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# Journal Club Theme of March 2014: 3D Experimental Measurements in Soft Materials

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**3D Experimental Measurements in Soft Materials - from measuring cell displacement fields to complex hierarchical material deformations and beyond**

**The Why? - Why do we need 3D material deformation measurement?**

This is an excellent question. If material surface deformation field descriptions are of prime interested, then Digital Image Correlation and interferometry techniques provide the user with a mature field available experimental approaches that might fit the bill [1]. However, if the material of interest presents itself with an internally non-homogenous, non-periodic or simply 3D microstructure, 3D internal deformation measurement techniques might be the only way to investigate the material’s deformation, and ultimately its constitutive behavior. One particularly important area where 3D local measurements are of great interest is in soft hierarchical biomaterials, or materials that mammalian cells interact with. The last two decades have highlighted the importance of the physical interactions between cells and their mechanical environment in controlling many aspects of human health from wound healing to cancer [2-7].

Investigations by our lab and others have highlighted the 3D nature of cellular deformation fields, and the need for experimental techniques being capable of both spatially and temporally resolving them [8-14]. Cellular deformation fields are highly localized, non-linear and as we recently reported finite (i.e. large strains) in character [15].

In a nutshell, cellular deformation signatures are not too different from deformation fields that can occur in locally anisotropic and heterogeneous or hierarchical materials. Given today’s capability to print such intricate hierarchical material structures via 3D printing or 2-photon lithography, understanding their constitutive behavior at a particular length scale will be important, which may necessitate the availability of quantitative 3D imaging and analysis modalities, such as DVC, that can capture these deformation fields.

**The What? Measuring 3D internal deformation fields using Digital Volume Correlation**

Digital volume correlation (DVC) is simply the volumetric extension of the popular surface displacement mapping technique of digital image correlation (DIC). While DIC is generally performed on a set of 2D images, DVC requires the usage of volumetric data, such as typically is acquired during a computer tomography (CT), magnetic resonance imaging (MRI) or laser scanning confocal microscopy (LSCM) scan. Since DVC is indeed the 3D, volumetric extension of DIC, the mathematical formulation of DVC can be derived using the same fundamental minimization equations used in DIC. This includes the same rules of thumb that apply to DIC, apply to DVC, e.g., speckles should be non-repetitive, of high contrast and isotropic, and typically between 2 - 4 pixels in size. Given the popularity of DIC in experimental mechanics, one might wonder why DVC has not quite gained the same momentum (although it is picking up speed very recently).

**DVC Grand Challenges**

The first challenge associated with DVC is the imaging modality. Although micro-CT, and confocal microscopes becoming more wide-spread, their total numbers are nowhere near CCDs and simpler 2D camera setups. The second major hurdle in moving DVC into the mechanics mainstream is to make it computationally cost-efficient, i.e., allowing a volume-by-volume frame analysis on the order of seconds to minutes.

Straight-up numerical implementation of the general homogenous deformation mapping approach as utilized in DIC was computationally prohibitive until very recently. Instead, historically, material deformation fields were approximated by localized rigid body motion only, which limited the amount of deformation a local DVC subset could undergo [16-19]. Nevertheless, this approximation was necessary to yield tractable computation times per deformation increment on the order of 30 - 60min or less. While ignoring local higher order terms would produce small errors for small or “gradual” material deformation gradients, as present in uniaxial stress-strain states for example, it can have a significant effect on the accuracy of the calculated displacements fields when large rotations and stretches are present [16, 18].

**Addressing these Challenges**

In 2011 Gates et al. addressed these shortcomings by presenting a complete, 12 degree of freedom DVC algorithm derived from the same mathematical principles as DIC. To handle the issue of computational cost the authors utilized high performance parallel computing, producing average run times just under six hour for 393 voxel correlation grids with interrogation window sizes 413 voxels [20]. Overall this was a significant improvement in terms of the spatial resolution capability for a DVC algorithm.

To further close the gap, and with the hope of providing virtually any user in our community with access to a full 12 DOF DVC algorithm, we recently developed a new DVC algorithm that is capable of capturing any general non-linear finite deformation with computation times on the order of 1-2min, termed fast iterative DVC, or FIDVC [8]. The advantage of the FIDVC algorithm is that it can be fairly easily implemented on a $300 graphics card GPU, allowing it to be run on any medium-sized personal computer with a graphics cards rather than on a high-performance computer or cluster.

