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On the in vivo function of the mitral heart valve leaflet: insights into tissue–interstitial cell biomechanical coupling

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Abstract

There continues to be a critical need for developing data-informed computational modeling techniques that enable systematic evaluations of mitral valve (MV) function. This is important for a better understanding of MV organ-level biomechanical performance, in vivo functional tissue stresses, and the biosynthetic responses of MV interstitial cells (MVICs) in the normal, pathophysiological, and surgically repaired states. In the present study, we utilized extant ovine MV population-averaged 3D fiducial marker data to quantify the MV anterior leaflet (MVAL) deformations in various kinematic states. This approach allowed us to make the critical connection between the in vivo functional and the in vitro experimental configurations. Moreover, we incorporated the in vivo MVAL deformations and pre-strains into an enhanced inverse finite element modeling framework (Path 1) to estimate the resulting in vivo tissue prestresses \((\sigma _\mathrm{CC}\cong \sigma _\mathrm{RR}\cong \, 30\,\hbox {kPa})\) and the in vivo peak functional tissue stresses \((\sigma _\mathrm{CC}\cong 510\, \hbox {kPa}, \sigma _\mathrm{RR}\cong 740\, \hbox {kPa})\). These in vivo stress estimates were then cross-verified with the results obtained from an alternative forward modeling method (Path 2), by taking account of the changes in the in vitro and in vivo reference configurations. Moreover, by integrating the tissue-level kinematic results into a downscale MVIC microenvironment FE model, we were able to estimate, for the first time, the in vivo layer-specific MVIC deformations and deformation rates of the normal and surgically repaired MVALs. From these simulations, we determined that the placement of annuloplasty ring greatly reduces the peak MVIC deformation levels in a layer-specific manner. This suggests that the associated reductions in MVIC deformation may down-regulate MV extracellular matrix maintenance, ultimately leading to reduction in tissue mechanical integrity. These simulations provide valuable insight into MV cellular mechanobiology in response to organ- and tissue-level alternations induced by MV disease or surgical repair. They will also assist in the future development of computer simulation tools for guiding MV surgery procedure with enhanced durability and improved long-term surgical outcomes.

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Abbreviations

\(\Psi _\mathrm{c} \) :

Strain energy function component of the collagen fiber network

\(\Psi _\mathrm{e} \) :

Strain energy function component of the elastin fiber networks

\(\Psi _\mathrm{m} \) :

Strain energy function component of non-fibrous matrix

\(\mathbf{C}=\mathbf{F}^{T}{} \mathbf{F}\) :

Right-Cauchy deformation tensor

\(\mathbf{E}=(\mathbf{C}-\mathbf{I})/2\) :

Green–Lagrange strain tensor

\(\mathbf{I}\) :

2nd-rank identity tensor

\(I_{4}=\mathbf{n}_\mathrm{c}\cdot \mathbf{Cn}_\mathrm{c}\) :

Square of the circumferential stretch associated with elastin fibers

\(I_{6}=\mathbf{n}_\mathrm{R}\cdot \mathbf{Cn}_\mathrm{R}\) :

Square of the radial stretch associated with elastin fibers

\({E}_\mathrm{ens}(\theta _{0})=\mathbf{n}(\theta _{0})\cdot \mathbf{En}(\theta _{0})\) :

Ensemble fiber strain of the individual collagen fiber in the direction of n(\(\theta )\)

\(S_\mathrm{ens}\) :

Ensemble fiber stress calculated based on the ensemble fiber stress–strain relation

\(E_\mathrm{cutoff}\) :

Cutoff limit of ensemble fiber strain beyond which the ensemble fiber stress–strain relationship becomes linear

\(\mathbf{n}(\theta _{0})\) :

Unit vector of collagen fiber orientation w.r.t. \(\beta _{0}\)

\(\mathbf{n}_\mathrm{c}=\mathbf{n}(0^{\circ })\) :

Unit vector of elastin fiber orientation along the circumferential direction w.r.t. \(\beta _{0}\)

\(\mathbf{n}_\mathrm{R}=\mathbf{n}(\pi )\) :

Unit vector of elastin fiber orientation along the radial direction w.r.t. \(\beta _{0}\)

