ISOTROPIC VIRTUAL FIELDS METHOD
IN this chapter, both analytic and optimised (numeric) versions of the isotropic virtual fields method are implemented to estimate a complex shear modulus, containing both real and imaginary components, from a 3D harmonic displacement field. The method requires a set of measured full-field displacements to directly solve for material properties. The primary advantage of this method over the boundary value FEMU method is that the subjective choice of boundary conditions is not required and all data points are weighted equally. Additionally, it is a direct (non-iterative) method, which is less computationally intensive than the FEMU method. The method was implemented with simulated displacements in an isotropic beam and left ventricular model. Further, the method was tested with MR elastography data collected from an isotropic phantom.
Analytic Virtual Fields Method
The virtual fields method (VFM) was developed by Gr´ediac et al.  as a method for the identification of material properties from mechanical tests using imaging data. The advantage of the VFM approach is that image data are used throughout the domain, resulting in a more direct estimation procedure compared with the traditional method of solving boundary value problems. The VFM utilises the principle of virtual work, which generally states that ”a continuous body is at equilibrium if the virtual work of all forces acting on the body is null for any kinematically admissible virtual displacement” . The principle of virtual work, identical to Equation 3.1, is written in this chapter denoting the virtual displacement field as u to be consistent with other VFM papers. is the density and a is the acceleration. and u are the virtual strains and virtual displacements, respectively. The first term represents the virtual work done by internal forces; the second and third terms represent virtual work done by external forces; and the right-hand side term represents the virtual work done by acceleration. The virtual fields must be 1) kinematically admissible (i.e. consistent with the boundary conditions of the problem) and 2) piecewise-C0 continuous. In biological tissue, body forces such as gravity are in equilibrium with reaction forces imposed by surrounding tissue. For most MRE problems, the body forces, b, can be assumed to be negligible since we assume that they don’t cause a resulting displacement, particularly in the time scale measured during one harmonic cycle.Additionally, one common strategy is to set the virtual displacement field to zero at any locations where the boundary traction forces are unknown, thereby eliminating the second term of the equation. In harmonic problems involving linear elastic media (such as elastography), the resultant displacement frequency is the same as the load frequency. The acceleration can be written as the product of the angular frequency squared and measured displacement field (!2u). where ! = 2f and f is the frequency of excitation in Hz. Equation (5.3) relies on four measurements: the internal stress field (which carries in it the unknown material parameters), density, angular frequency and the resulting displacement field.
The VFM was initially used to estimate material properties from MR elastography displacements by implementing analytic virtual displacement fields [117, 132]. Sinusoidal analytic virtual fields were applied to simulations of harmonic displacements in a cylindrical geometry and Lam´e parameters were estimated (normalised by density ), = and =. It was shown that the method was accurate except at interfaces between different materials within the cylinder . In another study, analytic virtual displacement fields were strategically chosen such that the bulk component of the internal stress term was zero, eliminating the need to estimate the longitudinal wavelength . Additionally, different analytic virtual displacement fields were investigated, which varied by k, the virtual wave number. The methods in  are outlined below and were implemented similarly to validate the method in a cantilever beam model in this thesis. An isotropic linear elastic material law was used to describe the constitutive behaviour where K represents the bulk modulus and G, the shear modulus. By substituting the constitutive equation into the left hand side of the principle of virtual work (Equation (5.3)), the term for internal stress can be written as K is dependent on the Poisson’s ratio , and consequently, the longitudinal wavelength. In elastography experiments, the longitudinal wavelength is much longer (approximately 300x) than the shear wavelength, and is therefore difficult to estimate since the geometry being imaged is much smaller than the longitudinal wavelength. Equation (5.4) is of a form that is advantageous in VFM applications since the internal stress is written as a linear combination of the bulk (K) and shear (G) components. With the VFM, a virtual displacement field can be chosen such that the internal virtual stress due to the bulk modulus is negligible. This condition was easily met for a simple rectangular geometry by ensuring that the trace of the virtual strain field was zero (Tr() =0).In , this was done by only including one shear term in the virtual displacement field which ensured that 11, 22 and 33 were zero so that Tr() = 11 + 22 + 33 = 0. In the cube geometric model in , the resulting virtual displacement field applied was where Ly was the length of the cube, y was the position in the y-direction and k was the virtual wave number, which controlled the spatial frequency. A sine function, polarised in the x-direction, was used since it closely matched the harmonic displacement field measured experimentally, in which an x-displacement was applied on one face of the cube. The only non-zero component of the resulting virtual strain field was A discrete sum was used to approximate the integrals in the equation of virtual work.
Equations (5.8) and (5.7) were substituted into (5.9) and the equation was written in terms of the homogeneous shear modulus, G.Overbars indicate spatial averaging. In , k was chosen to be six since there were approximately six wavelengths in the y-direction. An index of the sensitivity of the method to accurately estimate shear modulus from shear wave propagation was defined as the product of a and b, the numerator and denominator in Equation (5.10). At low sensitivities, the estimated shear moduli showed large variations. High sensitivity values indicated that the analytic virtual displacement field applied on the measured field at time point ni provided adequate information with which to estimate the shear wavelength. Spatial averages were computed for regions with sensitivity above specified thresholds. Testing thresholds between 0:1 max(a b) and 0:9 max(a b) resulted in shear moduli ranging between 1647 Pa – 1693 Pa. These initial studies showed promising results for a fast inversion method which does not rely on multiple iterations or application of boundary conditions in order to determine material properties from MRE data. Analytic virtual fields that a) enforce zero virtual displacement on the boundaries as well as b) dissociate shear from bulk stress can easily be developed in regular geometries.
1.2 Outline and Research Contributions
2.1 Role of Cardiac Tissue Stiffness
2.2 Measuring Cardiac Tissue Stiffness
2.4 Limitations and Gaps in Knowledge
3 Finite Element Model Update Method
3.1 Finite Element Methods for Steady State Harmonic Problems
3.2 FEMU Isotropic Method Validation
4 Identifiability of Transversely Isotropic Material Properties from MRE
5 Isotropic Virtual Fields Method
5.1 Analytic Virtual Fields Method
5.2 Optimised Virtual Fields Method
6 Anisotropic Optimised Virtual Fields Method
6.1 Formulation and Implementation
7.2 Contributions to the Field
7.4 Future Work
A Finite Element Strain Formulations
B Parameter Variance Matrices
C Constraints on Virtual Displacements
D Anisotropic Damping Coefficients
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