Skip to main content
SpringerLink
Log in

Measuring the Hyperelastic Response of Porcine Liver Tissues In-Vitro Using Controlled Cavitation Rheology

  • Research paper
  • Published:
Experimental Mechanics Aims and scope Submit manuscript

Abstract

Background

Measuring the mechanical properties of soft biological tissues in-vitro by using conventional methods (e.g. tension, compression, or indentation) is prone to external testing and sample preparation factors. For example, sample slippage affects the measurement accuracy of mechanical properties of soft biological tissues. In addition, samples have to go through a complex cutting process to be prepared in specific shapes, which increases the post-mortem time spent before testing.

Objective

The purpose of this study is to investigate the capability of a new technique to measure the mechanics of soft biological tissues without experiencing the conventional challenges. It measures the mechanical behavior of soft materials by introducing and expanding spherical cavity deformations within materials’ internal structure while generating two stresses simultaneously, radial and hoop stresses.

Methods

Two porcine livers were tested. The experimental data were used to calibrate the material parameters of three hyperelastic models, namely: Yeoh, Arruda-Boyce, and Ogden. Numerical simulations were performed to validate the material parameters. Also, Computed Tomography (CT) imaging technology was used to verify the configuration of the cavity deformations.

Results

The outcome of the numerical simulations showed that hyperelastic models were able to predict the material response to cavity loading. In addition, CT imaging confirmed the spherical configuration of the applied deformations.

Conclusions

From the experimental results, numerical simulations and the CT imaging, it can be concluded that the cavity expansion test is capable of measuring the mechanics of soft biological tissues without experiencing the difficulties encountered in conventional techniques.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9
Fig. 10
Fig. 11
Fig. 12
Fig. 13
Fig. 14

Similar content being viewed by others

References

  1. Krouskop TA, Wheeler TM, Kallel F, Garra BS, Hall T (1998) Elastic moduli of breast and prostate tissues under compression. Ultrason Imaging 20:260–274. https://doi.org/10.1177/016173469802000403

    Article  Google Scholar 

  2. Paszek MJ, Zahir N, Johnson KR, Lakins JN, Rozenberg GI, Gefen A, Reinhart-King CA, Margulies SS, Dembo M, Boettiger D, Hammer DA, Weaver VM (2005) Tensional homeostasis and the malignant phenotype. Cancer Cell 8:241–254. https://doi.org/10.1016/j.ccr.2005.08.010

    Article  Google Scholar 

  3. Samani A, Plewes D (2007) An inverse problem solution for measuring the elastic modulus of intact ex vivo breast tissue tumours. Phys Med Biol 52:1247–1260. https://doi.org/10.1088/0031-9155/52/5/003

    Article  Google Scholar 

  4. Yeh W-C, Li P-C, Jeng Y-M, Hsu HC, Kuo PL, Li ML, Yang PM, Lee PH (2002) Elastic modulus measurements of human liver and correlation with pathology. Ultrasound Med Biol 28:467–474. https://doi.org/10.1016/S0301-5629(02)00489-1

    Article  Google Scholar 

  5. Last JA, Pan T, Ding Y, Reilly CM, Keller K, Acott TS, Fautsch MP, Murphy CJ, Russell P (2011) Elastic Modulus determination of Normal and glaucomatous human trabecular meshwork. Investig Opthalmology Vis Sci 52:2147–2152. https://doi.org/10.1167/iovs.10-6342

    Article  Google Scholar 

  6. Eom J, Shi C, Xu XG, De S (2009) Modeling respiratory motion for cancer radiation therapy based on patient-specific 4DCT data. Med Image Comput Comput-Assist Interv MICCAI Int Conf Med Image Comput Comput-Assist Interv 12:348–355

    Google Scholar 

  7. Al-Mayah A, Moseley J, Brock KK (2008) Contact surface and material nonlinearity modeling of human lungs. Phys Med Biol 53:305–317. https://doi.org/10.1088/0031-9155/53/1/022

    Article  Google Scholar 

  8. Al-Mayah A, Moseley J, Hunter S, Brock K (2015) Radiation dose response simulation for biomechanical-based deformable image registration of head and neck cancer treatment. Phys Med Biol 60:8481–8489. https://doi.org/10.1088/0031-9155/60/21/8481

