Measuring the Mechanical Impedance of the
Isolated Tectorial Membrane
by
Betty Sheau-Jing Tsai
Submitted to the Department of Electrical Engineering and Computer
Science
in partial fulfillment of the requirements for the degree of
Master of Engineering in Electrical Engineering and Computer Science
at the
MASSACHUSETTS INSTITUTE OF TECHNOLOGY
May 2001
LOi.l
Al icroPPrvlPr
@Masssachussetts Institute of Technology, 2001. All r ih
MASSACHUSETTS
INSTITUTE
OF TECHNOLOGY
JUL 11 2001
IBR0ARIES
Author .........
.......................................
Departinent of Electrical Engineering and Computer Science
May 25, 2001
Certified by...
Dennis M. Freeman
W.M. Keck Career Development
Associate Professor in Biomedical Engineering
-Thesis Supervisor
Accepted by . . . . . . . . . . . . . . .- . . . . . . . . . . . . . . . ..
..
Arthur C. Smith
Chairman, Department Committee on Graduate Students
2
Measuring the Mechanical Impedance of the Isolated
Tectorial Membrane
by
Betty Sheau-Jing Tsai
Submitted to the Department of Electrical Engineering and Computer Science
on May 25, 2001, in partial fulfillment of the
requirements for the degree of
Master of Engineering in Electrical Engineering and Computer Science
Abstract
The tectorial membrane (TM) is believed to play an important role in cochlear micromechanics, yet relatively little is understood about its mechanical properties. A
novel in vitro technique to determine the transverse mechanical impedance of an
isolated mouse TM was developed. A broadband stimulus (10 Hz to 20 kHz) was
provided to a piezo-electric actuator, which vibrated the TM. A cantilever of known
impedance was placed on the surface of the TM, and the motion of the cantilever
was measured to determine the impedance of the TM. Impedances were measured
for a family of static indentations to test linearity for eight different TM preparations taken from six mice. Results show that the magnitude of the TM impedance
decreases with increasing frequency with a median slope of -4.6 dB/decade and a
constant phase of approximately -700. These results demonstrate the importance of
the viscous and elastic properties of the TM through the entire frequency range for
which data were obtained. The data supports the poroelastic theory for describing
the mechanics of polyelectrolyte gels. Additionally, measurements from various static
indentations show that the TM's impedance increases with indentation and grows
slightly faster than linearly. This suggests that although much of the applied indentation force deforms the TM, some of the force compresses its matrix, increasing its
stiffness with additional indentation.
Thesis Supervisor: Dennis M. Freeman
Title: W.M. Keck Career Development
Associate Professor in Biomedical Engineering
3
4
Acknowledgments
Thanks to my family, especially my parents who supported me in every one of my
academic endeavors. Their faith in my ability to tackle any challenge is what has
brought me here today and will carry me through the rest of my years.
Thanks to my advisor, Denny Freeman, who taught me so much over the past
three years. He is among the best teachers I have ever met. With many insightful
and clear comments on just about everything, my only regret is that I didn't take full
advantage of such a great learning opportunity. Nevertheless, I did learn a lot from
him, both in his Quantitative Physiology course and in lab. I entered my senior year
barely knowing how to conduct good research, and I will leave with more confidence
than I ever imagined possible.
Thanks to Werner Hemmert, the post-doc who helped me from my very first day
in lab. He was a friendly face that I felt comfortable asking as many stupid questions
as I needed to understand what I was doing. His constant smiles made long days of
running experiments in lab so much more bearable. His assistance in conducting the
experiments and analyzing the data were invaluable. Even though he's on the other
side of the Atlantic ocean now, I still appreciate all the help he continues to give on
this project.
Thanks to Professor Grodzinsky, who helped me understand poroelasticity and
saved me from being stuck for many more months.
Thanks to my office mate, Kinu Masaki, for all her support inside and outside of
lab. Her peppy personality and smile brightens everyone's day.
Thanks to AJ Aranyosi for being so patient and so willing to help me whenever
I needed it. He was always someone I knew I could turn to for guidance in the right
direction and also fixing all the computer problems I ran into.
Thanks to Durodami Lisk for all his prayers and words of encouragment when the
going go tough.
Thanks to UMech for all the suggestions, assistance, and help they've offered over
the past two and a half years.
5
Thanks to all my friends at MIT who made my six years here so much fun.
Thanks to the entity up above who has looked after me since the day I was born.
"I can do all things through Him who gives me strength." - Philippians 4:13
6
Contents
1
Introduction
17
1.1
The Cochlea . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
18
1.1.1
Anatomy. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
18
1.1.2
Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
20
The Tectorial Membrane . . . . . . . . . . . . . . . . . . . . . . . . .
20
1.2.1
Composition and Morphology . . . . . . . . . . . . . . . . . .
20
1.2.2
TM as a Polyelectrolyte Gel
. . . . . . . . . . . . . . . . . .
21
1.2.3
Mechanical Models of the TM . . . . . . . . . . . . . . . . . .
22
1.2.4
Previous measurements .
. . . . . . . . . . . . . . . . . .
24
1.2
.
2 Methods
27
2.1
Isolated TM Preparation . . . . . . . . . . . . . . . . . . . . . . . . .
27
2.2
Experimental Configuration.....
. . . . . . . . . . . . . . . . . .
28
2.3
Signal Generation . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
30
2.4
Amplifier Design
. . . . . . . . . . . . . . . . . . . . . . . . . . . . .
31
2.5
Impedance Measurement . . . . . . . . . . . . . . . . . . . . . . . . .
34
2.6
Cantilever Stiffness Calibration . . . . . . . . . . . . . . . . . . . . .
36
2.7
Animal Care . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
39
41
3 Poroelastic Theory
3.1
Constitutive Equations . . . . . . . .
42
3.1.1
Assumptions . . . . . . . . . .
42
3.1.2
Hooke's Law . . . . . . . . . .
43
7
3.1.3
Darcy's Law . . . . . . . . . . . . . . . . . . . . . . . . . . . .
45
3.1.4
Conservation of Mass . . . . . . . . . . . . . . . . . . . . . . .
45
3.1.5
Conservation of Momentum
. . . . . . . . . . . . . . . . . . .
45
3.1.6
Electrical Relationships . . . . . . . . . . . . . . . . . . . . . .
46
Application of Equations . . . . . . . . . . . . . . . . . . . . . . . . .
46
3.2.1
Boundary Conditions . . . . . . . . . . . . . . . . . . . . . . .
47
3.2.2
Mechanical Diffusion . . . . . . . . . . . . . . . . . . . . . . .
49
3.3
Extension of Theory to Unconfined Compression . . . . . . . . . . . .
50
3.4
Application to TM measurements . . . . . . . . . . . . . . . . . . . .
52
3.2
4
5
Results
55
4.1
TM Imaging . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
55
4.2
Velocity Measurements . . . . . . . . . . . . . . . . . . . . . . . . . .
55
4.3
TM Impedance
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
57
4.4
Cross Preparation Variation . . . . . . . . . . . . . . . . . . . . . . .
59
4.5
Effect of Increasing TM Indentation . . . . . . . . . . . . . . . . . . .
61
4.6
Other Dependences . . . . . . . . . . . . . . . . . . . . . . . . . . . .
63
Discussion
69
5.1
Preparation limitations . . . . . . . . . . . . . . . . . . . . . . . . . .
69
5.2
Implication of TM Measurements
. . . . . . . . . . . . . . . . . . . .
70
5.2.1
Point Stiffness . . . . . . . . . . . . . . . . . . . . . . . . . . .
70
5.2.2
Compression dependence . . . . . . . . . . . . . . . . . . . . .
70
5.2.3
Frequency Dependence . . . . . . . . . . . . . . . . . . . . . .
72
5.3
Summary
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
8
75
List of Figures
1-1
Schematic drawing of the cochlear anatomy by A.C. Greene. . . . . .
1-2
Schematic representation of the tectorial membrane. It is attached to
18
the organ of Corti at the spiral limbus and lies above three rows of
outer hair cells and one row of inner hair cells. Some believe that the
TM is anchored to the organ of Corti near the inner hair cells by the
Hensen stripe and at its outer zone by the marginal band (Lim, 1980).
Taken from Shah et. al, 1995. . . . . . . . . . . . . . . . . . . . . . .
1-3
19
Mechanical models of the TM and stereocilia motion frequency response relative to the basilar membrane motion. From left to right,
the first model depicts the TM as an infinitely stiff bar, while in the
second model both the stiffness and mass are important, and in the
third model, only the inertial mass is significant. Taken from Abnet,
1998 ........
....................................