\(\mu _\mathrm{m}\) :

Shear modulus of the non-fibrous matrix represented by a neo-Hookean material

\(\eta _\mathrm{c}\) :

Collagen fiber modulus

\(\eta _\mathrm{e}^{a} \) :

Elastin fiber modulus of the elastin fibers along the circumferential direction

\(\eta _\mathrm{e}^{b} \) :

Elastin fiber modulus of the elastin fibers along the radial direction

a :

Exponent of the elastin ensembles along the circumferential direction

b :

Exponent of the elastin ensembles along the radial direction

\(\Gamma (\theta _0 ;\mu _\mathrm{C} ,\sigma _\mathrm{C} )\) :

Orientation density function of the collagen fiber network with a mean fiber direction \(\mu _\mathrm{C}\) and a standard deviation \(\sigma _\mathrm{C}\)

\(D(x;\mu _\mathrm{D} ,\sigma _\mathrm{D},E_\mathrm{lb} ,E_\mathrm{ub})\) :

Fiber recruitment distribution function with a mean \(\mu _\mathrm{D}\), a standard deviation \(\sigma _\mathrm{D}\), , a lower-bound limit \(E_\mathrm{lb}\) at which collagen fiber recruitment begins, a lower-bound limit \(E_\mathrm{ub}\) at which collagen fiber is fully recruited

References

  • Aggarwal A, Sacks MS (2015) A framework for determination of heart valves’ mechanical properties using inverse-modeling approach. Lect Notes Comput Sci 9126:285–294

    Article  Google Scholar 

  • Aggarwal A, Pouch AM, Lai E, Lesicko J, Yushkevich PA et al (2016) In-vivo heterogeneous functional and residual strains in human aortic valve leaflets. J Biomech 49:2481–2490

    Article  Google Scholar 

  • Amini R, Eckert CE, Koomalsingh K, McGarvey J, Minakawa M et al (2012) On the in vivo deformation of the mitral valve anterior leaflet: effects of annular geometry and referential configuration. Ann Biomed Eng 40:1455–1467

    Article  Google Scholar 

  • Balachandran K, Konduri S, Sucosky P, Jo H, Yoganathan A (2006) An ex vivo study of the biological properties of porcine aortic valves in response to circumferential cyclic stretch. Ann Biomed Eng 34:1655–1665

    Article  Google Scholar 

  • Balachandran K, Sucosky P, Jo H, Yoganathan AP (2009) Elevated cyclic stretch alters matrix remodeling in aortic valve cusps: implications for degenerative aortic valve disease. Am J Physiol Heart Circ Physiol 296:H756–764

    Article  Google Scholar 

  • Bouma W, Lai EK, Levack MM, Shang EK, Pouch AM et al (2016) Preoperative three-dimensional valve analysis predicts recurrent ischemic mitral regurgitation after mitral annuloplasty. Ann Thorac Surg 101:567–575

    Article  Google Scholar 

  • Braunberger E, Deloche A, Berrebi A, Abdallah F, Celestin JA et al (2001) Very long-term results (more than 20 years) of valve repair with carpentier’s techniques in nonrheumatic mitral valve insufficiency. Circulation 104:I8–11

    Article  Google Scholar 

  • Cardamone L, Valentin A, Eberth JF, Humphrey JD (2009) Origin of axial prestretch and residual stress in arteries. Biomech Model Mechanobiol 8:431–446

    Article  Google Scholar 

  • Carpentier A (1983) Cardiac valve surgery-the “french correction”. J Thorac Cardiovasc Surg 86:323–337

    Google Scholar 

  • Carruthers CA, Alfieri CM, Joyce EM, Watkins SC, Yutzey KE et al (2012a) Gene expression and collagen fiber micromechanical interactions of the semilunar heart valve interstitial cell. Cell Mol Bioeng 5:254–265

    Article  Google Scholar 

  • Carruthers CA, Good B, D’Amore A, Liao J, Amini R et al (2012b) Alterations in the microstructure of the anterior mitral valve leaflet under physiological stress. In: ASME 2012 summer bioengineering conference, 2012b. American Society of Mechanical Engineers, pp 227–228

