2.3.1 Viscous model

This first model is applied to magnetic particles smaller than 100 nm, which means that we do not take into account the stiffness of the round window membrane. Like it has been defined in [34], we consider a single value of layers' viscosity, and the new dynamic model becomes

${\overrightarrow{F}}_d+{\overrightarrow{F}}_m+\overrightarrow{W}=m\overrightarrow{\gamma }.$

Replacing the forces by their analytical expressions, we obtain

$\gamma +\frac{6\eta \pi r}{m}v=\frac{V(M\mathrm{\nabla })B}{m}+g$

with η being the viscosity of the cytoplasm, r, V, and v respectively the radius of the particle, volume and particle velocity, M the magnetization of the particle, B the external magnetic field, and ∇ the gradient operator. At the equilibrium (when $\gamma =0$) Eq. (2.3) can be written as

$v=\frac{2(M\mathrm{\nabla })r^2}{9\eta }B+\frac{1}{6\eta \pi r}g.$

From Eq. (2.4) and knowing the thickness of the RWM, the injection time of the particles through the RWM is defined by

$t=\frac{h_{RWM}}{\frac{2(M\mathrm{\nabla })r^2}{9\eta }B+\frac{1}{6\eta \pi r}g}.$

In [32] the authors show that each layer of the RWM is characterized by its own viscosity and thickness. Eq. (2.5) can be written as follows:

$v_i=\frac{2(M\mathrm{\nabla })r^2}{9{\eta }_i}B+\frac{1}{6{\eta }_i\pi r}g$

where $i=[1,3]$ is the index of the RWM layer where the particle moves. We can also calculate the time required for a particle to cross each layer of the RWM:

$t=\sum _{i=1}^3\frac{h_i}{\frac{2(M\mathrm{\nabla })r^{2}B}{9{\eta }_i}+\frac{1}{6{\eta }_i\pi r}g}.$

2.3.2 Viscoelastic model

In this model, we will consider the effect of the stiffness of the inner and outer layers of the RWM. This model is applied to magnetic particles with a diameter greater than 100 nm. Using the Young's modulus as defined in [23], the stiffness of the RWM can be calculated as in [36]:

$k=\frac{24EI}{a^3}$

where E, I, and a are respectively the Young's modulus, inertia moment of the layer, and radius of the RWM. The dynamic model of the particle can be written as

$\left[\begin{array}{c}\hfill \gamma \hfill \\ \hfill v\hfill \end{array}\right]=\left[\begin{array}{cc}\hfill -\frac{6{\eta }_i\pi r}{m}\hfill & \hfill -\frac{k}{m}\hfill \\ \hfill 1\hfill & \hfill 0\hfill \end{array}\right]\left[\begin{array}{c}\hfill v\hfill \\ \hfill x\hfill \end{array}\right]+\left[\begin{array}{c}\hfill \frac{VM}{m}\hfill \\ \hfill 0\hfill \end{array}\right][\mathrm{\nabla }B]+\left[\begin{array}{c}\hfill g\hfill \\ \hfill 0\hfill \end{array}\right].$

The magnetic field gradient required to propel a magnetic particle through the RWM is computed at its equilibrium state ($v\ne 0$). We can calculate the time needed to cross the RWM as

$t=\sum _{i=1}^3\frac{h_i}{\frac{2(M\mathrm{\nabla })r^{2}B}{9{\eta }_i}+\frac{g+kx}{6{\eta }_i\pi r}}.$

2.4.1 Simulation of the viscous model

In this section we will simulate the dynamics of the magnetic particle in the RWM. Specifically, we will evaluate the time needed to cross the RWM. The simulations were performed for the particles with a radius $r=8\text{ nm}$ and magnetization $M=1.023⁎{10}^6{\text{ T}}^2/\text{A}$. The viscosity of the outer layers is the same ${\eta }_{1,3}\approx 0.001\text{ Pa\hspace{0.17em}s}$, and the viscosity of the middle layer is ${\eta }_2\approx 13\text{ Pa\hspace{0.17em}s}$. Fig. 2.4(A) shows the particle velocity without weight force. Fig. 2.4(B) shows the particle velocity in the case where its weight is taken into account. The minimum velocity obtained in this first case is $1.95⁎{10}^{-9}\text{ m/s}$, which corresponds to an injection time of 750 min, time to cross the membrane without magnetic actuation. Applying an external magnetic force, the particle velocity increases which strongly decreases the injection time. For a magnetic force of $2.61⁎{10}^{-19}\text{ N}$ the injection time is 300 min, and for a magnetic force of $4.98⁎{10}^{-19}\text{ N}$ the injection time is 200 min, which corresponds to a gain of 57% and 71%, respectively. When the magnetic force is 5 times the weight of the particle (about $9.25⁎{10}^{-19}\text{ N}$), the effect of weight is negligible as can be seen in Figs. 2.5(A) and 2.5(B).

Fig. 2.6 shows the velocity of a particle with a radius of 65 nm propelled with a magnetic gradient field $\mathrm{\nabla }B=0.5\text{ mT/m}$, which corresponds to a magnetic force of $5.8⁎{10}^{-16}\text{ N}$. The RWM is modeled by three layers, $h_1=h_3=25\text{μm}$ and $h_2=20\text{μm}$; these values are completely arbitrary because the only known information about the different layers of the RWM is that the thicknesses of the outer and inner layers are larger than the thickness of the inner layer [26,35]. This simulation shows the impact of viscosity change on the dynamics of the particle in the RWM. We clearly notice that the change in viscosity during the transition between the outer layer and inner layer results in a strong decrease of the velocity of the particle. When the particles pass through the second layer of the RWM, it is possible that they come into contact with substances, such as collagen which can significantly slow their movement. But in principle, the inner layer of the round window membrane contains essentially cytoplasm.

We also investigated the relationship between the magnetic force applied to a particle and its volume. Fig. 2.7 shows the evolution of the magnetic force and the particle velocity as functions of this radius. We can see that for a fixed value of the magnetic field, the magnetic force and velocity are proportional to the radius of the particle. However, from a certain radius, the particle is not able to penetrate the membrane due to its large volume. So in this case, it is assumed that the effect of the stiffness is taken into account, which increases the risk of perforation and leakage of the perilymph, and therefore a hearing loss.

2.4.2 Simulation of the viscoelastic model

This viscoelastic model is applied to a particle having radius of 100 nm in order to introduce the elastic effect of the membrane. As shown in Fig. 2.8, the simulation results obtained with this model are contradictory to those presented in [32–34]. Indeed, these results show that the nanoparticles completely crossed the RWM after a few hours under a magnetic field. The Young's modulus used to calculate the stiffness in the model gives results which indicate that the particle will never cross the round window. This could give us an indication of the parameters that we need to model the RWM, and estimate the time and the magnetic force to cross it. In addition, we can exclude this model of the round window membrane.