9
23
2-1
Experimental setup. (a) A computer-generated signal was fed to the
piezo-electric actuator, which provided the mechanical stimulus for the
tectorial membrane.
The laser-Doppler interferometer (LDV) mea-
sured the motion of the cantilever and its output was directed into
the data acquisition processor (DAP) in the computer.
(b) The ex-
perimental chamber was connected to the microscope stage via the
piezo-electric actuator. An LED at the base of the chamber provided
trans-illumination for the microscope. The TM sat on a coverslide attached to the chamber and was bathed in artificial endolymph. Raising
the microscope stage altered the distance between the support for the
cantilever and the coverslide that supported the TM and thus caused
the cantilever to exert a force on the TM. Stimulating the piezo-electric
actuator had a similar effect. . . . . . . . . . . . . . . . . . . . . . . .
29
2-2
Layout of 20 dB amplifier circuit. . . . . . . . . . . . . . . . . . . . .
32
2-3
Model of mechanical responses. Both the cantilever and the TM are
modeled as mechanical impedances and the piezo-electric actuator is
modeled as a velocity source . . . . . . . . . . . . . . . . . . . . . . .
3-1
35
(a) Poroelastic material compressed by a rigid permeable plate. Geometric constraints allow fluid flow in only the axial (z) direction. (b)
Poroelastic material compressed by a rigid impervious plate. This
allows unconstrained fluid and solid expansion radially. Taken from
Arm strong et al., 1984. . . . . . . . . . . . . . . . . . . . . . . . . . .
3-2
Displacement profile (u(x, t)) of the solid phase with increasing distance from the point of compression (x) and time for stress relaxation.
3-3
48
Displacement profile (u(x, t)) of the solid phase with increase distance
from the point of tension (x) and time for creep. . . . . . . . . . . . .
4-1
43
49
TM image obtained using transillumination. This was taken from one
of the preliminary measurements conducted while the experimental
protocol was being refined. . . . . . . . . . . . . . . . . . . . . . . . .
10
56
4-2
Magnitude and relative phase of the frequency response of the cantilever velocity measured for different TM indentations for preparation
3. The thick dots represent the motion measured on the cantilever due
to fluid coupling, whereas the small dotted line represents the maximum velocity measured by pressing the chamber against the cantilever.
The dashed line is the velocity of the chamber measured by reflecting
the laser beam off the glass coverslide. The solid lines represent the velocities for different TM indentations equally spaced at 10 pim. Phase
is plotted relative to the input stimulus. Gray-shaded regions show
where the velocities of the TM are within 6 dB of the fluid velocities.
4-3
58
Typical TM impedance. The solid black line shows the measured mechanical impedance of the TM (Table 4.1, preparation 1), when the
distnace between the TM and coverslide was 40 pm. The dotted and
dashed lines show the fluid "impedance" with the coverslide 200 pam
below the cantilever and at 50 ptm, just below the point where the TM
touches the cantilever. The gray solid line represents the cantilever
impedance, which has a resonant frequency in fluid of approximately
27 kHz. The shaded gray region shows where the TM impedance measurements lie within 6 dB of the fluid "impedance".
4-4
. . . . . . . . . .
60
Calculated TM impedance of all the preparations in Table 4.1 at the
point when the cantilever and TM first come in contact. The magnitude responses are separated by shade and line style such that each
individual trace can be distinguished. For TM impedances unaffected
by the chamber resonances between 300 and 500 Hz, the impedances
run parallel to one another. The remainder of the traces cross one
another at the resonant frequencies. . . . . . . . . . . . . . . . . . . .
11
62
4-5
Estimated slopes of the magnitude of the frequency response of the TM
impedance for each preparation in Table 4.1. Each point represents
the value for a different indentation within each preparation.
The
last column pools the data from all experiments. The box shows the
interquartile range, and the bar gives the median slope. . . . . . . . .
4-6
63
Phase of the frequency response of the TM impedance for each preparation in Table 4.1) at 10, 100, and 1000 Hz. Each point represents
the value for a different indentation within each preparation. The last
column pools the data from all experiments. The box shows the interquartile range and the horizontal bar provides the median phase. .
4-7
64
Effect of increasing the indentation of the cantilever into the TM for
preparation 3. The dotted lines represent fluid measurements, while
the solid lines represent the calculated impedances of the TM with
approximately 0, 10, and 20 pm of static compression with increasing impedance corresponding to increasing indentation. The dashed
line shows the measurement taken when the experimental chamber
was firmly pressed against the cantilever.
These are the calculated
impedances from the velocity data plotted in Figure 4-2.
4-8
. . . . . . .
65
TM stiffness dependence on final thickness after static compression.
The point stiffness at 1 kHz is plotted as a function of the estimated
thickness of the TM between the TM and coverslide. To distinguish
the trends between each of the 8 preparations, connecting lines between
points are of different line styles and shade. For comparison, dotteddashed lines show trends for functions that vary as t and 1/t 2 . The
boxes show the interquartile range for the point stiffnesses, with the
longer horizontal bar representing the median point stiffness for each
thickness and the shorter horizontal bars showing the extremes.
12
. . .
66
5-1
Calculated TM point stiffness values from various experiments. Solid
line is the median point stiffness of the TM when it first comes in contact with the cantilever and the gray region represents the interquartile
range for the data in this study. (a) Dotted line represents the median
radial stiffness (Abnet and Freeman, 2000). (b) Dotted line represents
the median longitudinal stiffness (Abnet and Freeman, 2000). (c) Radial stiffness at 20 Hz (Zwislocki and Cefaratti, 1989). (c) Transverse
stiffness at 20 Hz (Zwislocki and Cefaratti, 1989). (e) Static depression
stiffness (Bekesy, 1960). (f) Stiffness at 200 Hz (Bekesy, 1960). Bekesy
found the TM to be rigid when he probed it with a needle vibrating
at 200 H z. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5-2
71
Possible effects of indentation on a solid-fluid matrix. When the cantilever pushes into the gel-like structure, two things can happen. In
static compression, the probe compresses the solid matrix, increasing
the solid-to-fluid ratio in the volume directly beneath.
The second
possibility is pure deformation in which the applied force of the probe
pushes out the elastic network along with interstitial fluid, keeping the
density of the tissue constant. . . . . . . . . . . . . . . . . . . . . . .
13
73
14
List of Tables
4.1
Isolated TM preparations. The table lists the estimated TM thickness
under the cantilever tip and the time between the death of the mouse
to the first measurement. . . . . . . . . . . . . . . . . . . . . . . . . .
15
56
16
Chapter 1
Introduction
Many questions remain about the physical mechanism behind the phenomenon we
call hearing. How is an acoustic stimuli entering the ear converted to a series of electrical pulses in the brain? In many animals, sound enters the pinna, travels through
the ear canal, and eventually reaches the eardrum, where the acoustic waveform is
converted into mechanical vibrations. This signal then travels through the air spaces
and bones of the middle ear and finally reaches the inner ear. The inner ear acts as
a mechanoelectrical system that transduces the mechanical motion received from the
stapes footplate into an electrical signal, ultimately processed in the central nervous
system. While it has been established that the hair cells are responsible for the conversion from mechanical energy (Hudspeth and Corey, 1977; von Bekesy, 1960), the
actual mechanical coupling of the stapes bone in the middle ear to the stereocilia of
the hair cells has yet to be understood.
The tectorial membrane (TM) is a connective tissue that lies in the path that
links the two structures and is believed to play a key role in hearing. The TM is
thought to affect micromechanical tuning, which is determined by the length of the
hair bundles and the presence of an overlying tectorial structure in lizards (Manley,
2000). Hair cells covered by the tissue have shorter stereociliary bundles and higher
frequency selectivity than those without. However, very little is known about its
mechanical properties as demonstrated by the wide range in TM models in cochlear
micromechanics. The goal of this paper is to provide additional experimental evidence
17
Figure 1-1: Schematic drawing of the cochlear anatomy by A.C. Greene.
on TM mechanical properties to shed some light on the possible role of this enigmatic
structure.
1.1
The Cochlea
The cochlea lies in the inner ear and allows animals to perceive sound.
1.1.1
Anatomy
Embedded in the temporal bone of many animals, the cochlea is a highly specialized
frequency analyzer. It converts the acoustic mechanical stimuli that enters through
the oval window into nerve spikes that the brain can interpret as sound.
