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PDE-constrained shape registration to characterize biological growth and morphogenesis from imaging data

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Abstract

We propose a PDE-constrained shape registration algorithm that captures the deformation and growth of biological tissue from imaging data. Shape registration is the process of evaluating optimum alignment between pairs of geometries through a spatial transformation function. We start from our previously reported work, which uses 3D tensor product B-spline basis functions to interpolate 3D space. Here, the movement of the B-spline control points, composed of an implicit function describing the shape of the tissue, yields the total deformation gradient field. The deformation gradient is then split into growth and elastic contributions. The growth tensor captures the addition of mass, i.e. growth, and evolves according to a constitutive equation which is usually a function of the elastic deformation. Stress is generated in the material due to the elastic component of the deformation alone. The result of the registration is obtained by minimizing a total energy functional which includes: a distance measure reflecting similarity between the shapes, and the total elastic energy accounting for the growth of the tissue. We apply the proposed shape registration framework to study zebrafish embryo epiboly process and tissue expansion during skin reconstruction surgery. We anticipate that our PDE-constrained shape registration method will improve our understanding of biological and medical problems in which tissues undergo extreme deformations over time.

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Associated files to this publication are available at https://github.com/arpawar/StrainMin_Registration

References

  1. Ashburner J (2007) A fast diffeomorphic image registration algorithm. Neuroimage 38(1):95–113

    Article  Google Scholar 

  2. Beg MF, Miller MI, Trouvé A, Younes L (2005) Computing large deformation metric mappings via geodesic flows of diffeomorphisms. Int J Comput Vision 61(2):139–157

    Article  MATH  Google Scholar 

  3. Sederberg TW, Parry SR (1986) Proceedings of the 13th annual conference on computer graphics and interactive techniques. Association for Computing Machinery, New York, NY, USA. SIGGRAPH ’86, pp 151–160

  4. Szeliski R, Coughlan J (1997) Spline-based image registration. Int J Comput Vision 22(3):199–218

    Article  Google Scholar 

  5. Rueckert D, Sonoda LI, Hayes C, Hill DL, Leach MO, Hawkes DJ (1999) Nonrigid registration using free-form deformations: application to breast mr images. IEEE Trans Med Imaging 18(8):712–721

    Article  Google Scholar 

  6. Tustison NJ, Avants BB, Gee JC (2009) Directly manipulated free-form deformation image registration. IEEE Trans Image Process 18(3):624–635

    Article  MathSciNet  MATH  Google Scholar 

  7. Tustison NJ et al (2013) Explicit B-spline regularization in diffeomorphic image registration. Front Neuroinform 7:39

    Article  Google Scholar 

  8. Pawar A, Zhang YJ, Anitescu C, Rabczuk T (2019) Joint image segmentation and registration based on a dynamic level set approach using truncated hierarchical B-splines. Comp Math Appl 78:3250–3267

    Article  MathSciNet  MATH  Google Scholar 

  9. Veress AI, Phatak N, Weiss JA (2005) Handbook of biomedical image analysis. Springer, Berlin, pp 487–533

    Book  Google Scholar 

  10. Mang A, Gholami A, Davatzikos C, Biros G (2018) PDE-constrained optimization in medical image analysis. Optim Eng 19(3):765–812

    Article  MathSciNet  MATH  Google Scholar 

  11. Meier R, Knecht U, Loosli T, Bauer S, Slotboom J, Wiest R, Reyes M (2016) Clinical evaluation of a fully-automatic segmentation method for longitudinal brain tumor volumetry. Sci Rep 6(1):1–11

    Article  Google Scholar 

  12. Chien A, Callender RA, Yokota H, Salamon N, Colby GP, Wang AC, Szeder V, Jahan R, Tateshima S, Villablanca J et al (2019) Unruptured intracranial aneurysm growth trajectory: occurrence and rate of enlargement in 520 longitudinally followed cases. J Neurosurg 132(4):1077–1087

    Article  Google Scholar 

  13. Keller PJ, Schmidt AD, Wittbrodt J, Stelzer EH (2008) Reconstruction of zebrafish early embryonic development by scanned light sheet microscopy. Science 322(5904):1065–1069

    Article  Google Scholar 

  14. Zhang YJ (2016) Geometric modeling and mesh generation from scanned images. Chapman and Hall/CRC, Boca Raton

    Book  MATH  Google Scholar 

  15. Tepole AB, Ploch CJ, Wong J, Gosain AK, Kuhl E (2011) Growing skin: a computational model for skin expansion in reconstructive surgery. J Mech Phys Solids 59(10):2177–2190

    Article  MathSciNet  MATH  Google Scholar 

  16. Taber LA (1995) Biomechanics of growth, remodeling, and morphogenesis. Appl Mech Rev 48(8):487–545

    Article  Google Scholar 

  17. Cowin SC (2004) Tissue growth and remodeling. Annu Rev Biomed Eng 6:77–107

    Article  Google Scholar 

  18. Tepole AB (2017) Computational systems mechanobiology of wound healing. Comput Methods Appl Mech Eng 314:46–70

    Article  MathSciNet  MATH  Google Scholar 

  19. Göktepe S, Abilez OJ, Kuhl E (2010) A generic approach towards finite growth with examples of athlete’s heart, cardiac dilation, and cardiac wall thickening. J Mech Phys Solids 58(10):1661–1680