  • Chandran PL, Barocas VH (2006) Affine versus non-affine fibril kinematics in collagen networks: theoretical studies of network behavior. J Biomech Eng 128:259–270

    Article  Google Scholar 

  • Chuong CJ, Fung YC (1986a) On residual stress in arteries. J Biomech Eng 108:189–192

    Article  Google Scholar 

  • Chuong CJ, Fung YC (1986b) Residual stress in arteries. In: Schmid-Schonbein G, Woo SLY, Zweifach B (eds) Frontiers in biomechanics. Springer, New York, pp 117–129

    Chapter  Google Scholar 

  • Dahl KN, Ribeiro AJ, Lammerding J (2008) Nuclear shape, mechanics, and mechanotransduction. Circ Res 102:1307–1318

    Article  Google Scholar 

  • Eckert CE, Zubiate B, Vergnat M, Gorman JH 3rd, Gorman RC et al (2009) In vivo dynamic deformation of the mitral valve annulus. Ann Biomed Eng 37:1757–1771

    Article  Google Scholar 

  • Fan R, Sacks MS (2014) Simulation of planar soft tissues using a structural constitutive model: finite element implementation and validation. J Biomech 47:2043–2054

    Article  Google Scholar 

  • Flameng W, Herijgers P, Bogaerts K (2003) Recurrence of mitral valve regurgitation after mitral valve repair in degenerative valve disease. Circulation 107:1609–1613

    Article  Google Scholar 

  • Flameng W, Meuris B, Herijgers P, Herregods M-C (2008) Durability of mitral valve repair in Barlow disease versus fibroelastic deficiency. J Thorac Cardiovas Surg 135:274–282

    Article  Google Scholar 

  • Fung YC (1991) What are the residual stresses doing in our blood vessels? Ann Biomed Eng 19:237–249

    Article  Google Scholar 

  • Fung YC (1993) Biomechanics: mechanical properties of living tissues, 2nd edn. Springer, New York

    Book  Google Scholar 

  • Fung YC, Liu SQ (1991) Changes of zero-stress state of rat pulmonary arteries in hypoxic hypertension. J Appl Physiol 70:2455–2470

    Article  Google Scholar 

  • Gilbert TW, Sacks MS, Grashow JS, Woo SL, Badylak SF et al (2006) Fiber kinematics of small intestinal submucosa under biaxial and uniaxial stretch. J Biomech Eng 128:890–898

    Article  Google Scholar 

  • Gillinov AM, Blackstone EH, Nowicki ER, Slisatkorn W, Al-Dossari G et al (2008) Valve repair versus valve replacement for degenerative mitral valve disease. J Thorac Cardiovas Surg 135:885–893 e882

    Article  Google Scholar 

  • Gorman JH 3rd, Gupta KB, Streicher JT, Gorman RC, Jackson BM et al (1996) Dynamic three-dimensional imaging of the mitral valve and left ventricle by rapid sonomicrometry array localization. J Thorac Cardiovasc Surg 112:712–726

    Article  Google Scholar 

  • Grashow JS (2005) Evaluation of the biaxial mechanical properties of the mitral valve anterior leaflet under physiological loading conditions. Master’s Thesis, University of Pittsburgh

  • Grashow JS, Sacks MS, Liao J, Yoganathan AP (2006a) Planar biaxial creep and stress relaxation of the mitral valve anterior leaflet. Ann Biomed Eng 34:1509–1518

    Article  Google Scholar 

  • Grashow JS, Yoganathan AP, Sacks MS (2006b) Biaixal stress-stretch behavior of the mitral valve anterior leaflet at physiologic strain rates. Ann Biomed Eng 34:315–325

    Article  Google Scholar 

  • Jassar AS, Brinster CJ, Vergnat M, Robb JD, Eperjesi TJ et al (2011) Quantitative mitral valve modeling using real-time three-dimensional echocardiography: technique and repeatability. Ann Thorac Surg 91:165–171

    Article  Google Scholar 

  • Kunzelman KS, Cochran RP, Chuong C, Ring WS, Verrier ED et al (1993) Finite element analysis of the mitral valve. J Heart Valve Dis 2:326–340