The cochlea contains three fluid chambers (or scali) which are coiled up into a
spiral structure. A cross-section is shown in Figure 1-1. The scala media lies between
the scala vestibuli and scala tympani, separated by Reissner's membrane and the
basilar membrane respectively. Both the scala vestibuli and scala tympani are filled
with perilymph, which contains high concentrations of Na+, and communicate with
each other at the apex of the cochlea. The scala media, on the other hand, is filled
with endolymph, which has a high K+ concentration that is critical for the structures
18
covering net
marginal band
In-Situgeid
Hensen's stripe
Figure 1-2: Schematic representation of the tectorial membrane. It is attached to
the organ of Corti at the spiral limbus and lies above three rows of outer hair cells
and one row of inner hair cells. Some believe that the TM is anchored to the organ
of Corti near the inner hair cells by the Hensen stripe and at its outer zone by the
marginal band (Lim, 1980). Taken from Shah et. al, 1995.
in the scala media to function properly.
The hair cells sit above the basilar membrane and their apical surfaces define
the edge of the scala media. The basolateral surfaces of the hair cells are separated
from the basilar membrane by supporting cells. The TM overlays these mechanically
sensitive sensory cells, with the tips of the tallest stereocilia in the outer hair cells
protruding into the underside of the TM (Lim, 1980) as shown in Figure 1-2. The
hair bundles of the inner hair cells sit within the subtectorial fluid space just below
the TM. Therefore, the TM is believed to affect the motion of the stereocilia during
sound stimulation (Kimura, 1966; Lim, 1972; Lim, 1980).
19
1.1.2
Function
Sound stimuli enter the cochlea by the piston-like motion of the stapes motion through
the oval window. The mechanical vibrations due to sound enter the scala vestibuli
through the oval window, pass through the scala tympani and exit through the round
window. The pressure difference between the two fluid chambers generates transverse
vibrations of the basilar membrane in the form of a traveling wave (von Bekesy,
1953). The cochlea is tonotopic, with the basal regions more finely tuned to high
frequencies and the apical regions tuned to low frequencies.
The traveling wave
reaches a maximum amplitude at the place whose characteristic frequency matches
that of the stimulus and quickly attenuates as it passes the characteristic place.
The transverse motions of the basilar membrane push the hair cells towards the
TM, which rotates about its connection to the spiral limbus, and is believed to create
a shearing force on the stereocilia (Rhode and Geisler, 1967). This deflection opens
transduction channels, which depolarize the cells, which triggers nerve firings in the
peripheral auditory system.
1.2
The Tectorial Membrane
Nearly two decades ago, the TM was "arguably the most enigmatic part of the mammalian cochlea" (Steel, 1983). Today, much more is known about its various properties, in particular, its morphology and composition. This has provided some insight
into the mechanical properties of the tectorial membrane.
1.2.1
Composition and Morphology
Ultrastructural studies show that the TM is a translucent gel with increasing width
and thickness from the base to the apex and parallel fibrils radiating outwards from
the inner limbal zone (Steel, 1983; Kronester-Frei, 1978; Hasko and Richardson, 1988).
The TM contains two types of fibrils, type A and type B (Kronester-Frei, 1979).
Type A fibrils are mainly composed of collagen (Richardson et al., 1987; Slepecky
20
et al., 1992), whereas type B fibrils are believed to be composed of noncollageneous
structures that interact with the type A fibrils, possibly restricting their growth and
regulating their density (Tsuprun and Santi, 1996). Collagens make up 25-50% of the
protein in TM (Thalmann et al., 1986; Richardson et al., 1987; Kalluri et al., 1998)
and interact with type B fibrils and proteoglycans to create a rigid hydrated matrix
that can resist compressive forces (Thalmann et al., 1987; Santi et al., 1990; Tsuprun
and Santi, 1996).
Proteoglycans are negatively charged sugar macromolecules at
physiological pH; as a result, this causes mobile cations to flow into the TM matrix,
and osmotically drawing water with them, resulting in a highly hydrated gel (Munyer
and Schulte, 1991; Machiki et al., 1993; Thalmann et al., 1993).
1.2.2
TM as a Polyelectrolyte Gel
In addition to the macromolecules and water that make up the TM, like other connective tissues such as cartilage, cornea, and tendon, it also contains many small solutes
(Hart and Farrell, 1971; Maroudas, 1980; Hay, 1981; Weiss and Freeman, 1997). By
weight, the TM is 97% water and many of its properties could be described by the
polyelectrolyte gel model (Weiss and Freeman, 1996). A gel is "a matter intermediate
between a solid and liquid. It consists of polymers, or long-chain molecules, crosslinked to create a tangled network and immersed in a liquid medium" (Tanaka, 1981).
A gel's properties are determined by the way the fluid and solid components interact
with one another. The polymer network is often considered to be responsible for the
elastic properties while the friction generated by the fluid flowing through the matrix
give the viscous nature of the TM. Additionally, the presence of fixed charge groups,
such as proteoglycans, increases the TM's resistance to compressible forces (Santi
et al., 1990). The fixed charge draws in water and counterions into the solid network
to neutralize its environment. In exchange, the incompressibility of water allows the
TM to hold its structure. Therefore, the TM's size is dependent on how much fluid it
holds. In hypoosmotic solutions, it can be expected to swell, as opposed to shrinking
in hyperosmotic solutions. These two situations would result in very different mechanical properties. The amount a TM can swell is limited by the mechanical stiffness
21
of the tissue matrix while the amount to which a TM could shrivel up is limited by
the fixed charge repulsions. Although counterions and water enter or exit the TM to
achieve electroneutrality and osmotic equilibrium, the physical stiffness of the matrix
limits this ability, resulting in a gradient of ion concentrations and a negative potential in the TM (Steel, 1983; Santi and Anderson, 1986; Santi and Anderson, 1987).
In other words, the mechanical properties of the TM are intimately linked with its
osmotic properties, which are determined by its electrical properties.
Both electrical and osmotic properties have been fitted to the polyelectrolyte gel
model. Based on the electrical potential of the TM (Steel, 1983), the fixed charge
in the tissue was estimated to be between -6.4 and -8.4 mmol/L under physiological conditions (Weiss and Freeman, 1997). On the other hand, by fitting the gel
model to the biochemical composition of the TM (Thalmann et al., 1993), Weiss and
Freeman estimated the fixed charge to be approximately -18 mmol/L. Most recently,
experimental data in which the ionic strength of the bath was varied, the fixed charge
was estimated to fall between -5 and -20 mmol/L, consistent with the previous two
estimates (Masaki and Freeman, unpublished). Additionally, the data also showed
that non-ionic solutes such as glucose had no osmotic effect on the TM, consistent
with the gel model. While the osmotic and electrical properties can be described by
the polyelectrolyte gel model, it has yet to be extended to incorporate the mechanical
properties of the TM.
1.2.3
Mechanical Models of the TM
There has been a wide range of models to describe the mechanics of the TM in
the context of cochlear mechanics (Figure 1-3).
These conceptions vary from a
stiff bar rotating about its connection to the spiral limbus (Davis, 1958; Johnstone
and Johnstone, 1966; Rhode and Geisler, 1967) to acting solely as an inertial mass
(Zwislocki, 1988) while in other conceptions, the importance of the TM's mass and
damping creates a resonance that contributes to the frequency selectivity in the
cochlea (Zwislocki, 1979; Zwislocki and Kletsky, 1979; Allen, 1980). In the stiff bar
model, the TM has infinite radial and bending stiffness and negligible mass and damp22
-
--
-
oTM
0
TM
HHB&
HB
BM
TM rigid
IH(f) I)
ZH(f)
001
ETMV
HBI
TM mass only
TM resonant mass
IH(f)I
IH(f)
ZH(f)
ZH(f)
- 1800
0
-180e
Frequency
-
w
-360PI
Frequency
__________
Frequency
Figure 1-3: Mechanical models of the TM and stereocilia motion frequency response
relative to the basilar membrane motion. From left to right, the first model depicts
the TM as an infinitely stiff bar, while in the second model both the stiffness and mass
are important, and in the third model, only the inertial mass is significant. Taken
from Abnet, 1998
ing (Davis, 1958; von Bekesy, 1953; von Bekesy, 1960; Steel, 1983). The deflection
of the stereociliary bundles is in phase with the basilar membrane displacement. On
the other hand, for models in which the mass is the only significant property, at low
frequencies, the TM has very low impedance, and is very similar to a stiff bar at high
frequencies. This model contradicts many low-frequency microphonic measurements,
however (Patuzzi, 1996). The intermediate model, in which the mass and stiffness
are both important radially, suggests that the TM has a resonant frequency at which
its motion is optimal.
Despite all the disagreement on the mechanical properties of the TM, it has been
uniformly described as an anisotropic material with negligible longitudinal stiffness
(Allen, 1980; Mammano and Nobili, 1993; de Boer, 1996; Patuzzi, 1996).
23
1.2.4
Previous measurements
The mechanical properties of the TM have been previously characterized in several
investigations using a variety of different probes.
