2.1.

Synthesis of a Benchmark Dataset

Before application to biological datasets, a phantom dataset was synthesized to provide a benchmark for assessing the accuracy of the cellular tracking algorithms. The synthesized dataset was designed to replicate (as closely as possible) actual fluorescence endomicroscope images of displaced cells in tendons, but with a known underlying strain field.

To reproduce the influence of out-of-focus tissue structures, the dataset was generated by superimposing three reference images corresponding to cellular distributions within different tendon structures (Fig. 1 ). The replicated cells were modeled as disks with diameters of 6 pixels (corresponding to a cell diameter of $9\mathrm{\mu }\mathrm{m}$ ), and a mean intercellular spacing of 15 pixels $\left(12\mathrm{\mu }\mathrm{m}\right)$ . Cells within each image layer were assigned fluorescence intensity according to an assumed out-of-focus distance normal to the imaging plane. The structure represented by each image layer was designated a distinct (independent) functional loading axis. The images were incrementally distorted using an affine deformation to create a series of images representing the temporal transition from an undeformed state to a fully deformed state. The deformed images of each substructure were then superimposed at coincident time points to produce a final composite image frame. The in-focus layer, corresponding to higher intensity emissions from cells in the imaging plane, was assigned a purely dilatational strain of ${\epsilon }_x=-5\%$ and ${\epsilon }_y=20\%$ (strains in the local coordinate frame), with a functional loading axis rotated $45\mathrm{deg}$ counterclockwise from the vertical axis of the imaging plane. Out-of-focus layers were assigned similar strains but with the functional axis of each substructure rotated from the vertical axis by 0 and $45\mathrm{deg}$ (clockwise), respectively.

Fig. 1

Schematic representation of phantom dataset generation. Each component image represents an individual tendon substructure with a unique functional axis, with an applied tensile strain (major principal strain) on this axis and a corresponding perpendicular compressive strain (minor principal strain) due to the Poisson effect. Cells on structures that are out of focus appear with lower intensity.

In biological tissues, local variations in tissue stiffness (for instance, tissue weakness associated with injury) are identified by subregions of high or low tissue strains. Since B-spline smoothing of the displacement field can reduce sensitivity to identifying local variations in the strain field, the capacity of the algorithm to identify such localized variation in the engineering strain field was examined. A tissue strain concentration was simulated by distorting the cellular displacement field within a designated radius of a “strain raiser” specified at a discrete point. Here, the strain concentration was designed to represent a weak spot, where the magnitude of tissue distortion is twice that of the surrounding healthy tissue. The strain profile underlying the imposed strain concentration is illustrated graphically in Fig. 2 . To quantify the sensitivity of the method to local variations in the strain field, the radius of the strain concentration was progressively increased until the algorithm was able to uniquely identify it as the region of highest strain.

Fig. 2

Generating a local strain concentration. The “cells” shown here are distributed rectilinearly to better illustrate the imposed distortion. At the center of the defect, the tissue is stretched by a factor of two (40% maximum strain) compared to the surrounding tissue (20% maximum strain). The spatial transition from high strain to normal strain is exacted over a variable defect radius of $N$ pixels.

2.2.

Generating In Vivo Images of Cell Displacement Under a Defined External Load

A single female C3H/HeN mouse, aged 10 weeks, was used in this study. The mouse was anesthetized using an intraperitioneally injected Ketamine-Xylazine mixture. The Achilles tendon of the right hind leg was exposed, and the cells of the tendon (tenocytes) were stained in the elastic region using SYTO 13 green fluorescent nucleic acid stain with an excitation wavelength of $488\mathrm{nm}$ and emission wavelength of $509\mathrm{nm}$ (Invitrogen AG, Basel, Switzerland).

The anesthetized mouse was placed in a custom loading jig (Fig. 3 ) that was designed and constructed to permit the application of a repeatable load to the Achilles tendon. Specifically, the femur and tibia of the mouse were physically constrained using a soft foam pad while a rotational excursion was applied at the ankle joint. As load was applied to the tendon, the relative motions of the stained tenocytes were recorded at 12 frames per second and 5- $\mathrm{\mu }\mathrm{m}$ lateral resolution using a Cell- $\mathrm{Vizio}\circledR$ S-Series endoscopic microimaging system (Mauna Kea Technologies, USA). The $650\text{-}\mathrm{\mu }\mathrm{m}$ objective lens at the tip of the endoscope was held in light contact with the tendon surface, and the confocal imaging depth was adjusted such that the imaging plane fell just beneath the surface of the tendon sheath. The $y$ axis of the imaging plane was adjusted to align with the long axis of the tibia.

Fig. 3

Schematic representation of the experimental setup with fluorescence endomicroscope and an anesthetized mouse on the Achilles tendon stretching jig.