    Article  MathSciNet  MATH  Google Scholar 

  20. Himpel G, Kuhl E, Menzel A, Steinmann P et al (2005) Computational modelling of isotropic multiplicative growth. Comput Model Eng Sci 8:119–134

    MATH  Google Scholar 

  21. Goriely A, Amar MB (2007) On the definition and modeling of incremental, cumulative, and continuous growth laws in morphoelasticity. Biomech Model Mechanobiol 6(5):289–296

    Article  Google Scholar 

  22. Pawar A, Zhang Y, Jia Y, Wei X, Rabczuk T, Chan CL, Anitescu C (2016) Adaptive FEM-based nonrigid image registration using truncated hierarchical B-splines. Comput Math Appl 72(8):2028–2040

    Article  MathSciNet  MATH  Google Scholar 

  23. Pawar A, Zhang YJ, Anitescu C, Jia Y, Rabczuk T (2018) DTHB3D_Reg: dynamic truncated hierarchical B-spline based 3D nonrigid image registration. Commun Comput Phys 23(3):877–898

    Article  MathSciNet  MATH  Google Scholar 

  24. Lee EH (1969) Elastic-plastic deformation at finite strains. ASME J Appl Mech 36(1):1–6

    Article  MATH  Google Scholar 

  25. Gordon WJ, Riesenfeld RF (1974) Computer aided geometric design. Elsevier, Amsterdam, pp 95–126

    Book  Google Scholar 

  26. Farin G (2014) Curves and surfaces for computer-aided geometric design: a practical guide. Elsevier, Amsterdam

    MATH  Google Scholar 

  27. Giannelli C, Jüttler B, Speleers H (2012) THB-splines: the truncated basis for hierarchical splines. Comput Aided Geom Des 29(7):485–498

    Article  MathSciNet  MATH  Google Scholar 

  28. Pawar A, Zhang YJ (2020) Neuronseg_BACH: automated neuron segmentation using B-spline based active contour and hyperelastic regularization. Commun Comput Phys 28(3):1219–1244

    Article  MathSciNet  MATH  Google Scholar 

  29. Xie Z, Farin GE (2004) Image registration using hierarchical B-splines. IEEE Trans Visual Comput Graphics 10(1):85–94

    Article  Google Scholar 

  30. Wei X, Zhang Y, Liu L, Hughes TJR (2017) Truncated T-splines: fundamentals and methods. Comput Methods Appl Mech Eng 316:349–372

    Article  MathSciNet  MATH  Google Scholar 

  31. Bornemann P, Cirak F (2013) A subdivision-based implementation of the hierarchical b-spline finite element method. Comput Methods Appl Mech Eng 253:584–598

    Article  MathSciNet  MATH  Google Scholar 

  32. Rodriguez EK, Hoger A, McCulloch AD (1994) Stress-dependent finite growth in soft elastic tissues. J Biomech 27(4):455–467

    Article  Google Scholar 

  33. Holzapfel GA et al (2001) Biomechanics of soft tissue. Handbook Mater Behav Models 3(1):1049–1063

    Google Scholar 

  34. Leng J, Xu G, Zhang Y (2013) Medical image interpolation based on multi-resolution registration. Comput Math Appl 66(1):1–18

    Article  MathSciNet  MATH  Google Scholar 

  35. Jia Y, Zhang YJ, Rabczuk T (2015) A novel dynamic multilevel technique for image registration. Comput Math Appl 69(9):909–925

    Article  MathSciNet  MATH  Google Scholar 

  36. Bruce AE (2016) Zebrafish epiboly: spreading thin over the yolk. Dev Dyn 245(3):244–258

    Article  MathSciNet  Google Scholar 

  37. Lepage SE, Bruce AE (2010) Zebrafish epiboly: mechanics and mechanisms. Int J Dev Biol 54(8–9):1213–1228

    Article  Google Scholar 

  38. Campinho P, Behrndt M, Ranft J, Risler T, Minc N, Heisenberg CP (2013) Tension-oriented cell divisions limit anisotropic tissue tension in epithelial spreading during zebrafish epiboly. Nat Cell Biol 15(12):1405–1414

    Article  Google Scholar 

  39. Behrndt M, Salbreux G, Campinho P, Hauschild R, Oswald F, Roensch J, Grill SW, Heisenberg CP (2012) Forces driving epithelial spreading in zebrafish gastrulation. Science 338(6104):257–260

    Article  Google Scholar 

  40. Vladimirov N, Mu Y, Kawashima T, Bennett DV, Yang CT, Looger LL, Keller PJ, Freeman J, Ahrens MB (2014) Light-sheet functional imaging in fictively behaving zebrafish. Nat Methods 11(9):883–884

    Article  Google Scholar 

  41. Wu TC, Wang X, Li L, Bu Y, Umulis DM (2021) Automatic wavelet-based 3D nuclei segmentation and analysis for multicellular embryo quantification. Sci Rep 11(1):1–13