The main principle behind the proposed algorithm lies in an iterative mapping approach similar to schemes widely used in the particle image velocimetry (PIV) community. Similar to our previous DVC algorithm the proposed method is based on the standard cross-correlation formulation with the addition of an iterative image deformation method (IDM). This deformation method is applied to both deformed and undeformed volumetric images, which significantly improves the analysis accuracy, and since it utilizes the cross-correlation formulation at its core it can be represented by its Fast Fourier Transform (this is where the computation-savings come in).

One of the main advantages of the method is the multi-step interrogation window refinement during the iteration process allowing for the accurate capture of higher spatial frequencies including localized large displacement gradients. This allows the algorithm to take full advantage of the maximum intensity content in each image, reducing the local window subset sized below the usual 64 voxel size. For most of our confocal based images we can go as low as 32 voxels, but if the speckle intensity was even higher, one could go even lower.

Much more about our FIDVC method, its specifications, performance and implementation can be found in our paper [8] in the upcoming June special issue in Experimental Mechanics along with several other excellent DVC papers. Also, we are currently working on making our code freely available for download from our website (franck.engin.brown.edu), which we hope to post very soon.

**What’s next?**

Currently, most of our 3D DVC (or FIDVC) measurements pertain to analyzing the motility and deformation behavior of cells, in particular neurons in the brain and white blood cells during infection fighting. What was particularly interesting to us was the observation that with the new, FIDVC algorithm, we observed that cells are actually capable of generating finite material deformations, which previously had never been reported [15]. We think the reason lies in the highly non-linear, approximate Gaussian displacement field signature that cells generate. When such deformations were analyzed with our previous DVC algorithm, or other similar zeroth order algorithms, we only observed small, linear material strains, whereas the FIDVC clearly presents finite cell-generated material deformations. More detail can be found in our recent Plos One paper [15].

We also started using this algorithm to investigate cell-material interaction in hierarchical materials like collagen. Here we observe even larger cell-generated strains and displacement gradients. We hope to share some of those results with you soon.

In conclusion, I hope to have provided a brief overview of Digital Volume Correlation and what type of material application studies such an algorithm might be useful for (including some of the work we do in our lab). Besides the literature on confocal microscopy that uses visible wavelengths for imaging and DVC studies, there’s a steadily growing DVC literature on CT studies using X-rays, from which DVC actually originated from, which is very impressive [16, 17, 21-24].

**What other resources might be useful?**

First, I’d like to point the interested reader to some of the references I have cited here. Of course there’s a lot more great studies in the literature, and I’d be happy to send them along upon request.

Lastly, I would like to point folks interested in DIC and DVC to the Downloads section of our webpage (franck.engin.brown.edu) where you can find DIC and DVC simulators. These simulators are very simple image evaluation tools to allow the user to get a rough sense of whether a particular 2D or 3D image has sufficient spatial features (e.g., speckles) to work well for a DIC/DVC approach. The idea is replace some of the more time-consuming zero load, and rigid body translation tests, that we usually run to optimize our image preparation for DIC/DVC analysis. The simulators are written in Matlab, and have a fairly easy to use graphical user interface. Of course, I’m always grateful for improvement suggestions.

That’s all!

Thanks for reading!

Christian

**References**

1. Sutton, M., *Digital Image Correlation for Shape and Deformation Measurements*, in *Springer Handbook of Experimental Solid Mechanics*, W.N. Sharpe, Jr., Editor. 2008, Springer US. p. 565-600.

2. Chen, C.S., *Mechanotransduction - a field pulling together?* Journal of Cell Science, 2008. **121**(20): p. 3285-3292.

3. Chen, C.S., J. Tan, and J. Tien, *Mechanotransduction at cell-matrix and cell-cell contacts.* Annual Review of Biomedical Engineering, 2004. **6**: p. 275-302.

4. Corin, K.A. and L.J. Gibson, *Cell contraction forces in scaffolds with varying pore size and cell density.* Biomaterials, 2010. **31**(18): p. 4835-4845.

5. DuFort, C.C., M.J. Paszek, and V.M. Weaver, *Balancing forces: architectural control of mechanotransduction.* Nature Publishing Group, 2011. **12**(5): p. 308-319.

6. Ingber, D.E., *Can cancer be reversed by engineering the tumor microenvironment?* Seminars in Cancer Biology, 2008. **18**(5): p. 356-364.

7. Paszek, M.J. and V.M. Weaver, *The tension mounts: Mechanics meets morphogenesis and malignancy.* Journal of Mammary Gland Biology and Neoplasia, 2008. **9**(4): p. 325-342.

