elastograms of an inclusion geometry. They found that for the case of an inclusion perfectly embedded (bonded to its surroundings) in a homogeneous matrix material, the contrast between the strain in the inclusion, ${\epsilon }_{\text{inclusion}}$, and strain in the matrix, ${\epsilon }_{\text{matrix}}$, (at large distances from the inclusion), $C_{\mathrm{o}}={\epsilon }_{\text{inclusion}}/{\epsilon }_{\text{matrix}}$, is related to the true modulus contrast between the inclusion and matrix, $C_t=E_{\text{inclusion}}/E_{\text{matrix}}$, by

Note that $C_{\mathrm{o}}$ depends neither on the size of the inclusion nor on its depth below the surface, but only on the modulus contrast between the inclusion and matrix and on the Poisson’s ratio, $\upsilon$, which, as discussed above, falls in the range $\sim 0.49$ to 0.5 for most soft tissues. In this case, Eq. (1) may be approximated as $C_{\mathrm{o}}=1/2+C_t/2$. This indicates that for increasing elasticity contrast between a stiff inclusion and a soft matrix, the observed strain contrast is also expected to increase but that the CTE will plateau at just over 0.5 for large modulus contrast.^32,34 This is illustrated and explained in detail in Sec. 4.2. Equation 1 also applies to the case of a soft inclusion embedded in a stiff matrix; however, we limit the analyses in the present study to consider stiff inclusions.

The analysis of elastogram accuracy described above, first introduced in ultrasound elastography, is independent of the spatial resolution scale of the employed elastography technique and so may be applied to compression OCE. A key difference in OCE, however, is an increased sensitivity to surface effects (friction and coupling with the compressor) as images are generally limited by the penetration depth of OCT to the first 1 to 2 mm in dense tissues. Furthermore, optical interferometric detection in OCE will translate to a higher sensitivity to deformation than in ultrasound elastography, and the higher spatial resolution and smaller field of view in OCE will mean that a different scale of tissue structures is probed. We further examine some of these issues in the discussion in Sec. 5.

Finite element modeling is used in this study for addressing mechanical aspects of the third question proposed above: “What factors limit the accuracy and sensitivity?” Once shown to produce results that accurately model the physical system, FEMs enable ready variation of mechanical parameters, including sample geometry, mechanical properties, and surface friction. The model predicts the effects of these parameters on the resulting strain distribution in the absence of imaging system noise. Importantly, FEMs also provide estimates of the resulting stress distribution, allowing identification of sources of mechanical artifacts in strain elastograms.

2.3.

Strain Elastogram Performance Parameters in Compression OCE

In addition to artifacts arising from built-in aspects of mechanical behavior, elastogram contrast depends on imaging system noise and parameters used in signal processing, particularly in the strain estimation process. Our group recently defined performance parameters for strain elastograms in compression OCE, in terms of the mean and variance of the measured displacement values.¹⁵ Here, we summarize those definitions for reference in this article.

In phase-sensitive OCE, local displacement, $d(z)$, is calculated from the change in phase, $\mathrm{\Delta }\varphi (z)$, between A-scans or B-scans,²⁶ scaled by the mean wavelength of the source in the sample, $\lambda /n$, where $n$ is the refractive index of the sample, i.e., $d(z)=\mathrm{\Delta }\varphi (z)\lambda /4\pi n$. The local strain, ${\epsilon }_l$, is then estimated as the gradient of displacement over a range in depth: ${\epsilon }_l=\mathrm{\Delta }d/\mathrm{\Delta }z$. In this equation, $\mathrm{\Delta }z$ is the strain resolution, defined as $\mathrm{\Delta }z=mdz$, where $m$ is the number of OCT image pixels over which strain is estimated, and $dz$ is the axial pixel size.

The strain sensitivity, $S_{\epsilon }$, is defined as the standard deviation of strain (${\sigma }_{\epsilon }$) and, in phase-sensitive OCE, depends fundamentally on the phase stability of the OCT system. The strain dynamic range, ${\mathrm{DR}}_{\epsilon }$, is defined as the ratio of the maximum strain to the strain sensitivity. In this article, we assume that the maximum detectable strain is that due to a phase change of $\pi$ radians within one strain resolution (the axial distance over which the strain is estimated). Thus, the maximum measurable displacement, $\mathrm{\Delta }{d}_{\max }$, is equal to $\lambda /4n$. Under this assumption, the strain DR is defined as

The strain sensitivity and DR, as defined above, in combination with the contrast transfer functions detailed in the previous section, will determine the ultimate sensitivity of OCE strain elastograms to variations in elasticity within a sample. We present an analysis of this sensitivity in Sec. 4.3.

