MR-u: Material Characterization Using 3D Displacement-Encoded Magnetic Resonance and the Virtual Fields Method

Abstract

Background

Experimental, fully three-dimensional mechanical characterization of opaque materials with arbitrary geometries undergoing finite deformations is generally challenging.

Objective

We present a promising experimental method and processing pipeline for acquiring and processing full-field displacements and using them toward inverse characterization using the Virtual Fields Method (VFM), a combination we term MR-u.

Methods

Silicone of varying crosslinker concentrations and geometries is used as the sample platform. Samples are stretched cyclically to finite deformations inside a 7T MRI machine. Synchronously, a custom MRI pulse sequence encodes the local displacement in the phase of the MR image. Numerical differentiation of phase maps yields strains.

Results

We present a custom image processing scheme for this numerical differentiation of MRI phase-fields akin to convolution kernels, as well as considerations for gradient set calibration for data fidelity.

Conclusions

The VFM is used to successfully determine hyperelastic material properties, and we establish best practice regarding virtual field selection via equalization.

Appendices

Appendix A: Divolution

We illustrate our complex division-based numerical gradient approximation technique by the following example. Suppose first that we have an unwrapped scalar phase field θ = angle(Z) which is proportional to a single component of our induced displacement field u_k. We can smooth or numerically differentiate θ by convolution, or passing filter kernels w and d, respectively, over it as illustrated in Fig. 5. The phase function θ is defined at every pixel index i and j for the 2D example, which is henceforth denoted as θ^(i,j). By passing a symmetric smoothing filter w in the j direction over θ^(i,j), we produce a smoothed function

$$ \overline{\theta}^{(i,j)} = \sum\limits_{n=-H}^{H} w_{n} \theta^{(i,j+n)}, $$

(13)

while by passing an antisymmetric edge filter d (with a middle component of zero) in the j direction, we determine the average central phase difference

$$ \displaystyle \overline{{\Delta}_{j} \theta}^{(i,j)} = \sum\limits_{n=1}^{H} d_{n} \left( \theta^{(i,j+n)}-\theta^{(i,j-n)}\right), $$

(14)

where 2H + 1 is the length of the respective filter, and w and d follow normalizations of

$$ \sum\limits_{n=-H}^{H} w_{n} = 1 ~~\text{and}~~ \sum\limits_{n=1}^{H} d_{n} = 1 $$

(15)

If instead, the phases are wrapped on (−π/2,π/2), simple convolution operators will produce unwanted phase gradients or smoothing operations at the ±π/2 jumps. We can avoid smoothing wrap jumps by using the complex data Z directly. Notably, the values of Z are independent of the wrapping of θ, making Z itself a usable quantity to minimize unwrapping artifacts. Distinguishing the wrapped phase field $\widetilde {\theta } = \text {wrap}(\theta )$ from its unwrapped counterpart θ, the complex magnetic resonance output field Z at a pixel (i,j) is represented in polar form as

$$ Z^{(i,j)} = r^{(i,j)} ~\exp(i \theta^{(i,j)}) = r^{(i,j)} ~\exp(i \widetilde{\theta}^{(i,j)}), $$

(16)

where r^(i,j) represents the magnitude of the image signal. With equation (16), complex division can be used to produce the result of equation (14) in a similar way to a convolution filter - pixels on either side of the filter can be divided by one another, or divolved, to subtract phases. The divolution process is illustrated in Fig. 5(d), and the phase gradient can be expressed as

$$ \begin{array}{@{}rcl@{}} \displaystyle \overline{{\Delta}_{j} \theta}^{(i,j)} &=& \text{angle}\left[\prod\limits_{n=1}^{H}\left( \frac{Z^{(i,j+n)}}{Z^{(i,j-n)}} \right)^{d_{n}} \right] \\ &=& \text{angle}\left[\prod\limits_{n=1}^{H}\frac{r^{(i,j+n)}\exp{\left( id_{n}\theta^{(i,j+n)}\right)}}{r^{(i,j-n)}\exp{\left( id_{n}\theta^{(i,j-n)}\right)}} \right]. \end{array} $$

(17)

Appendix B: Gradient Coil Calibration

Correcting for the non-linearity in the imaging gradient set requires a set of two experiments: (a) a rigid translation of a large block of material to determine the higher order deviations from a linear gradient and (b) a spatial grid of material to determine both local and global dilation along the longitudinal direction, essentially the integration constant from the former. In general, we can describe the unwrapped phase field θ(X) for a sample in an ideal gradient coil, as in equation (7), as

$$ \boldsymbol{\theta}(\boldsymbol{X}) = \frac{\boldsymbol{u}(\boldsymbol{X})}{\boldsymbol{\Lambda}}, $$

(18)

where X is a position in our sample’s reference configuration, u(X) is the displacement in the sample, and Λ is the encoding length chosen by the user corresponding to how many phase wraps occur per 2π in the respective direction. However, as real gradient coils are designed to be very close to linear in the central region, the non-linearity at the periphery must be considered for calculating displacements and strains to a high degree of accuracy. We can describe the non-ideal gradient $\boldsymbol {G}^{\prime }$ with higher-order terms as G(1 + α(X)). Given the relation between our encoding length Λ and the applied magnetic field gradient,

$$ \boldsymbol{\Lambda} = \frac{1}{\gamma_{H} t_{\mathrm{e}} \boldsymbol{G}}, $$

(19)

we can write the phase in a non-ideal gradient coil as

$$ \boldsymbol{\theta}(\boldsymbol{X}) = \frac{\boldsymbol{u}(\boldsymbol{X})}{\boldsymbol{\Lambda}} \left( 1 + \boldsymbol{\alpha}(\boldsymbol{X}) \right), $$

(20)

where α is the higher order gradient coil correction function which is purely a vector function of the MR coordinate system χ (with all χ_i parallel to X_i). For cases of rigid translation of amplitude w, we can then rearrange for α(X),

$$ \boldsymbol{\alpha}(\boldsymbol{X}) = \frac{\boldsymbol{\Lambda}}{w}\left( \boldsymbol{\theta}(\boldsymbol{X}) - \boldsymbol{\theta_{0}} \right) + C, $$

(21)

where θ₀ is the value at the gradient coil center θ(χ = 0), and C is a constant taking into account the possibility of an offset error at the center.

