1. Introduction and Context

1.2. Medical Context

Under normal physiological conditions, sound is a mechanical wave that causes the tympano-ossicular complex to vibrate. Through the articulation between the stapes footplate and the oval window, sound is transmitted from the middle ear to the cochlea (1). The round window, on the other hand, is widely considered to be the anatomical structure that allows for pressure relief by releasing the mechanical energy provided by the ossicular chain to the labyrinthine fluid of the inner ear (2). Hearing encompasses the mechanisms that allow us to perceive the sound waves around us and is made possible by the conversion of these mechanical waves into nerve signals by the basilar membrane, which are then transmitted to the brain (3). Any pathologies or abnormalities in these structures (external, middle, and inner ears) and in the central auditory neural pathways can lead to hearing impairment. Minor was the first to describe, in 1998, a case series of sound- and/or pressure-induced vertigo in patients with superior semicircular canal dehiscence. The dehiscences were identified through computed tomographic scans, which revealed the absence of bone overlying the superior semicircular canal on the surface of the temporal bone (4). In addition to sound/pressure-induced vertigo, patients may experience air-conduction hearing loss observed at low frequencies with dehiscences larger than 3 mm. The significance of this air-bone gap is correlated with the size of the dehiscence (5). The most likely hypothesis is that the dehiscence acts as a "third window" in the inner ear, shunting acoustic energy away from the cochlea at low frequencies, thereby causing hearing loss (6). Some studies have attempted to understand the underlying biomechanical mechanism generating this shunt and have also tried to quantify it. One study specifically investigated the sound-induced displacement velocities of the stapes, umbo, and round window in a temporal bone bank using laser-Doppler vibrometry (6). These measurements were performed on healthy bones considered normal, on the same bones after creating a dehiscence in the superior semicircular canal, and also with the dehiscence patched. They found that the presence of a dehiscence in the superior semicircular canal led to a significant reduction in sound-induced round window velocity at low frequencies.

Figure 1. Modeling semi-circular canals for a normal ear (figure extracted from the reference 18)

Figure 1. Modeling semi-circular canals for a normal ear (figure extracted from the reference 18)

This allows us to better understand why there can be a air-bone gap result at low frequencies in the presence of a dehiscence, and also why patients might experience vertigo with loud sounds due to the shunting of acoustic energy toward the posterior labyrinth. Sound energy enters the inner ear via the oval window, causing fluid movement at the location of the bony defect. This generates propagating waves that subsequently lead to mechano-electrical transduction in the vestibular sensory organs through vibration and nonlinear fluid pumping (1). This same team also recorded changes in neural activity in the vestibular pathways in an animal model, evoked by auditory-frequency stimulation (1). Despite these findings, to our knowledge, no study currently exists to explain the purely biomechanical mechanism of the fluids, and this is what we aimed to address here.

2. Beam Closed Ring Model

3. Canal Segmented Model

3.1. Model of a Healthy Ear

The anterior semicircular canal is modelled as a succession of 3 beams that are closed topologically. The first one corresponds to the utricle and ampulla system. The second one corresponds to the anterior SCC and the third one models the common crus between the anterior and posterior SCCs.

We also have the following relation between the natural pulses of different parts of the canal :

$$\forall (i,j)\in {\left\{1;2;3\right\}}^2,\left\{\begin{array}{c}{E}_i=E_j\hfill \\ {\rho }_i={\rho }_j\hfill \end{array}\right.\Rightarrow \begin{array}{|c|}\hline {\Omega }_i={\Omega }_j\\ \hline\end{array}$$

(7)

which will be the case for our model.

The contribution of the posterior SCC is modelled by a normal force discontinuity at beam 3 (common crus) :

$$\epsilon N_1\left(0\right)=N_3\left(L_3\right);\delta N_3\left(0\right)=N_2\left(L_2\right).$$

(8)

In the same way as in the previous section, we will apply the boundary conditions for the model of the healthy inner ear, and analyse the analytical solutions – indeed, in this more complicated model, the majority of solutions are purely numerical.

