Membrane Curvature and Connective Fiber Alignment in Guinea Pig Round Window Membrane

Abstract

The round window membrane (RWM) covers an opening between the perilymph fluid-filled inner ear space and the air-filled middle ear space. As the only non-osseous barrier between these two spaces, the RWM is an ideal candidate for aspiration of perilymph for diagnostics purposes and delivery of medication for treatment of inner ear disorders. Routine access across the RWM requires the development of new surgical tools whose design can only be optimized with a thorough understanding of the RWM’s structure and properties. The RWM possesses a layer of collagen and elastic fibers so characterization of the distribution and orientation of these fibers is essential. Confocal and two-photon microscopy were conducted on intact RWMs in a guinea pig model to characterize the distribution of collagen and elastic fibers. The fibers were imaged via second-harmonic-generation, autofluorescence, and Rhodamine B staining. Quantitative analyses of both fiber orientation and geometrical properties of the RWM uncovered a significant correlation between mean fiber orientations and directions of zero curvature in some portions of the RWM, with an even more significant correlation between the mean fiber orientations and linear distance along the RWM in a direction approximately parallel to the cochlear axis. The measured mean fiber directions and dispersions can be incorporated into a generalized structure tensor for use in the development of continuum anisotropic mechanical constitutive models that in turn will enable optimization of surgical tools to access the cochlea.

Graphical Abstract

1. Introduction

The organ of hearing and balance — the ear — consists of three regions shown schematically in Fig. 1a. The outer ear encompasses the auricle and external auditory canal (ear canal) which terminates at the tympanic membrane (eardrum) that collects acoustic signals. The middle ear is an air-filled cavity containing the ossicular chain — the malleus, incus and stapes — that transmits acoustic energy from the tympanic membrane to the inner ear (cochlea), which is a fluid-filled spiral-shaped cavity encased within bone (osseous labyrinth). The cochlea contains several chambers, including the perilymph-filled scala tympani and scala vestibuli. The cochlea transduces acoustic signals to electrical impulses that are transmitted via the vestibulocochlear nerve to the brain [1].

Figure 1:

Anatomy of ear: a) Schematic representation; b) Cochlea of guinea pig obtained through μ-CT (Adapted from [2] with permission). Arrows represent turns of the spiral-shaped cochlear cavity from the apex to the RWM and dashed line represents the axis of the cochlea; c) Round window membrane (in rose) connects to the round window bone structure (in brown) at the sulcus. Scale is approximately 1 mm from lateral wall to medial wall.

Two openings — the oval window and the round window — in the osseous labyrinth allow interchange of acoustic energy and molecules between the middle ear and inner ear spaces [3]. The oval and round windows are covered, respectively, by the oval window membrane (OWM) and the round window membrane (RWM). The OWM, in turn, is covered by the osseous footplate of the stapes. Importantly, however, the RWM has no osseous covering [4]. As a biological membrane separating the perilymph fluid-filled inner ear from the air-filled middle ear cavity, the RWM plays an important role in modulating changes in perilymphatic pressure induced by movement of the stapes at the oval window [5–8]. In addition, the RWM serves as both a barrier and a portal, protecting the inner ear from middle ear pathology while also facilitating active molecular transport between the two compartments [4, 9–11].

The general inaccessibility of the cochlea has significantly hindered the diagnosis and treatment of common pathologies of the inner ear, such as Ménière’s disease and sensorineural hearing loss (SNHL) [12–16]. One current approach for therapeutic delivery to the inner ear is intratympanic injection, whereby the middle ear cavity is filled with a fluid containing the chosen drug, some of which will then diffuse across the intact RWM into the inner ear [17]. However, due to low permeability of the RWM and natural variation between patients, control of dosage administered into the cochlea in this manner is difficult [9–11].

Since the RWM is the only non-osseous barrier between the middle ear space and the cochlea, it represents a potential portal for the direct aspiration of perilymph from the cochlea for diagnostic purposes as well as for direct delivery of therapeutics into the cochlea [18, 19]. However given the small size as well as the presence of tensile pre-stress in the RWM [20], attempts to perforate the RWM for medical access using standard surgical tools typically result in the tearing of the membrane, with severe consequences for hearing [21].

Our group has recently developed a microneedle that can perforate the RWM safely [22–25]. The microneedle can have shaft diameter ranging from 100 μm to 150 μm that tapers to an ultra-sharp tip with radius of curvature of 0.5 μm to 1.5 μm. The microneedles perforate the RWM with a mean force of 1.2 mN in fresh guinea pig RWMs [22] and a mean force of 54 mN in human cadaveric RWMs [23]. The resulting perforation is lens-shaped or slit-like with the length of major axis established by the microneedle diameter and the direction of major axis aligned with the predominant orientation of the collagen and elastic fibers in the RWM [22–24]. Importantly, an ultra-sharp microneedle tip predominantly pushes aside — rather than cuts — the connective fibers within the RWM, thus introducing a minimal degree of trauma during perforation [22]. Indeed, perforations introduced in vivo in a guinea pig model have no long-term consequence to hearing and heal within 72 h. The presence of such microperforations through the RWM can significantly increase the rate of delivery of therapeutic across a membrane [11, 26] to such an extent that the naturally occurring permeability variation between patients is expected not to affect significantly the final delivered dose.

Very recently we developed a hollow microneedle with the same external dimensions but with a 35 μm lumen and aspirated 1 μL of fluid in vivo from the guinea pig cochlea across the RWM, again without structural or functional consequence. Proteomics analysis of the aspirant identified 620 proteins, including the perilymph-marker cochlin [27]. Furthermore, proteomics analysis of guinea pig perilymph aspirated via the hollow microneedles allowed the detection of differences in the proteome as a consequence of dexamethasone administered systemically and intratympanically [28]. That microneedles are also able to introduce microperforations across the larger human cadaveric RWM [23] demonstrates the potential for diagnostic aspiration and therapeutic delivery directly across the human RWM [18, 19].

The goal of this study is to understand the shape and structure of the guinea pig RWM and to lay the groundwork for characterizing and describing the mechanical properties of the RWM. These results will allow further optimization of individual microneedles and of microneedle arrays, as well as other new surgical tools to allow safe access to the inner ear through the RWM. The long-term objective is to extend this research to the shape, structure and properties of the human RWM and to translate the microneedle technology to humans in order to allow routine diagnosis and treatment of hearing and balance disorders.

This study represents the first step toward a thorough experimental characterization of the size, shape and structure of the RWM in a guinea pig model, as well as the distribution, density, orientation and dispersion of collagen and elastic fibers within the RWM. The paper is organized as follows. Section 2 presents background information on the guinea pig cochlea and round window membrane, as well as a brief review of relevant mechanics concepts. Section 3 presents the materials and methods used in the experiments and subsequent analysis. Statistical methods are presented in Section 4. The results are presented and discussed in Section 5. Conclusions are stated in Section 6.

2. Background

2.1. RWM Geometry and Structure

The round window is located at the basal end of the cochlea where the scala tympani terminates, as shown in Fig. 1b in a guinea pig model [3]. The RWM covers the round window and attaches at the sulcus of the bony structure surrounding the round window [4], as shown in Fig. 1c. In the guinea pig, the RWM spans about 1 mm from the lateral wall to the medial wall of the sulcus, and the RWM has an average thickness of about 30 μm. The guinea pig cochlea has an internal hydrostatic perilymph pressure ranging from 0.12 kPa to 0.46 kPa and cochlear function remains stable over the hydrostatic pressure range of −0.1 kPa to 0.7 kPa, as reviewed in [29]. The RWM supports a tensile pre-stress under physiological conditions, but the stress level has not been quantified either in its physiological or its unloaded (i.e. cochlea drained of perilymph) state. The degree of anisotropy of the pre-stress is not known.

Despite its name, the RWM is not round and previous studies have demonstrated that it adopts a complex configuration. Much of the RWM can be approximated as a hyperbolic paraboloid [30, 31]. However its shape deviates significantly at the basal terminus of the RWM (cf. Fig. 1c) where it connects to the cochlear aqueduct via the Round Window Membrane Extension [30]. Here, the RWM is heavily recessed into the cochlea and approximates a tapered parabolic cylinder. These geometric characteristics significantly impact the morphology and behavior of the RWM. The hyperbolic paraboloid shape renders the RWM mechanical compliance properties advantageous to the mechanics of hearing [31].

