High-resolution analysis of the mechanical behavior of tissue

Historically, researchers have studied diseases and disease progression from a chemical perspective, mapping cellular pathways, and genetic markers. Recently, efforts have begun to understand tissue behavior by combining chemical analyses from traditional molecular biology research with information about the mechanical behavior of the system.^1,2 However, characterizing tissue is challenging because biomaterials are heterogeneous and viscoelastic. Due to the complexity of viscoelastic material responses, materials science and mechanical engineering communities typically characterize viscoelastic materials at multiple strain rates and analyze both the loading and unloading curves. From these results, they determine the Young's Modulus (elasticity at a single point) and energy absorbed by the material (area between the loading and unloading curves).^3,4

Initial research using existing methods of mechanical characterization to determine the relative stiffness of cancer indicates that tumors are stiffer than their normal tissue counterparts.^5,6 Unfortunately, traditional methods of viscoelastic material analysis like load frames and rheometers do not accurately capture the mechanics of biological materials because the sensors are larger than the length scale of heterogeneity within biological samples. To overcome this limitation, researchers have explored alternative methods to quantify the mechanical properties of biomaterials such as Atomic Force Microscopy (AFM) and sonoelastography.^7,8 For example, atomic force microscopy (AFM) analysis has been conducted on individual cell lines such as HCT-8 colon cancer cells to determine if the metastatic potential can be correlated with the individual cellular mechanical properties based on changes in the stiffness of the growth substrate.⁶ 

While AFM is able to characterize the individual cells, it is not able to capture the heterogeneous structure of the tissue. Single cell and whole tissue measurements provide very different information, as single cell measurements do not include the extracellular matrix and in situ intercellular interactions. However, frequently, the results are directly compared.^1,9 This leads to great variability and confusion in the scientific literature. Sonoelastography can be used to characterize samples within the body, but it has a very low resolution and is thus primarily used for location rather than quantitative measurements.⁸ Several new methods for studying the mechanical behavior of individual cells have been developed.^9,10 The broad adoption of these systems is limited by portability and cost, which creates a specific use case for each system. Therefore, the currently available strategies are not able to meet the requirements of the biological and medical communities.

In the present work, a two-dimensional high-resolution optical polarimetric elastography system for analyzing the mechanical properties of biomaterials is developed and demonstrated. By comparing several different pieces of salmon skeletal muscle from different regions of the same fish, we are able to determine the ability of the optical polarimetric elastography system to reliably detect sub-mm, naturally occurring structures. The loading-unloading curves of several unprocessed salmon skeletal muscle samples with different numbers of collagen membranes are measured at multiple strain rates. The number of collagen membranes was determined by inspection and controlled by selecting different regions of the muscle with different spacings between the membranes. The same dataset is analyzed using two different data analysis approaches: the absolute change in polarization and the energy absorbed by the sample. Five consecutive tests of each condition were performed, and similar results were obtained, verifying system reproducibility. By analyzing the difference in energy absorbed by the samples with and without collagen membranes, the micron scale resolution is demonstrated.

The instrument is based on an optical polarimetric elastography instrument previously used to analyze viscoelastic polymers.¹¹ A polarization maintaining optical fiber is placed under a sample, and the polarization state is recorded as the sample is compressed. While the theoretical operating mechanism is the same, significant modifications have been made to the instrument to improve the portability while automating data analysis and material compression. A schematic of the device used for testing is shown in Fig. 1(a). Instead of using an Instron to apply the compression, a custom compressive stage is created from a linear stage with a motorized micrometer. The sample is located between the baseplate and an aluminum piston, with the optical fiber under the sample and above the baseplate [Fig. 1(a)]. The applied strain is varied by increasing the change in the thickness of the sample (Δh) due to the compression. This value is set directly, using the micrometer (Δz). Therefore, the micrometer directly controls the strain (Δh/h) and the strain rate (Δh/time) applied to the tissue sample and the polarization-maintaining fiber. The time interval remains consistent throughout testing, and therefore, the strain rate is increased to increase the strain.

The 250 μm diameter polarization maintaining (Newport, F-SPS) optical fiber sensor is located under the tissue sample and is connected to a 2 mW 1550 nm laser with an inline polarizer. 1550 nm is chosen based on previous work that studied the different contributions to the signal and the noise in the system.¹¹ To detect the polarization state, the fiber is connected to a polarimeter. Data are taken every 30 ms over a 30 s interval for a total of 1024 data points per loading-unloading curve. The three Stokes parameters are measured using the polarimeter. A representative dataset is shown in Fig. 1(b).