    Article  Google Scholar 

  9. Wittek A, Miller K, Kikinis R, Warfield SK (2007) Patient-specific model of brain deformation: application to medical image registration. J Biomech 40:919–929. https://doi.org/10.1016/j.jbiomech.2006.02.021

    Article  Google Scholar 

  10. Cash DM, Miga MI, Glasgow SC, Dawant BM, Clements LW, Cao Z, Galloway RL, Chapman WC (2007) Concepts and preliminary data toward the realization of image-guided liver surgery. J Gastrointest Surg 11:844–859. https://doi.org/10.1007/s11605-007-0090-6

    Article  Google Scholar 

  11. Al-Mayah A, Moseley J, Velec M, Brock K (2009) Deformable modeling of human liver with contact surface. In: 2009 IEEE Toronto international conference science and Technology for Humanity (TIC-STH). IEEE, Toronto, pp 137–140

    Chapter  Google Scholar 

  12. Carter FJ, Frank TG, Davies PJ, McLean D, Cuschieri A (2001) Measurements and modelling of the compliance of human and porcine organs. Med Image Anal 5:231–236. https://doi.org/10.1016/S1361-8415(01)00048-2

    Article  Google Scholar 

  13. Tay BK, Kim J, Srinivasan MA (2006) In vivo mechanical behavior of intra-abdominal organs. IEEE Trans Biomed Eng 53:2129–2138. https://doi.org/10.1109/TBME.2006.879474

    Article  Google Scholar 

  14. Han L, Noble JA, Burcher M (2003) A novel ultrasound indentation system for measuring biomechanical properties of in vivo soft tissue. Ultrasound Med Biol 29:813–823. https://doi.org/10.1016/S0301-5629(02)00776-7

    Article  Google Scholar 

  15. Fowlkes JB, Emelianov SY, Pipe JG, Skovoroda AR, Carson PL, Adler RS, Sarvazyan AP (1995) Magnetic-resonance imaging techniques for detection of elasticity variation. Med Phys 22:1771–1778. https://doi.org/10.1118/1.597633

    Article  Google Scholar 

  16. Muthupillai R, Lomas D, Rossman P, Greenleaf J, Manduca A, Ehman R (1995) Magnetic resonance elastography by direct visualization of propagating acoustic strain waves. Science 269:1854–1857. https://doi.org/10.1126/science.7569924

    Article  Google Scholar 

  17. Nightingale K (2011) Acoustic radiation force impulse (ARFI) imaging: a review. Curr Med Imaging Rev 7:328–339. https://doi.org/10.2174/157340511798038657

    Article  Google Scholar 

  18. Franchi-Abella S, Elie C, Correas J-M (2013) Ultrasound elastography: advantages, limitations and artefacts of the different techniques from a study on a phantom. Diagn Interv Imaging 94:497–501. https://doi.org/10.1016/j.diii.2013.01.024

    Article  Google Scholar 

  19. Varghese T, Ophir J, Konofagou E, Kallel F, Righetti R (2001) Tradeoffs in Elastographic imaging. Ultrason Imaging 23:216–248. https://doi.org/10.1177/016173460102300402

    Article  Google Scholar 

  20. Bouchard R, Dahl J, Hsu S, Palmeri M, Trahey G (2009) Image quality, tissue heating, and frame rate trade-offs in acoustic radiation force impulse imaging. IEEE Trans Ultrason Ferroelectr Freq Control 56:63–76. https://doi.org/10.1109/TUFFC.2009.1006

    Article  Google Scholar 

  21. Kemper AR, Santago AC, Stitzel JD et al (2010) Biomechanical response of human liver in tensile loading. Ann Adv Automot Med Assoc Adv Automot Med Annu Sci Conf 54:15–26

    Google Scholar 

  22. Nguyễn NH, Dương MT, Trần TN, Phạm PT, Grottke O, Tolba R, Staat M (2012) Influence of a freeze–thaw cycle on the stress–stretch curves of tissues of porcine abdominal organs. J Biomech 45:2382–2386. https://doi.org/10.1016/j.jbiomech.2012.07.008