Bekesy performed the first set of studies in the 1950s (von Bekesy, 1953).
By
pressing the tip of a small hair onto the surface of the TM, he inferred that it was
mechanically more rigid in the radial direction than in the longitudinal direction.
Additionally, using small compliant hairs to apply forces normal to the TM, he estimated the stiffness to be approximately 0.1 N/m (von Bekesy, 1960). His experiments
also showed that the TM's elasticity was small relative to its internal friction using a
needle vibrating at 200 Hz.
The first live in situ TM stiffness measurements were carried out in anesthetized
Mongolian gerbils nearly three decades later (Zwislocki and Cefaratti, 1989). With
the tip of glass micropipettes, the TM was deflected in the radial and transverse
directions and the point stiffnesses were determined to be 0.116 N/m and 0.125 N/m
respectively.
Because the TM is connected to other compliant structures in the cochlea, in situ
measurements could easily have included the TM's coupling to the organ of Corti
via the hair cell bundles or the spiral lamina. In addition, the described studies
involved displacements greater than 10 pm, while actual displacements from intense
sounds in hearing are smaller than 1 ptm (Rhode, 1981; Sellick et al., 1982; Cooper
and Rhode, 1997; Ruggero et al., 1997).
Most recently, magnetic beads were used to measure the point stiffness of isolated
TMs in the radial and longitudinal directions (Abnet and Freeman, 2000).
Also,
the forces applied on the TM were comparable in magnitude to those generated
from stimuli at frequencies up to 1 kHz. Previous studies could only quantify static
depressions and at low frequencies. Consistent with values from prior investigations,
the median point stiffness in Abnet and Freeman was 0.18 N/m. The measured phase
lag in the displacement of the isolated TM relative to the applied forces demonstrated
the importance of both viscous and elastic properties in the TM.
24
To further quantify the TM's material properties, a novel approach to determine
the TM's mechanical impedance in the transverse direction was developed. Not only
does it complement the measurements of Abnet and Freeman (2000), using compressive forces of reasonable amplitude, but the new technique encompasses a larger
frequency range than previous studies. Valid measurements in this study were obtained between 10 Hz and 4 kHz, a major portion of the hearing range.
25
26
Chapter 2
Methods
This chapter describes the use of an atomic force cantilever to probe the transverse
impedance of the TM. Basic setup and fundamental theory are explained below.
2.1
Isolated TM Preparation
The methods to isolate the TMs from adult male white mice (strain ICR, 30-50
grams, Taconic) have been described in previous papers (Shah et al., 1995; Abnet
and Freeman, 2000; Wei, 2001). The mice were sacrificed by CO 2 asphyxiation and
decapitated (MIT CAC#98-038-1). The cochlea was dissected away from the temporal bone and submerged in artificial endolymph (AE: 174 mmol/L KCL, 2 mmol/L
NaCl, 0.02 mmol/L CaCl 2 , 3 mmol/L dextrose, 5 mmol/L HEPES, titrated to pH 7.3,
osmolarity: 330 mOsm). When the round window became visible, a 27-1/2G needle
was used to flush the interior of the cochlea, including the scala vestibuli and scala
tympani, through the oval window with artificial endolymph. Since the TM is very
sensitive to changes in its ionic environment (Freeman et al., 1994; Shah et al., 1995),
the purpose of AE is to provide the TM with an environment similar to its natural surroundings in vivo. Further preparation was conducted under a dissection
microscope (Stemi SV11, Zeiss) featuring a ring light and a bright-field/dark-field
trans-illuminating base (TLB 6000, Diagnostic Instruments, Michigan, USA). Using
a fresh scalpel and forceps, most of the cochlear bony casing was carefully chipped
27
away. After the stria vascularis was removed, the apical 1.5 turns of the organ of
Corti was visible while it was still attached at the limbal edge of the modiolus. The
modiolus was broken off at the cochlear base with the tweezers and transferred to
an AE-filled petri dish. Dark-field trans-illumination provided the best view for the
almost transparent TM under a dissection microscope. While in some preparations,
the TM detached itself from the organ of Corti during the transfer, in many preparations, an eyelash glued to the tip of a steel holder was required to gently peel the
attached portions of the TM.
The TM was transferred with a 10 pL glass micropipette to the experimental
chamber, which was then filled with AE. An eyelash was used to generate convection
currents to draw the TM towards the edge of the glass coverslide, which was coated
with 1 [IL of Cell-Tak (Cellular Biomedical Inc., Waltham, MA) at least 1 hour prior
to the TM transfer. No attempt was made to control the orientation of the TM fibrils
or whether the TM covering net was facing up or down. With a 20x water immersion
objective, TM structural features, including the radial fibrils and Hensen's stripe,
could be easily seen using trans-illumination.
2.2
Experimental Configuration
The mechanical properties of the TM were determined by probing it with a tipless
atomic force cantilever (Pointprobes, FM-16, Nanosensors, Wetzlar, Germany). The
cantilever beam acts as a force transducer generating a vertical displacement proportional to the exerted force. It was supported by a micro-manipulator mounted
directly on a vibration isolated table (Integrated Dynamics Engineering, Raunhein,
Germany) and adjusted to lie in the focal plane of the microscope (Axioplan2, Zeiss,
Oberkochen, Germany). The microscope had two optical ports, one for a CCD-camera
(TM-1010, Pulniz, Sunnyvale, CA) and another for a laser-Doppler interferometer
(LDV, OFV 511 fiber optic interferometer head, OFV 3001 interferometer controller,
and a c-mount coupling unit, Polytec, Waldbronn, Germany). The laser beam entered through the optical path of the microscope and was focused on the tip of the
28
Laser-Doppler
Interferometer
LDV
20x-
Coverslide
Cantilever
TM
LED
PC
-***
(D AP)
Piezo-ele ctric
Actuator
to piezo-electric
Stage
actuator
(a)
(b)
Figure 2-1: Experimental setup. (a) A computer-generated signal was fed to the piezoelectric actuator, which provided the mechanical stimulus for the tectorial membrane.
The laser-Doppler interferometer (LDV) measured the motion of the cantilever and
its output was directed into the data acquisition processor (DAP) in the computer.
(b) The experimental chamber was connected to the microscope stage via the piezoelectric actuator. An LED at the base of the chamber provided trans-illumination
for the microscope. The TM sat on a coverslide attached to the chamber and was
bathed in artificial endolymph. Raising the microscope stage altered the distance
between the support for the cantilever and the coverslide that supported the TM and
thus caused the cantilever to exert a force on the TM. Stimulating the piezo-electric
actuator had a similar effect.
29
cantilever. The interferometer used the light reflected off the probe to determine its
velocity.
Computer software controlled the height of the microscope stage with high precision (minimum increments of 0.5 pm). With the cantilever was already in focus,
the purpose of the stage was to bring the TM towards the cantilever as part of
the experimental configuration. A piezo-electric focusing unit (PiFoc P-721, Physik
Instrumente, Waldbronn, Germany), connected to the objective lens (20x water immersion, N.A.:0.5W, Zeiss) of the microscope, varied the focal depth to allow the
viewieng of different planes of sections throughout the tectorial membrane.
A piezo-electric actuator (Model AR0505D08, Thorlabs, Inc., Newton, NJ) supported the experimental chamber (Figure 2-1) on the microscope stage and provided
the mechanical stimulus for impedance measurements. The chamber was constructed
out of aluminum and coated with black spray paint to minimize light reflections. It
served as a bath containing artificial endolymph to maintain a constant chemical and
osmotic environment for the TM during the course of the experiment. To visualize
the TM, high-quality trans-illumination was realized by inserting a surface-mount
light emitting diode (Nichia) into the bottom of the chamber. Wires were soldered to
the electrical contacts of the LED and insulated with epoxy to prevent a short circuit
when the LED was immersed in artificial endolymph during the experiments. At least
one hour prior to each experiment, a new coverslide was attached with candle wax
to the surface of the chamber. It covered about half of the bath such that enough
interaction between the fluids on both sides of the chamber was permitted. The TM
was placed on the edge of the coverslide (Figure 2-1), to minimize hydrodynamic
coupling between the cantilever and coverslide.
2.3
Signal Generation
A broadband signal which permitted frequency responses with a high signal-to-noise
ratio was designed for TM impedance measurements. The stimulus consisted of repeating a signal frame 100 times. Each frame consisted of 4096 points presented at
30
a rate of 40 kHz, the equivalent of 9.76 frames per second. The signal was synthesized by adding sinusoids with a frequency spacing of approximately 9.1 frequencies
per octave for frequencies above 150 Hz. At lower frequencies, sinusoids were spaced
about 10 Hz apart. The phases were randomly distributed from 0 to 360 degrees.