    Google Scholar 

  • Lee CH, Amini R, Gorman RC, Gorman JH 3rd, Sacks MS (2014) An inverse modeling approach for stress estimation in mitral valve anterior leaflet valvuloplasty for in-vivo valvular biomaterial assessment. J Biomech 47:2055–2063

    Article  Google Scholar 

  • Lee C-H, Amini R, Sakamoto Y, Carruthers CA, Aggarwal A et al (2015a) Mitral valves: A computational framework. In: Multiscale modeling in biomechanics and mechanobiology. Springer, pp 223–255

  • Lee C-H, Carruthers CA, Ayoub S, Gorman RC, Gorman JH et al (2015b) Quantification and simulation of layer-specific mitral valve interstitial cells deformation under physiological loading. J Theor Biol 373:26–39

    Article  MATH  Google Scholar 

  • Lee CH, Rabbah JP, Yoganathan AP, Gorman RC, Gorman JH 3rd et al (2015c) On the effects of leaflet microstructure and constitutive model on the closing behavior of the mitral valve. Biomech Model Mechanobiol 14:1281–1302

    Article  Google Scholar 

  • Lee CH, Zhang W, Liao J, Carruthers CA, Sacks JI et al (2015d) On the presence of affine fibril and fiber kinematics in the mitral valve anterior leaflet. Biophys J 108:2074–2087

    Article  Google Scholar 

  • Liao J, Yang L, Grashow J, Sacks MS (2005) Collagen fibril kinematics in mitral valve leaflet under biaxial elongation, creep, and stress relaxation. In: Society for Heart Valve Disease Third Biennial Meeting, Vancouver. SHVD

  • Liao J, Yang L, Grashow J, Sacks MS (2007) The relation between collagen fibril kinematics and mechanical properties in the mitral valve anterior leaflet. J Biomech Eng 129:78–87

    Article  Google Scholar 

  • May-Newman K, Yin FC (1995) Biaxial mechanical behavior of excised porcine mitral valve leaflets. Am J Physiol 269:H1319–1327

    Google Scholar 

  • Nevo E, Lanir Y (1994) The effect of residual strain on the diastolic function of the left ventricle as predicted by a structural model. J Biomech 27:1433–1446

    Article  Google Scholar 

  • Pierlot CM, Lee JM, Amini R, Sacks MS, Wells SM (2014) Pregnancy-induced remodeling of collagen architecture and content in the mitral valve. Ann Biomed Eng 42:2058–2071

    Article  Google Scholar 

  • Pierlot CM, Moeller AD, Lee JM, Wells SM (2015) Pregnancy-induced remodeling of heart valves. Am J Physiol Heart Circ Physiol 309:H1565–1578

    Article  Google Scholar 

  • Prot V, Skallerud B (2016) Contributions of prestrains, hyperelasticity, and muscle fiber activation on mitral valve systolic performance Int J Numer Meth Biomed Eng. doi:10.1002/cnm.2806

  • Prot V, Skallerud B, Holzapfel G (2007) Transversely isotropic membrane shells with application to mitral valve mechanics. Constitutive modelling and finite element implementation. Int J Numer Methods Eng 71:987–1008

    Article  MathSciNet  MATH  Google Scholar 

  • Prot V, Haaverstad R, Skallerud B (2009) Finite element analysis of the mitral apparatus: Annulus shape effect and chordal force distribution. Biomech Model Mechanobiol 8:43–55

    Article  Google Scholar 

  • Rausch MK, Bothe W, Kvitting JP, Goktepe S, Miller DC et al (2011) In vivo dynamic strains of the ovine anterior mitral valve leaflet. J Biomech 44:1149–1157

    Article  Google Scholar 

  • Rausch MK, Famaey N, Shultz TO, Bothe W, Miller DC et al (2013) Mechanics of the mitral valve: A critical review, an in vivo parameter identification, and the effect of prestrain. Biomech Model Mechanobiol 12:1053–1071

