## Abstract

Collagen fibers are the primary structural elements that define many soft-tissue structure and mechanical function relationships, so that quantification of collagen organization is essential to many disciplines. Current tissue-level collagen fiber imaging techniques remain limited in their ability to quantify fiber organization at macroscopic spatial scales and multiple time points, especially in a non-contacting manner, requiring no modifications to the tissue, and in near real-time. Our group has previously developed polarized spatial frequency domain imaging (pSFDI), a reflectance imaging technique that rapidly and non-destructively quantifies planar collagen fiber orientation in superficial layers of soft tissues over large fields-of-view. In this current work, we extend the light scattering models and image processing techniques to extract a critical measure of the degree of collagen fiber alignment, the normalized orientation index (*NOI*), directly from pSFDI data. Electrospun fiber samples with architectures similar to many collagenous soft tissues and known *NOI* were used for validation. An inverse model was then used to extract *NOI* from pSFDI measurements of aortic heart valve leaflets and clearly demonstrated changes in degree of fiber alignment between opposing sides of the sample. These results show that our model was capable of extracting absolute measures of degree of fiber alignment in superficial layers of heart valve leaflets with only general *a priori* knowledge of fiber properties, providing a novel approach to rapid, non-destructive study of microstructure in heart valve leaflets using a reflectance geometry.

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## References

Allen, A. C., E. Barone, O. Cody, K. Crosby, L. J. Suggs, and J. Zoldan. Electrospun poly(

*N*-isopropyl acrylamide)/poly(caprolactone) fibers for the generation of anisotropic cell sheets.*Biomater. Sci.*5:1661–1669, 2017.Amoroso, N. J., A. D’Amore, Y. Hong, W. R. Wagner, and M. S. Sacks. Elastomeric electrospun polyurethane scaffolds: the interrelationship between fabrication conditions, fiber topology, and mechanical properties.

*Adv. Mater.*23:106–111, 2011.Ayoub, S., K. C. Tsai, A. H. Khalighi, and M. S. Sacks. The three-dimensional microenvironment of the mitral valve: insights into the effects of physiological loads.

*Cell. Mol. Bioeng.*11:291–306, 2018.Bodenschatz, N., P. Krauter, A. Liemert, J. Wiest, and A. Kienle. Model-based analysis on the influence of spatial frequency selection in spatial frequency domain imaging.

*Appl. Opt.*54:6725–6731, 2015.Bohren, C. F., and D. R. Huffman. Absorption and Scattering of Light by Small Particles. New York: Wiley, 2008.

Chenault, D. B., and R. A. Chipman. Measurements of linear diattenuation and linear retardance spectra with a rotating sample spectropolarimeter.

*Appl. Opt.*32:3513–3519, 1993.Courtney, T., M. S. Sacks, J. Stankus, J. Guan, and W. R. Wagner. Design and analysis of tissue engineering scaffolds that mimic soft tissue mechanical anisotropy.

*Biomaterials*27:3631–3638, 2006.Cuccia, D. J., F. Bevilacqua, A. J. Durkin, F. R. Ayers, and B. J. Tromberg. Quantitation and mapping of tissue optical properties using modulated imaging.

*J. Biomed. Opt.*14:024012–024013, 2009.Cuccia, D. J., F. Bevilacqua, A. J. Durkin, and B. J. Tromberg. Modulated imaging: quantitative analysis and tomography of turbid media in the spatial-frequency domain.

*Opt. Lett.*30:1354–1356, 2005.Cuccia, D. J., F. Bevilacqua, A. J. Durkin, and B. J. Tromberg. Depth-sectioned imaging and quantitative analysis in turbid media using spatially modulated illumination. In:

*Biomedical Topical Meeting*. Optical Society of America, 2004, p. FF5.D’Amore, A., J. A. Stella, W. R. Wagner, and M. S. Sacks. Characterization of the complete fiber network topology of planar fibrous tissues and scaffolds.

*Biomaterials*31:5345–5354, 2010.Deitzel, J., J. Kleinmeyer, D. Harris, and N. B. Tan. The effect of processing variables on the morphology of electrospun nanofibers and textiles.

*Polymer*42:261–272, 2001.Doshi, J., and D. H. Reneker. Electrospinning process and applications of electrospun fibers. In:

*Industry Applications Society Annual Meeting, 1993. Conference Record of the 1993 IEEE.*IEEE, 1993, pp. 1698–1703.Eckert, C. E., R. Fan, B. Mikulis, M. Barron, C. A. Carruthers, V. M. Friebe, N. R. Vyavahare, and M. S. Sacks. On the biomechanical role of glycosaminoglycans in the aortic heart valve leaflet.

*Acta biomater.*9:4653–4660, 2013.Fratzl, P. Collagen: structure and mechanics, an introduction. In:

*Collagen*. New York: Springer, 2008, pp. 1–13.Gelse, K., E. Pöschl, and T. Aigner. Collagens—structure, function, and biosynthesis.