Since the piezo-electric actuator generates a displacement proportional to input voltage and the frequency components of its velocity are calculated by multiplying the
displacement with frequency, to achieve a constant velocity amplitude, the signal was
low-pass filtered at a slope of 6 dB/octave with a cutoff frequency of 100 Hz. Below
100 Hz, the displacement magnitudes were limited to be less than to 30 nm per frequency; therefore, the velocity amplitude increased with a slope of 6 dB/octave up
to 100 Hz and remained constant for higher frequencies. The total peak amplitude of
the stimulus signal was 250 nm.
The signal was generated in Matlab, loaded into the memory of a data acquisition
processor (DAP) board (Microstar 3200a/415, Microstar Labs, Bellevue, WA) and
delivered to the piezo-actuator via a 6 dB attenuator. Simultaneously, the velocity
signal from the laser-Doppler vibrometer was acquired by the input port of the DAP
board and discretized with 10 bit resolution. The velocity signal was preconditioned
with a custom-built 20 dB amplifier (LT1028, Linear Technology) and a 20 kHz antialiasing filter (LE1192-20k-3k-720B, TTE, Los Angeles, CA). During data collection,
a continuous stream of signal frames were presented, of which the first and those
with large noise (> 5 mm/s) were discarded. One hundred valid input frames were
acquired and averaged in time before the signal was turned off.
2.4
Amplifier Design
An amplifier was built to boost the signal from the laser-Doppler interferometer to
the DAP board. An ultra low noise precision operational amplifier (LT1028, Linear
Technology) was selected and a circuit was designed to provide a 20 dB gain of the
input signal. The resistors and capacitors were chosen to minimize the total noise
generated by the operational amplifier.
31
+15V
R3=1 OOQ
C1=100tF
C2=1 OOnF
R1=1000
INR
R2=100O
R5=1OQ
+
0=
LT1 028
OUT
-~C
-C=00nF
C4 1
1
R4= 1 OQ
-15V
R6=10kQ
C5=100pF
Figure 2-2: Layout of 20 dB amplifier circuit.
The non-inverting amplifier circuit (Figure 2-2) was built by soldering the parts
onto a circuit board. For this circuit, the gain, G, is
Vout
G
-+
R6
R2
(2.1)
A gain of 20 dB is the equivalent of amplifying a signal by a factor of 10. Therefore,
R6 must be 9 times larger than R2. Since it is difficult to find resistors to provide
exact gain values, a 10 kQ potentiometer was gradually adjusted until the measured
output to input ratio was approximately 10.
At a first glance, the resistor R1 is unnecessary for many amplifier circuits. However, the LT1028 operational amplifier is made up of bipolar transistors and has a
32
relatively high input bias current (25 nA) compared to that of many operational
amplifiers, typically on the order of pA for FET-input operational amplifiers. This
current would cause a voltage drop across the positive and negative input terminals,
which then gets amplified. Unless the input resistances are matched, when the input
is connected to ground, the operational amplifier would generate a non-zero output.
As a result, RI was chosen to equal R2.
Determining the appropriate sizes for R1 and R2 was not trivial. Larger resistor
loads (R6) reduced the amount of distortion at the output. This would require larger
source resistances (RI and R2) to maintain a constant gain of 20 dB. However, the
amount of noise injected into the system increases with increasing source resistance.
The total input noise of an operational amplifier is given by (Horowitz and Hill, 1989):
et2
=
(2.2)
en 2 + 4kTReq + (inReq) 2
where et is the total input noise of an operational amplifier, en is the voltage noise, k
is Boltzmann's constant, T is the temperature in Kelvins, in is the current noise, and
Req is the equivalent input resistance. At a room temperature of 25'C, the middle
term, 4kTReq simplifies to 0.01 6 9 Req in nV/
Hz.
Noise specifications for the LT1028 were provided at 10 Hz and 1000 Hz. Since
the noise is higher at 10 Hz than at 1000 Hz, noise calculations were based on typical
values for 10 Hz: en=1.00nV/
Hz and in=4.7pA//Hz. When the two inputs are
tied to ground, the equivalent resistance, Req = RI + R2 11R6 ~~RI + R2 = 2R1,
since R6 >> R2 and RI = R2. Therefore, at 10 Hz, the typical noise can be calculated
by substituting the appropriate values into equation 2.2, which gives
et 2 = (1.00 x 10- 9 V/
Hz) + (0.13/2R1 x 10-9V/V/iii)2 + 88.36R1 2
x
10-
24
V2 /Hz.
(2.3)
Since the resistor values affect current noise more than it does resistor noise, it
was more important to cap the maximum resistance for RI and R2 based on current
noise. Setting an upper limit of 106Q on RI ensured that the current noise never
33
exceeds the voltage noise, which was a constant. Hence, RI and R2 were chosen to be
, even though the resistor noise does exceed voltage noise for small resistances.
100
The power supply nodes were connected to voltage generators set at +15 and
-15V. To prevent large currents from the voltage sources from damaging the operational amplifier, resistors R3 and R4 were inserted between the power supply and the
terminals of the operational amplifier. Additionally capacitors C1 through C4 were
added to prevent high frequency variations in supply voltage such as DC step from
overloading the amplifier chip.
To ensure that the amplifier could drive a large capacitive load (Horowitz and
Hill, 1989), a small series resistor R5 was added into the feedback loop. Although
this degrades the high frequency performance of the circuit, it sufficiently handles the
desired frequency range of 10Hz to 20 kHz.
When the circuit was first tested, a lot of high frequency noise was observed at
the output. This was fixed by adding capacitor C5 to serve as part of a low-pass filter
in the circuit itself.
2.5
Impedance Measurement
The measurement configuration is shown in Figure 2-3. The mechanical properties of
the TM and the cantilever are represented by their respective mechanical impedance,
defined as the amount of force [N] required to compress a material at a specific rate
(velocity [m/s]). When the TM presses against the cantilever, an equal and opposite
force is exerted by the cantilever, thereby compressing the TM. The piezo-electric
actuator provides the velocity stimulus that pushes against the bottom of the TM.
For a known cantilever impedance, ZC, the force applied can be calculated from the
measured cantilever velocity, Uc, by
Zc
F =_ .
UC
34
(2.4)
Uc
Z
zC
Gan
TM
UP
piezo
i
Coverslide
)n chamber
+
ZTM
Uc
Up+
not drawn to scale
Figure 2-3: Model of mechanical responses. Both the cantilever and the TM are
modeled as mechanical impedances and the piezo-electric actuator is modeled as a
velocity source.
From basic circuit theory, the impedance of the TM, ZTM, is
ZTM
=
.
UP - UC
(2.5)
During the experiments, the TM was brought up towards the cantilever, creating
the same effect as lowering a cantilever on a TM while maintaining focus of the
microscope on the cantilever. The piezo-electric actuator vibrated the chamber on
which the specimen was located, and the motions were measured using the laser light
reflected off the cantilever beam. Chamber velocities, Up, were measured by two
different methods. In all experiments, the microscope stage was raised in steps until
by the end of the experiment the cantilever touched the coverslide. Velocity of the
cantilever was then equal to that of the coverslide. In some of the initial experiments,
the velocity of the chamber was separately measured by reflecting the laser beam off
the glass coverslide.
35
2.6
Cantilever Stiffness Calibration
Tipless cantilevers (Pointprobes FM-16, Nanosensors, Wetzlar, Germany) with nominal length of 225 ± 5pm, a width of 28 ± 5pm and a thickness of 3± 0.5pm were used
for the impedance measurements. After the mechanical impedance of the cantilever
was calibrated, they could be used for multiple experiments.
The stiffness of a cantilever beam is
3EI
k = L3
(2.6)
where E is the elastic modulus of crystalline silicon (169 GPa) (Gere and Timoshenko,
1990), I is the moment of inertia of its cross-section, and L the length of the cantilever.
The cross-section of the cantilevers were trapezoidal and dimensions from both
sides were measured from images taken from the CCD camera with a dry objective
(10x Epiplan, Zeiss). The stiffness of the cantilever is proportional to the cube of
its thickness, h3 . However, the images from a light microscope did not permit us
to measure the thickness with sufficient precision. Instead, we measured the first
resonant frequency of the cantilever in air to calibrate its stiffness.
The resonant frequency in a vacuum is given by (Elmer and Dreier, 1997):
c27r El
,A4
f~ 2 L2r
where
fn
(2.7)
is the nth resonant frequency of the cantilever, A is the cross-sectional area
of the cantilever, p is the density of silicon (2.33g/cm 3), and an is a solution to
cos an +
1
cosh a,
= 0
(2.8)
where the lowest values are a, = 1.8751, a 2 = 4.6491, and a 3 = 7.8578 (Elmer and
Dreier, 1997). The area, A, of the cantilever is
A =(a+b)h
2
36
(2.9)
where a and b are its two widths.
