3.1.

Imaging in a Mammalian Cochlea

The image resolution of the DOCM system is derived from both the OL and the light source. Transverse resolution is determined by the OL in a confocal imaging configuration and can be described by a transverse point-spread function²⁷

where $v$ is a transverse optical coordinate given by $v=(2\pi ∕\lambda )r\sin \left(\mathrm{NA}\right)$ and $J_1$ is a Bessel function of the first kind. In contrast, axial resolution is determined by both the OL and the coherence properties of the light source. The axial resolution of the OL can be described by an axial point-spread function²⁷ where $u$ is an axial optical coordinate given by $u=(8\pi ∕\lambda )z{\sin }^2(\mathrm{NA}∕2)$ . If the confocal gate and coherence gate are aligned and scanned in synchrony (achieved in the DOCM system by mounting the OL and retroreflector on a common translation stage), the interferometric term in the detector current can be written as²⁸ where $l_r$ and $l_s$ denote, respectively, the reference and sample arm lengths; $R_s$ is the reflectivity of the sample; and $R_{ij}$ is the source autocorrelation function. The response of the DOCM system to a point scatterer can be written as a simple productNote that with a low NA OL, axial resolution is determined almost entirely by the coherence length of the light source.

Figure 2 shows the cochlea of a Mongolian gerbil (Meriones unguiculatus) imaged using the DOCM system. The cochlea was isolated and fixed in an artificial perilymph solution²⁹ containing 2.5% glutaraldehyde. The apical turn was opened by removing the bony wall enclosing scala vestibuli using the tip of a scalpel. The image shows a cross section of the exposed apical turn. Light from the DOCM system is incident from the top; the cochlear partition is imaged through Reissner’s membrane. The central axis of the cochlea is tilted approximately $30\mathrm{deg}$ relative to the optical axis. Backscattered intensity is shown on a logarithmic scale in order to display a wide dynamic range.

Although the DOCM image in Fig. 2 is degraded by speckle noise, numerous features relevant to cochlear mechanics are readily identified including the fluid spaces scala vestibuli (SV), scala media (SM), and scala tympani (ST) as well as the inner sulcus (IS) and tunnel of Corti (TC), Reiss-ner’s membrane (RM), the tectorial membrane (TM), the reticular lamina (RL), the arcuate and pectinate zones of the basilar membrane (BMa and BMp, respectively), an outer pillar (OP) cell, and three outer hair cells (OHCs). The integrity of the sample was verified by subsequent conventional visible-light imaging. The most strongly backscattering structures in the cochlear partition are the OHCs. Note that the region beneath the OHCs appears shadowed. The apparent increased thickness of RM in Fig. 2 is an artifact due to the high reflectivity of the membrane and the normalization of the logarithmic scale. The FWHM thickness of RM in Fig. 2 is approximately $12\mathrm{\mu }\mathrm{m}$ .

3.2.

Measuring Motion in a Mammalian Cochlea

Motion of the sample can be measured using the DOCM system by monitoring the difference in optical path length between the sample and reference arms. In Doppler optical coherence tomography, this can be achieved by monitoring the instantaneous phase of the heterodyne signal using a Hilbert transform.³⁰ In the DOCM system, the difference in instantaneous phase $\varphi \left(n\right)$ between the heterodyne signal $V_{het}\left(n\right)$ and reference signal $V_{ref}\left(n\right)$ can be estimated accordingly as

where $H$ denotes the discrete Hilbert transform. Sample displacement $d\left(n\right)$ along the optical axis can be estimated from $\varphi \left(n\right)$ aswhere $k=2\pi ∕\lambda$ and $\tilde{n}$ is the refractive index.

The sample shown in Fig. 2 was mechanically stimulated using a piezoelectric transducer (nominal unloaded resonant frequency, $261\mathrm{kHz}$ ) attached to the sample mount. The transducer was driven at $2.0\mathrm{kHz}$ . Motion artifacts are not apparent in Fig. 2 due to the low amplitude of the motions (tens of nanometers). However, motion is readily detected in the spectrum of the heterodyne signal. For example, Fig. 3 shows the spectrum of the heterodyne signal measured at the top of the first OHC. The plot shows the magnitude of the $\mathrm{32,768}$ -point discrete Fourier transform of $V_{het}\left(n\right)$ (corresponding to a 6.6-ms measurement time). For clarity, 10 measurements were averaged to reduce the spectral noise variance. Note that the amplitude of the 500-kHz peak in Fig. 3 corresponds to the brightness of $1\mathrm{pixel}$ in Fig. 2, while the amplitude and phase of the $500\pm 2\text{-}\mathrm{kHz}$ side peaks correspond respectively to the amplitude and phase of motion measured at that pixel. Every pixel in Fig. 2 similarly represents a simultaneous measurement of backscattered intensity and motion.

