Right Ventricular Strain, Torsion, and Dyssynchrony in Healthy Subjects Using 3D Spiral Cine DENSE Magnetic Resonance Imaging
Jonathan D SueverGregory J WehnerLinyuan JingDavid K. PowellSean M HamletJonathan D. GrabauDimitri MojsejenkoKristin N. AndresChristopher M. HaggertyBrandon K. Fornwalt
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Mechanics of the left ventricle (LV) are important indicators of cardiac function. The role of right ventricular (RV) mechanics is largely unknown due to the technical limitations of imaging its thin wall and complex geometry and motion. By combining 3D Displacement Encoding with Stimulated Echoes (DENSE) with a post-processing pipeline that includes a local coordinate system, it is possible to quantify RV strain, torsion, and synchrony. In this study, we sought to characterize RV mechanics in 50 healthy individuals and compare these values to their LV counterparts. For each cardiac frame, 3D displacements were fit to continuous and differentiable radial basis functions, allowing for the computation of the 3D Cartesian Lagrangian strain tensor at any myocardial point. The geometry of the RV was extracted via a surface fit to manually delineated endocardial contours. Throughout the RV, a local coordinate system was used to transform from a Cartesian strain tensor to a polar strain tensor. It was then possible to compute peak RV torsion as well as peak longitudinal and circumferential strain. A comparable analysis was performed for the LV. Dyssynchrony was computed from the standard deviation of regional activation times. Global circumferential strain was comparable between the RV and LV (-18.0% for both) while longitudinal strain was greater in the RV (-18.1% vs. -15.7%). RV torsion was comparable to LV torsion (6.2 vs. 7.1 degrees, respectively). Regional activation times indicated that the RV contracted later but more synchronously than the LV. 3D spiral cine DENSE combined with a post-processing pipeline that includes a local coordinate system can resolve both the complex geometry and 3D motion of the RV.Keywords:
Infinitesimal strain theory
Finite strain theory
Radial stress
It is widely accepted that any finite strain recorded in the field may be interpreted in terms of the simultaneous combination of a pure shear component with one or several simple shear components. To predict strain in geological structures, approximate solutions may be obtained by multiplying successive small increments of each elementary strain component. A more rigorous method consists in achieving the simultaneous combination in the velocity gradient tensor, but solutions already proposed in the literature are valid for some special cases only and cannot be used, e.g., for the general combination of a pure shear component and six elementary simple shear components. In this paper, we show that the combination of any strain components is very simple, both analytically and numerically. The finite deformation tensor is given by D = exp ( L Δ t ), where L Δ t is the time‐integrated velocity gradient tensor. This method makes it possible to predict finite strain for any combination of strain components. Reciprocally, L Δ t = ln ( D ), which allows us to unravel the simplest deformation history that might have generated a given finite deformation. Given the strain ellipsoid only, it is still possible to constrain the range of compatible deformation tensors and thus the range of strain component combinations. Interestingly, certain deformation tensors, though geologically sensible, have no real logarithm and so cannot be explained by a deformation history implying strain rate components with a common time dependence. This implies significant changes of stress field or material rheology during deformation.
Finite strain theory
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Finite strain theory
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Infinitesimal strain theory
Finite strain theory
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Hyperelastic material
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Cauchy elastic material
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This chapter contains sections titled: Deformation Tensors Strain Tensors Compatibility Condition Strain Rate and Spin Tensors Representations of Strain Rate and Spin Tensors in Lagrangian and Eulerian Triads Decomposition of Deformation Gradient Tensor into Isochoric and Volumetric Parts
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The generalised deformation of strain has major importance for deformation of rubbers where the strains are generally not small. The role of rigid body rotations, polar decomposition and principal extension ratios are explained, together with examples of elementary strain fields, logarithmic strain and the stress tensor. Stress-strain relationships are developed for finite strain analogous to the generalised Hooke's Law for small strains. The use of a strain function for finite deformation requires thermodynamic considerations i.e. the relationship to Helmholtz and Gibbs free energies. Finally, consideration of the formulation of the strain energy in terms of strain invariants or, more directly, extension ratios.
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Abstract. Image correlation techniques have provided new ways to analyze the distribution in space and time of deformation in analogue models of tectonics. Here we demonstrate how the correlation of successive time-lapse images of a deforming model allows not only to evaluate the components of the strain-rate tensor at any time in the model but also calculate the finite displacements and finite strain tensor. We illustrate, using synthetic images, the ability of the algorithm to produce maps of the velocity gradients, small-strain tensor components, but also incremental or instantaneous principal strains and maximum shear. The incremental displacements can then summed up using a Eulerian or a Lagrangian summation, and the components of the 2-D finite strain tensor can be calculated together with the finite principal strain and maximum finite shear. We benchmark the measures of finite displacements using specific synthetic tests for each summation mode. The deformation gradient tensor is calculated from the deformed state, and decomposed into the finite rigid-body rotation and left or right finite stretch tensors, allowing the deformation ellipsoids to be drawn. The finite strain has long been the only quantified measure of strain in analogue models. The presented software package allows producing these finite strain measures while also accessing incremental measures of strain. The more complete characterization of the deformation of tectonic analogue models will facilitate the comparison with numerical simulations and geological data, and help produce conceptual mechanical models.
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Finite strain theory
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Abstract. Image correlation techniques have provided new ways to analyse the distribution of deformation in analogue models of tectonics in space and time. Here, we demonstrate, using a new version of our software package (TecPIV), how the correlation of successive time-lapse images of a deforming model allows not only to evaluate the components of the strain-rate tensor at any time in the model but also to calculate the finite displacements and finite strain tensor. We illustrate with synthetic images how the algorithm produces maps of the velocity gradients, small-strain tensor components, incremental or instantaneous principal strains and maximum shear. The incremental displacements can then be summed up with Eulerian or Lagrangian summation, and the components of the 2-D finite strain tensor can be calculated together with the finite principal strain and maximum finite shear. We benchmark the measures of finite displacements using specific synthetic tests for each summation mode. The deformation gradient tensor is calculated from the deformed state and decomposed into the finite rigid-body rotation and left or right finite-stretch tensors, allowing the deformation ellipsoids to be drawn. The finite strain has long been the only quantified measure of strain in analogue models. The presented software package allows producing these finite strain measures while also accessing incremental measures of strain. The more complete characterisation of the deformation of tectonic analogue models will facilitate the comparison with numerical simulations and geological data and help produce conceptual mechanical models.
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