Abstract
There continues to be a critical need for developing datainformed computational modeling techniques that enable systematic evaluations of mitral valve (MV) function. This is important for a better understanding of MV organlevel 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 populationaveraged 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 prestrains 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 crossverified 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 tissuelevel kinematic results into a downscale MVIC microenvironment FE model, we were able to estimate, for the first time, the in vivo layerspecific 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 layerspecific manner. This suggests that the associated reductions in MVIC deformation may downregulate 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 tissuelevel 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 longterm 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 nonfibrous matrix
 \(\mathbf{C}=\mathbf{F}^{T}{} \mathbf{F}\) :

RightCauchy deformation tensor
 \(\mathbf{E}=(\mathbf{C}\mathbf{I})/2\) :

Green–Lagrange strain tensor
 \(\mathbf{I}\) :

2ndrank 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 nonfibrous matrix represented by a neoHookean 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 lowerbound limit \(E_\mathrm{lb}\) at which collagen fiber recruitment begins, a lowerbound limit \(E_\mathrm{ub}\) at which collagen fiber is fully recruited
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Acknowledgements
Support from the National Institutes of Health (NIH) Grants R01 HL119297, HL63954, HL103723, and HL73021 is gratefully acknowledged. Dr. ChungHao Lee was in part supported by the startup 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:

(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.

(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 tissuelevel deformation gradient tensor.

(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 pressureloaded 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:
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 tissuelevel 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):
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):
The modified collagen fiber ODF \(\Gamma _2 (\theta _2 )\) due to the change in the referential configurations, resulting in the prestrains \(_0^2 \mathbf{F}\), 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:
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
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 prestrains \(_0^2 \mathbf{F}\) 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
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
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/s1023701709084
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DOI: https://doi.org/10.1007/s1023701709084