Marco Fiorito^{1}, Jack Lee^{1}, Daniel Fovargue^{1}, Adela Capilnasiu^{1}, David Nordsletten^{1,2}, and Ralph Sinkus^{1,3}

Solid tumour growth is often associated with the accumulation of mechanical stresses acting on the surrounding host tissue. These forces alter the biomechanics of the adjacent soft tissue, generating a variation in stiffness resulting in a signature pattern that can be probed through MR-Elastography. The probed stiffness, however, is strongly dependent on the direction of propagation of the employed shear waves, leading to the reconstruction of anisotropic mechanical properties of the peri-tumoural tissue. Here we present, using theoretical and experimental means, a closed theoretical understanding of the observed alteration of tangent stiffness of soft tissue generated by pressurised tumour expansion.

**Introduction**

**Methods**

Due to nonlinear mechanical properties of tissue, the shear modulus of the tumour environment will change according to the local deformation created by the pressurised tumour, increasing when undergoing stretch and decreasing when compressed[3]. This renders the mechanical properties of the peri-tumoral tissue anisotropic. Shear waves can sense these apparent changes in stiffness associated to their direction of propagation, hence ideally probing a softening at the leading edge of the inflated object and a stiffening along the lateral area. This pattern was previously shown experimentally, employing an inflated balloon-catheter inserted inside a PVC-based hyperelastic isotropic phantom[4].

Analytically, tumour expansion can be idealised with a thick-walled sphere subjected to an axisymmetric macro-deformation (Fig.1-A). The pressure generated by the inflation of the inner sphere is analytically given by

$$p_i=\int_a^b\frac{d\sigma_{rr}}{dr}\,dr$$

where the radial stress $$$\sigma_{rr}$$$ depends on the chosen material law. Through a pure-stretch rheological
investigation we identified a modified Mooney-Rivlin law in the form $$$W=\frac{\mu_1}{2}\left(I_{\hat{\mathbf{C}}}-3\right)+\frac{\mu_2}{2}\left(II_{\hat{\mathbf{C}}}-3\right)^2$$$ that describes the viscoelastic behaviour of the above-mentioned
tissue-mimicking phantom (Fig.1-B), yielding an N^{th}-order polynomial
expression of $$$p_i$$$. Fig.1-C shows that, while $$$\mu_1$$$ purely scales the curve produced by the linear
term of the constitutive equation, $$$\mu_2$$$ is responsible for the more dramatic variations at higher inflations.
Once combined, the two terms result in a characteristic S-shaped pressure
curve.

Superimposing the perturbation generated by the low amplitude waves used in MRE on top of the existing macro-deformation, a solution to the wave equation can be found after a linearization process, yielding[3]

$${\rho}J\partial_{t}^{2}\mathbf{u}_{\varepsilon}-\nabla_{\mathbf{x}}\cdot\left[\boldsymbol{\mathcal{C}}:\nabla_{\mathbf{x}}\mathbf{u}_{\varepsilon}+p_{\varepsilon}\mathbf{I}\right]=0$$

with the stiffness tensor being

$$\mathcal{C}_{ijkl}=\frac{1}{J}\frac{{\partial}P_{is}}{{\partial}F_{mn}}F_{ln}F_{js}$$

where the first Piola-Kirchhoff stress tensor $$$\mathbf{P}$$$ will depend on the deformation gradient $$$\mathbf{F}$$$ and the constitutive law describing the phantom. In the simple case of a plane shear wave $$$\mathbf{u}_{\varepsilon}=u_{\varepsilon}(x)\mathbf{\hat{e}_y}$$$ (Fig.2-A), the sensed stiffness $$$\boldsymbol{\mathcal{C}}:\nabla_{\mathbf{x}}\mathbf{u}_\varepsilon$$$ will change according to the applied deformation, producing the expected decrease in stiffness when the waves approach the inclusion head-on (Fig.2-B). In contrast, along the peripheral interface of the inner sphere, a stiffening is detected. This matches our expectations from simple considerations of waves in anisotropic media and our previous phantom experiments. The numerically calculated associated signature pattern is shown in Fig.2-C.

We experimentally validated this theory by fitting the model parameters to pressure measurements obtained at different inflation states and comparing the numerically generated apparent stiffness patterns in the implemented model with those acquired from MRE data.