Table 1 shows the measured motion at several locations in Fig. 2. The motion measurements were generated by calculating the displacement $d\left(n\right)$ at each pixel in the $10\times 10\mathrm{pixel}$ region centered at the location of interest. Displacement signals corresponding to low-intensity pixels contained phase wrapping errors and were subsequently disregarded. The displacement signals were least-squares fit to 2.0-kHz sinusoids, and Table 1 shows the means and standard deviations of the fit amplitudes and phases. Note that the coherence length of the SLD ( $4.5\mathrm{\mu }\mathrm{m}$ in water) is sufficiently short to differentiate the motion of the top and the bottom of each OHC, regardless of the NA of the OL. Also, the DOCM system is sufficiently sensitive to measure the motions of weakly backscattering structures such as the TM. Note that in contrast to stapes-driven acoustic stimulation, mechanical stimulation does not create a pressure differential across the cochlear partition. Consequently, large relative motions are not expected in Table 1.

Table 1

Motion of cochlear structures measured with DOCM.

The motion measurement accuracy of the DOCM system was compared with that of a commercial laser Doppler vibrometer system (nominal accuracy, 1%) by measuring the motion of a cover slip mounted on a piezoelectric actuator. The actuator was driven at $100\mathrm{Hz}$ . Simultaneous measurements using the two systems agreed to within 3% in amplitude and $0.04\mathrm{rad}$ in phase.

The ability to discriminate motions at different depths was examined by measuring the motions of two glass/water interfaces separated by a variable gap; the first interface was stationary, while the second interface was mechanically driven. As illustrated in Fig. 4 , the bottom surface of a microscope cover glass served as the first interface, while the top surface of an uncoated half-ball lens (radius, $1\mathrm{mm}$ ) served as the second interface. The half-ball lens was attached to a piezoelectric transducer (nominal unloaded resonant frequency, $261\mathrm{kHz}$ ) driven at $2.0\mathrm{kHz}$ . The use of a ball lens allowed the two interfaces to come in close proximity without contact. Scattering medium was not simulated as direct line of sight to the organ of Corti is prerequisite to most studies of cochlear mechanics and the fluid spaces in the cochlea are optically transparent. Figure 5 shows the motion of the two interfaces measured using the DOCM system as the size of the gap was varied from 100 to $0\mathrm{\mu }\mathrm{m}$ . The results show that the motions of the two interfaces are clearly resolved with a separation of more than $10\mathrm{\mu }\mathrm{m}$ and difficult to differentiate with a separation of less than $3\mathrm{\mu }\mathrm{m}$ .

The noise characteristics of the DOCM system were examined by imaging and measuring the motion of a stationary mirror with a neutral density filter (optical density, 3) inserted in the sample arm between the OL and the DM. The measured imaging sensitivity of the DOCM system is approximately $85\mathrm{dB}$ . Figure 6 shows the magnitude of the $\mathrm{32,768}$ -point discrete Fourier transform of $d\left(n\right)$ (corresponding to a $150\text{-}\mathrm{Hz}$ measurement bandwidth). One hundred measurements were averaged to reduce the spectral noise variance. With a sample reflectivity of ${10}^{-6}$ (which is comparable to the reflectivities of structures within the organ of Corti³¹), the motion measurement noise floor is less than $30\mathrm{pm}∕{\mathrm{Hz}}^{0.5}$ at frequencies above $1\mathrm{kHz}$ . The motion amplitude of the basilar membrane is on the order of $100\mathrm{pm}$ at the threshold of hearing,³ and the upper limit on the frequency range of hearing is between 10 and $100\mathrm{kHz}$ for most mammals.

The optical access requirements of DOCM are identical to those of heterodyne laser interferometry. Consequently, the technique is expected to find immediate application in both in vitro and in vivo studies of hearing. In a clinical setting, DOCM could be used to measure the motions of outer- and middle-ear structures to diagnose disorders such as otosclerosis. In a research setting, DOCM could clarify the relation between OHC somatic motility³² and the sharpness of cochlear tuning, one of the largest unresolved issues in cochlear mechanics.