$$\mathrm{Conditions}\text{:}\begin{array}{|c|}\hline \begin{array}{cc}\hfill f_1\left(0\right)=f_3\left(L_3\right)& \epsilon N_1\left(0\right)=N_3\left(L_3\right)\hfill \\ \hfill f_2\left(0\right)=f_1\left(L_1\right)& N_2\left(0\right)=N_1\left(L_1\right)\hfill \\ \hfill f_3\left(0\right)=f_2\left(L_2\right)& \delta N_3\left(0\right)=N_2\left(L_2\right)\hfill \\ \hfill E_1=E_2=E_3& {\rho }_1={\rho }_2={\rho }_3\hfill \end{array}\\ \hline\end{array};$$

(9)

$$\mathrm{where}{N}_i=E_i{S}_i{\displaystyle \frac{\partial U_i}{\partial x}}=E_i{S}_i{g}_i{f}_i^{\prime }.$$

3.2. Pathological Configuration

Pathology can be modelled by a discontinuity within the anterior SCC. A discontinuity in displacement and normal effort is applied :

$$\mu f_{2b}\left(0\right)=f_{2a}\left(L_{2a}\right);\eta N_{2b}\left(0\right)=N_{2a}\left(L_{2a}\right)$$

(10)

The dehiscence is generally located between a third and two thirds of the anterior SCC, starting from the ampulla.

We will apply the following boundary conditions :

$$\begin{array}{|c|}\hline \begin{array}{c}\begin{array}{cc}\hfill f_1\left(0\right)=f_3\left(L_3\right)& \epsilon N_1\left(0\right)=N_3\left(L_3\right)\hfill \\ \hfill f_{2a}\left(0\right)=f_1\left(L_1\right)& N_{2a}\left(0\right)=N_1\left(L_1\right)\hfill \\ \hfill \mu f_{2b}\left(0\right)=f_{2a}\left(L_{2a}\right)& \eta N_{2b}\left(0\right)=N_{2a}\left(L_{2a}\right)\hfill \\ \hfill f_3\left(0\right)=f_{2b}\left(L_{2b}\right)& \delta N_3\left(0\right)=N_{2b}\left(L_{2b}\right)\hfill \\ \hfill E_1=E_{2a}=E_{2b}=E_3& {\rho }_1={\rho }_{2a}={\rho }_{2b}={\rho }_3\hfill \end{array}\end{array}\\ \hline\end{array}$$

3.3. Anatomical Data

The data we are using comes from references [1] to [4]. These are approximate values for an anterior SCC. We take the density of water, as we are studying biological tissues. The radius of curvature is approximately $R=4.0$ mm while the slender duct cross-sectional radius is approximately $r=0.19$ mm. We will therefore assume that $R\gg r$.

In the physiological case, REF the pressure variation amplitude of the SCC are on the order of $1nPa\sim 1\mu Pa$ and contribute very little to the dynamics of the inner ear, and can thus be mostly discarded. The SCC also does not contribute to the ear’s sound perception, as sound is only perceived in the cochlea.

In the case of the dehiscence, the dynamics of the SCC become much more prevalent, with pressure variations in the order of $10\mu Pa\sim 100\mu Pa$ REF. In this case, because of the third window, vibrations in the SCC have a much greater impact on the dynamics of the inner ear and notably the cochlea, meaning vibrations caused by sound outside sound will cause vibration in the SCC (which could lead to conditions like vertigo, loss of balance, ...).

In addition, with the opening of the third window, addition vibrations will come from the interior of the body, which will cause vibrations in the SCC, thus affecting the cochlea and creating a phenomenon of autophony. The dynamics of the SCC thus become fundamental in the dynamics of the whole inner ear, as the amplified vibrations could create tinnitus. On the other hand, vibration damping in the dynamics of the SCC could create a loss of hearing in certain frequency bands.

Related pathologies are described in the following table :

4. Application and Discussion

4.1. Model Implementation in Physiological Configuration

Applying the boundary conditions, we have the following system :

$$\left\{\begin{array}{cccccccccccccc}& A_1& & & & & & & -& A_{3}cos(\Omega L_3)& -& B_{3}sin(\Omega L_3)& =& 0\\ -& A_{1}cos(\Omega L_1)& -& B_{1}sin(\Omega L_1)& +& A_2& & & & & & & =& 0\\ & & & & -& A_{2}cos(\Omega L_2)& -& B_{2}sin(\Omega L_2)& +& A_3& & & =& 0\\ & & & B_1{S}_1\Omega \epsilon & & & & & +& A_3{S}_3\Omega sin(\Omega L_3)& -& B_3{S}_3\Omega cos(\Omega L_3)& =& 0\\ & A_1{S}_1\Omega sin(\Omega L_1)& -& B_1{S}_1\Omega cos(\Omega L_1)& & & +& B_2{S}_2\Omega & & & & & =& 0\\ & & & & & A_2{S}_2\Omega sin(\Omega L_2)& -& B_2{S}_2\Omega cos(\Omega L_2)& & & +& B_3{S}_3\Omega \delta & =& 0\end{array}\right..$$