**Results & Discussion**

Fig.3 shows how three out of four experimental pressure measures, obtained at different inflations through a pressure sensor (left), fit the curve produced using the modified Mooney-Rivlin model (right). Furthermore, the linearised shear modulus of the undeformed material, calculated from the model using

$$\mu=\lim_{a{\longrightarrow}0}2\left(\frac{{\partial}W}{{\partial}I_{\hat{\mathbf{C}}}}+\frac{{\partial}W}{{\partial}II_{\hat{\mathbf{C}}}}\right)=\mu_1$$

matches the background stiffness reconstructed from the MRE data ($$$\mu=11.9\pm2.9$$$ kPa).

The material parameters were used to numerically calculate the relative
stiffness variation around the inclusion. The patterns presented in Fig.4 were radially
unravelled following the perimeter of the balloon, with the 0° angle associated
to the mean k-vector of the waves. The modelled pattern in Fig.4-A, obtained
simulating simple uni-directional shear waves, shows a qualitative and
quantitative agreement with the voxel-wise projection of the right Cauchy-Green
tensor associated to the real deformation along the local direction of the
k-vector, presented in Fig.4-B. Given the more complex wave behaviour and
deformation field in the experimental case, $$$\mathbf{K}\cdot\mathbf{C}\cdot\mathbf{K}$$$ gives an understanding of the type and intensity of deformation probed
by the waves. Similar patterns are found in the relative stiffness variation
experimentally measured; however the model does not predict the shift generated
in the region immediately adjacent to the inflated balloon, possibly due to the
impact of the phantom/balloon discontinuity on the reconstructed stiffness.

[1] T. Stylianopoulos, “The Solid Mechanics of Cancer and Strategies for Improved Therapy,” J. Biomech. Eng., vol. 139, no. 2, p. 021004, Jan. 2017.

[2] R. Sinkus, M. Tanter, T. Xydeas, S. Catheline, J. Bercoff, and M. Fink, “Viscoelastic shear properties of in vivo breast lesions measured by MR elastography.,” Magn. Reson. Imaging, vol. 23, no. 2, pp. 159–65, Feb. 2005.

[3] A. Capilnasiu et al., “Magnetic resonance elastography in nonlinear viscoelastic materials under load,” Biomech. Model. Mechanobiol., pp. 1–25, Aug. 2018.

[4] D. Fovargue et al., “Non-linear Mechanics Allows Non-invasive Quantification of Interstitial Fluid Pressure,” ISMRM #3832, 2018.

Figure 1. Section of the idealised
thick-shelled sphere used to model balloon inflation is soft PVC-phantom (A).
The phantom was subjected to uniaxial-sinusoidal micro- and macro-oscillations
at different frequencies and compression levels to identify a suitable material
law. The proposed modified Mooney-Rivlin model produced a good fit of the
experimental data (B). The generated family of pressure curves reveals the
scaling effect of μ_{1} on
the linear term of the constitutive equation (C-left), as well as the impact of
μ_{2} at higher inflations on
the quadratic term (C-middle). Once combined, the two terms produce a
characteristic S-shaped curve (C-right).

Figure 2. Simple shear wave propagation through thick-shelled sphere.
Due to local spherical coordinate orientation, at position A the wave will
appear as θ-polarised, travelling in the radial direction. At B the opposite is
found (A). Using the modified Mooney-Rivlin law, the tangent stiffness will logarithmically
decrease and increase at location A and B, respectively, proportionally to the
applied inflation. While the stiffening is enhanced by a higher μ_{2}, the same values produce a
less pronounced softening in the vicinity of the inner sphere under larger
deformations (B). Such analytical considerations result in a signature pattern
around the inclusion (C).

Figure 3. The inflation pressure generated by the balloon
onto the surrounding phantom was measured using a pressure sensor directly
connected to the catheter (left). The analytically derived pressure shows a
good agreement with three of the four experimental observations, producing a
linearised shear modulus for the phantom material of 12.395 kPa (right). For an
accurate estimate of the radial expansion, the balloon volume was segmented
from high-resolution T2-weighted MR images of the phantom/balloon ensemble.

Figure 4. Numerical calculation of the relative stiffness
variation generated by several balloon inflations. The phantom stress-strain
response was modelled using the material parameters obtained from the fitting
of the pressure measurements. Radially unravelled images centre on the mean k-vector direction offer a
better visualisation. In this case, simple uni-directional shear
waves have been considered (A). The modelled pattern strongly correlates with the
deformation sensed by the waves in the corresponding phantom cases (B). The
model, however, does not match the intensity of the stiffness variation in the
vicinity of the inclusion, although the pattern is recovered (C).