3.1.

Phantom Fabrication and Characterization

Tissue-mimicking phantoms with controllable structure and mechanical properties were fabricated using a range of silicone elastomers, namely, combinations of Elastosil® RT601, Elastosil® P7676, and AK50 Silicone Fluid (Wacker, Germany).³⁵ Three types of phantoms were constructed for this study: homogeneous, bilayer, and inclusion. All phantoms were cylindrical with 15-mm diameter and were made $\sim 1\mathrm{mm}$ thick such that OCT could readily image the entire depth of the phantom to facilitate comparison to the FEM. Representative OCT B-scans of each phantom type are shown in Fig. 1. All OCT images have been scaled to physical dimensions using a refractive index of 1.4 for the silicone materials.³⁵ The bilayer phantoms comprise two layers (soft and stiff) of approximately equal thickness. The inclusion phantoms consist of a soft silicone matrix containing a stiff silicone inclusion in the form of a rectangular prism embedded 300 to 350 μm below the surface. The inclusions were cut by hand from a bulk of cured silicone to sizes in the range of 300 to 500 μm. This range was set by the achievable tolerances. (Smaller inclusions than those used here could be fabricated using soft lithography techniques previously used to fabricate 3-D structured phantoms.³⁶) To ensure that layers and inclusions were distinguishable in OCT images, titanium dioxide particles were added to the uncured silicones in concentrations of $0.8\mathrm{mg}/\mathrm{mL}$ for the homogeneous phantoms, matrix materials, and top layers, and $2.5\mathrm{mg}/\mathrm{mL}$ for the inclusions and bottom layers.

Fig. 1

Optical coherence tomography (OCT) B-scans of (a) homogeneous, (b) bilayer, and (c) inclusion phantoms. Scale bar applies to both dimensions. Dashed lines in (b) and (c) indicate feature boundaries.

The Young’s modulus of the cured silicone was controlled by varying the volumetric ratio of crosslinker, catalyst, and noncrosslinking silicone fluid.³⁵ To measure the Young’s modulus of each silicone, stress–strain curves were obtained using standard compression tests (Instron, Norwood, Massachusetts). The Young’s modulus was then estimated by linear regression against the stress-strain curves, as described in further detail in Sec. 3.3.2.

3.2.

Compression OCE System and Measurements

Compression OCE measurements were performed using a fiber-based, Fourier-domain OCT system. The light source is a superluminescent diode with a central wavelength of 835 nm and bandwidth of 50 nm. The measured axial resolution is 8.5 μm. The lens in the sample arm has a working distance of 25 mm and provided a measured lateral resolution of 11 μm. The measured sensitivity is 102 dB at an exposure time of 36 μs. B-scans were acquired with $1000\times 2048$ ($x\times z$) pixels over a $6\times 3\mathrm{mm}$ field of view and at a line rate of 10 kHz.

The sample arm comprises an imaging window fixed to a ring-actuator set-up described previously,^14,15 enabling loading and imaging from the same side. A preload was applied to each phantom using an upper brass plate, of larger surface area than the phantom, to ensure uniform contact between the phantom, the upper plate, and the imaging window (lower plate). The amount of preload required was dependent on the magnitude of variations in the surface topography in each phantom, which arose due to imperfections in the manual fabrication process of the phantoms, and typically ranged between 50 and 150 μm, as measured using OCT. The preload served to ensure optimal transfer of the load from the ring actuator during measurements.

The ring actuator introduced displacements of 80 to 120 nm to the sample surface using a square wave function at 5 Hz, synchronized to the OCT B-scan acquisition (at 10 Hz), as described in Ref. 15. This loading frequency was chosen to remain in the quasi-static loading regime, i.e., low enough to avoid wave propagation in the sample. The local displacement in the sample was measured by taking the phase difference between consecutive B-scans, i.e., between the compressed and uncompressed states. Displacement data were derived from the average of 50 sets of these B-scans. Temporal averaging served to reduce the variance of the measured displacement at each point and was performed in the complex plane in order to reduce systematic underestimation of displacement due to averaging of phase differences obtained at positions with low OCT signal-to-noise ratio (SNR).³⁷ The phase sensitivity of the system in a scanning configuration (calculated as the standard deviation of the measured phase difference between sequential B-scans of a stationary sample, over 50 pairs of B-scans) was 25 mrad at an OCT SNR of 50 dB, corresponding to a displacement sensitivity of 1.2 nm. Discrepancy between the measured sensitivity and the predicted shot noise–limited sensitivity (3.2 mrad at 50 dB)³⁸ is attributed to galvanometer lateral positioning error between B-scans.

