Simultaneous Wide-Field Planar Strain–Fiber Orientation Distribution Measurement Using Polarized Spatial Domain Imaging

Abstract

In the present study, we demonstrate that soft tissue fiber architectural maps captured using polarized spatial frequency domain imaging (pSFDI) can be utilized as an effective texture source for DIC-based planar surface strain analyses. Experimental planar biaxial mechanical studies were conducted using pericardium as the exemplar tissue, with simultaneous pSFDI measurements taken. From these measurements, the collagen fiber preferred direction $\theta _{\text {p}}$ was determined at the pixel level over the entire strain range using established methods (https://doi.org/10.1007/s10439-019-02233-0). We then utilized these pixel-level $\theta _{\text {p}}$ maps as a texture source to quantify the deformation gradient tensor ${\mathbf {F}}({\mathbf {X}},t)$ as a function of spatial position ${\mathbf {X}}$ within the specimen at time t. Results indicted that that the pSFDI approach produced accurate deformation maps, as validated using both physical markers and conventional particle based method derived from the DIC analysis of the same specimens. We then extended the pSFDI technique to extract the fiber orientation distribution $\Gamma (\theta ,{\mathbf {X}},t)$ as a function of ${\mathbf {F}}({\mathbf {X}},t)$ from the pSFDI intensity signal. This was accomplished by developing a calibration procedure to account for the optical behavior of the constituent fibers for the soft tissue being studied. We then demonstrated that the extracted $\Gamma (\theta ,{\mathbf {X}},t)$ was accurately computed in both the referential (i.e. unloaded) and deformed states. Moreover, we noted that the measured $\Gamma (\theta ,{\mathbf {X}},t)$ agreed well with affine kinematic deformation predictions. We also demonstrated this calibration approach could also be effectively used on electrospun biomaterials, underscoring the general utility of the approach. In a final step, using the ability to simultaneously quantify ${\mathbf {F}}({\mathbf {X}},t)$ and $\Gamma (\theta ,{\mathbf {X}},t)$, we examined the effect of deformation and collagen structural measurements on the measurement region size. For pericardial tissues, we determined a critical length of $\sim$ 8 mm wherein the regional variations sufficiently dissipated. This result has immediate potential in the identification of optimal length scales for meso-scale strain measurement in soft tissues and fibrous biomaterials.

Appendix

Appendix 1: Surface Strain Analysis

From the post-processed imaging data, DIC analysis was carried out using the open-source software $\mu$DIC.⁴⁶ In brief, $\mu$DIC takes pixel-level measurements of the gray-scale intensities of a reference image and registers their positions to a finite element surface (Fig. 11). B-splines are used to discretize the deformation field into a bi-directional grid of ${\mathbf {m}} \times {\mathbf {n}}$ control points. The control points are used to manipulate the spline field and the coordinates ${\mathbf {x}}(u,v)$ on the B-spline surface are

$$\begin{aligned} \mathbf{x }(u,v) = \sum _{i=0}^{m}\sum _{j=0}^{n} \, N_{i,p}(u) N_{j,q}(v) \, \mathbf{m }_{ij} \end{aligned}$$

(15)

where $N_{i,p}$ and $N_{j,q}$ are the B-spline basis functions, $u,v \in [0, 1]$ are the spline coordinates, p, q denote the polynomial order, ${\mathbf {m}}_{ij}$ are the control points’ coordinates. The resulting 2D B-spline surface convects (deforms) and the mapped $I_{\text {pSFDI}}({\mathbf {X}})$ is compared with the deformed image. A modified Newton scheme was then used to minimize the sum of squared differences between the reference and current images.

This method produced pixel-level resolution spatial maps of ${\mathbf {F}}({\mathbf {X}})$ for each applied deformation step. To allow direct comparison to the physical marker (Fig. 1) ${\mathbf {F}}$ results, the positions of each of the four ROI corners for each loading step (Fig. 11) were extracted and used to compute an equivalent ’virtual’ ${\mathbf {F}}$. Finally, to thoroughly validate pSFDI as a texture source for DIC-based strain measurement, we conducted extensive synthetic deformation validation studies. Reference images of both the pSFDI and particle textures were synthetically deformed in relevant modes. Details of the synthetic testing methods and results can be found in the Appendix.

Figure 11

Schematic of the finite element surface of µDIC in the (a) reference and (b) deformed states showing the displacement of the measurement points (red dots) within elements (black squares) as the control points (line intersections) are moved.