As shown in Fig. 2a and schematically in Fig. 2b, the RWM is structured in three layers through its thickness: 1) an outer epithelial layer that interfaces with the air-filled middle ear; 2) a central stroma (or layer) with an abundance of elastic fibers and collagen connective tissue; and, 3) an inner epithelial layer that interfaces with the fluid-filled scala tympani of the inner ear. The outer epithelium consists of a single layer of interdigitating cells that connects to the promontory (a bony protuberance in the middle ear, caused by the basal turn of the cochlea, and situated between the round window and the oval window) as well as a continuous underlying basement membrane [32, 33]. The density of elastic fibers and collagen in the central layer increases gradually from the outer epithelial layer to the inner epithelial layer [6]. Collagen fiber bundles and elastic fibers adopt specific orientations within the central layer, but it is unclear whether the directions of these bundles are arranged in distinct layers [6, 34]. In addition, near the bone, the connective tissue components intermingle with each other and the collagen fibers are linked directly to the fibers of the bone matrix [34]. Finally, the inner epithelium is constituted by squamous cells [35].

Figure 2:

Round Window Membrane cross-section from 70-year-old human: a) Histological image of the RWM. OE–outer epithelium; IE–inner epithelium; CTl–loose connective tissue; CTd–dense connective tissue; F–fibroblasts; LEF–large elastic fibers; CF–collagen fibers; GS–ground substance. RWM thickness approximately 70 μm. Original magnification 3000×. (Adapted from [6] with permission) b) Schematic representation of cross-section; the circles in the central stroma represent cross-sections of connective fibers.

2.2. Connective Tissue of the RWM Middle Layer

Connective tissues are found throughout the body in many different forms with diverse functions, such as connection, defense, support, storage, transport, metabolism, and repair following injury. Connective tissues are composed of isolated cells and the extra-cellular matrix (ECM) [36]. The extra-cellular matrix (ECM) consists of four principal components: collagens, elastin, glycoproteins, and proteoglycans. These components form a dynamic and heterogeneous structure, continuously altered by pathological conditions. The stiffness of the tissue, its shape and porosity are determined by the four ECM components as well as water content. Tensile strength, elasticity, hydration, and the ability to withstand mechanical forces are attributed to these four components and the suprastructures they form [37].

In particular, proteoglycans and glycosaminoglycan (GAG) comprise the ground substance² of the ECM, which is interspersed between the collagen and elastic fibers [37]. They have space-filling properties, and are responsible for cell-matrix interactions and binding of other matrix components. Collagen fibers are the main load-bearing elements in a variety of soft tissues. They serve to store, transfer or dissipate elastic energy during deformation, and prevent mechanical failure of tissues. They are fibrous structures that provide support for the tissue and enable high tensile strength. The basic building element of collagen fibers is the fibril with thickness ranging from 50 nm to 500 nm. The collagen fibrils are assembled into varied multiscale and hierarchical structures. Due to this structural hierarchy the mechanical behavior of collagen is nonlinear, hyperelastic, anisotropic and tissue-specific. In addition, collagen fibers exhibit viscoelastic and time-dependent mechanical behavior [37–40]. Lastly, the elastic fibers are made predominately of elastin in the form of an insoluble core. Their role is to provide tissue with elastic recoil and resilience and can withstand repeated distension and recoil [37, 39]. Overall, the orientation and reorganization of the collagen fibers is responsible for the nonlinearity and anisotropy of the tissue; whereas, the elasticity of the tissue is attributed mostly to the elastic fibers. The middle layer of the guinea pig RWM consists of both collagen and elastic fibers [6], which are known to affect the mechanical behavior of the RWM [41].

2.3. Fiber Orientation and Dispersion

Collagen reinforced biological tissues are ubiquitous, and many previous studies have characterized the properties and configurations of the reinforcement fibers. In addition to the RWM, other thin biological membranes include: heart valve leaflets [42–45], fetal membranes [46], skin [47, 48], scleral tissue [49], and the pericardium [50]. Tissues such as arterial walls — too thick to be considered membranes — have complex fiber reinforced layered structures [51, 52]. The myocardium is a thick-walled tissue with a highly complex configuration and reinforcement structure [53, 54]. The uterus also is thick-walled but with a less complex geometrical configuration than the myocardium. Complexity of the fiber reinforcement within the uterus arises due to differences in size, configuration and function under pregnant and non-pregnant conditions [55–57]. A common conclusion from these studies is that the primary fiber reinforcement direction correlates locally with the principal tensile load bearing direction.

In seeking potential principal tensile load bearing directions in the RWM, we note that a hyperbolic paraboloid is a doubly ruled surface with the geometric property that each point on the surface has two directions of zero curvature, and that the trajectories of zero curvature follow straight lines across the surface. This property makes the hyperbolic paraboloid a popular shape for reinforced-concrete shell saddle roofs in structural engineering because steel reinforcements can be placed in the directions of zero curvature, allowing for extremely slender shells (span to thickness ratios up to 700) with very good response to transverse pressure loads [58]. The directions of zero curvature are the principal tensile load bearing directions. Given that the RWM is a slender membrane, with reinforcement-like fibers and with physiological loads being almost exclusively transverse pressure loads, such a shape could be beneficial for the RWM.

2.4. Experimental Overview

Identification of elastic and collagen fibers in the RWM of the guinea pig has been established in the literature by histological sectioning. Herein this connective tissue is studied in a whole mount (i.e. without histological slicing). Experimental characterization of fiber reinforcement direction and dispersion are accompanied by experimental characterization of the profile and local curvatures of the RWM. We seek correlations between fiber directionality and dispersion with properties of the local RWM curvature at several characteristic points of the RWM.

The connective tissue is first characterized using standard techniques like histochemistry, immunofluorescence staining and two-photon microscopy (TPM) taking advantage of autofluorescence (AF) and second harmonic generation (SHG) effects. These results are then used to validate the approach for fiber visualization of the tissue herein, where Rhodamine B staining is imaged through confocal and two-photon excited microscopy.

2.5. Constitutive Modeling

The experimental measurements are made with the goal of incorporating structural information about the RWM in a mechanical constitutive model. We therefore discuss the elements of a constitutive model to place the experimental measurements in context. We assume that the mechanical response of the RWM is dominated by the mechanical response of the middle layer; hence we neglect contributions from the two epithelial layers. Fiber direction and dispersion play important roles in establishing anisotropic mechanical properties of the RWM middle layer.

In order to model finite deformations and large strains, we employ here a hyperelastic constitutive framework that accounts for fiber direction and dispersion within the middle layer of the RWM. A free energy density function is defined of the form [59]

$$\mathbf{\Psi }={\mathbf{\Psi }}_{\text{g}}\left(\mathbf{\text{C}}\right)+{\mathbf{\Psi }}_{\text{f}}\left(\mathbf{\text{C}},\mathbf{\text{H}}\right)$$ (1)

where Ψ_g and Ψ_f are the individual free energy density functions for the ground substance and the fibers, respectively; they are functions of the deformation of the RWM through the right Cauchy-Green strain tensor C. A quantity that relates the current configuration to the reference configuration of a solid body is the deformation gradient tensor F = I + ∇u, where u is the displacement vector. For these experiments, the periphery of the RWM remains attached to the round window sulcus with the cochlea drained of fluid, which defines the reference configuration. The right Cauchy-Green strain tensor C is related to the deformation gradient tensor F by C = F^TF, where the superscript T indicates the transpose.

In addition to C, the mechanical constitutive behavior is formulated to depend on the fibers’ specific geometrical characteristics, such as mean orientation and dispersion. This dependence comes through a generalized structure tensor, H, that models the dispersion of the fibers and is defined as

$$\mathbf{\text{H}}=\frac{1}{4\pi }\underset{\text{Ω}}{\int }\rho (\mathbf{\text{n}})\mathbf{\text{n}}\otimes \mathbf{\text{n}}\text{dΩ}$$ (2)

with tr(H) = 1, where ρ is the orientation distribution function of the fibers in space Ω, and n is a unit vector denoting fiber direction [51, 60].

Measurements reported below will show that the lengths of collagen and elastic fibers typically far exceed the thickness of the RWM, indicating that fibers are oriented primarily in the local tangent plane of the RWM. Therefore, we assume the distribution of fibers possesses independent in-plane dispersion and out-of-plane dispersion [61]. Thus when considering in-plane dispersion, n in Eq. 2 is confined locally to lie within the tangent plane of the RWM. Since the in-plane orientation distribution function, ρ(n), depends upon the in-plane angle, denoted φ, in the local tangent plane of the RWM, its functional dependence can equivalently be written ρ(ϕ).