The moment of inertia for a bar with a trapezoidal cross-sectional area is (Gere
and Timoshenko, 1990)
h 3 (a 2 + 4ab + b2 )
36(a + b)
(2.10)
Plugging equations 2.9 and 2.10 into equation 2.7, the unknown thickness can be
determined by
18 p
E(a2 + 4ab + b2 )
h=27f1L2(a + b)
a,2
(2.11)
using the first resonant frequency, fi of the cantilever. Hence, the stiffness can be
calculated by
36v 27r3 L 3 (a + b) 2 p 3/ 2 fI
a' VE(a2 + 4ab + b2 )
(2.12)
The relationship between the resonant frequencies in air and in vacuum is given
by (Elmer and Dreier, 1997)
f air
A_
1
+
4
(2.13)
PairW
where Pair is the density of air (10-3 g/cm3 ), W is the average width of the cantilever
and f is a master function that is approximated by (Weigert et al., 1996)
~ )0.( 5.20//,.
(2.14)
Using the nominal width and thickness values provided by Pointprobes, the ratio of
37
the first resonant frequency in vacuum, f""', to that in air, f 1 air is
fhvac
= 1+
4PairW 0.2056L
Wa1
ph
flair
4(10-3g/cm 3 ) (28pm)
(2.33g/cm 3 )(3gm)
= 1.0077.
(0.2056) (255pm)
(2.15)
(28pim)(1.7227)
Therefore, the resonant frequency of the cantilever in vacuum can be approximated
to its resonant frequency in air.
To determine the first resonant frequency of the cantilever, a piezo-electric bimorph on which the cantilever was mounted was stimulated with a chirp signal with
a frequency range from 62.5 Hz to 256 kHz. The velocity of the cantilever was mea-
sured at its tip and close to the silicon substrate (100 averages each) using the DAP
card and a 100 kHz anti-aliasing filter (LE1182-100k-3k-720B, TTE). At low frequencies, the ratio of complex velocities at the cantilever tip to its substrate was about
1, while there was a clear maximum ratio at the resonance, which peaked between
30 kHz and 40 kHz. The first resonant frequency was determined to be the frequency
at which the imaginary part of the velocity ratio changed signs (same location where
phase changes sign) to a precision of 62.5 Hz. Assuming that the error in the measured dimensions of the cantilever are within ±1,pm, the precision of the cantilever
calibration is within ±1%.
The cantilever's resonant frequencies decrease dramatically when it is immersed in
liquid instead of air. The liquid creates additional loading on the probe and reduces
its resonant frequency by almost an order of magnitude.
Furthermore, the friction
adds a damping component. Therefore, the cantilever was also calibrated in saline.
For frequencies below the second resonant frequency, the mechanical impedance of
the cantilever can be approximated as follows:
Z =
k
+ RfIind + jwm fluid
(2.16)
where Rf1lid and jWmf luid are terms representing the viscous effect of the surrounding
38
endolymph and the mass of the cantilever with fluid loading. The stiffness k should
remain the same as that calibrated in air as in equation 2.12.
Two calibration measurements in fluid were taken to determine the friction and
mass loading on the cantilever. The piezo-electric bimorph which held the cantilever
was stimulated as velocities of the cantilever tip and cantilever substrate were measured. The relative velocities of the cantilever tip and its substrate are proportional
to their relative impedances. Therefore,
VTip
__
Zair
1
1 + jW Rf jid
Zfluid
VSubstrate
.(2.17)
k
-
W2 M
luid(
k
The resonant frequency in fluid, ffluid, and quality factor, Qfluid, could also be used
to describe the ratio of the impedances,
Vrip
_ I
VSubstrate
2.
1+
(ued
(2.18)
2lidQfIlid
From this, the effective mass of the cantilever is
k
mfluid =
)2
(27ffluid)
(2.19)
and the fluid resistance is
Rf uid =
2.7
k
k
.
27r ffguidQfid'a
(2.20)
Animal Care
The care and use of the mice in this study were approved by the Massachusetts
Institute of Technology Committee on Animal Care (#98-038-1).
39
40
Chapter 3
Poroelastic Theory
Biological tissues, such as the TM, comprise a solid network and interstitial fluid. Viscous behavior has often been ascribed to fluid motion within the elastic extracellular
matrix. Traditionally, viscoelastic models with lumped elements (springs, dashpots,
and masses) have been used to model tissue mechanics, which are dominated by the
solid phase of the tissue.
However, the TM is mostly water (Thalmann et al., 1987). As a result, it can be
modeled as a porous network, whose mechanics are dominated by the flow of fluid
that fills its spaces. The frictional interaction between the collagen network and the
fluid in the TM would seem to be better described by poroelastic theory, in which
fluid flow dominates tissue mechanics, as opposed to viscoelastic theory.
First proposed by Maurice Biot in the 1940's and 50's (Biot, 1941; Biot, 1956b;
Biot, 1956a), poroelastic theory was used to characterize soils and geophysical materials. It was later applied to the mechanical deformations of gels (Tanaka and Fillmore, 70). Most recently, this theory has successfully described the behavior of soft
connective tissues, in particular articular cartilage (Armstrong et al., 1984; Grodzinsky et al., 1981; Lai et al., 1991; Lee et al., 1981; Mow et al., 1980) and the cornea,
all of which have been successfully modeled by their own respective polyelectrolyte
gel models.
This chapter will describe the basis of the theory for poroelasticity in an effort to
understand the TM's mechanical properties.
41
3.1
Constitutive Equations
Unlike soil, the structure for biological tissues is maintained by a equilibrium balance
of swelling and elastic recoil (Grodzinsky, 2000). The swelling is caused by electric
repulsion from the charged proteoglycan aggregates. Elastic recoil results from the
collagen networks. It is well-known that the chemical environment also affects the
mechanical and electromechanical properties of the TM (Freeman et al., 1994; Shah
et al., 1995; Weiss and Freeman, 1997). However, in this thesis, we will be measuring
effects of mechanical stimulation in the absence of chemical changes, and no attempt
is made to model chemical effects.
3.1.1
Assumptions
Although the TM is far from homogeneous (Kronester-Frei, 1978; Lim, 1986; Steel,
1983) and has anisotropic mechanical properties (von Bekesy, 1953; Abnet and Freeman, 2000), in developing and obtaining the general feel of the poroelastic theory, we
will assume that it is a linear, homogeneous, isotropic material, as in the polyelectrolyte gel model. In general, for sufficiently small deformations, materials are linearly
elastic. Also, we will assume that the area of the TM covered by the cantilever applying the force is so small that the volume below could be considered homogeneous.
Additionally, since the forces were applied to a tacked down TM in the vertical direction, the calculations will be specific to one-dimension, although the core equations
for three-dimensions for an isotropic material will be presented.
Due to the complexity of the theory, the most simplified case, the continuum
model for uniaxial confined compression, Figure 3-1(a) (Frank and Grodzinsky, 1987;
Grodzinsky, 2000), will first be presented.
As shown in Figure 3-1(a), the tissue
is bound in all directions except for the top surface, which is a frictionless, rigid,
permeable plate that allows fluid to pass through. As a result, during compression,
since neither the fluid nor the solid are compressible, fluid must be forced through the
plate, in the z-direction only. There is no net flow of fluid in the horizontal direction.
On the other hand, in unconfined compression a rigid impervious plate applies the
42
CONFINED COMPRESSION
i
E xudation
Oi1
UNCONFINED COMPRESSION
Rigid
Impervious
smooth
r
I IIPlate
laxi,
Exudation
conlining
.* *
,
e'
COMPRESSION DIRECTION
FLOW DIRECTION
~specimen/+-,-*e*
i
+
COMPRESSION DIRECTION
T
FLOW DIRECTION
+-
Figure 3-1: (a) Poroelastic material compressed by a rigid permeable plate. Geometric
constraints allow fluid flow in only the axial (z) direction. (b) Poroelastic material
compressed by a rigid impervious plate. This allows unconstrained fluid and solid
expansion radially. Taken from Armstrong et al., 1984.
compressive force. The interface between the tissue and its surroundings is assumed
to be frictionless.
Since the fluid cannot move upward, it must exit through the
sides of the specimen, demonstrated in Figure 3-1(b). Because of the similarity of the
cantilever compression experiments to unconfined compression experiments, Figure 31(b), the poroelastic theory will be extended to cover this scenario and a discussion
of how this relates to the TM will follow.