(11)

The nullity of the determinant in this system allows us to find simple solutions. Here are a few of them:

We can further find through relation 7,

$$\forall (i,j)\in {\left\{1;2;3\right\}}^2,{\displaystyle \frac{{\Omega }_i{L}_i}{L_i}}={\displaystyle \frac{{\Omega }_j{L}_j}{L_j}}\Rightarrow {\displaystyle \frac{L_i}{L_j}}={\displaystyle \frac{{\Omega }_i{L}_i}{{\Omega }_j{L}_j}}={\displaystyle \frac{n_i+\frac{1\pm 1}{4}}{n_j+\frac{1\pm 1}{4}}},$$

(12)

which will provide another relation between the ratio of the lengths $L_i$ and a ratio of integers with varying parities depending on the case being treated. This will limit the amount of cases which would actually be viable in practice given the set lengths $L_i$, and even more so were we to only consider low frequency cases – the higher frequency pulses being damped by the various low-pass filters created by the body’s tissues.

$$\begin{array}{|c|}\hline \begin{array}{ccc}\hfill \left(1\right)& \mathrm{following}& \left\{\begin{array}{c}{n}_1+n_2+n_3\equiv 0(mod2)\hfill \\ \epsilon \delta =1\hfill \\ {\displaystyle \forall (i,j)\in {\left\{1;2;3\right\}}^2,\frac{L_i}{L_j}=\frac{n_i}{n_j}\Rightarrow n_1\approx 4k,n_2\approx 8k,n_3\approx 2k,k\in \mathbb{N}}\hfill \end{array}\right.\hfill \\ \\ \hfill \left(2\right)& \mathrm{following}& \left\{\begin{array}{c}\epsilon \delta =-1\hfill \\ {\displaystyle \forall (i,j)\in {\left\{1;2;3\right\}}^2,\frac{L_i}{L_j}=\frac{2n_i+1}{2n_j+1}}\hfill \end{array}\right.\hfill \\ \hfill \left(3\right)& \mathrm{following}& \left\{\begin{array}{c}\epsilon \delta =-1\hfill \\ \begin{array}{c}{\displaystyle \frac{L_1}{L_2}=\frac{n_1}{n_2}}\hfill \\ {\displaystyle \frac{L_1}{L_3}=\frac{2n_1}{2n_3+1}}\hfill \end{array}\Rightarrow n_1\approx 3,n_2\approx 6,n_3\approx 1\hfill \end{array}\right.\hfill \\ \hfill \left(4\right)& \mathrm{following}& \left\{\begin{array}{c}{\displaystyle \frac{S_1}{S_3}=\delta {(-1)}^{k_1+k_2+k_3+1}}\hfill \\ \epsilon \delta =1\hfill \\ {\displaystyle \frac{L_1}{L_2}=\frac{2n_1+1}{2n_2}}\hfill \\ {\displaystyle \frac{L_2}{L_3}=\frac{2n_2}{2n_3+1}}\hfill \end{array}\right.\hfill \\ \end{array}\\ \hline\end{array}$$

Shown above in Figure 3 are screen captures of the cases (1) and (3) animated in Julia, for arbitrary $A_1$ and $B_1$. We can notice discontinuities in the slopes of the normal displacement, which are a result of the varying cross sections of the different beams, regardless of the values of $\delta$ and $\epsilon$.

We ought to be careful using these visualisations as the displacement represented here radially is actually a longitudinal displacement. This representation was chosen for the sake of legibility.