The von Mises circular probability distribution function, f_m, valid on the semi-circle −π/2 ≤ ϕ < π/2, describes a fiber distribution with a symmetric dispersion. It is defined as

$$f_{\text{m}}\left(\varphi ;\widehat{\varphi },c\right)=\frac{e^{c\cos 2\left(\varphi -\widehat{\varphi }\right)}}{\pi {\mathit{I}}_0\left(c\right)}$$ (3)

where $-\pi /2\le \widehat{\varphi }<\pi /2$ is the mean direction of the in-plane fiber angle distribution, the scalar concentration parameter is c ≥ 0, and I₀(c) is the modified Bessel function of the first kind and zeroth order. As c → ∞, the distribution tends to a point distribution concentrated in the mean direction $\widehat{\varphi }$. As c → 0, the distribution converges to the uniform distribution f_u, defined as

$$f_{\text{u}}\left(\varphi \right)=\frac{1}{\pi }.$$ (4)

The orientation distribution function is then defined as

$$\rho \left(\varphi \right)=p_{\text{u}}{f}_{\text{u}}+p_{\text{m}}{f}_{\text{m}}\left(\varphi ;\widehat{\varphi },c\right)$$ (5)

where p_u ≥ 0 is the proportion of the uniform distribution, and p_m ≥ 0 is the proportion of the von Mises distribution, with p_u + p_m = 1. The values of p_u, p_m, $\widehat{\varphi }$ and c are determined by fitting Eq. 5 to the experimentally measured fiber orientation distribution. A more general model for ρ(ϕ) that accounts for additional von Mises distributions associated with other potential fiber distribution families is discussed in the Supplementary Material.

The Circular Standard Deviation (CSD) represents one-half the angle in radians from the mean direction, $\widehat{\varphi }$, that contains roughly 70% of the distribution. For the von Mises semi-circular distribution the CSD is

$$\text{CSD}\left(c\right)\text{=}\frac{1}{2}\sqrt{-2\ln \left(\frac{I_1\left(c\right)}{I_0\left(c\right)}\right)}$$ (6)

where I₁ (c) is the modified Bessel function of first kind and first order.

A dispersion parameter, k, is used in the generalized structure tensor for the constitutive modeling of the fibers [61]. It is expressed as

$$k=\frac{1}{2}\left(1-\exp [-2{\left(\text{CSD}\right)}^2]\right)$$ (7)

and thus is related to the concentration parameter, c, as

$$k=\frac{1}{2}\left[1-\frac{I_1\left(c\right)}{I_0\left(c\right)}\right]$$ (8)

with values ranging in k ∈ [0, 0.5] for c ≥ 0. A value of c = 0 or k = 0.5 corresponds to a uniform, isotropic fiber distribution, while c → ∞ or k = 0 corresponds to all fibers aligned with the mean direction $\widehat{\varphi }$.

The RWM is deformed predominantly in tension within the local tangent plane and out-of-plane deformation is compressive. Hence any out-of-plane contribution to fiber direction will not be in tension, and will not contribute to the accumulation of strain energy. Therefore we further assume that all fibers are oriented only in the plane of the RWM. The generalized structure tensor, H, then reduces to [61]

$$\mathbf{\text{H}}=k\mathbf{\text{I}}+\left(1-2k\right){\mathbf{\text{v}}}_{\text{f}}\otimes {\mathbf{\text{v}}}_{\text{f}}+k{\mathbf{\text{m}}}_{\text{n}}\otimes {\mathbf{\text{m}}}_{\text{n}}$$ (9)

where m_n is the unit normal vector to the local tangent plane, v_f denotes the unit vector in the mean in-plane fiber direction, I is the 3-dimensional identity matrix, and k is the concentration parameter.

3. Materials and Methods

3.1. Harvesting Guinea Pig Cochleae

The guinea pigs (Hartley, Charles River, Massachusetts) — weighing between 200 g to 300 g with no history of middle ear disease — were euthanized using pentobarbital overdose (100 mg kg⁻¹) in compliance with a protocol approved by the Institutional Animal Care and Use Committee (IACUC) at Columbia University and consistent with the NIH Guide for Care and Use of Laboratory Animals. Immediately following euthanasia, the intact temporal bone of the guinea pig was harvested with blunt dissection. A Stryker S2 πDrive Drill (Stryker, Kalamazoo, Michigan) was used to remove the surrounding bone, exposing a clear, wide-angle view of the RWM. The fluid in the cochlea was drained during dissection. The RWM remained attached to the sulcus of the bone for all subsequent processing and imaging steps. The resulting specimen was rinsed in 0.9% phosphate buffered saline (PBS - Sigma Aldrich D5652) and inspected for gross membrane perforations and fractures of the RWM sulcus and niche. The RWM was then either fixed overnight in 10% buffered formalin or stored in PBS and refrigerated.

3.2. Coordinate Frames and Reference Configuration

The global coordinate frame is taken to be a set of x-, y- and z-axes, defined by orthonormal basis vectors e_x, e_y and e_z, respectively, corresponding to directions of the motion stage of a microscope. A local coordinate frame in the local RWM tangent plane is defined by the orthonormal basis set ${\mathbf{e}}_x^{\prime }$, ${\mathbf{e}}_y^{\prime }$ and ${\mathbf{e}}_z^{\prime }$. The positive sense of m_n is chosen such that the angle between m_n and e_z is acute (i.e. m_n · e_z > 0), after which we take ${\mathbf{e}}_z^{\prime }={\mathbf{\text{m}}}_{\text{n}}$. Then ${\mathbf{e}}_x^{\prime }$ is taken as the direction of e_x projected onto the local tangent plane, and ${\mathbf{e}}_y^{\prime }={\mathbf{e}}_z^{\prime }\times {\mathbf{e}}_x^{\prime }$. The in-plane angle, ϕ, is measured from ${\mathbf{e}}_x^{\prime }$ in a CCW sense about ${\mathbf{e}}_z^{\prime }$.

Under physiological conditions, the RWM supports a small positive perilymph pressure within the cochlea, which defines the physiological reference configuration. However, all experimental measurements reported herein are made in the non-physiological drained condition.

3.3. Experimental Approaches for Imaging Connective Tissue

Several techniques were used for the imaging of collagen and elastic fibers: histology and immunofluorescence; second harmonic generation and autofluorescence; and finally confocal and 2-photon imaging of Rhodamine B staining.

Histological work was performed at the Molecular Pathology Shared Resource of the Herbert Irving Comprehensive Cancer Center of Columbia University with the goal to identify the fibers present in the membrane. Formalin-fixed RWMs were decalcified, paraffin-embedded and cut into 5 μm sections. A Russell-Movat Pentachrome stain was used to identify collagen and elastic fibers. Fig. 3a shows a schematic image of the three RWM layers, and Fig. 3b shows results of the imaging with Russell-Movat Pentachrome stain.

Figure 3:

Characterization of the connective tissue in the RWM: a) Schematic of images; b) Histochemistry Movat staining, with the stroma showing staining for elastic fibers (black) and collagen fibers (yellow) and the outer epithelial layers stained in blue and red; c) Immunohistochemistry staining, with anti-collagen antibody in green and anti-elastin antibody in red. Both colors are present in the stroma, while the epithelial layers were also heavily stained by the collagen marker; d) Endogenous fluorescent signals, with second harmonic generation of collagen in blue and autofluorescence of elastic fibers in green. Epithelial layers also show some autofluorescence in green.

Both paraffin-embedded histological sections and formalin-fixed whole RWMs were studied via immunofluorescence. A double immunofluorescence staining was done using primary antibodies specific for elastin (mouse monoclonal [BA-4] to elastin, Abcam ab9519) and collagen I (rabbit polyclonal to Collagen I, Abcam ab34710). This approach also successfully identified the collagen and elastin fibers (cf. Fig. 3c).

Histological techniques are well-established but their effects on RWM fiber tension and geometry are unclear. Tissue swelling and shrinking during fixation and paraffin embedding may change fiber orientation from that in live RWM tissue [62]. Furthermore, 3D reconstruction remains a difficult problem for histological studies of detailed tissue anatomy such as in this study [63]. Because immunofluorescence staining requires tissue processing that can drastically disrupt RWM shape and fibers, our study of fiber alignment within the membrane requires an alternative technique for imaging fresh tissue with minimal pre-treatment or processing.