While recording movements of the tenocytes in the exposed tendon (Fig. 4 ), the foot was first held at a neutral posture $\left(0\mathrm{deg}\right)$ and then slowly flexed from 0- to 45-deg dorsiflexion. The foot was then held at 45-deg flexion for a few seconds before being slowly returned to neutral. To investigate reproducibility, the experiment was repeated an additional four times on the same animal at five-minute intervals.

Fig. 4

Sequential images of tendon cells as the tissues are stretched by imposing an angular displacement at the ankle joint. Strains are inferred by measuring the relative change in distance between cells on the same structure.

As described in the following sections, the videos of cellular movements in the imaging plane were analyzed using a cross-correlation-based tracking algorithm that extracts and derives the tissue strain field. To verify the accuracy of the algorithm, as well as the reproducibility of the tendon loading device, cell displacements and the corresponding local strains were manually determined for a readily identifiable reference tendon structure that spanned the width of the imaged field of view.

2.3.

Tracking of Cellular Displacements by Normalized Cross-Correlation

Normalized cross-correlation (NCC) was implemented within custom software designed to track the displacements of fluorescently labeled cells over time. While the theory, merits, and shortcomings of NCC as a method for feature tracking and optical elastography are treated thoroughly in the literature, ^{13, 14, 15, 16, 17} we provide a short synopsis here for the sake of completeness and the interested reader.

Normalized cross-correlation is a robust, nonparametric method well suited for dealing with noisy image data, and is based on the block search of a grayscale pattern from one frame of a video sequence to the next. The mathematical definition of the cross-correlation coefficient $c(u,v)$ is based on a similarity measure and is given by

The concept of NCC as applied to fluorescently labeled cell tracking is illustrated in Fig. 5 , whereby a reference pattern of grayscale values that correspond to cells in frame $i$ is to be located in frame $i+1$ . After the reference template has been designated (our experience is that the template should include at least two partial cells), a search window is created in frame $i+1$ that encompasses all regions to which the template might plausibly move.

Fig. 5

Graphical representation of the principle of normalized cross-correlation. A template is created from a region of interest, including parts of several cells in frame $i$ . In frame $i+1$ , a search matrix (shown here as a white rectangle) is then iteratively examined to find the spatial offset of the best correlated (highest NCC coefficient) subimage match to the template. This procedure is incrementally repeated over the entire recorded sequence to determine the cumulative cellular displacement.

Since NCC is performed on the basis of pixel by pixel comparison, the resulting spatial resolution is limited to whole pixels. Thus, the rounding error may range up to 0.5 pixels. This error can be particularly problematic when the displacement data are later differentiated to generate tissue strain maps.¹⁸ One way to address this issue is by subsampling (interpolating) the raw images before NCC is applied. However, increasing spatial resolution in this manner is associated with exponential increases in computational expense.

A computationally efficient, alternative method for attaining subpixel resolution is to instead perform the interpolation step after the cross-correlation operation has been performed.¹⁹ Once the correlation peak is identified at whole pixel resolution, a Gaussian surface may be fitted to the correlation coefficients in the neighborhood of the whole pixel peak, and the subpixel location of the surface maximum is then used to determine the displacement of the reference template. This concept is illustrated one dimensionally in Fig. 6 , and described mathematically in Eqs. 3, 4, where $u$ and $v$ are the total displacement calculated with subpixel accuracy, and $u_{\mathrm{pix}}$ and $v_{\mathrm{pix}}$ are the calculated whole pixel displacements, as indicated by the peak correlation coefficient $R$ , at pixel location $i$ and $j$ .

Fig. 6

1-D representation of Gaussian interpolation to determine ROI displacement with subpixel accuracy.

2.4.

Mapping Engineering Strains from the Cellular Displacement Field

The ultimate goal of the method is to determine the engineering strain field of loaded soft tissues in the imaging plane. By definition, the strain field is the gradient of the deformation field, and a common approach for determining engineering strain from the displacement field relies on mapping an interpolated mathematical surface over the displacement field and calculating the spatial gradient of that surface.

In the current study, after loading the tendon and tracking the displacements of target cells, the resultant displacement field was fitted with a bicubic spline interpolated surface. As described in Eq. 5, we determine the smoothest B-spline function $f$ that lies within a specified tolerance of the given data points $x\left(i\right)$ and $\widehat{y}\left(i\right)$ , where $x\left(i\right)$ are the Cartesian coordinates of cell $i$ , and $\widehat{y}\left(i\right)$ is the calculated cumulative displacement vector. To help minimize the effects of spurious displacement values, $f$ is determined using weights assigned to each displacement data value according to its associated NCC coefficient $w\left(i\right)$ . The total difference of the calculated function from the given data is measured by summing over $n$ tracked cells.