    Article  Google Scholar 

  23. Dương MT, Nguyễn NH, Trần TN, Tolba RH, Staat M (2015) Influence of refrigerated storage on tensile mechanical properties of porcine liver and spleen. Int Biomech 2:79–88. https://doi.org/10.1080/23335432.2015.1049295

    Article  Google Scholar 

  24. Lu Y-C, Untaroiu CD (2012) Freezing and decay effects on material properties of porcine kidney and liver. Biomed Sci Instrum 48:275–281

    Google Scholar 

  25. Brands DW, Bovendeerd PH, Peters GW, Wismans JS (2000) The large shear strain dynamic behaviour of in-vitro porcine brain tissue and a silicone gel model material. Stapp Car Crash J 44:249–260

    Google Scholar 

  26. Ng BH, Chou SM, Krishna V (2005) The influence of gripping techniques on the tensile properties of tendons. Proc Inst Mech Eng [H] 219:349–354. https://doi.org/10.1243/095441105X34239

    Article  Google Scholar 

  27. Matthews GL, Keegan KG, Graham HL (1996) Effects of tendon grip technique (frozen versus unfrozen) on in vitro surface strain measurements of the equine deep digital flexor tendon. Am J Vet Res 57:111–115

    Google Scholar 

  28. Herzog W, Gal J (1999) Tendon. In: Biomechanics of musculo-skeletal system, 2nd edn. John Wiley, New York, pp 127–147

    Google Scholar 

  29. Nafo W, Al Mayah A (2018) Mechanical investigations of biological tissues using tensile loading and indentation. In: biomechanics of soft tissues, 1st ed. CRC Press, pp 27–54

  30. Soden PD, Kershaw I (1974) Tensile testing of connective tissues. Med Biol Eng 12:510–518. https://doi.org/10.1007/BF02478609

    Article  Google Scholar 

  31. Riemersa DJ, Schamhardt HC (1982) The cryo-jaw, a clamp designed for in vitro rheology studies of horse digital flexor tendons. J Biomech 15:619–620

    Article  Google Scholar 

  32. Bergstrom J (2015) Mechanics of solid polymers: theory and computational modeling. Elsevier, Amsterdam

    Google Scholar 

  33. Nafo W, Al-Mayah A (2020) Characterization of PVA hydrogels’ hyperelastic properties by uniaxial tension and cavity expansion tests. Int J Non-Linear Mech 124:103515. https://doi.org/10.1016/j.ijnonlinmec.2020.103515

    Article  Google Scholar 

  34. Nafo W, Al-Mayah A (2020) Mechanical characterization of PVA hydrogels’ rate-dependent response using multi-axial loading. PLoS One 15:e0233021. https://doi.org/10.1371/journal.pone.0233021

    Article  Google Scholar 

  35. Barney CW, Dougan CE, McLeod KR et al (2020) Cavitation in soft matter. Proc Natl Acad Sci 117:9157–9165. https://doi.org/10.1073/pnas.1920168117

    Article  Google Scholar 

  36. Chui C, Kobayashi E, Chen X, Hisada T, Sakuma I (2004) Combined compression and elongation experiments and non-linear modelling of liver tissue for surgical simulation. Med Biol Eng Comput 42:787–798. https://doi.org/10.1007/BF02345212

    Article  Google Scholar 

  37. deBotton G, Bustamante R, Dorfmann A (2013) Axisymmetric bifurcations of thick spherical shells under inflation and compression. Int J Solids Struct 50:403–413. https://doi.org/10.1016/j.ijsolstr.2012.10.004

    Article  Google Scholar 

  38. Yeoh OH (1993) Some forms of the strain energy function for rubber. Rubber Chem Technol 66:754–771. https://doi.org/10.5254/1.3538343

    Article  Google Scholar 

  39. Arruda EM, Boyce MC (1993) A three-dimensional constitutive model for the large stretch behavior of rubber elastic materials. J Mech Phys Solids 41:389–412. https://doi.org/10.1016/0022-5096(93)90013-6

    Article  MATH  Google Scholar 

  40. Ogden RW (1972) Large deformation isotropic elasticity - on the correlation of theory and experiment for incompressible rubberlike solids. Proc R Soc Math Phys Eng Sci 326:565–584. https://doi.org/10.1098/rspa.1972.0026

