AbstractThis study presents the first observation of shear wave induced remotely within soft tissues. It was performed through the combination of a transcranial magnetic stimulation device and a permanent magnet. A physical model based on Maxwell and Navier equations was developed. Experiments were performed on a cryogel phantom and a chicken breast sample. Using an ultrafast ultrasound scanner, shear waves of respective amplitude of 5 and 0.5 micrometers were observed. Experimental and numerical results were in good agreement. This study constitutes the framework of an alternative shear wave elastography method.

Propagation of elastic waves in solids has been described in various fields of physics, including geophysics, soft matter physics or acoustics. Elastic waves can be separated in two components in a bulk: compression waves, corresponding to a curl-free propagation; and shear waves, corresponding to a divergence-free propagation. Shear waves have drawn a strong interest in medical imaging with the development of shear wave elastography methods (Muthupillai 1995), (Sarvazyan 1998). These methods use shear waves to measure or map the elastic properties of biological tissues. Shear wave speed measurement permits calculation of the tissue shear modulus. Shear wave elastography techniques have been successfully applied to several organs such as the liver (Sandrin 2003), the breast (Berg 2012), the arteries (Schmitt 2010) and the prostate (Cochlin 2002), to name a few examples. The brain has also been studied, and its elasticity is of strong interest for clinicians (Mariappan 2010), (Kruse 2008). For example, it has been shown that Alzeihmer’s disease, hydrocephalus or multiple sclerosis are associated with changes in brain elastic properties (Murphy 2011), (Taylor 2004), (Wuerfel 2010).

Clinical shear wave elastography techniques currently rely on an external vibrator (Muthupillai 1995), (Sandrin 2003) or on a focused acoustic wave (Nightingale 2002), (Sarvazyan 1998) as the shear wave source. However, these techniques are limited in situations where the organ of interest is located behind a strongly attenuating medium like the brain behind the skull and surrounded by the cerebrospinal fluid. While external shakers are able to transmit some shear waves, using acoustic, pneumatic, piezoelectric or electromagnetic actuators (Kruse 2008), (Latta 2011), (Weaver 2001), (Braun 2003), this approach can be uncomfortable for patients. Alternatively, acoustic waves may be transmitted through the skull to induce shear waves inside the brain, but the skull attenuates and deforms the acoustic beam, preventing efficient transmission of energy. Recently, it has also been shown that physiological body motion can be used, via blood pulsation (Hirsch 2013), (Weaver 2012) or noise correlation (Gallot 2011), (Zorgani 2015), but these methods still require further development before clinical application in the context of brain elastography.

Recently, it was demonstrated that the combination of an electrical current and a magnetic field could create displacements which propagate as shear waves in biological tissues (Basford 2005), (Grasland-Mongrain 2014). If the electrical current is induced using a coil, this would allow the technique to remotely induce shear waves. In the case of brain elastography, this would allow inducing shear waves directly inside the brain.

To achieve this objective, we propose to use a transcranial magnetic stimulation (TMS) device (Hallett 2000). This instrument is used to induce an electrical current directly inside the brain by using an external coil. TMS is currently employed by neurologists to study brain functionality (Ilmoniemi 1999) and by psychiatrists to treat depression (Sakkas 2006). TMS is occasionally combined with magnetic resonance imaging (MRI) (Devlin 2003), (Bohning 1997); however, no study has yet reported the production of shear waves when combining TMS and magnetic fields.

This article first presents the physical model describing the generation of shear waves resulting from the combination of a remotely induced electrical current and a magnetic field. It describes experiments performed in poyvinyl alcohol cryogel and biological tissue samples. A numerical study of the experiments is then presented. Results section shows a good consistency between experimental and numerical displacement maps. Some critical excitation parameters were investigated as well as dependence of the shear wave amplitude with the magnetic field and electrical current intensity. Practical implementation in a context of shear wave elastography of the brain is finally discussed.

We set up the experiment illustrated in Figure \ref{Figure1}-(A). The key components are as follows: a coil induces an electrical current \(\mathbf{j}\) in the sample; a magnet creates a magnetic field \(\mathbf{B}\); an ultrasound probe tracks displacements \(\mathbf{u}\) propagating as shear waves in the sample. X is defined as the main magnetic field axis, Z as the main ultrasound propagation axis, and Y an axis orthogonal to X and Z following the right-hand rule. The origin of coordinates (0,0,0) is located in the middle of the coil (i.e., between the two loops).

For a circular coil centered in (0,0,0) of linear element \(d\mathbf{l}\) crossed by an electrical current \(I(t)\), using Coulomb gauge (i.e., \(\nabla . \mathbf{A} = 0\) where \(\mathbf{A}\) is the magnetic potential vector), and negligible propagation time of electromagnetic waves, the electrical field \(\mathbf{E}(\mathbf{r},t)\) along space \(\mathbf{r}\) and time \(t\) is equal to (Jackson 1998): \[\mathbf{\mathbf{E(\mathbf{r},t)}} = - \nabla \Phi - \frac{d I}{d t} \frac{N \mu_0}{4\pi}\int{\frac{\mathbf{dl}}{r}} \label{Equation1}\] where \(\Phi\) is the electrostatic scalar potential, \(N\) is the number of turns of the coil and \(\mu_0\) is the magnetic permeability of the coil material. In an unbounded medium, \(\Phi\) is only due to free charges (Grandori 1991), that we supposed negligible in our case. Being additive, the total electrical field created by two or more coils is simply the sum of the contribution of each coil. The induced electrical current density \(\mathbf{j}\) is retrieved using the local Ohm’s law \(\mathbf{j}=\sigma \mathbf{E}\), where \(\sigma\) is the electrical conductivity of the medium.

The body Lorentz force \(\mathbf{f}\) can then be calculated using the relationship \(\mathbf{f} = \mathbf{j} \times \mathbf{B}\), where \(\mathbf{B}\) is the magnetic field created by the permanent magnet. Considering the tissue as an elastic, linear and isotropic solid, Navier’s equation governs the displacement \(\mathbf{u}\) at each point of the tissue submitted to an external body force \(\mathbf{f}\) (Aki 1980): \[\rho\frac{d^2\mathbf{u}}{dt^2} = (K + \frac{4}{3}\mu) \nabla (\nabla . \mathbf{u}) + \mu \nabla \times (\nabla \times \mathbf{u}) + \mathbf{f} \label{Equation3}\] where \(\rho\) is the medium density, \(\mathbf{u}\) the local displacement, \(K\) the bulk modulus and \(\mu\) the shear modulus.

Using Helmholtz decomposition \(\mathbf{u}=\nabla \phi + \nabla \times \mathbf{\psi}\), where \(\phi\) and \(\mathbf{\psi}\) are respectively a scalar and a vector field, two elastic waves can be retrieved: a compression wave, propagating at a celerity \(c_k = \sqrt{(K+\frac{4}{3}\mu)/\rho}\), and a shear wave, propagating at celerity \(c_s = \sqrt{\mu/\rho}\) (Sarvazyan 1998). As \(\rho\) varies typically by a few percent between different soft tissues (Cobbold 2007), we can suppose an homogeneous density, and measuring \(c_s\) allows to compute the shear modulus \(\mu\) of the tissue.