The resulting data can be analyzed to determine the material properties using two different mathematical methods [Fig. 1(b)]. The first approach evaluates the total change in the polarization upon compression of the sample. The biomaterial community commonly uses this strategy for analyzing an absolute signal change.¹² However, for a viscoelastic material, this value is dependent on the strain rate. Additionally, because the value is the difference between two end points, the error can be large. The second approach determines the energy loss within the sample by evaluating the hysteresis in the measurement. Although this approach is also dependent on the strain rate, it provides additional insight into changes in the sample microstructure due to the measurement process. The material science and mechanical engineering communities commonly use this strategy.^3,4

When stress is applied to the optical fiber sensor, the polarization state of the propagating optical field in the fiber defined by the Stokes parameters ⟨s₁, s₂, s₃⟩ changes according to the photoelastic effect, which is detected by the polarimeter. To analyze the results, two parameters are calculated: β and γ. β is the angle of offset between the polarization state as aligned and the fast and slow axis of the fiber. γ is the angle of offset between the alignment of the fiber coupled into the polarimeter and the fast and slow axes of the fiber. These are related to ϕ through the series of transfer matrices

where $Ex$ and $Ey$ are the x and y components of the electric field recorded by the polarimeter and $Ex0$ is the initial state of polarization. The length of the section under compression is defined by l, and the polarization from before and after the compressed section of the fiber is included in δ. The modified fast and slow axes of the fiber are N_f and N_s. These are related to the stressed beat length (L_b) by N_s−N_f = 2π/kL_b. The Young's Modulus (E) is the stress (σ) divided by the strain (ε) of a material at a specific point (E = σ/ε). Therefore, in our measurements, ΔPol is directly proportional to the Young's Modulus. However, since we have the entire compressive loading and unloading curves for our sample, we perform several comparative analyses of the change in material behavior rather than the Young's Modulus at a single point. The full derivation of this equation is published in previous works on optical fiber polarimetry and optical polarimetric elastography.^11,13 This equation enables the change in polarization to be plotted as it is related to the time or strain. Therefore, we can characterize the ΔPol and energy loss of our biomaterials.

The energy absorbed or energy loss is determined by analyzing both the loading and unloading curves. Hysteresis in a loading-unloading curve is indicative of the energy loss by the sample during the entire loading-unloading interval. This value is typically calculated using the following expression:³ 

where U is the energy, σ is the stress, and ε is the strain. In the present work, to determine the relative energy loss to the sample, the fact that polarization is proportional to stress (σ) is used. Therefore, the energy loss is proportional to the area under the unloading curve subtracted from the area under the loading curve [Fig. 1(b)].

Finally, residual strain in a material after a single loading-unloading cycle arises when a material does not return to its initial state. In the experimental data, this strain appears as an offset in the x-axis in the unloading curve.

As a testbed, three samples of fresh raw salmon muscle are used. Salmon is the ideal material because it has well-defined muscle grain that is clearly delineated by naturally occurring collagen membranes, which are more elastic than the bulk muscle tissue.¹⁴ Therefore, the mechanical properties of these samples vary in a defined way on micron length scales. From the same region of the salmon skeletal muscle, samples with different numbers of collagen membranes are identified and tested to determine if differences between these samples can be resolved. It is possible to find three samples from the same muscle with different numbers of collagen membranes because of the natural structure of the muscle as it tapers.

The samples are cut into 9 mm × 9 mm × 5 mm (l × w × h) rectangles with zero, one, or two parallel collagen membranes dividing the sample (Fig. 2). The salmon samples are aligned, and so, the collagen membranes are oriented perpendicular to the baseplate. The samples with zero collagen membranes are used to determine the influence of the intrinsic muscle grain on the signal. N > 20 sample sets were tested, and the results from one sample set are presented. If the collagen membranes are oriented parallel to the fiber sensor and do not interact with the sensor, the results are classified as “parallel” in the subsequent discussion. Similarly, if the collagen membranes are perpendicular to the fiber sensor or cross the sensor, they are classified as “perpendicular.” For the measurements without a membrane, the muscle grain is closely inspected and aligned based on the grain of the muscle.

Each sample is first tested at 10%, 20%, and 30% strain parallel to the collagen membranes. Then, the sample is rotated 90° and tested at 10%, 20%, and 30% strain perpendicular to the collagen membranes. Several complementary studies, including an investigation into the dependence of the generated signal on the sample length, were performed to determine whether this configuration provides the most consistent results. These results are included in the supplementary material.

The results of subsequent analysis of the loading-unloading curves are shown in Fig. 2. Within a measurement, the primary source of error is environmental noise, whereas across multiple measurements, sample degradation begins to play a role, particularly at high strain values. There is high reproducibility between multiple runs with the same parameters across all strain values when ΔPol is used as the detection metric. However, when energy loss is evaluated, this consistency changes at 30% strain. This change is indicative of damage to the salmon muscle sample and most likely arises from the destruction of the microarchitecture of the connective fibers of the muscle. Damage to these muscle fibers impacts the sample elasticity.