4.2. Model Implementation in Pathological Configurations

As in the physiological case, there are simple modal solutions which are only possible under certain conditions (see table below).

$$\begin{array}{|c|}\hline \begin{array}{ccc}\hfill \left(1\right)& \mathrm{following}& {(-1)}^{n_1+n_{2a}+n_{2b}+n_3}=\mu =\delta \epsilon \eta \mathit{with}\left\{\begin{array}{c}{n}_1\approx 4,n_{2a}\approx 4,n_{2b}\approx 4,n_3\approx 2\mathrm{if}\mu =1\hfill \\ n_1\approx 2,n_{2a}\approx 2,n_{2b}\approx 2,n_3\approx 1\mathrm{if}\mu =-1\hfill \end{array}\right.\hfill \\ \\ \hfill \left(2\right)& \mathrm{following}& \delta \epsilon \eta \mu =-1\mathit{with}{n}_1\approx 3,n_{2a}\approx 3,n_{2b}\approx 3,n_3\approx 1\hfill \\ \\ \hfill \left(3\right)& \mathrm{following}& {\displaystyle {(-1)}^{n_1+n_{2a}+n_{2b}+n_3+1}=\epsilon \eta \frac{S_{2b}}{S_3}=\delta \mu \frac{S_3}{S_{2b}}}\hfill \\ \\ \hfill \left(4\right)& \mathrm{following}& \delta \epsilon \eta \mu =-1\hfill \end{array}\\ \hline\end{array}$$

Cases (1) and (2) are represented in the visualisation above. As expected, the analytical solutions provide a way to model the discontinuities both in displacement and pressure (through displacement derivative). We can now use these solutions to model different pathologies which correspond themselves to different discontinuities.

To fully understand the effect of these pathologies we can now exploit the frequency spectra of the model.

4.3. Frequency Response Function

The following results show the impact of the different conditions, described above, on the SCC’s dynamics modeled as a discontinuity on either or both pressure or displacement.

Figure 5. Modeling graph of semi-circular canals for a healthy ear

Figure 5. Modeling graph of semi-circular canals for a healthy ear

We traced frequency response function modeling the resonant frequencies of semicircular canals. We then obtain graphs of the log of the determinant of the system as a frequency response function in Hz, as a way to detect the zeros of the determinant corresponding to spikes towards infinity – the spikes not seeming to shoot to infinity being a simple artifact of the generously discrete nature of the frequency. We have chosen to include in its graphs 2 cursors: one on the discontinuity in viscosity and one on the spatial discontinuity. For a healthy ear, we thus get the graph above. A first deduction would be that semicircular canals react to audible frequencies.

Figure 6. Modeling graph of semi-circular canals for a sick ear

Figure 6. Modeling graph of semi-circular canals for a sick ear

We notice that if we vary only slightly the $\eta$ (pressure discontinuity) nothing happens at the graph level but if this change is more important we have a loss of the lower frequencies. We have linked this graph to dehiscence. Indeed in the physiological case, the sound waves have no impact on the ciliated cells of the semicircular channels because their amplitudes are insufficient (some $\mu$ to mPa) which is modeled here by the small variation of the $\eta$. Whereas in the pathological case, the sound waves are amplified (0.01Pa - 0.1Pa) by the hole present above the channels. The variation of the $\eta$ is therefore more important. The sound waves impact the hair cells of the channels.

The second graph is when we change both of the parameters.

Figure 7. Modeling graph of semi-circular canals for a sick ear

Figure 7. Modeling graph of semi-circular canals for a sick ear

For this first sub graph, we varied the $\eta$ (discontinuity on pressure) and the $\mu$ (discontinuity of displacement). This has been associated with labyrinthitis since we have assimilated edema to a more or less important obstruction and have coupled this obstruction to the increase in pressure that causes this pathology.

And if this phenomenon increases, this third graph is obtained.

In this case, no matter the external vibrations, nothing reacts.

The last graph represents the case where there is a loss of pressure within the system.

Figure 8. Modeling graph of semi-circular canals for a sick ear

Figure 8. Modeling graph of semi-circular canals for a sick ear

This graph was plotted for a $\eta$ negative which means there would be a loss of pressure. We have connected this with a perilymphatic fistula since it is due to a loss of fluid from the inner ear to the middle ear and therefore a loss of pressure in the inner ear.

5. Conclusions and Perspectives

Abbreviations

6.