Appendix 2: Synthetic Deformation Validation

Overview

This section covers the synthetic image testing that was used to confirm that the pSFDI texture source could be utilized to carry-out DIC analysis. Reference images of soft-tissue specimens, one with a physically applied particle texture and another imaged with pSFDI were synthetically deformed in modes that were relevant to the tissue testing performed on the specimens. The synthetic deformations were administered by applying a known F and the effectiveness of the texture for the application of DIC analysis was verified by the successful extraction of synthetic F. The synthetic deformations were both homogeneous (uniaxial stretch, biaxial stretch, simple shear, pure shear, subsimple shear) and heterogeneous(uniaxial and biaxial sinusoidal displacement fields). Rigid body transformations were also applied (translation and rotation).

Software Pipeline

An in-house code was used to generate the synthetic image sets that operates as follows; The type of synthetic deformation is selected, then the number of deformation steps and their individual extent is chosen. Once the synthetic images have been generated, they are passed to the µDIC solver where an ROI and interpolation degree are applied by the user before finally, after DIC analysis, the individual deformation gradient component tensor fields are outputted (for each synthetic deformation step) as a spatial contour map, the displacement field is outputted as a vector plot and the average deformation gradient tensor components at each step are outputted (Fig. 12).

Figure 12

Software pipeline for synthetic image validation.

Rigid Body Motion

Rigid body displacement and rotation (Fig. 13) synthetic image sequences were generated and their direction and magnitudes were successfully ascertained by the DIC analysis. Moreover, both textures produce identical results, which is consistent as the same translation was applied in each case.

Figure 13

Displacement fields (black arrows) measured from (a) particle textured and (b) pSFDI($\uptheta$_p) textured image sets with a 10 pixel rigid body displacement in the $X_1$ direction synthetically applied. Measured rigid body rotation data for (c) particle and (d) pSFDI($\uptheta$_p).

Stretch and Shear

Next, synthetic stretch and synthetic simple shear images were produced from both the particle texture (Fig. 14) and the pSFDI($\uptheta$_p) reference images (Fig. 15). For the homogeneously deformed synthetic images, it can be seen that the measured deformation was well within 0.5 % of the the applied deformation.

Figure 14

Synthetically applied (a) F₁₁ unidirectional stretch, (b) F₁₂ simple shear, (c) F₂₁ simple shear and (d) F₂₂ unidirectional stretch simple shear generated from an image of the particle textured surface. The measured deformation gradient field is appended to the image and show an excellent agreement between the applied deformation and the measured deformation.

Figure 15

Synthetically applied (a) F₁₁ unidirectional stretch, (b) F₁₂ simple shear and (c) F₂₁ simple shear and (d) F₂₂ unidirectional stretch simple shear generated from an image of the pSFDI textured surface. The measured deformation gradient field is appended to the image and show an excellent agreement between the applied and the measured deformations.

Heterogeneous Deformation Modes

Finally, sinusoidal deformation fields were applied to the reference images to test the ability of the DIC software’s tracking of heterogeneity. These were applied in a single direction (Figs. 16 and 17) and bidirectionally (Figs. 18 and 19). It can be seen that in each case, the undulating deformation is followed effectively by µDIC, using both textures.

Figure 16

(a) A plot of the measured F₁₁ deformation gradient component (red dots) co-plotted with the equivalent synthetic data (black line). (b) Measured deformation gradient component fields for a synthetically applied strip-biaxial deformation of the particle textured reference image.

Figure 17

(a) A plot of the measured F₁₁ deformation gradient component (red dots) co-plotted with the equivalent synthetic data (black line). (b) Measured deformation gradient component fields for a synthetically applied strip-biaxial deformation of the pSFDI($\uptheta$_p) textured reference image.

Figure 18

(a) Measured deformation gradient component fields for a synthetically applied bidirectional sinusoidal deformation of the particle textured reference image. (b) A plot of the measured F₁₁ deformation gradient component (red dots) co-plotted with the equivalent synthetic data (black line).

Figure 19

(a) A plot of the measured F₁₁ deformation gradient component (red dots) co-plotted with the equivalent synthetic data (black line). (b) Measured deformation gradient component fields for a synthetically applied bidirectional sinusoidal deformation of the pSFDI($\uptheta$_p) textured reference image.