    Article  Google Scholar 

  • Rego BV, Wells SM, Lee CH, Sacks MS (2016) Mitral valve leaflet remodelling during pregnancy: Insights into cell-mediated recovery of tissue homeostasis. J R Soc Interface 13(125):20160709. doi:10.1098/rsif.2016.0709

    Article  Google Scholar 

  • Sacks MS, Smith DB, Hiester ED (1997) A small angle light scattering device for planar connective tissue microstructural analysis. Ann Biomed Eng 25:678–689

    Article  Google Scholar 

  • Sacks MS, He Z, Baijens L, Wanant S, Shah P et al (2002) Surface strains in the anterior leaflet of the functioning mitral valve. Ann Biomed Eng 30:1281–1290

    Article  Google Scholar 

  • Sacks MS, Enomoto Y, Graybill JR, Merryman WD, Zeeshan A et al (2006) In-vivo dynamic deformation of the mitral valve anterior leaflet. Ann Thorac Surg 82:1369–1377

    Article  Google Scholar 

  • Sacks MS, Zhang W, Wognum S (2016) A novel fibre-ensemble level constitutive model for exogenous cross-linked collagenous tissues. Interface focus 6(1):20150090. doi:10.1098/rsfs.2015.0090

    Article  Google Scholar 

  • Sakamoto Y, Buchana RM, Sanchez-Adams J, Guilak F, Sacks MS (2017) On the functional role of valve interstitial cell stress fibers: a continuum modeling approach. J Biomech Eng 139:021007

    Article  Google Scholar 

  • Votta E, Caiani E, Veronesi F, Soncini M, Montevecchi FM et al (2008) Mitral valve finite-element modelling from ultrasound data: a pilot study for a new approach to understand mitral function and clinical scenarios. Philos Trans A Math Phys Eng Sci 366:3411–3434

    Article  Google Scholar 

  • Wang Q, Sun W (2013) Finite element modeling of mitral valve dynamic deformation using patient-specific multi-slices computed tomography scans. Ann Biomed Eng 41:142–153

    Article  Google Scholar 

  • Wells SM, Pierlot CM, Moeller AD (2012) Physiological remodeling of the mitral valve during pregnancy. Am J Physiol Heart Circ Physiol 303:H878–892

    Article  Google Scholar 

  • Zhang W, Ayoub S, Liao J, Sacks MS (2016) A meso-scale layer-specific structural constitutive model of the mitral heart valve leaflets. Acta Biomater 32:238–255

    Article  Google Scholar 

Download references

Acknowledgements

Support from the National Institutes of Health (NIH) Grants R01 HL119297, HL63954, HL103723, and HL73021 is gratefully acknowledged. Dr. Chung-Hao Lee was in part supported by the start-up funds from the School of Aerospace and Mechanical Engineering (AME) at the University of Oklahoma, and the American Heart Association Scientist Development Grant Award (16SDG27760143).

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Appendix: Modifications of the full structural constitutive model to account for a change in the referential configurations for structural model parameters

Appendix: Modifications of the full structural constitutive model to account for a change in the referential configurations for structural model parameters

The formulation described in this appendix allows for handling the changes in tissue geometry at the reference state due to mechanisms, such as preconditioning, layer separation, removal of tissue component via enzyme degradation and others. The framework is also applicable to employed in vitro measured mechanical properties for in vivo modeling of the MV. Several key assumptions were considered:

  1. (i)

    Changes are due to alterations in the reference configuration only; the mass fractions of tissue constituents remain unchanged and the internal mechanical energy remains zero.

  2. (ii)

    Changes of tissue dimensions and internal architecture are in accordance with the collagen fiber affine assumption (Chandran and Barocas 2006; Lee et al. 2015d). Thus, the configurational change is homogeneous and can be described by tissue-level deformation gradient tensor.

  3. (iii)

    Changes associated with alterations of referential configurations result only in collagen fiber orientation, degree of fiber undulation (fiber recruitment), and tissue dimensions.

Structural parameters of the tissue are most conveniently obtained in state \(\beta _{0}\), which refers to in vitro reference configuration, and the tissue Helmholtz strain energy function \(\Psi \) is defined in state \(\beta _{2}\) for in vivo modeling of the MVAL tissue at any pressure-loaded state \(\beta _{t}\). Hence, our goal here is to seek the expressions with the quantified structural parameters at state \(\beta _{2}\).