The local strain in the sample was estimated from the measured displacements using a weighted-least squares (WLS) algorithm and represented in a strain elastogram. In WLS strain estimation, the measured displacement values are weighted based on the underlying OCT SNR at each point, since the phase variance and, therefore, the displacement measurement accuracy, depends on the OCT signal intensity. (We demonstrated in Ref. 15 that WLS strain estimation improves the strain sensitivity and DR over previously reported strain estimation techniques.) We applied Gaussian smoothing to the strain elastograms, using an $18\times 7\text{-}\mu \text{m}$ window ($\text{width}\times \text{height}$) to further improve strain sensitivity. The axial strain resolution (axial distance over which the slope of axial displacement versus depth is calculated) was 90 pixels, corresponding to a physical length of 92 μm, using a group refractive index of 1.4. The lateral strain resolution was equal to the lateral resolution of the OCT system (11 μm).

3.3.

Finite Element Model

3.3.2.

Determination of model inputs

Inputs to the models of each experiment included:

• • phantom geometry (total thickness, thickness of layers, inclusion dimensions, depth of inclusion below sample surface), as measured using OCT before application of preload;

• • modulus contrast between features, determined from estimates of Young’s modulus from the stress-strain curve of each silicone (further details below);

• • amount of displacement introduced to the sample by the preload, as measured using OCT;

• • amount of nanometer-scale displacement introduced to the sample surface by the actuator; and

• • the friction conditions present in the experiment (further details below).

Compression testing of the bulk silicones typically revealed a nonlinear stress-strain relationship, as illustrated by the representative stress-strain curve in Fig. 2. However, as a relatively small range of strain ($\le 0.12$) was used in OCE experiments, the silicones were modeled as linear elastic. The Young’s modulus of each material was estimated based on the bulk strain due to the preload. For example, in an inclusion sample, the change in thickness of the matrix was measured from OCT images taken before and after preload, from which a bulk strain on the matrix was estimated. The Young’s modulus was estimated by the linear regression from 0 to this bulk strain on the stress-strain curve of the matrix material (See Fig. 2). Similarly, the change in an inclusion thickness due to preload was determined and its modulus approximated using the same fitting procedure on the stress-strain curve of the inclusion material. This same procedure was used to estimate Young’s modulus of the silicones in the bilayer phantom. Table 1 lists the estimated Young’s modulus of each silicone as well as the modulus contrast between features (inclusion:matrix or bottom:top layer) for all phantoms used in this study.

Fig. 2

Representative stress-strain curve obtained from compression testing of the silicones used in phantom fabrication. Inset shows an example of a linear fit (black line) used to estimate Young’s modulus.

Table 1

Bulk mechanical characterization of silicone phantom materials.

Phantom number	Feature	Young’s modulus (kPa)	Modulus contrast
1	Homogeneous	20	N/A
2	Bottom layer	520	$37:1$
	Top layer	14
3	Inclusion	837	$45:1$
	Matrix	19
4	Inclusion	153	$10:1$
	Matrix	15
5	Inclusion	100	$5:1$
	Matrix	20
6	Inclusion	34	$2:1$
	Matrix	17
7	Inclusion	20	$1.1:1$
	Matrix	19

Unless otherwise stated, the coefficients of friction used were 0.1 at the phantom-glass imaging window interface and 0.5 at the phantom-brass plate interface. These values were empirically determined to give the best quantitative matches between the modeled and measured displacements. The difference in the degree of friction between the upper and lower surfaces is expected, as the imaging window is smoother than the brass upper plate, and because silicone fluid (Wacker AK50) was applied to lubricate the phantom-imaging window interface. Exceptions to these conditions are specified, where relevant, in Secs. 4.1 and 4.2. Finally, a Poisson’s ratio of 0.49 was assumed for all silicones.⁴⁰

4.1.

Comparison of Measured and Simulated Deformation in Compression OCE

Figures 3(a)–3(c) present the structural OCT images, along with the measured and simulated displacement maps resulting from compression OCE experiments and FEMs, respectively, for (top to bottom) homogeneous, bilayer, and inclusion phantoms (Phantoms 1 to 3 in Table 1). Figure 3(d) is a comparison of the measured and simulated displacement versus depth taken from the positions indicated by the dashed lines in Figs. 3(b) and 3(c). Silicone fluid was used to lubricate the upper and lower surfaces to minimize friction, and these surfaces were modeled as frictionless in the FEM, in order to simplify the mechanical system in the first instance.