    Article  MATH  Google Scholar 

  41. Shahzad M, Kamran A, Siddiqui MZ, Farhan M (2015) Mechanical characterization and FE Modelling of a Hyperelastic material. Mater Res 18:918–924. https://doi.org/10.1590/1516-1439.320414

    Article  Google Scholar 

  42. Rashid B, Destrade M, Gilchrist MD (2014) Mechanical characterization of brain tissue in tension at dynamic strain rates. J Mech Behav Biomed Mater 33:43–54. https://doi.org/10.1016/j.jmbbm.2012.07.015

    Article  Google Scholar 

  43. Zimberlin JA, McManus JJ, Crosby AJ (2010) Cavitation rheology of the vitreous: mechanical properties of biological tissue. Soft Matter 6:3632. https://doi.org/10.1039/b925407b

    Article  Google Scholar 

  44. Zimberlin JA, Sanabria-DeLong N, Tew GN, Crosby AJ (2007) Cavitation rheology for soft materials. Soft Matter 3:763–767. https://doi.org/10.1039/b617050a

    Article  Google Scholar 

  45. Brostow W, Pietkiewicz D, Wisner SR (2007) Polymer tribology in safety medical devices: retractable syringes. Adv Polym Technol 26:56–64. https://doi.org/10.1002/adv.20084

    Article  Google Scholar 

  46. De Bardi M, Müller R, Grünzweig C et al (2018) Clogging in staked-in needle pre-filled syringes (SIN-PFS): influence of water vapor transmission through the needle shield. Eur J Pharm Biopharm 127:104–111. https://doi.org/10.1016/j.ejpb.2018.02.016

    Article  Google Scholar 

  47. Zhang Q, Fassihi MA, Fassihi R (2018) Delivery Considerations of Highly Viscous Polymeric Fluids Mimicking Concentrated Biopharmaceuticals: Assessment of Injectability via Measurement of Total Work Done “WT.”. AAPS PharmSciTech 19:1520–1528. https://doi.org/10.1208/s12249-018-0963-x

    Article  Google Scholar 

  48. Kasem H, Shriki H, Ganon L, Mizrahi M, Abd-Rbo K, Domb AJ (2019) Rubber plunger surface texturing for friction reduction in medical syringes. Friction 7:351–358. https://doi.org/10.1007/s40544-018-0227-5

    Article  Google Scholar 

  49. volumegraphics.com. https://www.volumegraphics.com/. Accessed 3 Sep 2018

  50. ABAQUS/CAE (2013) Abaqus/CAE v6.13 User’s Manual

  51. Samur E, Sedef M, Basdogan C, Avtan L, Duzgun O (2007) A robotic indenter for minimally invasive measurement and characterization of soft tissue response. Med Image Anal 11:361–373. https://doi.org/10.1016/j.media.2007.04.001

    Article  Google Scholar 

  52. Gao Z, Lister K, Desai JP (2010) Constitutive modeling of liver tissue: experiment and theory. Ann Biomed Eng 38:505–516. https://doi.org/10.1007/s10439-009-9812-0

    Article  Google Scholar 

  53. Nafo W, Al-Mayah A (2019) Measuring Hyperelastic properties of hydrogels using cavity expansion method. Exp Mech 59:1047–1061. https://doi.org/10.1007/s11340-019-00504-4

    Article  Google Scholar 

  54. Dương MT, Nguyễn NH, Trần TN, Tolba RH, Staat M (2015) Influence of refrigerated storage on tensile mechanical properties of porcine liver and spleen. Int Biomech 2:79–88. https://doi.org/10.1080/23335432.2015.1049295

    Article  Google Scholar 

  55. Lu Y-C, Kemper AR, Untaroiu CD (2014) Effect of storage on tensile material properties of bovine liver. J Mech Behav Biomed Mater 29:339–349. https://doi.org/10.1016/j.jmbbm.2013.09.022

    Article  Google Scholar 

  56. Tan Z, Dini D, Rodriguez y Baena F, Forte AE (2018) Composite hydrogel: a high fidelity soft tissue mimic for surgery. Mater Des 160:886–894. https://doi.org/10.1016/j.matdes.2018.10.018