In the formulations below, we first adopted the following notations:

$$\begin{aligned}&{}_0^t \mathbf{F}=\left( {_2^t \mathbf{F}} \right) \left( {_0^2 \mathbf{F}} \right) \nonumber \\&{}_0^t E_\mathrm{ens} =\mathbf{n}^{T}\left( {_0^t \mathbf{E}} \right) \mathbf{n} \nonumber \\&{} \mathbf{E} = 1/2\left( {\mathbf{F}^{T}{} \mathbf{F}-\mathbf{I}} \right) , \end{aligned}$$
(6)

where \(_0^t E_\mathrm{ens} \) is the effective Green strain of the collagen fiber ensemble at state \(\beta _{t}\) with respect to \(\beta _{0}\), and \(\mathbf{n}=[\hbox {cos}(\theta _{0}),\hbox {sin}(\theta _{0})]^{T}\) is the unit vector in which the collagen fiber orients in the in vitro reference configuration.

1.1 Changes in the collagen fiber orientation density function (ODF)

The assumption of fiber affine deformation can be described by the relation \(d\mathbf{x}=\mathbf{F}d\mathbf{X}\), which maps the material line segment \(d\mathbf{X}\) at the undeformed configuration to the corresponding deformed configuration \(d\mathbf{x}\) via the homogeneous tissue-level deformation gradient tensor F. Let us restrict our discussion on planar soft biological tissues, the fiber ODF \(\Gamma _2 (\theta _2 ;{\mu }'_\mathrm{c} ,{\sigma } '_\mathrm{c} )\) at state \(\beta _{2}\) can be related to the measured ODF \(\Gamma _0 (\theta _0 ;\mu _\mathrm{c} ,\sigma _\mathrm{c} )\) at state \(\beta _{0}\) by adopting Nanson’s relation in finite strain theory (Fan and Sacks 2014):

$$\begin{aligned} \Gamma _2 \left( {\theta _2 ;\mu _\mathrm{c} ,\sigma _\mathrm{c}} \right) \,=\,\Gamma _0 \left( {\theta _0 ;\mu _\mathrm{c} ,\sigma _\mathrm{c}} \right) \frac{\mathbf{n}\left( {\theta _0} \right) \cdot \left[ {\left( {_0^2 \mathbf{C}} \right) \mathbf{n}\left( {\theta _0} \right) } \right] }{\det \left( {_0^2 \mathbf{F}} \right) }, \end{aligned}$$
(7)

where \(_0^2 \mathbf{C}=(_0^2 \mathbf{F})^{T}(_0^2 \mathbf{F})\); the statistical distribution parameters are \(\mu _\mathrm{c} \) and \(\sigma _\mathrm{c} \) at state \(\beta _{0}\). The fiber orientation angle at state \(\beta _{2}\) is determined by the relation below (Gilbert et al. 2006; Lee et al. 2015c, d):

$$\begin{aligned} \theta _2 =\tan ^{-1}\left[ {\frac{\left( {_0^2 \mathbf{F}} \right) _{21} \cos \left( {\theta _0} \right) +\left( {_0^2 \mathbf{F}} \right) _{22} \sin \left( {\theta _0} \right) }{\left( {_0^2 \mathbf{F}_{11}} \right) \cos \left( {\theta _0} \right) +\left( {_0^2 \mathbf{F}} \right) _{12} \sin \left( {\theta _0} \right) }} \right] . \end{aligned}$$
(8)

The modified collagen fiber ODF \(\Gamma _2 (\theta _2 )\) due to the change in the referential configurations, resulting in the pre-strains \(_0^2 \mathbf{F}\), is depicted in Fig. 5b.