*Adv. Drug Deliv. Rev.*55:1531–1546, 2003.Ghosh, N., and I. A. Vitkin. Tissue polarimetry: concepts, challenges, applications, and outlook.

*J. Biomed. Opt.*16:110801–11080129, 2011.Ghosh, N. , I. A. Vitkin, and M. F. Wood. Mueller matrix decomposition for extraction of individual polarization parameters from complex turbid media exhibiting multiple scattering, optical activity, and linear birefringence.

*J. Biomed. Opt.*13:044014–044036, 2008.Gilbert, T. W., S. Wognum, E. M. Joyce, D. O. Freytes, M. S. Sacks, and S. F. Badylak. Collagen fiber alignment and biaxial mechanical behavior of porcine urinary bladder derived extracellular matrix.

*Biomaterials*29:4775–4782, 2008.Gioux, S., A. Mazhar, D. J. Cuccia, A. J. Durkin, B. J. Tromberg, and J. V. Frangioni. Three-dimensional surface profile intensity correction for spatially modulated imaging.

*J. Biomed. Opt.*14:034045, 2009.Goth, W., J. Lesicko, M. S. Sacks, and J. W. Tunnell. Optical-based analysis of soft tissue structures.

*Annu. Rev. Biomed. Eng.*2016. https://doi.org/10.1146/annurev-bioeng-071114-040625.Guo, X., M. F. Wood, and A. Vitkin. A Monte Carlo study of penetration depth and sampling volume of polarized light in turbid media.

*Opt. Commun.*281:380–387, 2008.Holzapfel, G. A. Biomechanics of soft tissue.

*Handb. Mater. Behav. Models*3:1049–1063, 2001.Hotaling, N. A., K. Bharti, H. Kriel, and C. G. Simon. DiameterJ: a validated open source nanofiber diameter measurement tool.

*Biomaterials*61:327–338, 2015.Hulst, H. C., and H. Van De Hulst. Light Scattering by Small Particles. Mineola: Courier Dover Publications, 1957.

Jacques, S. L., and J. C. Ramella-Roman.

*Polarized Light Imaging of Tissues*. Royal Society of Chemistry, 2004, pp. 591–607.Jammalamadaka, S. R., and A. Sengupta. Topics in Circular Statistics. Singapore: World Scientific, 2001.

Joyce, E. M., J. Liao, F. J. Schoen, J. E. Mayer, Jr., and M. S. Sacks. Functional collagen fiber architecture of the pulmonary heart valve cusp.

*Ann. Thorac. Surg.*87:1240–1249, 2009.Kemp, N., H. Zaatari, J. Park, H. G. Rylander, III, and T. Milner. Form-biattenuance in fibrous tissues measured with polarization-sensitive optical coherence tomography (PS-OCT).

*Opt. Express*13:4611–4628, 2005.Liu, B., M. Harman, S. Giattina, D. L. Stamper, C. Demakis, M. Chilek, S. Raby, and M. E. Brezinski. Characterizing of tissue microstructure with single-detector polarization-sensitive optical coherence tomography.

*Appl. Opt.*45:4464–4479, 2006.Lu, S.-Y., and R. A. Chipman. Interpretation of Mueller matrices based on polar decomposition.

*JOSA A*13:1106–1113, 1996.Mark, J. E. Physical Properties of Polymers Handbook. New York: Springer, 2007.

Martin, C., and W. Sun. Biomechanical characterization of aortic valve tissue in humans and common animal models.

*J. Biomed. Mater. Res. A*100:1591–1599, 2012.Mega, Y., M. Robitaille, R. Zareian, J. McLean, J. Ruberti, and C. DiMarzio. Quantification of lamellar orientation in corneal collagen using second harmonic generation images.

*Opt. Lett.*37:3312–3314, 2012.Misfeld, M., and H.-H. Sievers. Heart valve macro- and microstructure.

*Philos. Trans. R. Soc. Lond. B*362:1421–1436, 2007.Oppenheim, A. V., and R. W. Schafer. Discrete-Time Signal Processing. Upper Saddle River: Prentice Hall, pp. 86–87, 1989.

Parry, D. A. The molecular fibrillar structure of collagen and its relationship to the mechanical properties of connective tissue.

*Biophys. Chem.*29:195–209, 1988.Qi, J., and D. S. Elson. Mueller polarimetric imaging for surgical and diagnostic applications: a review.

*J. Biophotonics*10:950–982, 2017.Sacks, M. S. Incorporation of experimentally-derived fiber orientation into a structural constitutive model for planar collagenous tissues.

*J. Biomech. Eng.*125:280–287, 2003.Sacks, M. S., D. B. Smith, and E. D. Hiester. A small angle light scattering device for planar connective tissue microstructural analysis.

*Ann. Biomed. Eng.*25:678–689, 1997.Samuels, R. J. Small angle light scattering from deformed spherulites. Theory and its experimental verification.

*J. Polym. Sci. C*1966. https://doi.org/10.1002/polc.5070130105.Stella, J. A., and M. S. Sacks. On the biaxial mechanical properties of the layers of the aortic valve leaflet.