The RWM can be imaged in a non-destructive way by whole mount confocal or two-photon microscopy. Elastic fibers and collagen fibers can be successfully imaged simultaneously by capturing the autofluorescence signal from elastin and the second-harmonic generation signal from collagen when a near-infrared laser is used in a two-photon microscope [64–71]. Second harmonic generation (SHG) is a non-linear optical phenomenon of frequency doubling of an incident light source and in biological tissues is specific to collagen due to its macroscopic fibrillar structure [64, 72]. SHG microscopy is therefore a very attractive alternative to image collagen since the phenomenon is naturally occurring and does not require staining or treatment of the tissue. SHG signals of collagen are generated both in backscattering and transmission; the transmitted signal is less diffuse (i.e. generates better images) but impractical for our whole mount studies [64]. Autofluorescence (AF) is another naturally occurring optical phenomenon where intrinsic fluorescence of elastin is used to image elastic fibers without pretreatment [65, 66]. It has been noted that collagen also exhibits autofluorescence in the same wavelength as elastic fibers, but its intensity is much lower such that if elastic fibers are present, the autofluorescence signal of collagen will be virtually undetectable [70]. Both these techniques are non-destructive and can be used in vivo [67, 68].

Two-photon excited fluorescence imaging and second-harmonic generation imaging were performed at the CSMSR³ using a Nikon A1RMP laser scanning system on an Eclipse Ti stand. Either a 25×/NA1.1 ApoLWD water-immersion objective, a 10×/0.5 Plan-Apo dry objective, or a 40×/1.3 Plan-Fluor oil-immersion objective was used. Pulsed excitation at 800 nm was provided by a Chameleon Vision II tunable laser (Coherent, Santa Clara, CA). Fluorescence emission or back-propagated SHG was collected using non-descanned detectors. Second harmonic generation and autofluorescence signals were detected for the fresh RWM (cf. Fig. 3d).

These imaging techniques provide valuable fiber information without the need for special dyes. However, the signal to noise ratios are low in our specimens and the autofluorescence signal is not exclusive to the elastic fibers but is also present on the collagen fibers and the epithelial cells. This led us to consider the alternative approach using Rhodamine B.

Rhodamine B has been used by other researchers to stain elastic fibers [73] and in our case this technique shows a significant improvement in the signal to noise ratio. The RWM is first immersed in a 1 mM solution of Rhodamine B in PBS overnight and then rinsed with PBS and immersed in 100 mL of PBS 1 h before imaging. The RWM is then rinsed a final time with PBS and placed in a MatTek glass bottom dish (No. 1.5). Imaging was done in the CSMSR microscope mentioned earlier and also on an inverted confocal laser scanning microscope Zeiss LSM 880, Axio Observer with a 10× objective (EC Plan-Neofluar 10×/0.30 M27) or a 20× objective (Plan-Apochromat 20×/0.8 M27).

A stack of images was generated at several focal heights spaced 1 μm and 5 μm apart for the 20× objective and the 10× objective, respectively, to obtain 3D images of the RWM. These images were then projected in the stacking direction (maximum intensity z-projection) using an algorithm described later in Section 3.5 to obtain a global image with the visible fibers.

Fig. 4 compares stacked images obtained by combinations of SHG, AF, and Rhodamine B. The collagen is shown in blue (due to SHG) and elastic fibers in green (due to AF) in Fig. 4a. An image of the same location obtained by a combination of AF and Rhodamine B imaging, is shown in Fig. 4b, where the Rhodamine B imaging appears yellow. Both collagen and elastic fibers are imaged via Rhodamine B staining. A qualitative comparison of collagen and elastic fiber alignment in Fig. 4a shows several regions of close alignment, but also regions of misalignment.

Figure 4:

Comparison of second harmonic generation, autofluorescence, and Rhodamine B imaging: a) Collagen fibers appear in blue and elastic fibers in green; b) Both collagen and elastic fibers appear in yellow in Rhodamine B stain image.

The goal of this study is to characterize the overall connective fiber orientation and dispersion to incorporate into a mechanical constitutive model. Therefore in what follows, we report images from Rhodamine B staining alone to quantify the combined characteristics of both sets of fibers. Also, Fig. 4 shows that the length of the collagen and elastic fibers extend many tens of micrometers within the RWM. Thus fiber lengths can be much greater than the RWM thickness. This justifies the assumption of independence of the in-plane dispersion and out-of-plane dispersion.

3.4. Geometry and Curvature of the RWM

One of the main observations of this work is that fiber directionality varies significantly between different regions of the RWM. With the goal of providing a rationale for such a diverse set of fiber directions, the geometry of the membrane was measured from the confocal images with Rhodamine B and studied in relation to the fiber orientations.

3.4.1. Mid-Surface Idealization

For the purpose of computational modeling, the RWM can be idealized as a two-dimensional membrane whose position in space is defined by the mid-surface of the RWM. This idealized membrane can be endowed with an effective thickness that is consistent with the measured thickness. The geometry of the membrane is extracted by identifying the mid-surface of the membrane based on the 10× image stack from Rhodamine B imaging, using a Java plugin developed in-house for the image processing software ImageJ [74, 75]. As depicted schematically in Fig. 5a, the mid-surface is obtained by scanning the 3D image stack for each position in the x-y plane and computing the average pixel height (z-coordinate) weighted by the pixel intensity. A lower intensity threshold is enforced to remove spurious noise in regions without membrane such as shadows and the contour of the membrane.

Figure 5:

Mid-surface and Numerical Flattening: a) red boxes represent all voxels with constant x- and y-coordinate, blue and green layers represent epithelial layers, the dashed line represents the mid-surface; b) voxel columns are shifted in the z-direction so as to numerically flatten the mid-surface.

Taking an average position along the z-direction is an approximation of taking the average in the direction normal to the surface. However, this methodology is easier to implement and a simple analysis (not shown) of the error due to this approximation reveals that the error for the worst case with our imaging setup would be at most 0.45 μm, which is two orders of magnitude smaller than the thickness of the membrane, and therefore negligible. The final 2D representation is of the z-coordinate of the membrane mid-surface as a function of independent values of x and y, denoted z_ms(x, y).

Imaging with a different contrast agent would change the through-thickness pixel intensity distribution. However, given the similarity in through-thickness structure, the change would on average be uniform across the entire RWM. This would induce a systematic shift of the computed mid-surface, but is not expected to affect significantly the measured profile.

Since a dry 10× objective is used due to an impractical working distance, but the membrane is immersed in PBS, a correction factor is applied to the height of the membrane to account for refraction effects [76]. The correction factor is defined as the ratio of the refractive index of the specimen to the refractive index of the immersion fluid of the objective. Here, the refractive index of the specimen refers to the medium above the cover glass which is PBS with refractive index 1.335; the immersion fluid of the objective is air with refractive index of 1.0. This results in a correction factor of 1.335.

3.4.2. Principal Curvatures

To compute the curvature properties of the RWM, the 2D mid-surface representation, z_ms (x, y), is converted into a triangular mesh by means of a Python script. Since the resulting mesh has roughly twice as many triangles as the number of pixels, the mesh is simplified by quadratic edge collapse decimation [77] and smoothed with the Taubin filter [78] to remove spurious local curvature effects.

The curvature shape operator, S, is computed from the dihedral angles of the triangle edges [79]. The eigenvectors of the shape operator are the two principal curvature directions denoted by orthonormal vectors v_min and v_max as well as the normal to the surface (which is part of the null space of S). Both v_min and v_max are tangent to the surface at the point under consideration.

The RWM curvature in a general direction v is given by

$$\kappa \left(\mathbf{\text{v}}\right)={\mathbf{\text{v}}}^{\text{T}}\cdot \mathbf{\text{S}}\cdot \mathbf{\text{v}}$$ (10)

where κ is a scalar value of curvature, v a unit vector tangent to the surface at the point being considered and S is a rank-2 tensor representing the shape operator of the surface at that point.