References

1. Johnson Chacko, L. et al., Analysis of Vestibular Labyrinthine Geometry and Variation in the Human Temporal Bone. Front. Neurosci. 2018; 12, 107. [Google Scholar] [CrossRef]
2. Lee, J.-Y. et al, A Morphometric Study of the Semicircular Canals Using Micro-CT Images in Three-Dimensional Reconstruction Anat. Rec. 296:834–839. 2013. [Google Scholar]
3. Iversen, M. , Rabbitt R., Wave Mechanics of the Vestibular Semicircular Canals, Biophys. J.,vol. 113, Issue 5, pp.1133-1149. 2017; ISSN 0006-3495. [Google Scholar] [CrossRef]
4. David R. et al., Supplementary Information for: Assessing morphology and function of the semicircular duct system: introducing new in-situ visualization and software toolbox. Sci. Rep. 2016, Supplementary information for the article "Assessing morphology and function of the semicircular duct system: introducing new in-situ visualization and software toolbox." by David et al. which are available online: https://www.nature.com/articles/srep32772.
5. Piton, J. , Négrevergne M., Portmann D., Dehiscence of the superior semicircular canal: Approach and CT scan classifications. Rev. Laryngol. Otol. Rhinol. 2008.
6. Philippe Herman, Fistules périlymphatiques. Available online: https://traitesdemedecine.fr/table-des-matieres/s26-oto-rhino-laryngologie/s26-p01-c08-fistules-perilymphatiques/ (accessed on 10 July 2024).
7. Mickie Hamiter, Purulent labyrinthitis. Available online: https://www.msdmanuals.com/fr/accueil/troubles-du-nez,-de-la-gorge-et-de-l%E2%80%99oreille/troubles-de-l%E2%80%99oreille-interne/labyrinthite-purulente (accessed on 10 July 2024).
8. Werner Graf, François Klam, The vestibular system: comparative and functional anatomy, evolution and development. Available online: https://www.sciencedirect.com/science/article/pii/S1631068305001867 (accessed on 17 July 2024).
9. Docteur Catherine Vidal, Dizziness and disorders of balance. Available online: https://www.docteurvidal-iec.com/otoneurologie/vertiges.html (accessed on 18 July 2024).
10. S. KHARRAT et al., DEHISCENCE OF THE SUPERIOR SEMI-CIRCULAR CANAL LA RABTA hospital. TUNIS. 2010, which is available online : https://www.ajol.info/index.php/jtdorl/article/download/64252/52047.
11. Arnaud ATTYEMichael ELIEZER, Towards a generalization of late acquisition for the exploration of cochlear-vestibular disorders in MRI? Available online: https://ebulletin.radiologie.fr/e-quotidien-jfr-samedi/generalisation-lacquisition-tardive-lexploration-troubles-cochleo (accessed on 18 July 2024).
12. M.DELIAT.,Post-traumatic perilymphatic fistulas: etiopathogenesis-diagnosis-treatment University of Limoges 1967, which is available online :https://aurore.unilim.fr/theses/nxfile/default/0a3e3eca-464f-4895-851d-75a8c9adab07/blobholder:0/M1998158.pdf.
13. Iversen MM, Zhu H, Zhou W, Della Santina CC, Carey JP, Rabbitt RD., Sound abnormally stimulates the vestibular system in canal dehiscence syndrome by generating pathological fluid-mechanical waves. Sci Rep. 6 juill 2018;8(1):10257.
14. Goycoolea MV, Muchow D, Schachern P., Experimental studies on round window structure: Function and permeability. The Laryngoscope. juin 1988;98(S44):1-20.
15. Buck, A. , The Mechanism of Hearing. Edinb Med J. nov 1874;20(5):468.
16. Minor LB, Solomon D, Zinreich JS, Zee DS., Sound- and/or Pressure-Induced Vertigo Due to Bone Dehiscence of the Superior Semicircular Canal. Arch Otolaryngol Neck Surg. 1 mars 1998;124(3):249.
17. Yuen H, Boeddinghaus R, Eikelboom RH, Atlas MD., The Relationship Between the Air-Bone Gap and the Size of Superior Semicircular Canal Dehiscence. Otolaryngol Neck Surg. déc 2009;141(6):689-94.
18. Chien W, Ravicz ME, Rosowski JJ, Merchant SN., Measurements of Human Middle- and Inner-Ear Mechanics With Dehiscence of the Superior Semicircular Canal. Otol Neurotol. févr 2007;28(2):250-7.