    Google Scholar 

  42. LoGiudice J, Gosain AK (2004) Pediatric tissue expansion: indications and complications. Plast Surg Nurs 24(1):20–26

    Article  Google Scholar 

  43. Radovan C (1984) Tissue expansion in soft-tissue reconstruction. Plast Reconstr Surg 74(4):482–492

    Article  Google Scholar 

  44. Bertozzi N, Pesce M, Santi P, Raposio E (2017) Tissue expansion for breast reconstruction: methods and techniques. Ann Med Surg 21:34–44

    Article  Google Scholar 

  45. Rivera R, LoGiudice J, Gosain AK (2005) Tissue expansion in pediatric patients. Clin Plast Surg 32(1):35–44

    Article  Google Scholar 

  46. Chun JT, Rohrich RJ (1998) Versatility of tissue expansion in head and neck burn reconstruction. Ann Plast Surg 41(1):11–16

    Article  Google Scholar 

  47. Lee T, Vaca EE, Ledwon JK, Bae H, Topczewska JM, Turin SY, Kuhl E, Gosain AK, Tepole AB (2018) Improving tissue expansion protocols through computational modeling. J Mech Behav Biomed Mater 82:224–234

    Article  Google Scholar 

  48. Han T, Lee T, Ledwon J, Vaca E, Turin S, Kearney A, Gosain AK, Tepole AB (2022) Bayesian calibration of a computational model of tissue expansion based on a porcine animal model. Acta Biomater 137:136–146

    Article  Google Scholar 

  49. Zöllner AM, Tepole AB, Kuhl E (2012) On the biomechanics and mechanobiology of growing skin. J Theor Biol 297:166–175

    Article  MathSciNet  MATH  Google Scholar 

  50. Bauer S, Seiler C, Bardyn T, Buechler P, Reyes M (2010) 2010 annual international conference of the IEEE engineering in medicine and biology (IEEE), pp 4080–4083

  51. Genet M, Stoeck CT, Von Deuster C, Lee LC, Kozerke S (2018) Equilibrated warping: finite element image registration with finite strain equilibrium gap regularization. Med Image Anal 50:1–22

    Article  Google Scholar 

  52. Ortiz-Puerta D, Cox A, Hurtado DE (2022) Snakes isogeometric analysis (siga): towards accurate and flexible geometrical models of the respiratory airways. Comput Methods Appl Mech Eng 394:114,841

    Article  MathSciNet  MATH  Google Scholar 

  53. Eskandari M, Kuschner WG, Kuhl E (2015) Patient-specific airway wall remodeling in chronic lung disease. Ann Biomed Eng 43(10):2538–2551

    Article  Google Scholar 

  54. Budday S, Raybaud C, Kuhl E (2014) A mechanical model predicts morphological abnormalities in the developing human brain. Sci Rep 4(1):1–7

    Google Scholar 

  55. Eskandari M, Kuhl E (2015) Systems biology and mechanics of growth. Wiley Interdiscip Rev Syst Biol Med 7(6):401–412

    Article  Google Scholar 

  56. Valentin A, Humphrey JD, Holzapfel GA (2013) A finite element-based constrained mixture implementation for arterial growth, remodeling, and adaptation: Theory and numerical verification. Int J Numer Methods Biomed Eng 29(8):822–849

    Article  MathSciNet  Google Scholar 

  57. Tepole AB, Gart M, Purnell CA, Gosain AK, Kuhl E (2016) The incompatibility of living systems: characterizing growth-induced incompatibilities in expanded skin. Ann Biomed Eng 44(5):1734–1752

    Article  Google Scholar 

  58. Tepole AB, Vaca EE, Purnell CA, Gart M, McGrath J, Kuhl E, Gosain AK (2017) Quantification of strain in a porcine model of skin expansion using multi-view stereo and isogeometric kinematics. JoVE (J Visual Exp) (122):e55,052

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Acknowledgements

The research at Purdue University was supported in part by the NIHR01GM132501-02 and NIHR01AR074525-01A1.

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Authors and Affiliations

  1. School of Mechanical Engineering, Purdue University, 585 Purdue Mall, West Lafayette, IN, 47907, USA

    Aishwarya Pawar & Adrian Buganza Tepole

  2. Weldon School of Biomedical Engineering, Purdue University, 206 S Martin Jischke Dr, West Lafayette, IN, 47907, USA

    Linlin Li, David M. Umulis & Adrian Buganza Tepole

  3. Lurie Children’s Hospital, Northwestern University, 225 East Chicago Ave, Chicago, IL, 60611, USA

    Arun K. Gosain

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  1. Aishwarya Pawar
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Correspondence to Aishwarya Pawar or Adrian Buganza Tepole.

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Pawar, A., Li, L., Gosain, A.K. et al. PDE-constrained shape registration to characterize biological growth and morphogenesis from imaging data. Engineering with Computers 38, 3909–3924 (2022). https://doi.org/10.1007/s00366-022-01682-x

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  • DOI: https://doi.org/10.1007/s00366-022-01682-x

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