The principal values of curvature are given by

$${\kappa }_{\max }=\kappa \left({\mathbf{\text{v}}}_{\max }\right)={\mathbf{\text{v}}}_{\max }^{\text{T}}\cdot \mathbf{\text{S}}\cdot {\mathbf{\text{v}}}_{\max }$$ (11a)

$${\kappa }_{\min }=\kappa \left({\mathbf{\text{v}}}_{\min }\right)={\mathbf{\text{v}}}_{\min }^{\text{T}}\cdot \mathbf{\text{S}}\cdot {\mathbf{\text{v}}}_{\min }$$ (11b)

where κ_min, κ_max and v_min, v_max are the corresponding minimum and maximum eigenvalues and eigenvectors, respectively, of the shape operator S.

3.4.3. Directions of Zero Curvature

A direction of zero curvature is a direction tangent to the surface for which Eq. 10 has a result of zero. Here we derive an expression for the direction of zero curvature. An explicit quadratic surface in the context of differential geometry of surfaces locally approximates to second order any regular surface in the neighborhood of a point p defined as having height z above an x-y plane. Mathematically, it is defined by the position vector $\mathbf{r}(x,y)={[x,y,z=\frac{1}{2}({\kappa }_{\text{I}}{x}^2+{\kappa }_{\text{II}}{y}^2)]}^{\text{T}}$ where −κ_I and −κ_II are the principal curvatures at the point p of the regular surface [e.g. 80, 81]. Conceptually, an explicit quadratic surface allows consideration of a continuous transition between different general classes of surfaces.

We consider two special surface classes that each has one or at most two directions of zero curvature: 1) a hyperbolic paraboloid with principal curvatures of equal magnitude but opposite sense; and, 2) a parabolic cylinder in which one of the principal curvatures is zero.⁴ Since the stationary point of the hyperbolic paraboloid has principal curvatures with equal non-zero magnitude and opposite signs, its directions of zero curvature bisect the principal directions. Defining θ as the angle between the principal curvature direction with minimum magnitude and the direction of zero curvature, θ = 45° for a hyperbolic paraboloid. The stationary point of a parabolic cylinder has one principal curvature equal to zero, so the angle between this principal direction and the direction of zero curvature is trivially θ = 0°. Fig. 6a shows these two surfaces as limiting cases of the range of θ.

Figure 6:

Direction of zero curvature and angle θ: a) Limiting values of θ, ranging from 0° in the case of a parabolic cylinder (left) to 45° in the case of a hyperbolic paraboloid (right), where green line represents local direction of minimum principal curvature and red line represents local direction of maximum principal curvature; b) Directions of zero curvature V₀₁ and V₀₂ make an angle ±θ with respect to the principal direction with minimum magnitude v_min; and, c) Value of curvature varies continuously between the two principal directions, so a direction with zero curvature must exist if the principal curvatures have opposite signs.

We now consider the case where a surface adopts a value 0° < θ < 45° between the limiting values. To define precisely the value of θ we recall the value of curvature in a general direction v from Eq. 10. Since S is diagonalizable through its eigenvectors in Eq. 11, the value of the curvature is computed as

$$\kappa \left(\mathbf{\text{v}}\right)={\left(\mathbf{\text{v}}\cdot {\mathbf{\text{v}}}_{\min }\right)}^2{\kappa }_{\min }+{\left(\mathbf{\text{v}}\cdot {\mathbf{\text{v}}}_{\max }\right)}^2{\kappa }_{\max }.$$ (12)

Referring to Figs. 6b, since the orthonormal v_min and v_max are tangent to the surface, v is represented in terms of v_min, v_max and θ as

$$\mathbf{\text{v}}=\left(\mathbf{\text{v}}\cdot {\mathbf{\text{v}}}_{\min }\right){\mathbf{\text{v}}}_{\min }+\left(\mathbf{\text{v}}\cdot {\mathbf{\text{v}}}_{\max }\right){\mathbf{\text{v}}}_{\max }={\mathbf{\text{v}}}_{\min }\cos \theta +{\mathbf{\text{v}}}_{\max }\sin \theta .$$ (13)

The sign, or sense, of curvature is arbitrary and depends on the definition of the normal of the surface. For convenience of notation we define v_min as the principal direction with minimum magnitude of curvature.

To obtain the direction of zero curvature, κ(v) in Eq. 12 is set to zero and upon substitution into Eq. 13 and solving for θ there results

$$\theta =\arctan \sqrt{-{\kappa }_{\min }/{\kappa }_{\max }}.$$ (14)

This analysis applies to surfaces locally intermediate (and at the limits) between a hyperbolic paraboloid and a parabolic cylinder, so that −κ_min/κ_max ≥ 0. As will be seen for the guinea pig RWM, all measured values of κ_max have a positive sense and all but three measured values of κ_min have a negative sense. The three positive κ_min values occur at the basal terminus of the RWM where the mean κ_max is an order of magnitude larger than the mean κ_min. A statistical test failed to detect a significant non-negative κ_min in this region. Therefore the RWM in this region has a strong parabolic cylinder character. It bears emphasis that the curvature measurements are made in the drained state, and that deformations induced by the positive pressure in the perilymph under physiological conditions will lead κ_min at the basal terminus to have a more negative numerical value.

Nonetheless, to process these three measurements, we generalize the definition of θ to include cases of principal curvatures with same sign by taking the absolute value of the radicand in Eq. 14, resulting in the following definition of θ

$$\theta =\arctan \sqrt{\left|{\kappa }_{\min }/{\kappa }_{\max }\right|}.$$ (15)

When both principal curvatures have the same sign, the meaning of θ is no longer the angle to the direction of zero curvature but instead a measure of the relative strength between the principal curvatures.

3.5. Numerical Flattening of the 3D Image Stack

To obtain a 2D image to process fiber directionality based upon 20× Rhodamine B imaging, we need to flatten the stack from 3D to 2D. The membrane is very thin, so each slice of the 3D stack (i.e. all voxels in a plane of constant z) contains only a very small strip of membrane (cf. Fig. 7a), which is impractical to analyze.

Figure 7:

Numerical flattening of a 20× membrane stack: a) Single slice of image stack before the numerical flattening, with both epithelial layers and connective tissue layer visible; b) Result of applying standard z-projection to the stack; c-d) After numerically flattening, the different regions are z-projected independently; c) connective tissue layer; d) outer epithelial layer.

The common approach is to perform a so called z-projection, whereby a 2D image is generated for each x-y position by taking the maximum voxel intensity for that z-column (i.e. all the voxels spanning the z-direction with the same x- and y-coordinate). This approach removes the z dimension of the stack and depicts the brightest structures in the membrane.

A simple z-projection, as shown in Fig. 7b, does not produce interpretable results because epithelial cells of the RWM are stained at higher intensity than the fibers in the central stroma, so the projected image is dominated by the epithelial cells, overshadowing the fibers in the membrane.

Thus we developed a method to “remove” the epithelial layers prior to projecting to the x-y plane. Simply put, our methodology is able to flatten the membrane such that all layers will be roughly parallel to the x-y plane. Then by simply removing all the slices above a certain z-value in the flattened image, the epithelial layer can be removed.

Our method, hereby called “numerical flattening” was developed as a Java plugin for ImageJ [74, 75]. The first step is to compute the mid-surface, z_ms(x, y), of the RWM, as described in Section 3.4.1 and shown in Fig. 5a. Then, as depicted in Fig. 5b, each voxel in the stack is moved in the z-direction such that all membrane layers become parallel to the horizontal plane. More precisely, the mid-surface of the membrane is moved pixel-by-pixel to the central slice of the stack and all voxels in that z-column are moved so as to maintain their relative vertical position with respect to the mid-surface. This is expressed as

$$z_{\text{new}}\left(x,y\right)=z_{\text{old}}\left(x,y\right)-\left[z_{\text{ms}}\left(x,y\right)-z_{\text{cs}}\left(x,y\right)\right]$$ (16)

where x, y and z are coordinates of the columns of voxels, z_old and z_new correspond to the z-coordinate before and after the transformation, and z_cs is the z-coordinate of the central slice of the stack.

Since the 3D stack is a discretized representation of the intensity field, voxels can only occupy a z-coordinate that is a multiple of the z-distance between slices, denoted Δz. However, since most voxels will be shifted by a value not restricted to this discretization (i.e. z_ms(x, y) − z_cs(x, y) is not necessarily a multiple of Δz), a linear interpolation is used to obtain the voxel intensity at each discrete slice. The algorithm is referred to as “numerical flattening” since the result resembles a stack imaged from a flat membrane. Finally, in this “flattened” configuration we select only the slices parallel to the x-y plane that contain the connective tissue and apply a standard z-projection to generate a 2D image of the fibers in the x-y plane as shown in Fig. 7c.