Fig. 3

Comparison of measured and simulated displacement maps in (top to bottom): homogeneous; bilayer; and inclusion phantoms. (a) OCT image; (b) experimentally measured displacement map; (c) simulated displacement map; and (d) displacement versus depth for the lateral positions indicated by the blue and black dashed lines in (b) and (c), respectively.

Qualitatively, the measured and simulated displacement maps show good correspondence, and the displacement traces in Fig. 3(d) verify a good quantitative match. These traces also allow a clear interpretation of the displacement trends in each phantom. In each case, compression was introduced from the top of the image, resulting in maximum displacement at the top surface and zero displacement at the opposite surface, where the phantom was compressed against a rigid, unmoving plate. For the homogeneous case, a linear decrease in displacement with depth is observed, corresponding to a uniform local strain throughout the phantom.

In the bilayer phantom, two slopes are observed, with greater slope, corresponding to higher strain, in the softer top layer. The location of the change in slope corresponds to the physical interface between the layers. The slight discrepancy between the experiment and simulation in the displacement at the layer interface is likely to be due to an imperfect interface between the two silicone layers that arose during curing.

Finally, in the inclusion phantom, large displacement gradients, corresponding to higher strain, are observed in the matrix material, surrounding a region of bulk displacement, or low strain, in the stiff inclusion. The slight discrepancy in the displacement of the inclusion seen here is likely due to a slight discrepancy in the depth at which the inclusion was situated in the model versus in the actual phantom.

4.2.

Accuracy of Strain Elastograms in Representing Elasticity

The agreement between measured and simulated displacement demonstrated in the previous section confirms the validity of the assumption of linear elastic behavior of these materials over the range of employed strains. In this section, we compare measured and simulated strain elastograms for varying geometries, friction conditions, and modulus contrasts. We also use FEMs to simulate the corresponding stress distributions and, thus, aid our analysis of mechanical artifacts in elastograms and their impact on accuracy.

4.2.1.

Nonuniform stress due to friction

As described in Sec. 2.1, axial compression of a soft tissue sample is accompanied by lateral expansion (the Poisson effect). In the ideal case of zero friction between the sample and compressor, the sample is free to slip along this interface and expand laterally under axial compression. This is the case for the phantoms shown in Fig. 3, as the surfaces were well lubricated to minimize friction. When friction is present, this lateral expansion is restricted. This is illustrated in the plot of displacement versus depth, Fig. 4(a), for a homogeneous phantom in which no lubrication was used at the surfaces. The silicone used here, like many soft tissues, is inherently adhesive and is expected to undergo little or no slipping at the boundary in the absence of lubricant. In the experimental elastogram, Fig. 4(b), this restricted motion manifests as bands of low strain at the top and bottom surfaces, which are also seen in the simulated elastogram, Fig. 4(c), for which a no-slip condition was defined at the surfaces in the model.

Fig. 4

Effect of friction on strain and stress distribution in a homogeneous sample. (a) Measured (blue) and simulated (black) displacement versus depth along the central vertical axis of the sample, (b) measured strain elastogram, (c) simulated strain elastogram, and (d) simulated stress map for a homogeneous sample with friction present at both surfaces.

The simulated stress, Fig. 4(d), reveals a nonuniform distribution, with increasing stress toward the center of the sample. This trend is attributed to the increasing restriction on lateral motion of the sample with distance from the free vertical edges.

4.2.2.

Nonuniform stress due to inhomogeneity

Figure 5 presents the experimental and simulated elastograms, and simulated stress distribution of the same bilayer and inclusion phantoms presented in Sec. 4.1. The experimental strain elastograms were generated using the displacement maps in Figs. 3(c) and 3(d).

Fig. 5

Measured strain elastogram, simulated strain elastogram, and simulated stress map, respectively, for (a)–(c) a bi-layer sample with modulus contrast $37:1$ and (d)–(f) an inclusion sample with modulus contrast $45:1$.

The elastograms of the bilayer phantom, Figs. 5(a) and 5(b), show layers of high and low strain, corresponding to the soft and stiff layers, respectively. Importantly, the stress distribution, Fig. 5(c), is uniform in this sample. There are no stress concentrations or strain artifacts apparent at the interface of the two materials. This confirms that the observed strain contrast should match the true modulus contrast between the layers as predicted by the mechanical analysis presented in Sec. 2.1.