3.1.2
Hooke's Law
Most materials, including the TM, have elastic properties.
Because we assume it
to be linear, we would expect that the relationship between the stress and strain to
exhibit a linear relationship of the following form:
T =
Ec
(3.1)
where T is the stress (tension or compression) applied to the material in a given
direction on a particular face, c is the strain that results from the stress. The constant
of proportionality, E, is referred to as the modulus of elasticity or bulk modulus,
which remains constant throughout the linear portion of the stress-strain curve. This
43
equation is often referred to as Hooke's law.
However, to include shear stresses and external fluid pressure, a more generalized
form of Hooke's law must be used. The relationship between the total stress,
Tij,
and total strain, Eij, can be expressed with the addition of Poisson's ratio, v, which
represents the ratio of the strain in the lateral direction to the strain in the axial
direction. i represents the direction of the applied force, while j represents the face
to which the force is applied. Thus
1+tV
1+ 2v
V
E Tj EE(XX
+ T,
+
T,)
zz
+
E
Pig ij
(3.2)
where 6oj is the Kroenecker function and p is the normal pressure applied by the
surrounding fluid to the material (Armstrong et al., 1984).
For our purposes, we will use the alternate form of generalized Hooke's law:
rij = 2Gcij + A(Ell+
622 +
C33)63 j -
Pijij
(3.3)
where G is the shear modulus of elasticity and A is the Lame constant.
The relationship between the elastic modulus, E, and the shear modulus, G, is
G =
2(1 + v)'
.
(3.4)
Since the experiments deal with normal forces in the z direction, equation 3.3 can
be simplified to
T=
(2G + A)E6z - pf.
(3.5)
The term 2G + A is also known as HA, the equilibrium confined compression modulus
for the uniaxial confined case (Mow et al., 1984).
44
3.1.3
Darcy's Law
As both the fluid and solid matrix in biological tissues is incompressible, any gradient
in pressure across them would result in fluid flow at a proportional velocity. In other
words, Darcy's law states
U = -kVp
(3.6)
where k is the hydraulic permeability of the material and U is the relative fluid
velocity to the solid matrix, i.e.
U=
where
(Uf --
)
(3.7)
#
is defined as the porosity, a ratio of the fluid volume to the total volume, Vf
and T are the fluid and solid velocities respectively (Armstrong et al., 1984).
3.1.4
Conservation of Mass
The incompressibility of the solid and fluid components necessitates that any net flow
of fluid out of a specific volume must be replaced by an inward "flow" of solid and
vice versa. Therefore,
V -U = -V
3.1.5
FS.
(3.8)
Conservation of Momentum
The last core equation to understand the mechanics of the TM is based on the conservation of momentum. Because the tissue is not moving in space, all the forces
applied to a unit volume must cancel each other out. Another way of looking at it is
that the divergence of stress tensor equals 0,
V - rg = 0.
45
(3.9)
3.1.6
Electrical Relationships
As the TM is inherently charged, motions affecting its solid network would result in
charge rearrangement to satisfy electroneutrality. This causes a current and streaming
potential to flow through the tissue. To model the TM more accurately, we would
have to modify Darcy's law and add in Ohm's law and the conservation of current to
the basic equation list. The following linear, macroscopic laws used to characterize
the electrokinetics of isotropic media (DeGroot and Mazur, 1969) relate relative fluid
velocity, U, and current density, J, to fluid pressure, pf, and electrical potential, V,
gradients:
-k ilV~ kf(32
L J
L
k 21
k22
.10)
VV
where k1 l is the hydraulic permeability, k, in equation 3.6, k12 and k2 1 are electrokinetic coupling coefficients, and k22 is the electrical conductivity, -, typically found in
Ohm's law.
Conservation of current states that all the current entering the medium must sum
to zero,
V J=0.
(3.11)
While it is clear that the fixed charge on the tectorial membrane would affect
electromechanics from the coupling in equation 3.10, to simplify the theory, only the
mechanical properties will be considered, and the electrical effects will be considered
negligible.
3.2
Application of Equations
The equations listed above provide the fundamentals to understanding poroelasticity.
However, as with any problem involving an interface between two different materials,
one needs to understand the conditions at such boundaries.
46
3.2.1
Boundary Conditions
Depending on characteristics of the object applying the force and the conditions to
which the poroelastic network is subject, boundary limits for the above equations
differ.
Recall that the mechanics of a poroelastic material are governed by viscous fluidsolid interactions. Neither the fluid nor the solid will move instantaneously to its
most "relaxed", or uniform, state after a step compression or extension. Instead, it
will undergo a process known as stress relaxation or creep to reach its final state
with a time constant that is frequency dependent. In both cases, because one end
of the tissue is anchored, at x = L, where L is the poroelastic body thickness, the
displacement is always 0.
Stress Relaxation
After a compressive force is applied, stress relaxation occurs. The initial solid displacement of the poroelastic material is zero everywhere except at the surface of the
applied force, which results in a displacement of no. Since neither the fluid, nor solid
is compressible, to be able to decrease the volume, fluid must exit the system. In
uniaxial confined compression, the fluid is immediately displaced from the surface
into the porous filter. Meanwhile, the solid network moves to overcome the resistive
force from the fluid to obtain more uniform strain. The displacement profile can be
expressed as follows (Grodzinsky and Mahadevan, 2000):
u(x, t) = uo(1
L
) - EA si
7rnx
L
_
-
with time constants (Tanaka and Fillmore, 70)
7n =-
L
r2Hkn2
47
(3.13)
u(x,t)
increasing time
t = 0+
It
L
Figure 3-2: Displacement profile (u(x, t)) of the solid phase with increasing distance
from the point of compression (x) and time for stress relaxation.
and coefficients
An =
2uO
7rn
(3.14)
Figure 3-2 shows the displacement profile for stress relaxation as a function of
time.
Creep
Creep is the relaxation phase after tension is applied to poroelastic material. Unlike
compression in which the initial displacement at the free end, u(O), is identical to
its final value, the displacement of the material increases with time. At time t
u(0)
=
0.
0,
As the object continues to be stretched by a constant force, the solid
and fluid phases of the tissue "flow" into the new "volume" at a constant rate, such
that the displacement profile has a uniform slope for most of the time as shown in
figure 3-3.
48
u(x,t)
A,4
increasing time
t=
0+
x
L
Figure 3-3: Displacement profile (u(x, t)) of the solid phase with increase distance
from the point of tension (x) and time for creep.
The time constants for creep are also (Tanaka and Fillmore, 70)
Tn = 7r 2 Hkn2
3.2.2
(3.15)
Mechanical Diffusion
The phenomena described above can be derived by combining the four fundamental
equations - 3.3, 3.6, 3.8, 3.9. The mechanics of linear, homogeneous, poroelastic materials can be summarized in one equation. Assuming uniaxial, confined compression
(i.e. one-dimensional motion only),
eDu
at
2_
_
=
Ak at2 .2
(3.16)
This has the form of a simple diffusion equation
Ou
at
=D
49
02u
at2
(3.17)
where D = HAk. Thus, this problem can be likened to many known problems such as
the chemical diffusion of solutes or the diffusion of magnetic fields into a conductor. In
magneto diffusion, D = op where the electrical conductivity of the material, or, is analogous to the equilibrium confined compression modulus, HA, and
jt,
the permittivity,
corresponds to k, the permittivity in the poroelastic medium. While in poroelasticity,
the diffusion variable is displacement, the strength of the magnetic field, H, is the
variable in magnetodiffusion. Using the similarity between the two situations, one
can derive an intuitive understanding of the effects of a dynamic mechanical force on
a poroelastic material.
Any periodic stimulus can be represented as a Fourier series. Assuming the system is linear, the resulting transformation from a given stimulus is the sum of the
individual transformations from the frequency components that make up the series.
Therefore, we will discuss the response of a sinusoidal steady-state signal and use
superposition to understand the overall response.
For a sinusoidal steady-state dynamic load, the effects of the force are only felt
up to a certain distance from the location of the applied force, which we will refer to
as the boundary layer. Therefore, as long as the thickness of the specimen is greater
than the boundary layer the measured response for a poroelastic body would not
depend on the thickness of the material. Likewise, if the boundary layer is greater
than the thickness of the material, the measured characteristics would differ between
objects of different heights.