It is important to note that the stack is not deformed in a length preserving fashion, as one would generally assume when flattening something in the physical world. Instead, this algorithm is equivalent to shearing the membrane along the z direction, introducing a distortion in the directions with non-zero slope. Nevertheless, this distortion is the same as the one exhibited by a z-projection, and in our case, is accounted for and corrected.

A direct consequence of this approach is that information regarding the out-of-plane distribution of the fibers is lost, both in the collapse of fibers oriented in the z-direction as well as the merger of all potential layers of fibers within the stroma. This seems an acceptable approximation due to two main factors. First, the membrane is very thin and measurements show that the fibers are predominantly oriented in the in-plane direction. Second, due to the highly curved nature of the membrane, the segregation of the membrane imaging data into different layers has proven to be a significant challenge.

3.6. Quantification of Fiber Orientation and Dispersion

The process by which the fiber directionality and dispersion are quantitatively extracted from the numerically flattened images is described. While the directionality and dispersion can be quantified, the constraints on working distance preclude the characterization of the length and diameters of individual fibers. Therefore, it is not possible to weight the contribution of individual fibers by their relative volumes when calculating the overall fiber orientation distribution, as discussed in [82].

3.6.1. Fourier Transform

To quantify fiber orientation and dispersion, we apply the Fast Fourier Transform (FFT) algorithm to the flattened image of the fibers in the x-y plane, producing a power spectrum, which is then processed through a wedge filtering technique. This approach follows closely the methodology described in [83], with additional details in the Supplementary Material and references cited therein [84–91].

The final result of this process is a histogram (cf. Fig. 8a) where each bin corresponds to a 1° angular sector and the ordinate corresponds to the integral of the intensity of the power spectrum when filtered by the corresponding wedge. This histogram represents the angular distribution of the fiber density in the flattened image in the x-y plane.

Figure 8:

Fiber directionality: a) Example of histogram corresponding to the angular distribution of fiber density in x-y plane. Red curve represents fitting of von Mises plus a uniform distribution of Eq. 5 to the histogram data; and, b) Reversal of numerical flattening for the fiber direction where v_min and v_max are the principal curvature directions of the membrane at point p, where v_fp is the computed projected fiber direction and ${\mathbf{v}}_{\text{f}}^{\ast }$ points in the corresponding fiber direction in the local tangent plane of the RWM at point p.

3.6.2. Distribution Fitting

The overall fiber orientation distribution function, ρ(ϕ), in Eq. 5 consists of the sum of a uniform distribution, f_u, and a von Mises distribution f_m(ϕ; $\widehat{\varphi }$, c) on the semi-circle −π/2 ≤ ϕ < π/2 in the local tangent plane of the RWM. However the histogram quantifies fiber directionality in the flattened image, so we define

$${\rho }_{\text{p}}\left(\varphi \right)=p_{\text{u}}{f}_{\text{u}}+p_{\text{m}}{f}_{\text{m}}\left({\varphi }_{\text{p;}}{\widehat{\varphi }}_{\text{p}},c\right)$$ (17)

where ρ_p (ϕ) is the orientation distribution function measured in the x-y plane, ${\widehat{\varphi }}_{\text{p}}$ is the mean fiber direction projected to the x-y plane and ϕ_p is the angle in the x-y plane measured relative to the x-axis in a CCW sense about the z-axis. The values of ${\widehat{\varphi }}_{\text{p}}$, c, p_u and p_m are determined by fitting Eq. 17 to the histogram using the maximum likelihood estimation (MLE) method [92–94].

3.6.3. Fiber Direction within RWM

The mean fiber orientation in the direction of ${\widehat{\varphi }}_{\text{p}}$ in the x-y plane is represented by the unit vector v_fp in Fig. 8b. Upon projection to the local RWM tangent plane there results

$${\mathbf{\text{v}}}_{\text{f}}^{*}={\mathbf{\text{v}}}_{\text{fp}}-\frac{{\mathbf{\text{v}}}_{\text{fp}}\cdot {\mathbf{\text{m}}}_{\text{n}}}{{\mathbf{\text{e}}}_z\cdot {\mathbf{\text{m}}}_{\text{n}}}{\mathbf{\text{e}}}_z$$ (18)

that indicates the mean fiber direction vector tangent to the RWM surface. The unit vector in the local plane of the RWM in the direction of the mean fiber direction is then given as ${\mathbf{\text{v}}}_{\text{f}}={\mathbf{\text{v}}}_{\text{f}}^{*}/\Vert {\mathbf{\text{v}}}_{\text{f}}^{*}\Vert$, which is used in the calculation of the generalized structure tensor, H, in Eq. 9.

We now express the mean fiber direction relative to two other directions in the local RWM tangent plane. For convenience of incorporating into a computational model, $\widehat{\varphi }$ is defined as the angle between v_f and ${\mathbf{\text{e}}}_x^{\prime }$

$$\widehat{\varphi }=\arccos \left({\mathbf{\text{v}}}_{\text{f}}\cdot {\mathbf{\text{e}}}_x^{\prime }\right).$$ (19)

As shown schematically in Fig. 8b, the angle between v_f and v_min, can be calculated as

$$\widehat{\mathbf{\Phi }}=\arccos \left({\mathbf{\text{v}}}_{\text{f}}\cdot {\mathbf{\text{v}}}_{\min }\right)$$ (20)

to allow a direct comparison of $\widehat{\mathbf{\Phi }}$ and θ.

4. Statistical Analysis

RWM curvatures and connective fiber angles are reported as mean, median, 25th percentile (denoted Q₁), 75th percentile (denoted Q₃), outliers and most extreme data not taken to be outliers. Interquartile range is defined as IQR = Q₃ − Q₁, and outlier measurements are defined as having value less than Q₁ − 1.5 × IQR or greater than Q₃ + 1.5 × IQR.

Linear regression results report R² as the variance explained by the linear relationship normalized by total variance and the two-sided p-value as the probability of falsely rejecting the null hypothesis of zero slope. The confidence interval about the regression line is taken to be 95%. Statistical significance of the sign of RWM curvature is determined via a one sample t-test with p < 0.05.

5. Results and Discussion

A total of ten RWMs were processed and imaged from eight animals. Both temporal bones containing the RWMs were harvested from each guinea pig and images were taken from four right RWMs and six left RWMs. The other six RWMs were damaged during the harvesting process. We arbitrarily chose one RWM from a left ear to be the “reference” membrane. Subsequently all membranes from a right ear were transformed by a mirror symmetry operation equivalent to reflecting the RWM about the guinea pig’s mid-sagittal plane. This does not affect the local properties of the measurements, like curvature and fiber direction, but it allows for the comparison of these measurements between different membranes. Then all triangular meshes of the RWMs underwent rigid body translations and rotations to minimize the average distance between the membrane in question and the reference membrane. The average distance between all membranes and the reference membrane is less than 10 μm that in turn is less than the thickness of the membrane of about 30 μm, which is not inconsistent with all guinea pigs being from the Hartley strain.

5.1. Geometry of the membrane

As shown in Fig. 9, the geometry of the RWM is complex, having apparently two main regions: a relatively flat shallow region with small curvatures of opposite sign as well as a deeply recessed tapered region with one curvature much smaller in magnitude than the other. The specimen global coordinate frame is also shown in Fig. 9.

Figure 9:

Geometry of the RWM from three different perspectives shown relative to global specimen coordinate frame: a) Top view from z-axis as seen from the middle ear. Bottom region (smaller y-values) corresponds to the flatter region of the membrane while the top region (larger y-values) corresponds to a deeper and more conically shaped region. b) Side view from y-axis. Conical shape of membrane is evident; c) Side view from x-axis. Transition from flat to conical shape is seen as y-value increases. Images obtained with inverted confocal laser scanning microscope Zeiss LSM 880, Axio Observer with a 10× objective (EC Plan-Neofluar 10×/0.30 M27).

To illustrate how closely the membranes overlap, Fig. 10 shows the profiles of all membranes, after the rigid-body rotations for alignment, for specific horizontal and vertical sections. The profile plots demonstrate that the membranes fit very well to each other. The small variability between specimens shows that the techniques developed to compute the mid-surface and the rigid-body fitting produce consistent results.