8. Bar-Kochba E, T.J., Andrews E,, Kim K S, Franck C, *A Fast Iterative Digital Volume Correlation Algorithm for Large Deformations.* Experimental Mechanics, 2014. **accepted**.

9. Franck, C., S.A. Maskarinec, D.A. Tirrell, and G. Ravichandran, *Three-Dimensional Traction Force Microscopy: A New Tool for Quantifying Cell-Matrix Interactions.* Plos One, 2011. **6**(3): p. e17833.

10. Franck, C., S.A. Maskarinec, D.A. Tirrell, and G. Ravichandran, *Quantifying cellular traction forces in three dimensions.* Proceedings of the National Academy of Sciences of the United States of America, 2009. **106**(52): p. 22108-22113.

11. Hur, S.S., J.C. del Alamo, J.S. Park, Y.S. Li, H.A. Nguyen, D. Teng, K.C. Wang, L. Flores, B. Alonso-Latorre, and J.C. Lasheras, *Roles of cell confluency and fluid shear in 3-dimensional intracellular forces in endothelial cells.* Proceedings of the National Academy of Sciences of the United States of America, 2012. **109**(28): p. 11110-11115.

12. Hur, S.S., Y. Zhao, Y.-S. Li, E. Botvinick, and S. Chien, *Live Cells Exert 3-Dimensional Traction Forces on Their Substrata.* Cellular and Molecular Bioengineering, 2009. **2**(3): p. 425-436.

13. Koch, T.M., S. Münster, N. Bonakdar, J.P. Butler, and B. Fabry, *3D Traction Forces in Cancer Cell Invasion.* Plos One, 2012. **7**(3): p. e33476.

14. Legant, W.R., J.S. Miller, B.L. Blakely, D.M. Cohen, G.M. Genin, and C.S. Chen, *Measurement of mechanical tractions exerted by cells in three-dimensional matrices.* Nature Methods, 2010. **7**(12): p. 969-U113.

15. Toyjanova J, B.-K.E., Lopez-Fagundo C, Reichner J, Hoffman-Kim,D, Franck C, *High Resolution, Large Deformation 3D Traction Force Microscopy.* Plos One, 2014. **accepted**.

16. Smith, T.S., B.K. Bay, and M.M. Rashid, *Digital volume correlation including rotational degrees of freedom during minimization.* Experimental Mechanics, 2002. **42**(3): p. 272-278.

17. Roeder, B.A., K. Kokini, J.P. Robinson, and S.L. Voytik-Harbin, *Local, three-dimensional strain measurements within largely deformed extracellular matrix constructs.* Journal of Biomechanical Engineering-Transactions of the Asme, 2004. **126**(6): p. 699-708.

18. Franck, C., S. Hong, S.A. Maskarinec, D.A. Tirrell, and G. Ravichandran, *Three-dimensional full-field measurements of large deformations in soft materials using confocal microscopy and digital volume correlation.* Experimental Mechanics, 2007. **47**(3): p. 427-438.

19. Bay, B.K., T.S. Smith, D.P. Fyhrie, and M. Saad, *Digital volume correlation: Three-dimensional strain mapping using X-ray tomography.* Experimental Mechanics, 1999. **39**(3): p. 217-226.

20. Gates, M., J. Lambros, and M.T. Heath, *Towards High Performance Digital Volume Correlation.* Experimental Mechanics, 2011. **51**(4): p. 491-507.

21. Leclerc, H., J.N. Perie, F. Hild, and S. Roux, *Digital volume correlation: what are the limits to the spatial resolution?* Mechanics & Industry, 2012. **13**(6): p. 361-371.

22. Leclerc, H., J.N. Perie, S. Roux, and F. Hild, *Voxel-Scale Digital Volume Correlation.* Experimental Mechanics, 2011. **51**(4): p. 479-490.

23. Hild, F., A. Fanget, J. Adrien, E. Maire, and S. Roux, *Three-dimensional analysis of a tensile test on a propellant with digital volume correlation.* Archives of Mechanics, 2011.**63**(5-6): p. 459-478.

24. Rethore, J., N. Limodin, J.Y. Buffiere, F. Hild, W. Ludwig, and S. Roux, *Digital volume correlation analyses of synchrotron tomographic images.* Journal of Strain Analysis for Engineering Design, 2011. **46**(7): p. 683-695.

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