Studying Dynamic Myofiber Aggregate Reorientation in Dilated Cardiomyopathy Using In Vivo Magnetic Resonance Diffusion Tensor Imaging

Supplemental Digital Content is available in the text.


Supplemental Material B Overview of the biomechanical model
The biomechanical model is based on the classical continuum approach combined with finite element discretisation. This supplement provides a brief overview of the mathematical and computational model. Further particulars of the model and methods used can be found in [2,3].
Let the reference configuration of the myocardium be denoted by Ω 0 with the coordinate X, and a deformed configuration at time t  (0,T], T>0, be denoted by Ω t with the coordinate x. At a given time t domain deformations can be defined as u = x -X, while the hydrostatic pressure and the boundary tractions (described below) are denoted by p and λ respectively. The principle of stationary potential energy [4] states that at any time t this deformation can be found by minimising the total energy of the system: The total energy can be separated into internal and external energy terms: We assume that myocardium can be modelled as a hyperelastic incompressible tissue, so that the internal energy can be written in terms of the strain energy function Ψ, the hydrostatic pressure p and the determinant of the deformation gradient J = det(F), where F =  X u + I, and I is the identity matrix: The strain energy function incorporates orthotropic passive and active behaviour of the tissue: The model employs the Holzapfel-Ogden passive constitutive law [5], defined as follows: where a, b, a f , b f , a s , b s , a fs , b fs represent material parameters (values given in Table S2 below), and I 1 , I 4f , I 4s , I 8fs are the strain invariants associated with fibre and sheet directions. Specifically, if at a given location in the reference domain the myocyte aggregate (referred to as fibre in the model) and sheet orientation vectors are f 0 and s 0 respectively, and C = F T F is the right Cauchy-Green strain tensor, then Active response is produced with a simplified version of Kerchoffs-type length-dependent constitutive laws [6,7] with reference myocyte compressive strain l 0 =0.8, global active tension value AT, and added transverse activation at 30% of the fibre activation as discussed in [8,9]: The external energy Π ext comes from any forces acting on the boundaries of the domain. Ventricular cavity volume is set via an endocardial energy term where λ endo is a scalar endocardial Lagrange multiplier representing cavity pressure, V the cavity volume produced by the simulation, and V data the prescribed cavity volume. The cavity volume can be approximated as follows [2]: where Γ endo denotes the deformed configuration of the endocardial surface of the ventricle, n b is the base normal vector and n the outward endocardial normal vector.
A simpified base condition allowing sliding in plane only was imposed due to the generic nature of the model. The base plane was aligned with the z = 0 plane, and the centre of the base coincided with the origin. The external energy term on the base could be written as follows: where Γ base denotes the deformed configuration of the base surface of the ventricle, λ base is a spatially varying Lagrange multiplier enforcing no longitudinal motion of the base plane, n b is the base normal vector, λ 0,1 and λ 0,2 are scalars ensuring no translation of the base centre, and λ 2 is a scalar enforcing no rotation around the axis.
With this definition of the energy terms, and combining all Lagrange multipliers in a vector λ=(λ endo ,λ base ,λ 0,1 ,λ 0,2 ,λ 2 ), the full state of the system (u,p,λ) at a given time t is found as the critical point of the total energy Π: This equation provides the weak form of the problem, which can be solved numerically using the finite element method. The reference domain is discretised into quadratic hexahedral elements, with Q2-Q1 approximation for the displacement-pressure pair, quadratic approximation on quadrilateral surface elements for λ base , and constant approximations for λ endo , λ 0,1 , λ 0,2 and λ 2 . The resulting nonlinear system is solved via Newton-Raphson iteration with line search, with linear solve steps carried out by direct matrix inversion.
All simulation results were obtained using CHeart [10], a parallel multiphysics software engine.

Test specifications
Two reference geometries were produced: one to represent a generic healthy ventricle, and another to represent a generic DCM ventricle. Both shapes were simplified as ellipsoids cropped in the short axis plane below the base of the ventricle. The shapes were adjusted in such a way that passive inflation to prescribed end-diastolic volume produced representative short and long axis dimensions and wall thickness. The prescribed cavity volume, both at end diastole and end systole, was set to values lower than the data averages to account for ventricle truncation.
The end-systolic state for each geometry was obtained by prescribing end-systolic volume, and gradually increasing active tension in the tissue to reach end-systolic cavity pressures of ~100 mmHg. It should be noted that the ventricle dimensions, as well as the HA/E2A values undergo significant changes at the first few activation steps (AT < 25 kPa), and then stabilise, meaning that the precise cut-off pressure (with AT > 100 kPa) has virtually no effect on these measurements. The prescribed and observed metrics are presented in Table S1.

Supplemental Material C Mean Diffusivity (MD) and Fractional Anisotropy (FA) Maps
Upon tensor reconstruction, MD and FA maps were computed for DCM and control ( Figure S2).
MD and FA are defined as follows:  Representative Raw Data Figure S3 shows example raw data images for DTI (b=0, b=400 s/mm 2 ) and 3D tagging data. Figure S3: Example raw data images for Control (a,b) and DCM (c,d).

Supplemental Material D
Axial (  ) and radial (   ) diffusivities were computed in the left ventricle for both cohorts and heart phases. They are defined as follows: With the three eigenvalues of the diffusion tensor: 1 2 3 , , Table S3 reports  and   as median and interquartile ranges across both groups. DCM diffusivities were found to be increased or at least equal to corresponding diffusion metrics of the control group. Diastolic diffusion data is in very good agreement with previous data [12], while systolic data shows an increase in axial and radial diffusivity for DCM compared to controls. Differences in diffusivities between DCM and control were determined by Wilcoxon rank sum testing.
According to Tsagalou et al [13], DCM patients suffer from depressed coronary flow reserve and reduced capillary density compared to healthy subjects. Hence the impact of perfusion on radial and axial diffusivities is expected to be reduced in DCM relative to controls. It is speculated that actual diffusivities (without perfusion bias) for the control group may result in clearer statistical significances between both cohorts.  Table S3. Axial and radial diffusivities for control and DCM.