Since the image plane may not be necessarily aligned with a physiologically relevant coordinate system, tissue strains are best represented using primary and secondary (maximum and minimum) principal strains. The principal strains incorporate both dilatational strains (stretching or compression along the axes of the imaging plane) and distortional (shear) strains. Principal strain magnitudes ${\epsilon }_{\max }$ and ${\epsilon }_{\min }$ and direction are calculated according to Eqs. 6 through 11, where $x$ and $y$ represent the Cartesian coordinates of the image field, $u$ and $v$ represent the calculated displacement field, ${\epsilon }_x$ and ${\epsilon }_y$ are dilatational strains in the imaging plane, ${\gamma }_{xy}$ is the shear strain in the imaging plane, and $\theta$ is the angle between the principal strain vector and the vertical axis of the imaging plane.

3.1.

Synthetic Benchmark Dataset

The synthetic dataset was analyzed to isolate potential errors related to the automatic tracking of cell displacements and the subsequent mathematical reduction of displacements to tissue strains. A known strain field was imposed on the in-focus layer of the synthetic data (maximum and minimum principal strains of 20 and $-5\%$ , respectively, oriented 45% ccw to the vertical imaging axis and including a local strain concentration of peak strain 40%). The apparent strains in the phantom dataset were analyzed by deriving the spatial gradient of the displacement field as approximated by an interpolated B-spline surface.

The accuracy of the algorithm in correctly detecting the local strain field depended heavily on the degree of smoothing applied to the deformation field (Fig. 7 ). An optimal balance between reducing effects of spurious displacement values and maintaining sensitivity to actual differences in the local strain field was achieved with B-spline surface grid spacing set to half of the mean intercellular spacing (here 8 and 16 pixels, respectively). At this setting, the peak strain concentration (40% local strain) was uniquely identifiable as the region of highest strain, albeit with a reduced apparent magnitude (31% calculated). For a strain concentration of this magnitude, the algorithm was able to identify imposed defects of radius 2.5 times the mean intercellular spacing (i.e., 40 pixels). This equates to an experimental equivalent of $50\mathrm{\mu }\mathrm{m}$ or 13% of the image field width of 600 pixels.

Fig. 7

Local engineering strain distributions as dependent on B-spline mesh density. Spline smoothing reduces the effects of spurious displacement values, but also reduces sensitivity in identifying real local variations in the material strain field.

With this B-spline mesh grid setting, the algorithm was able to accurately assess engineering strains in regions distorted by a homogeneous strain field. The uniformly imposed maximum and minimum principal strains of 20 and $-5\%$ were determined from the local strain field to be 19.6 ± 2.5% and $-5.0$ $\pm$ 1.9%, respectively. The reference strain orientation of $45\mathrm{deg}$ counterclockwise from the vertical imaging axis was assessed as $46.6\pm 1.5\mathrm{deg}$ .

3.2.

Biological Dataset

A series of five experiments were performed on the Achilles tendon of an anesthetized mouse. In vivo tissue strains were inferred from endomicroscope video of tenocytes moving in response to an applied external load. Reproducibility of the tendon loading protocol was assessed using gross structural tendon strain. This structural strain was determined manually by measuring the distance between tenocytes located at distant ends of a prominent superficial tendon structure that spanned the entire field of view (as illustrated in Fig. 4). Over the five trials, manually measured engineering strain in the reference tendon structure was determined to be $20.3\pm 3.1\%$ , indicating that the externally induced tendon load was reproducible.

To gain insight into the heterogeneity of the local tendon strain field, tissue strains were calculated using the algorithm described in the previous sections. With algorithm settings optimized based on the results of phantom testing, the cellular deformation field was mapped using a B-spline mesh grid with a spacing of half the mean intercellular distance. Local strains estimated by the tracking algorithm ranged from 5 to 55%, indicating a highly inhomogeneous tissue strain state in the imaging plane (Fig. 8 ).

Fig. 8

(a) Full-field results of tracked 2-D cell displacements. (b) The resulting principal engineering strain distribution.

Cross-referencing of the cellular displacement field as tracked by the algorithm with manual inspection of the raw video confirmed that the algorithm was able to correctly track cellular displacements, indicating that the large variation in local strain did not primarily arise from tracking error. Further, visual inspection confirmed that the imaged tendon structure was complex, with both coplanar and stratified tissue substructures that alternately loaded and unloaded according to the degree of ankle rotation. While only a superficial region of the tendon was thoroughly analyzed with respect to local tissue strains, it was apparent that more superficial layers of tendon were generally stretched along the long axis of the tibia, with deeper tissue layers moving in various directions deviating from the tendon loading axis. Thus, these results revealed a complicated distribution of internal loads among the constituent structures, and indicated limitations to the assumption that all observed cells in the focal layer resided on a single structure.