Figure 10:

Surface profiles across all membranes. a) Profiles of each membrane along vertical lines with x-coordinates of 400 μm, 700 μm and 1000 μm (top row) and along horizontal lines with y-coordinates of 400 μm, 700 μm and 1000 μm (bottom row); b) Representation of the mid-surface of reference membrane (20180308LK), where the lightness of the color corresponds to decreasing z-coordinate. White lines represent positions from which profiles are extracted; c) Five different positions on the membrane: Bottom, Left, Center, Right and Top. The Top position corresponds to the location furthest into the inner ear, towards the cochlear aqueduct. The Bottom position corresponds to the position on the membrane closest to the outer wall of the cochlea and most exposed during surgical interventions. Referring to the global specimen z-axis from Fig. 9, the Left, Center and Right are three positions with the same value of z, which is roughly the mean z between the bottom and top positions. The Left position is closest to the Basal wall and the Right position is closest to the Apical wall.

The two regions previously evident in Fig. 9 can be identified. By looking at the profiles with constant x-coordinate, the profile lines remain relatively straight as the y-coordinate increases. However, the membrane becomes more curved along the x-axis as the value of y-coordinate increases.

5.2. Curvature

The RWM has a complex geometry with various regions having significantly different values of curvature. To quantify the curvature and fiber distribution across the membrane, Fig. 10c shows several points within each membrane — denoted as Bottom, Left, Center, Right and Top — where we report detailed measurements and comparisons.

Fig. 11a compiles the values of the principal curvatures in all membranes at the five points.⁵ Two logical groupings of the RWM shape emerge: hyperbolic-type and cylindrical-type. The hyperbolic-type corresponds to principal curvatures of similar magnitude and opposite sign. The cylindrical-type corresponds to one of the principal curvatures being substantially larger in magnitude than the other. It is expected that each point in the membrane is in some intermediate state between the two idealized limits of these cases, as quantified by θ in Eq. 14 and shown schematically in Fig. 6. Then, Fig. 11b plots θ at the five sample points in the membrane. The hyperbolic-type regions (Bottom, Left and Center) have on average a higher value of θ while the cylindrical-type regions (Right and Top) have a lower value of θ.

Figure 11:

Directions of principal curvatures and zero curvatures: a) For regions indicated on left side of the figure (Bottom, Left and Center) the magnitudes of principal curvatures are similar but with opposite sign (hyperbolic-type). Regions on right side of the figure (Right and Top) have very distinct magnitudes of principal curvature (cylindrical-type), with the minimum close to zero and the maximum much higher than the maximum in the region with hyperbolic-type curvatures; and, b) Values of θ at the different regions in the membrane.

5.3. Fiber Directionality

Fig. 12a compares the three directions: Fiber direction, Zero curvature direction and Minimum principal curvature direction; the directions are represented by values of ϕ_p in the x-y plane. The relatively small spread of the points suggests that each region has comparable fiber directions between membranes. The correspondence between the fiber direction and zero curvature direction is strong in the Bottom, Left and Top regions, and weaker in the Center and Right regions. The type of curvature can be inferred from the magnitude of the difference between the zero curvature angle and the principal direction angle, where hyperbolic-type points have a large difference and cylindrical-type points have a small difference.

Figure 12:

Fiber and curvature directions: a) Comparison between the fiber direction, the zero curvature direction and the minimum principal curvature direction, expressed by the respective values of ϕ_p. There is a relatively small spread of the points, which suggests that each region has comparable fiber directions between membranes; b) Angles of zero curvature in selected regions of the membranes, θ, and of mean fiber direction, $\widehat{\mathbf{\Phi }}$, both relative to the direction of v_min at different regions in the membrane.

The angle $\widehat{\mathbf{\Phi }}$ is compared to the value of θ in Fig. 12b. We observe that in the Bottom, Left and Top points, the mean fiber direction is close to the mean direction of zero curvature, while for the Center and Right points the mean directions are separated by more than 20°.

A linear regression was done between θ and $\widehat{\mathbf{\Phi }}$ to quantify the level of correlation (cf. Fig. 13a). The results suggest there to be a highly significant correlation between the fiber direction and the direction of zero curvature (R² = 0.466, p-value = 9.21E–8). However, it is clear that the points of the Center and Right positions are not varying around the solid line, but are instead systematically deviating towards lower values of fiber angle. In fact, if we remove these points from the regression analysis, the explaining power of θ markedly increases, as shown in Fig. 13b (R² = 0.619, p-value = 6.91E–7). This suggests that there might exist another variable related to the position of the measured point within the membrane that could explain some of the variation.

Figure 13:

Linear regression between θ on abscissa and $\widehat{\mathbf{\Phi }}$ on ordinate. Solid line corresponds to idealized case in which fibers exactly follow the direction of minimum curvature. Dashed line corresponds to the regression line and the shaded area to the 95% confidence interval of the regression. a) Regression using all points in the membrane; b) Regression using only the Bottom, Left and Top points.

To better visualize how fiber directions relate to directions of zero curvature across the membrane, a field with the directions of zero curvature at each node was computed for each membrane mesh, represented by arrows in Fig. 14a. To enhance the clarity of the visualization of this field, a “streamline” or zero-curvature trajectory representation was computed as represented in Fig. 14b. To achieve this, a randomly seeded surface line is integrated numerically as if the zero curvature field corresponded to a velocity field. One family of zero-curvature trajectories is shown in red and the other family is shown in blue. The direction of v_min bisects the acute angle and the direction of v_max bisects the obtuse angle between the red and blue trajectories of zero curvature at any point. Finally, the fiber direction at each of the five points of each RWM is superimposed atop the trajectory plot in Fig. 14b, which shows significant alignment between the direction of zero curvature and the fiber directions. The zero-curvature trajectories and measured fiber directions for all other membranes are shown in Appendix A.

Figure 14:

Representation of the directions of zero curvature and fiber directions for one membrane (20180427RA): a) Arrows representing both directions of zero curvature at each node of the mesh; b) Numerically computed trajectories were determined by the field of zero curvature. Fiber direction at each point is represented as a black arrow and superimposed on the trajectories; c) Representation of fiber directions from all membranes superimposed to the trajectories of directions of zero curvature. Legend for Arrow locations: Orange at “Top”; Red at “Left”; Green at “Center”; Blue at “Right”; Purple at “Bottom”; Yellow at “Bottom_Right”. Points labeled “Bottom_Right” were merged into the label “Bottom” for quantitative analysis.

In Fig. 14c are represented all the measured fiber direction points for all membranes allowing for a global view of the membrane and its fiber orientations. It is clear by observing Fig. 14c that the clustering of the different position labels (Top, Bottom, etc.) is indicative that each label corresponds to a specific and unique region of the membrane. Nevertheless, even though they are self-consistent in each membrane, two neighboring points may be labeled with different positions (e.g. “Center” and “Bottom”) due to experimental difficulties like a different angle of imaging, shadows from debris and variation in the mounting height for the sample (that constricts the available working distance). This provides an explanation for why there is some variation of the different angles even within a region, since each region encompasses a relatively large range of positions within the membrane.

However, this does not explain why locally each point does not follow the direction of zero curvature. To understand this phenomenon we refer again to Fig. 14c where we can observe that even though the average fiber direction can be significantly different between regions, the direction of the fibers varies smoothly from one region to the next. We verified this by looking at the correlation between the y-coordinate of a particular point and the fiber angle ${\widehat{\varphi }}_{\text{p}}$ in the x-y plane, where a significant correlation is evident (R² = 0.829 and p-value = 3.05E–19).

We note that there is no particular reason why the y-coordinate would correlate so well with the angle of the fiber direction; this happened rather fortuitously due to the fact that the reference membrane was imaged in this orientation. We therefore developed an algorithm to search for the angle with respect to the x-axis that yields a maximum R². According to Fig. 15a, the maximum of the R² function is relatively broad. The best fitting direction is at the angle of ϕ_p = 95.42° from the x-axis (R² = 0.833 and p-value = 1.68E–19), as seen in Fig. 15b, as opposed to 90° corresponding to the y-axis direction. Thus, while the fiber angle of each individual point of the membrane may vary significantly (roughly 50° to 150°), this angle correlates well with a linear function of the coordinate of the point with respect to the position along the line making a ϕ_p = 95.42° angle with the x-coordinate. The biological mechanism underlying this correlation is not known.