    Article  Google Scholar 

  57. Fu YB, Chui CK (2014) Modelling and simulation of porcine liver tissue indentation using finite element method and uniaxial stress–strain data. J Biomech 47:2430–2435. https://doi.org/10.1016/j.jbiomech.2014.04.009

    Article  Google Scholar 

  58. Umale S, Chatelin S, Bourdet N, Deck C, Diana M, Dhumane P, Soler L, Marescaux J, Willinger R (2011) Experimental in vitro mechanical characterization of porcine Glisson’s capsule and hepatic veins. J Biomech 44:1678–1683. https://doi.org/10.1016/j.jbiomech.2011.03.029

    Article  Google Scholar 

  59. Miller K, Chinzei K (1997) Constitutive modelling of brain tissue: experiment and theory. J Biomech 30:1115–1121. https://doi.org/10.1016/S0021-9290(97)00092-4

    Article  Google Scholar 

  60. Nava A, Mazza E, Furrer M, Villiger P, Reinhart WH (2008) In vivo mechanical characterization of human liver. Med Image Anal 12:203–216. https://doi.org/10.1016/j.media.2007.10.001

    Article  Google Scholar 

  61. Tan Z, Dini D, Rodriguez y Baena F, Forte AE (2018) Composite hydrogel: a high fidelity soft tissue mimic for surgery. Mater Des 160:886–894. https://doi.org/10.1016/j.matdes.2018.10.018

    Article  Google Scholar 

  62. Wan WK, Campbell G, Zhang ZF, Hui AJ, Boughner DR (2002) Optimizing the tensile properties of polyvinyl alcohol hydrogel for the construction of a bioprosthetic heart valve stent. J Biomed Mater Res 63:854–861. https://doi.org/10.1002/jbm.10333

    Article  Google Scholar 

  63. Roan E, Vemaganti K (2007) The nonlinear material properties of liver tissue determined from no-slip uniaxial compression experiments. J Biomech Eng 129:450–456. https://doi.org/10.1115/1.2720928

    Article  Google Scholar 

  64. Chau KT (1997) Young’s Modulus interpreted from compression tests with end friction. J Eng Mech 123:1–7. https://doi.org/10.1061/(ASCE)0733-9399(1997)123:1(1)

    Article  Google Scholar 

  65. Chen J, Patnaik SS, Prabhu RK, Priddy LB, Bouvard JL, Marin E, Horstemeyer MF, Liao J, Williams LN (2019) Mechanical response of porcine liver tissue under high strain rate compression. Bioengineering 6:49. https://doi.org/10.3390/bioengineering6020049

    Article  Google Scholar 

  66. Al-Mayah A, Moseley J, Velec M et al (2010) Deformable image registration of heterogeneous human lung incorporating the bronchial tree: deformable image registration of heterogeneous human lung. Med Phys 37:4560–4571. https://doi.org/10.1118/1.3471020

    Article  Google Scholar 

  67. Chui C, Kobayashi E, Chen X, Hisada T, Sakuma I (2007) Transversely isotropic properties of porcine liver tissue: experiments and constitutive modelling. Med Biol Eng Comput 45:99–106. https://doi.org/10.1007/s11517-006-0137-y

    Article  Google Scholar 

  68. Li L, Maccabi A, Abiri A, Juo YY, Zhang W, Chang YJ, Saddik GN, Jin L, Grundfest WS, Dutson EP, Eldredge JD, Benharash P, Candler RN (2019) Characterization of perfused and sectioned liver tissue in a full indentation cycle using a visco-hyperelastic model. J Mech Behav Biomed Mater 90:591–603. https://doi.org/10.1016/j.jmbbm.2018.11.006

    Article  Google Scholar 

  69. Miller K (2000) Constitutive modelling of abdominal organs. J Biomech 33:367–373. https://doi.org/10.1016/S0021-9290(99)00196-7

    Article  Google Scholar 

  70. Chui C, Kobayashi E, Chen X, Hisada T, Sakuma I (2004) Combined compression and elongation experiments and non-linear modelling of liver tissue for surgical simulation. Med Biol Eng Comput 42:787–798. https://doi.org/10.1007/BF02345212