*J. Biomech. Eng.*129:757–766, 2007.Stoller, P., K. M. Reiser, P. M. Celliers, and A. M. Rubenchik. Polarization-modulated second harmonic generation in collagen.

*Biophys. J.*82:3330–3342, 2002.Sun, M., H. He, N. Zeng, E. Du, Y. Guo, S. Liu, J. Wu, Y. He, and H. Ma. Characterizing the microstructures of biological tissues using Mueller matrix and transformed polarization parameters.

*Biomed. Opt. Express*5:4223–4234, 2014.Tower, T. T., M. R. Neidert, and R. T. Tranquillo. Fiber alignment imaging during mechanical testing of soft tissues.

*Ann. Biomed. Eng.*30:1221–1233, 2002.Van Krevelen, D. W., and K. Te Nijenhuis. Properties of Polymers: Their Correlation with Chemical Structure; Their Numerical Estimation and Prediction from Additive Group Contributions. Amsterdam: Elsevier, 2009.

Wiest, J., N. Bodenschatz, A. Brandes, A. Liemert, and A. Kienle. Polarization influence on reflectance measurements in the spatial frequency domain.

*Phys. Med. Biol.*60:5717, 2015.Yang, B., J. Lesicko, M. Sharma, M. Hill, M. S. Sacks, and J. W. Tunnell. Polarized light spatial frequency domain imaging for non-destructive quantification of soft tissue fibrous structures.

*Biomed. Opt. Express*6:1520–1533, 2015.York, T., L. Kahan, S. P. Lake, and V. Gruev. Real-time high-resolution measurement of collagen alignment in dynamically loaded soft tissue.

*J. Biomed. Opt.*19:066011, 2014.Zhou, W.-S., and X.-Y. Su. A direct mapping algorithm for phase-measuring profilometry.

*J. Mod. Opt.*41:89–94, 1994.

## Acknowledgments

This work was supported by funding from the National Heart, Lung, and Blood Institute of the National Institutes of Health (awards RO1-HL108330 and RO1-HL129077), the National Institute of Biomedical Imaging and Bioengineering of the National Institutes of Health (Award T32-EB007505), and the Cancer Prevention and Research Institute of Texas (Award RP-130702). The authors would also like to thank Mason Dana for his contributions to data collection and instrumentation troubleshooting, and acknowledge the Microscopy and Imaging Facility of the Institute for Cellular and Molecular Biology at The University of Texas at Austin for use of their electron microscope facilities. There are no conflicts of interest from financial or other commercial benefits related to the development of this manuscript.

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## Appendices

### Appendix A

The full derivation of our polarized light model begins from Eq. (4):

The initial Stokes vector describing the incident light (\(\vec{S}_{\text{in}}\)), along with the Mueller matrix components representing the polarizer (*M*_{p}) and rotational transformations (*R*_{p}), are defined as follows:

The Mueller matrix for the sample (*M*_{s}) is given as the special case scattering *T*-matrix derived for normally incident light scattering from infinitely long cylinders:

The full solution for the *T*-matrix elements, along with efficient computational algorithms, has been described extensively by Bohren and Huffman.5 The inputs required to solve for *T*_{11}, *T*_{12}, *T*_{33}, and *T*_{34} are the relative refractive index of the cylinder and the medium (*m*), the size parameter (*x*), and the system collection angles (*ψ*). Plugging (A1)–(A4) into Eq. (4) can be shown to simplify to:

(A5) shows that the intensity response detected by the camera is now entirely dependent on the linear polar response, and the Stokes vector can therefore be collapsed into Eq. (6).

### Appendix B

To allow more rapid fitting, a modified but mathematically identical form of Eq. (6) is used. Each sinusoidal term includes a non-linear phase offset. For linearized fitting, it is transformed using the identity \(a \cdot \sin (\theta ) + b \cdot \cos (\theta ) = c \cdot \cos (\theta + \varphi ),\) where \(c = \sqrt {a^{2} + b^{2} }\) and \(\varphi = a\tan 2(a,\;b)\). This results in a Fourier expansion form of Eq. (6):

In this form, a linearized representation of the reflectance is *I* = *Sb*, where *I* is the detected reflectance intensity, *S* is the Fourier expansion representation of the model in (A6), and ** b** is a vector containing the five transformed model coefficients from (A6). Solving this system of equations by

**=**

*b**S*\

*I*allows extraction of the coefficients by Gaussian elimination (Matlab function

**). Subsequently, a 1 s fitting time was achieved for a 1.5-megapixel image, compared to several hours with the**

*mldivide***fitting algorithms for the original equation containing a non-linear phase offset term. After fitting, the original form of the model coefficients and phase offset were recovered using the same identities.**

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Goth, W., Potter, S., Allen, A.C.B. *et al.* Non-Destructive Reflectance Mapping of Collagen Fiber Alignment in Heart Valve Leaflets.
*Ann Biomed Eng* **47**, 1250–1264 (2019). https://doi.org/10.1007/s10439-019-02233-0

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DOI: https://doi.org/10.1007/s10439-019-02233-0