In the elastograms of the inclusion phantom, Figs. 5(d) and 5(e), the stiff inclusion is clearly distinguishable as a region of very low strain, and higher strain is observed in the soft matrix. However, the strain in the matrix is not uniform throughout the elastogram even though this is a mechanically uniform material. This is due to variations in stress due to the presence of the stiff inclusion as seen in the simulated stress map in Fig. 5(f). Higher stresses above and below the inclusion manifest as regions of higher strain in the elastogram. In addition, at the vertical edges of the inclusion, a region of high stress is observed, adjacent to a region of lower stress in the matrix material. This can be attributed to a stress-shielding effect, in which the two materials at this location experience similar loads transferred from the material above, but the deformation of the matrix is restricted by that of the stiff inclusion. As the stress at this interface is not effectively distributed to the matrix, the inclusion experiences higher stress. This manifests in the elastograms as regions of low strain in the matrix adjacent to the sides of the inclusion.

As made apparent in Figs. 5(d) and 5(e), strain artifacts degrade the fidelity of the elastogram to the true distribution of modulus. The severity of these artifacts, however, depends on the modulus contrast between the inclusion and matrix. This is demonstrated in Fig. 6, where the modeled stress and strain are shown for Phantoms 3 to 7 listed in Table 1, from highest to lowest modulus contrast (top to bottom). Note that the color scale for the stress maps was adjusted in each case, depending on the preload strain applied to the sample, to facilitate comparison of stress distribution due solely to changes in modulus contrast. The inclusion sizes in each set of images in Fig. 6 correspond to those used in the actual experiments (in the range of 300 to 500 μm in height and width); however, recall that the resulting strain contrast is independent of the inclusion size [Eq. (1)]. This sequence of images illustrates that although the stress concentrations and strain artifacts decrease with decreasing modulus contrast, the contrast between inclusion and background in the elastogram also diminishes. In the following sections, we consider the limits on this contrast imposed by both mechanical and imaging parameters.

Fig. 6

Simulated strain elastograms and stress maps for varying degrees of contrast between inclusion and matrix. Modulus contrasts (inclusion:matrix): (a) $45:1$, (b) $10:1$, (c) $5:1$, (d) $2:1$, (e) $1.1:1$.

4.2.3.

Contrast transfer efficiency

Figure 7 compares, using Eq. (1), the true modulus contrast, $C_t$, and predicted strain contrast, $C_{\mathrm{o}}$, to the strain contrast observed between the inclusion and matrix in the OCE experiments. The curve for the predicted strain contrast was generated for a Poisson’s ratio of 0.49. The average strains in the elastograms were calculated for regions both within the inclusion and in the matrix at least 500 μm away from the inclusion. Fifty strain pixels were used to calculate the average and standard deviation of strain in each feature, all at the same depth in the sample to minimize the effects of decreasing OCT SNR with depth on the standard deviation of strain. Error bars reflect the standard deviation of contrast between the inclusion and matrix. The contrast values are plotted as amplitude ratios on a logarithmic scale, $C_{\mathrm{o}}(\mathrm{dB})=20\log ({\epsilon }_{\text{inclusion}}/{\epsilon }_{\text{matrix}})$, following the convention used in ultrasound elastography studies of contrast.^32,34,41,42

Fig. 7

True modulus contrast, $C_t$ (black dashed line), predicted strain contrast, $C_{\mathrm{o}}$ (blue solid line) from Eq. (1), and experimental strain contrast versus true modulus contrast for a stiff inclusion in a soft matrix. All contrasts are plotted on a 20 log scale.

Note that for modulus contrasts $>10:1$ (20 dB), the observed contrast maintains an approximately constant offset of $\sim 6\mathrm{dB}$ from the true modulus contrast. In other words, the CTE reaches a maximum of just over 0.5 for modulus contrasts of $10:1$ and greater. This can be explained by considering the interaction of the inclusion and matrix under compression. A stiff inclusion resists deformation, and because it is mechanically coupled to the surrounding matrix, this causes a perturbation to the stress and strain in the matrix. This manifests as stress and strain concentrations, as illustrated in Figs. 5 and 6. Increasing inclusion modulus increases these perturbations on the matrix, also illustrated in Fig. 6. Above a certain contrast, the inclusion can be regarded as a rigid body, i.e., it undergoes very little strain; thus, the perturbations it causes to the matrix remain fixed regardless of increases in modulus contrast.³² Low contrast inclusions, which more easily deform with the matrix under compression and cause little perturbation on the matrix strain, will result in strain elastograms that more accurately represent the true mechanical contrast. This reiterates that a fundamental limitation on strain elastogram accuracy, previously described in Refs. 32 and 34, also applies to OCE.