1.2 Changes in the collagen fiber ensemble formulation

Next, let us consider an ensemble collection of collagen fibers with varying fiber slack strain and use a fiber recruitment function \(D_0 (_0^t E_\mathrm{ens} ;\mu _\mathrm{D} ,\sigma _\mathrm{D} ,{} _0E_\mathrm{lb} ,{} _0E_\mathrm{ub} )\) at the ensemble level to account for a range of fiber slack strains as referred to state \(\beta _{0}\) by:

$$\begin{aligned}&D_0 \left( {_0^t E_\mathrm{ens} ;\mu _\mathrm{D} ,\sigma _\mathrm{D} ,{} _0E_\mathrm{lb} ,{} _0E_\mathrm{ub}} \right) \nonumber \\&\quad =\left\{ {{\begin{array}{ll} \frac{y^{\alpha -1}\left( {1-y} \right) ^{\beta -1}}{\hbox {Beta}\left( {\alpha ,\beta } \right) \left( {_0 E_\mathrm{ub} -_0 E_\mathrm{lb}} \right) } &{}\quad y\in \left[ {0,1} \right] \nonumber \\ 0 &{}\quad \hbox {otherwise} \\ \end{array}} } \right. \\&y=\frac{{} _0^t E_\mathrm{ens} -{} _0E_\mathrm{lb}}{{} _0E_\mathrm{ub} -{} _0E_\mathrm{lb}} ,\quad \bar{{\mu } }_\mathrm{D} =\frac{\mu _\mathrm{D} -{} _0E_\mathrm{lb}}{{} _0E_\mathrm{ub} -{} _0E_\mathrm{lb}} ,\nonumber \\&\bar{{\sigma } }_\mathrm{D} =\frac{\sigma _\mathrm{D}}{{} _0E_\mathrm{ub} -{} _0E_\mathrm{lb}} ,\nonumber \\&\alpha =\frac{\bar{{\mu } }_\mathrm{D}^2 -\bar{{\mu } }_\mathrm{D}^3 -\bar{{\sigma } }_\mathrm{D}^2 \bar{{\mu } }_\mathrm{D}}{\bar{{\sigma } }_\mathrm{D}^2} ,\quad \beta =\alpha \frac{1-\bar{{\mu } }_\mathrm{D}}{\bar{{\mu } }_\mathrm{D}}, \end{aligned}$$
(9)

where \(\hbox {Beta}\left( {\alpha ,\beta } \right) \) is the Beta distribution function with shape parameters \(\alpha \) and \(\beta \).

For the recruitment parameters to be recast with respect to state \(\beta _{2}\), we seek for the modified fiber recruitment function \({D}_{2}\) with the same Beta distribution parameters and by using the mapping from in vitro reference configuration \((\beta _{0})\) to in vivo reference configuration \((\beta _{2})\) below

$$\begin{aligned}&D_2 \left( {_2^t E_\mathrm{ens} ;\mu _\mathrm{D} ,\sigma _\mathrm{D}} \right) \nonumber \\&\quad =\left\{ {{\begin{array}{ll} \frac{{y}'^{\alpha -1}\left( {1-{y}'} \right) ^{\beta -1}}{\hbox {Beta}\left( {\alpha ,\beta } \right) \left( {_2 E_\mathrm{ub} -_2 E_\mathrm{lb}} \right) }&{}\quad {y}'\in \left[ {0,1} \right] \\ 0&{}\quad \hbox {otherwise} \\ \end{array}} } \right. \nonumber \\&{y}'=\frac{{} _2^t E_\mathrm{ens} -{} _2E_\mathrm{lb}}{{} _2E_\mathrm{ub} -{} _2E_\mathrm{lb}},\quad {} _2^t E_\mathrm{ens} =\frac{{} _0^t E_\mathrm{ens} -{} _0^2 E_\mathrm{ens}}{1+2{} _0^2 E_\mathrm{ens}} , \nonumber \\&{} _2E_\mathrm{lb} =\frac{{} _0E_\mathrm{lb} -{} _0^2 E_\mathrm{ens}}{1+2{} _0^2 E_\mathrm{ens}} ,\quad {} _2E_\mathrm{ub} =\frac{{} _0E_\mathrm{ub} -{} _0^2 E_\mathrm{ens}}{1+2{} _0^2 E_\mathrm{ens}}. \end{aligned}$$
(10)

The modified fiber recruitment function associated with collagen fiber oriented along the circumferential direction \((\theta _{0}=0{^\circ })\) due to the change in the referential configurations as described by the pre-strains \(_0^2 \mathbf{F}\) is depicted in Fig. 5c.