    Article  Google Scholar 

  71. Marchesseau S, Chatelin S, Delingette H (2017) Nonlinear biomechanical model of the liver. In: Biomechanics of Living Organs. Elsevier, pp. 243–265

  72. Chitnis GD, Verma MKS, Lamazouade J, Gonzalez-Andrades M, Yang K, Dergham A, Jones PA, Mead BE, Cruzat A, Tong Z, Martyn K, Solanki A, Landon-Brace N, Karp JM (2019) A resistance-sensing mechanical injector for the precise delivery of liquids to target tissue. Nat Biomed Eng 3:621–631. https://doi.org/10.1038/s41551-019-0350-2

    Article  Google Scholar 

  73. DeCarlo PF, Slowik JG, Worsnop DR et al (2004) Particle morphology and density characterization by combined mobility and aerodynamic diameter measurements. Part 1: theory. Aerosol Sci Technol 38:1185–1205. https://doi.org/10.1080/027868290903907

    Article  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Civil and Environmental Engineering Department, University of Waterloo, Waterloo, Ontario, Canada

    W. Nafo & A. Al-Mayah

  2. Mechanical and Mechatronics Engineering Department, University of Waterloo, Waterloo, Ontario, Canada

    A. Al-Mayah

  3. Centre for Bioengineering and Biotechnology, University of Waterloo, Waterloo, Ontario, Canada

    A. Al-Mayah

Authors
  1. W. Nafo
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. A. Al-Mayah
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to A. Al-Mayah.

Ethics declarations

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper. In addition, the authors acknowledge that they did not receive financial support from any funding agencies to carry out the work.

Conflict of Interest

There are no conflicts to declare.

Additional information

Publisher’s Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Appendix 1

Appendix 1

The definition of R is very critical to determine the deformation term in the circumferential direction, λ. In this work, R was estimated to be 1.42 mm; this radius is derived from an initial volume which is expressed as

$$ Vi={V}_{n-b}+{V}_p $$
(24)

Where Vn-b is the volume of the balloon region which is 2.837 mm3, The dimensions of this region are described in Fig. 2 of the main manuscript; Vp is the volume that is introduced into the balloon before the tissues start to resist the cavity expansion, which is 9.177 mm3 as indicated by the pressure sensor data in Fig.7, see [72] for more details about the mechanism of injecting fluids into soft biological tissues. When the needle was inserted inside the tissues, it created a channel along the needle’s shaft. This channel made the tissues loose resulting into a negligible resistance against initially introduced volumes. A FE simulation was conducted for a process similar to that of a flat punch indentation test to calculate the measured initial volume (Vi) numerically. It was found that the magnitude of the volume induced due to a displacement equal to the thickness of the needle-balloon region was around 12 mm3, similar to the experimental Vi. This indicates that the insertion of the needle created a cut that allows the tissues to displace during initial volumes of inflation before resisting expansion.

Although, CT images have shown that the inflated balloon was nearly spherical in the injected volumes, the initial volume of the balloon has a cylindrical configuration. However, the stretch ratio (λ) can be calculated based on the volumetric ratio. Therefore, the exact geometry of the cavity does not affect the stretch (λ) calculation. The initial cylindrical volume can be represented by an equivalent spherical volume by using the concept of Equivalent-Volume Diameter (EVD) which was adopted in this study to define the initial radius, see [73] for more details about EVD. The EVD is based on equating irregular-shaped volumes to an equivalent sphere volume with a diameter of

$$ {\mathrm{D}}_{\mathrm{e}}=\sqrt[3]{\frac{6\ast {\mathrm{V}}_{\mathrm{i}}}{\pi }} $$
(25)

R is then calculated as \( \frac{{\mathrm{D}}_{\mathrm{e}}}{2} \)

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Nafo, W., Al-Mayah, A. Measuring the Hyperelastic Response of Porcine Liver Tissues In-Vitro Using Controlled Cavitation Rheology. Exp Mech 61, 445–458 (2021). https://doi.org/10.1007/s11340-020-00674-6

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11340-020-00674-6

Keywords

Search

Navigation