4.3.

Sensitivity of Strain Elastograms to Variations in Elasticity

The CTE limits the accuracy of mechanical contrast in strain elastograms, but to determine the smallest detectable mechanical contrast, i.e., the elastogram sensitivity, we must consider the limitations of the employed imaging system and strain estimation process. Using the definitions for elastogram performance parameters presented in Sec. 2.2, we can estimate the strain sensitivity ($S_{\epsilon }$) and DR for the measurements performed in this article. These values are summarized in Table 2. Note that these values are specific to the phase stability (25 mrad) and wavelength (835 nm) of the employed OCT system, the axial strain resolution (92 μm), and the strain estimation technique (WLS with Gaussian smoothing).

Table 2

Estimated elastogram performance parameters (dB calculated using 20 log).

The reported strain DR of 56 dB is based on a theoretical maximum displacement corresponding to a phase change of $\pi$ radians within one strain resolution (in this case, $1600\mu \epsilon$ over 92 μm). However, the actual maximum displacement used in the experiments corresponded to a phase change of $\pi$ radians over the entire depth of the phantom ($\sim 1\mathrm{mm}$) to avoid the need for any phase unwrapping. This decreases the actual strain DR for these measurements to $\sim 36\mathrm{dB}$ and points to the potential improvement to be gained in the strain DR through implementation of robust phase unwrapping algorithms.

$S_{\epsilon }$ was calculated as the standard deviation of 500 strain pixel values over a $50\times 600\mu \text{m}$ region in an elastogram of a homogeneous phantom, and at a depth with an average OCT SNR of 20 dB. $S_{\epsilon }$ degrades with decreasing OCT SNR. For example, in the same phantom, $S_{\epsilon }$ of 3.7 and $5.6\mu \epsilon$ was calculated at depths of average OCT SNR of 12 and 8 dB, respectively. $S_{\epsilon }$ also depends on the strain axial resolution and is expected to improve with a tradeoff of lower resolution.¹⁵

$S_{\epsilon }$ limits the minimum detectable contrast in strain elastograms, i.e., to distinguish two features in a strain elastogram, the difference in strain between two features must be greater than $S_{\epsilon }$. Figure 8 illustrates this problem of feature detectability. In Figs. 8(a)–8(c), experimental elastograms of inclusion phantoms with decreasing modulus contrast are shown, along with plots, Fig. 8(d), of the simulated (solid lines) and experimental (dotted lines) strain values through the locations indicated by the dashed lines in the elastograms. This plot illustrates the degradation of strain contrast due to noise in the experimental versus simulated strain. Still, the inclusions are readily detectable in each elastogram, including the inclusion with small modulus contrast of $1.1:1$, Fig. 8(c). The measured strains in the inclusion and matrix for this phantom were 90 and $120\mu \epsilon$, respectively, a difference sufficiently greater than the strain sensitivity to make the inclusion readily detectable.

Fig. 8

Impact of strain sensitivity on detectable contrast. Experimental strain elastograms for modulus contrasts (a) $45:1$, (b) $5:1$, and (c) $1.1:1$. (d) Plots of experimental (dotted lines) and simulated (solid lines) strain in the samples at depths specified by the dashed lines in the elastograms.

The strain sensitivity, in combination with the CTE of a particular geometry, ultimately determines the sensitivity to variations in elasticity. For layered samples, as well as for inclusion samples with low modulus contrast between the inclusion and matrix, such as that in Fig. 8(c), the strain contrast closely approximates modulus contrast (CTE close to 1). It follows that the strain sensitivity is approximately equal to the modulus sensitivity in these cases. Considering our best-case scenario, with $S_{\epsilon }$ of $2.4\mu \epsilon$ and maximum strain of $1600\mu \epsilon$, a modulus contrast of ${\epsilon }_{\max }/({\epsilon }_{\max }-S_{\epsilon })$, corresponding to $\sim 1.002:1$, could theoretically be detected in a strain elastogram. In the phantoms shown in Fig. 8, where the matrix material has Young’s modulus $\sim 18\mathrm{kPa}$, this would translate to the ability to detect a change in elasticity as small as 36 Pa.