Finally, the resulting expression for the ensemble fiber stress–strain relationship with respect to state \(\beta _{2}\) can be obtained

$$\begin{aligned}&S_\mathrm{ens} \left( {_2^t E_\mathrm{ens}} \right) \nonumber \\&\quad =\eta _\mathrm{c} \int _0^{_2^t E_\mathrm{ens} } {\frac{D_2 \left( {x;\mu _\mathrm{D} ,\sigma _\mathrm{D}} \right) }{\left( {1+2x} \right) }\left( {1-\frac{\sqrt{1+2x}}{\sqrt{1+2_2^t E_\mathrm{ens}} }} \right) \hbox {d}{x}}.\nonumber \\ \end{aligned}$$
(11)

1.3 Complete formulation of the tissue strain energy function with respect to state \(\beta _{2}\)

Therefore, we can facilitate in vivo modeling and stress estimation using the final form of tissue stress–strain relationship

$$\begin{aligned} {}_2^t \mathbf{S}_\mathrm{c} \left( {_2^t \mathbf{C}} \right)= & {} \eta _\mathrm{e}^{a} \left( {{I}'_4 -1} \right) ^{a} \mathbf{{n}'}_\mathrm{C} \otimes \mathbf{{n}'}_\mathrm{C} +\eta _\mathrm{e}^{b} \left( {{I}'_6 -1} \right) ^{b}{} \mathbf{{n}'}_R \otimes \mathbf{{n}'}_R \nonumber \\&+\, \mu _\mathrm{m} \left[ {\mathbf{I}-\left( {_2^t C_{33}} \right) \left( {_2^t \mathbf{C}^{-1}} \right) } \right] \nonumber \\&+\, \eta _\mathrm{c} \int _{-\frac{\pi }{2}}^{\frac{\pi }{2}} \Gamma _2 \left( {\theta _2 ;\mu _\mathrm{C} ,\sigma _\mathrm{C}} \right) \nonumber \\&\left[ \int _0^{_2^t E_\mathrm{ens}} \frac{D_2 \left( {x;\mu _\mathrm{D} ,\sigma _\mathrm{D}} \right) }{\left( {1+2x} \right) }\right. \nonumber \\&\left. \left( {1-\frac{\sqrt{1+2x}}{\sqrt{1+2_2^t E_\mathrm{ens}} }} \right) \hbox {d}x \right] \mathbf{{n}'}\left( {\theta _2} \right) \otimes \mathbf{{n}'}\left( {\theta _2} \right) d\theta _2,\nonumber \\ \end{aligned}$$
(12)

where \({I}'_4 = \mathbf{{n}'}_\mathrm{C} \cdot \left( {_2^t \mathbf{C}} \right) \mathbf{{n}'}_\mathrm{C} \), \({I}'_6 = \mathbf{{n}'}_R \cdot \left( {_2^t \mathbf{C}} \right) \mathbf{{n}'}_R \), \( \mathbf{{n}'}_\mathrm{C} ={ \left( {_0^2 \mathbf{F}} \right) \mathbf{n}_\mathrm{C}} /{\left| { \left( {_0^2 \mathbf{F}} \right) \mathbf{n}_\mathrm{C}} \right| }\), \( \mathbf{{n}'}_R ={ \left( {_0^2 \mathbf{F}} \right) \mathbf{n}_\mathrm{C}} /{\left| { \left( {_0^2 \mathbf{F}} \right) \mathbf{n}_\mathrm{C}} \right| }\). \( \mathbf{{n}'}(\theta _2 )=[\cos (\theta _2 ),\sin (\theta _2 )]^{T}\) with the fiber orientation angle with respect to state \(\beta _{2}\) as determined by Eq. (8).

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Lee, CH., Zhang, W., Feaver, K. et al. On the in vivo function of the mitral heart valve leaflet: insights into tissue–interstitial cell biomechanical coupling. Biomech Model Mechanobiol 16, 1613–1632 (2017). https://doi.org/10.1007/s10237-017-0908-4

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