For calibration of our setup, we ran two rigid translation experiments of 5mm and 7.5mm, respectively. As described in Section “Loading and Imaging Procedure”, phase maps were put through the processing procedure shown in Fig. 5. Importantly, phase maps were first unwrapped using the procedure described in [41], which prioritizes unwrapping of pixels based on the flatness of the second central difference. Surface fits for the components α_i(χ) (i.e. using θ_i(χ)) were then performed to values on symmetric planes in the gradient coil (χ₁-χ₂, χ₁-χ₃). Furthermore, due to symmetry in gradient set design and winding, α₁ is comprised of even terms, while α₂ and α₃ are comprised of odd terms in χ. The expressions α_i were found to be well-described by the analytical forms:

$$ \begin{array}{@{}rcl@{}} \alpha_{1} &= &(n_{2} {\chi_{2}^{2}} + C_{2})(n_{3} {\chi_{3}^{2}}+C_{3})\\ &&\times \left[a_{1} (\chi_{1}-e_{1})^{4} + b_{1}(\chi_{1}-e_{1})^{2} + d_{1} \right] + c_{1} \end{array} $$

(22)

$$ \begin{array}{@{}rcl@{}} \alpha_{2} &= &\chi_{2} \left[a_{2} (\chi_{1}-e_{2})^{3} + b_{2}(\chi_{1}-e_{2}) + d_{2} \right] + c_{2} \end{array} $$

(23)

$$ \begin{array}{@{}rcl@{}} \alpha_{3} &= &\chi_{3} \left[a_{3} (\chi_{1}-e_{3})^{3} + b_{3}(\chi_{1}-e_{3}) + d_{3} \right] + c_{3}, \end{array} $$

(24)

where n_i, C_i, a_i, b_i, c_i, d_i, and e_i are all fitting constants.

To correct displacement gradient tensor data in practice, we can take analytical derivatives of the gradient coil correction function α and apply them to components,

$$ F^{\prime}_{ij}(\boldsymbol{X}) = F_{ij}(\boldsymbol{X})-\frac{\partial \alpha_{i}}{\partial \chi_{j}}(\boldsymbol{X}) \cdot u_{1}(\boldsymbol{X}), $$

(25)

where $F^{\prime }_{ij}$ is the correction of the deformation gradient tensor with components F_ij, and $\frac {\partial {\alpha _{i}}}{\partial {\chi _{j}}}(\boldsymbol {X})\equiv \alpha _{i,j}(\boldsymbol {X})$, for completeness, are,

$$ \begin{array}{@{}rcl@{}} \alpha_{1,1} &=& (n_{2} {\chi_{2}^{2}} + C_{2})(n_{3} {\chi_{3}^{2}}+C_{3})\\ &&\times\left[4 a_{1} (\chi_{1}-e_{1})^{3} + 2 b_{1}(\chi_{1}-e_{1}) \right] \end{array} $$

(26)

$$ \begin{array}{@{}rcl@{}} \alpha_{1,2} &=& 2 n_{2} \chi_{2}(n_{3} {\chi_{3}^{2}}+C_{3})\\ &&\times\left[a_{1} (\chi_{1}-e_{1})^{4} + b_{1}(\chi_{1}-e_{1})^{2} + d_{1} \right] \end{array} $$

(27)

$$ \begin{array}{@{}rcl@{}} \alpha_{1,3} &=& 2 n_{3} \chi_{3}(n_{2} {\chi_{2}^{2}} + C_{2})\\ &&\times\left[a_{1} (\chi_{1}-e_{1})^{4} + b_{1}(\chi_{1}-e_{1})^{2} + d_{1} \right] \end{array} $$

(28)

$$ \begin{array}{@{}rcl@{}} \alpha_{2,1} &=& \chi_{2} \left[3 a_{2} (\chi_{1}-e_{2})^{2} + b_{2} \right] \end{array} $$

(29)

$$ \begin{array}{@{}rcl@{}} \alpha_{2,2} &=& a_{2} (\chi_{1}-e_{2})^{3} + b_{2}(\chi_{1}-e_{2}) + d_{2} \end{array} $$

(30)

$$ \begin{array}{@{}rcl@{}} \alpha_{2,3} &=& 0 \end{array} $$

(31)

$$ \begin{array}{@{}rcl@{}} \alpha_{3,1} &=& \chi_{3} \left[3 a_{2} (\chi_{1}-e_{2})^{2} + b_{2} \right] \end{array} $$

(32)

$$ \begin{array}{@{}rcl@{}} \alpha_{3,2} &=& 0 \end{array} $$

(33)

$$ \begin{array}{@{}rcl@{}} \alpha_{3,3} &=& a_{3} (\chi_{1}-e_{3})^{3} + b_{3}(\chi_{1}-e_{3}) + d_{3}. \end{array} $$

(34)