Figure 15:

Direction of best fit: a) Value of R² of the linear fit for each direction from 0° to 180°. The maximum value is indicated by the circular marker. b) Correlation between mean fiber direction, ${\widehat{\varphi }}_{\text{p}}$, and distance along the line in the direction of ϕ_p = 95.42° in the x-y plane. The dashed line corresponds to the regression line and the shaded area to the 95% confidence interval of the regression.

We did not measure the direction of the cochlear axis (cf. Fig. 1) before removing the bone surrounding the RWM from the cochleae. However we note that the direction of highest correlation aligns — at least qualitatively — with the axis of the cochlea.

The relevance of this analysis lies on the observation that, while the fibers following the direction of zero curvature works well for the larger peripheral regions of the membrane, the interior regions are also constrained by the fact that the fiber directionality transitions smoothly between different regions. Since the direction of zero curvature and the fiber direction are correlated, the direction of zero curvature will also correlate well with the putative direction of the cochlear axis (cf. Fig. 16a), albeit less so due to the fact that the curvature is a local quantity and does not require as much continuity as the fibers do.

Figure 16:

a) Correlation between angle θ and the position along the line in the direction of ϕ_p = 95.42° in the x-y plane; and b) Correlation between fiber dispersion as measured by the circular standard deviation of the fiber distribution and the angle θ. For both, the dashed line corresponds to the regression line and the shaded area to the 95% confidence interval of the regression.

5.4. Fiber Dispersion

Finally, we check the correlation between the geometric properties of the membrane and the amount of dispersion in the fiber distribution histogram. In Fig. 16b the fiber dispersion, as measured by the circular standard deviation of the distribution, is compared to the angle θ, as defined in Equation 14. The circular standard deviation ranges from 15° to 45°, with an average value of 27.5°. The higher values of dispersion are in the hyperbolic-type region of the membrane. Fig. 16b shows a statistically significant linear correlation between these two quantities (R² = 0.298 and p-value = 6.06E–5), revealing an increase in the fiber dispersion as the type of curvature changes from parabolic cylinder (θ → 0°) to hyperbolic paraboloid (θ → 45°). However, unlike the fiber direction, there is no discernible correlation with the cochlear axis direction (R² = 0.033 and p-value = 2.15E–1, figure not shown). This suggests that dispersion is dictated by local quantities of the membrane geometry to a larger degree than by its global properties.

6. Conclusions

Herein we describe experimental measurements of the geometry of the RWM in guinea pigs. The geometry from specimen to specimen is surprisingly regular with an average difference between membranes of roughly 1 % of the total size of the membrane (10 μm average profile difference between membranes normalized by a width of 1000 μm). The results show that the RWM can be classified as having three distinct regions characterized by local curvature: 1) a hyperbolic-type with principal curvatures of opposite sign and two directions of zero curvature; 2) a cylindrical-type, with only one direction of zero curvature coincident with a principal direction; and, 3) a transition region between the hyperbolic-like and cylindrical-like regions. Trajectories of directions of zero curvature on the RWM were calculated from the geometrical measurements.

Fiber direction and dispersion were quantified using confocal microscopy coupled with Rhodamine B staining of the collagen and elastic fibers. The shape of the RWM transitions from a hyperbolic paraboloid configuration in the apical direction to a tapered parabolic cylinder in the basal direction near the cochlear aqueduct. The predominant fiber directions correlate with a direction of zero curvature in regions where the two principal curvatures of opposite sign have approximately the same magnitude, such as in the Bottom, Left and Center positions. There also is some correspondence between fiber direction and zero curvature direction in the region that is strongly cylinder-type in the top region. However there is no correlation with direction of zero curvature in the transition region (i.e. Right) between the hyperbolic-type and the cylindrical-type regions.

Quantitatively, there is a significant correlation between the fiber alignment and the geometric properties of the membrane, through the parameter that quantifies the direction of zero curvature (R² = 0.466, p-value = 9.21E–8). The correlation increases when the transition between the hyperbolic-type and the cylindrical-type regions is taken into account (R² = 0.619, p-value = 6.91E–7). The observed transition regions indicate that fiber directions in the RWM vary smoothly between the two extreme cases with very different type of fiber alignment, demonstrated by the surprisingly strong correlation between the fiber direction and a linear function of the position along what appears to be the direction of the cochlear axis (R² = 0.833, p-value = 1.68E–19). Fiber dispersion, as quantified by the Circular Standard Deviation, is also correlated with local geometric properties (R² = 0.298, p-value = 6.06E–5), with values from 15° to 45°, and an average value of 27.5°, with greater dispersion in the hyperbolic-type region of the membrane. The mean fiber direction and dispersion measurements can be used to define material parameters for a generalized structure tensor that incorporates fiber direction and dispersion in a continuum anisotropic constitutive model.

While we make no claims of cause and effect, the results of this work suggest that the geometry and fiber distribution of the RWM may provide some optimization of its mechanical behavior. Specifically, previous studies [31] have shown that a hyperbolic paraboloid is more suitable than a simple flat disk geometry to maintain RWM compliance properties for physiological variations of the perilymphatic pressure. The guinea pig RWM approximates a hyperbolic paraboloid shape (i.e. principal curvatures of opposite sign and two directions of zero curvature) over the region of its greatest width. In this region, the collagen and elastic fibers tend to follow directions of zero curvature, which in turn allows them to remain as straight as possible in their undeformed pre-stressed configuration.

We comment on whether the shape of the RWM can be rationalized as being a minimal surface that would be adopted by a soap bubble spanning the RWM opening. The Young-Laplace equation describing such minimal surfaces requires that the pressure difference across the minimal surface be proportional to the mean curvature defined as ${\kappa }_{\text{m}}=\left({\kappa }_{\max }+{\kappa }_{\min }\right)/2$ at all points. Since the mean curvature at all points from Fig. 11 is not constant, this condition does not hold.

The results presented herein suggest many future directions of research.

1. Rhodamine B staining did not distinguish between collagen and elastic fibers. Individual measurements of direction, dispersion and through-thickness variations of the two fiber types would be of interest.

2. The technique used to image the RWM did not allow clear observation of the inner epithelium. Use of 2-photon microscopy to avoid the effect of signal loss with increasing depth would allow characterization of the inner epithelium.

3. Investigation in greater detail of the correlation between different layers in the stroma and their fiber orientations could give additional insight into connective fiber structure.

4. More closely spaced measurements of geometry and fiber directions in the transition region would allow further analysis of potential relationships between fiber directions and zero-curvature directions there.

5. Measurements of the mechanical response of the RWM under known loading conditions will be necessary for constitutive model development and validation.

6. Detailed measurement of the direction of the cochlear axis as well as additional fiber direction measurements with the purpose of tightening the range of high R² may provide rationalization for the high observed correlations between fiber orientation and direction along the RWM.

7. Past experimental work has suggested that both the thickness and the mechanical response of the RWM are affected by middle ear pathologies such as Otitis Media [95–97]. It would therefore be interesting to investigate if the geometry of the membrane is affected by such pathologies and whether the effect of Otitis Media in the dynamic response of the membrane is also affected by geometry differences other than morphological and local thickness changes.

8. The increased quality of the geometrical measurements reported herein allows for more accurate dynamic simulations, which in turn can facilitate the design of medical tools such as middle ear actuators [98, 99] and microneedles.

9. Since the mechanism of microneedle perforation across the RWM is fiber “separation not scission” and the major axis of a microperforation is aligned with the predominant fiber direction [22], the quantification of fiber directions herein as well as future investigations of the RWM mechanical properties may guide studies that seek the optimal locations for microperforations.

10. Since the RWM remained attached to the bone during imaging, it was not possible to characterize the RWM thickness across the entire membrane. Future measurements based upon micro-computed tomography would allow such measurements.

11. The structure and properties of the guinea pig RWM may serve as a starting point for characterizing and understanding the structure and properties of the human RWM, which will assist with translating the microneedle technology for treatment of hearing and balance disorders.

Appendix A. Zero-curvature Trajectories and Fiber Directions in Additional Membranes

Figure A.1:

Representation of fiber directions superimposed onto numerically computed trajectories of the direction of zero curvature. Fiber direction at each measured point is represented as a black arrow. Membrane IDs: a) 20180308LK; b) 20180308RK; c) 20180420L; d) 20180425R; e) 20180427LB; f) 20180511LE; g) 20180511LP; h) 20180516L; i) 20180516R.