The results from a 9 mm × 9 mm × 5 mm sample with no collagen membranes are shown in Figs. 2(a) and 2(b). The results are highly reproducible in five consecutive 30-s compression tests repeated at three different strains with two orientations. At low strain, the magnitude of the signal change in polarization (ΔPol) for both parallel and perpendicular muscle grain orientations is consistent [Fig. 2(a)]. This response indicates that the material is relatively isotropic and elastic across all strain regimes, as would be expected for the salmon muscle fiber.¹⁴ 

When the same dataset is evaluated for the energy loss of the sample, the consistency across all strain rates is lost [Fig. 2(b)]. Notably, at 20% strain in the perpendicular configuration, a slight upward trend is observed, indicating that the material is becoming more viscous. At 30% strain in the perpendicular configuration, this behavior becomes more pronounced. This behavior is not detected using the conventional data analysis method, yet it is a direct indicator that the sample's microarchitecture is being damaged by the measurement method.

The results from a 9 mm × 9 mm × 5 mm sample with a single bisecting collagen membrane are shown in Figs. 2(c) and 2(d). For both methods, there is a noticeable difference in the detected signal for the parallel and perpendicular samples, indicating that both methods detected the presence of the membrane. For low strain, the magnitude of the ΔPol signal change for both parallel and perpendicular orientations is consistent. However, at 20% and 30% strains for the parallel samples, the values within a given measurement exhibit slightly upward trends, indicating that the ΔPol signal detects sample damage during the measurement at moderate and high strains. In contrast, the energy loss approach detects sample damage across strain values and for all sample types.

The results from a 9 mm × 9 mm × 5 mm sample divided by two collagen membranes are shown in Figs. 2(e) and 2(f). There is a notable difference between the parallel and perpendicular sample configurations, indicating that the fiber sensor easily detects the collagen membrane at all strain rates. While an important finding, perhaps more significant is that the energy loss method detects irreversible material damage that occurs at moderate and high strain values that is not detected by the ΔPol method. This finding raises concerns regarding the accuracy of using 30% strain in analyzing biomaterials, which has been the assumed optimum applied strain for determining the mechanical properties in previous theoretical and experimental work.¹² If 30% strain is at the border of the linear elastic region for tissues with a relatively high Young's Modulus, it is likely to be outside of the linear elastic region for tissues like pancreas, kidneys, breast, and brain.

Figure 3(a) summarizes the ΔPol results. Upon comparing the differential polarization change at the maximum value between parallel and perpendicular measurements in all three samples, several trends are apparent. Across sample types, the change in polarization increases as the strain increases. Additionally, the total polarization change decreases as the number of membranes increases. However, the difference in ΔPol for the parallel and perpendicular alignment for a given sample type cannot resolve the difference between the uniform sample, one membrane, and two membranes.

Figure 3(b) summarizes all the energy loss results. The different samples are clearly identifiable unlike the ΔPol results. The energy loss is also strongly dependent on the orientation of the collagen, with the perpendicular configuration experiencing less energy loss. Therefore, if energy loss is used to analyze the data, it is possible to detect the presence of the 500 μm collagen membranes, indicating that we can resolve microstructures that are as small as 500 μm thick. This resolution using unprocessed tissue is unprecedented. The ultimate limit of the system is yet to be determined, but it will depend on the sensor surface area as well as the mechanical behavior contrast between the surrounding tissue and the item of interest.

In conclusion, a rapid, non-destructive polarimetry based, optical elastography method for analyzing biomaterials has been demonstrated using raw unprocessed salmon muscle tissue. By comparing the difference in the mechanical properties of samples with different numbers of intrinsic collagen membranes, we study the impact of heterogeneity on the mechanical properties of biomaterials. The data are highly reproducible across five consecutive compression tests over the 15–30 min testing interval at room temperature. The optical polarization results are analyzed using two different strategies: total signal change and total energy absorbed. It is evident that energy absorbed is a superior metric for measuring the properties of heterogeneous viscoelastic materials with nano- and microscale architectures, such as our salmon skeletal muscle samples. In addition to superior performance, this system has fewer components, making it cheaper and decreasing the setup time, enabling it to be more widely translatable. In the future, this device will be useful in many areas of biological research and clinical practice such as linking measurements of the mechanical properties to molecular approaches currently used to analyze the tissue stiffness.^1,2

See supplementary material for detailed information on the testing setup, loading-unloading curves, and discussion of the impact of the interaction length on the signal.

The authors thank Q. Wang, S. Mumenthaler, V. Sun, M. Anders, S. Liu, L. Fang, B. Hudnut, and L. Lash-Rosenberg. This work was supported by the Office of Naval Research [N00014-11-1-0910].