3.3
Extension of Theory to Unconfined Compression
As suggested earlier, there are distinct differences between the uniaxial confined compression situation discussed earlier and unconfined compression. Since the tissue is
not bound by its sides, it is free to expand radially when compressed. Like the confined compression scenario, there can be no instantaneous change in tissue volume at
50
time t = 0+. The stress in confined compression is isotropic since the tissue is bound
on all sides, while in unconfined compression, the stress on the tissue is completely
uniaxial as there is no pressure from the sides. The time constant for the stress relaxation and creep problems are similar to that for confined compression, except now
it depends on the radius (a) of the material under compression, as opposed to its
thickness (Equation 3.13). The new time constant is (Armstrong et al., 1984)
t =
a2
HAk
(3.18)
This is because the flow path of the fluid is now in the radial as opposed to the vertical
direction. While deformations in confined compression diffuse through the poroelastic
body as demonstrated in figures 3-2 and 3-3, in unconfined compression, there can be
an initial radial deformation in the tissue, whose elasticity causes it to recoil during
creep and stress relaxation as the fluid moves through the pores. This suggests that
the equilibrium response of the body is governed by the elastic properties of the solid
network (Armstrong et al., 1984). After the sudden lateral expansion of the tissue,
the fluid pressure also rises dramatically. Towards the center, where the deformations
are significantly smaller, the fluid pressure is the highest and decreases to 0 at the
edge of the tissue.
It can therefore be expected that the frequency response in unconfined compression
is very different from confined compression because of the added dimension of motion
for the elastic matrix. However, one can draw parallels between the two situations to
predict the response of the specimen in unconfined compression. The most important
time course in stress relaxation or creep in unconfined compression is the recoil of
the solid matrix from its initial deformation, with a time constant similar to that in
confined compression as the network tries to rearrange itself from the initial uniaxial
deformation. In unconfined compression, as fluid leaks out due to high interstitial
pressure from when a force is first applied, the elastic fibers begin to move to their
final position. It would be expected that the effects of recoil would be limited to a
boundary layer that is frequency dependent. The centermost areas would be able
51
to reach their final matrix state much faster than the outermost regions, which have
larger deformations. Simultaneously, the high interstitial fluid pressure in the center
forces it to redistribute throughout the poroelastic body during stress relaxation. As
the frequency of loading increases, the observed behavior of the tissue shows that it
becomes more incompressible as there is not enough time for the fluid to leave the
tissue.
3.4
Application to TM measurements
As mentioned earlier, the cantilever method is most similar to the unconfined compression scenario. The cantilever is a relatively rigid impermeable structure that
presses against the TM. When the mostly incompressible TM is squeezed, it is expected that there is an initial radial expansion that parallels the axial compression as
predicted by unconfined compression theory. However, there are limitations. First of
all, the unconfined compression case assumes frictionless interfaces between the TM
and cantilever and between the TM and coverslide. This would allow the TM to expand freely at the boundaries. However, there is a high degree of friction between the
TM and coverslide because the cell-tak that is used to glue the two surfaces together
would prevent such freedom of movement. Additionally, as shown in figure 3-1, both
surfaces above and below have a larger area than the sandwiched poroelastic body;
therefore, the compressive forces would be applied over the entire tissue. However,
with the cantilever being significantly smaller than the TM, such forces would be
limited to the tissue just below the surface of the cantilever. Also, the side boundaries would no longer be pure fluid with a pressure of 0. Instead, the surrounding
elastic matrix of the TM would generate a higher fluid pressure and forces to prevent
the TM directly below the silicon probe from freely expanding. This would liken the
situation to that of confined compression, with pressure from the sides. However, the
fluid is free to move upwards where the cantilever does not touch the TM and to the
sides that the situation is a combination of the confined and unconfined compression
problems. Because the fluid pressure to the sides is higher than on top, water will
52
most likely exit the TM from the top, similar to the confined compression problem
posed in this chapter.
53
54
Chapter 4
Results
A total of eight impedance measurements were conducted on tectorial membranes
obtained from six mice. Two pairs of measurements were each acquired from one
cochlea.
4.1
TM Imaging
The TM is almost transparent and barely visible under standard reflected light illumination. However, the LED transilluminator and a high-quality microscope objective
provided detailed images of the TM (Figure 4-1). Images were taken in 50 focal planes
by stepping the piezo-electric focusing unit in increments of 2 pm. Image contrast
was sufficient to discern Hensen's stripe and the direction of the radial fibrils in the
TM. By finding the focal planes of the coverslide and the upper surface of the tissue,
an estimated thickness was obtained. The values varied from 31 to 42 pm for the
TMs in this study (Table 4.1).
4.2
Velocity Measurements
After taking images of the TM, the laser-Doppler interferometer measured the velocity
of the cantilever's motion in response to mechanically stimulating the TM. The first
measurements were taken with the cantilever far above the surface of the TM (100
55
Preparation
Thickness (pm)
Time (min)
1
2
3
4
5
6
7
8
31
35
38
41
40
42
38
40
65
130
210
460
100
195
105
380
Table 4.1: Isolated TM preparations. The table lists the estimated TM thickness
under the cantilever tip and the time between the death of the mouse to the first
measurement.
Figure 4-1: TM image obtained using transillumination. This was taken from one of
the preliminary measurements conducted while the experimental protocol was being
refined.
56
- 200 pm) to estimate the fluid forces acting on the cantilever (Figure 4-2). The
motorized microscope stage was then raised in 10 pm increments and data were
acquired at each level. When the cantilever first touched the surface of the TM,
the velocity response changed significantly: the amplitude and phase traces became
significantly smoother and the phase of the velocity relative to the stimulus went from
looking random to +90' at low frequencies when the cantilever and the TM came
in contact. High frequency phase lags intrinsic to the laser-Doppler interferometer
contributed to the measured phase lag at high frequencies in figure 4-2.
As the
stage was elevated, the TM was further squeezed and the amplitude of the cantilever
velocity increased. There were no significant changes in the phase response (Figure 42). When the cantilever had thoroughly compressed the TM, the cantilever velocity
matched that of the chamber.
As the microscope stage was raised, the chamber
pushed the cantilever out of focus.
Because the chamber-actuator complex is many times stiffer than the probe, the
measured response on the cantilever was the velocity of the piezo-electric actuator,
Up, when the cantilever came in full contact with the coverslide. The good reflectivity
of the cantilever surface consistently provided reliable velocity measurements for Up;
on the other hand, it was difficult to obtain sufficient reflections of the laser beam off
the coverslide. To verify that the first technique og determining Up was not affected
by the presence of the TM and cantilever stiffnesses, both methods of obtaining
chamber velocity were used in early experiments. Both methods produced similar
results, except that there was significantly less noise when the laser-Doppler recorded
the response of the cantilever.
4.3
TM Impedance
From the velocity measurements of the cantilever's response to a broadband stimulus, TM impedances were calculated from Equation 2.5 derived from the mechanical
circuit shown in Figure 2-3. The impedance of the cantilever is approximated as that
57
10 -4
10
E
-6
1c.
10
180
V.
0
-18010
10
100
1000
10000
Frequency(Hz)
Figure 4-2: Magnitude and relative phase of the frequency response of the cantilever
velocity measured for different TM indentations for preparation 3. The thick dots
represent the motion measured on the cantilever due to fluid coupling, whereas the
small dotted line represents the maximum velocity measured by pressing the chamber
against the cantilever. The dashed line is the velocity of the chamber measured by
reflecting the laser beam off the glass coverslide. The solid lines represent the velocities
for different TM indentations equally spaced at 10 pum. Phase is plotted relative to
the input stimulus. Gray-shaded regions show where the velocities of the TM are
within 6 dB of the fluid velocities.
58
of a spring with negligible resistance and mass. Hence,
Zc =-
(4.1)
where kc is the spring constant of the cantilever derived from Equation 2.12. For
the three cantilevers used in this investigation, estimated stiffness values were 1.04,
1.27 and 2.04 N/m. They are approximately an order of magnitude stiffer than the
TM as determined from previous measurements (von Bekesy, 1953; Zwislocki and
Cefaratti, 1989; Abnet and Freeman, 2000).
For the first velocities obtained, the TM was far below the cantilever. As the microscope stage was raised, the TM eventually came in contact with the probe. In measurements for which the cantilever did not touch the TM, the calculated "impedance"
using Equation 2.5 was classified as noise due to the fluid coupling between the cantilever and chamber. Figure 4-3 shows a typical TM impedance frequency response
for 3 conditions: (1) when the cantilever first pressed into the tissue, (2) when the
cantilever was 10 pm above the TM, and (3) when the cantilever was far away from
(> 100pum) the surface of the TM. There was a distinct qualitative difference between
the response of TM and fluid that made finding the TM's point of first contact with
the probe possible. Because above 4 kHz the TM's impedance lies within 6 dB of the
fluid impedance, it was not possible to distinguish between the two. Therefore, only
data up to 4 kHz will be discussed.
4.4
Cross Preparation Variation
Because TM preparations were taken from different mice and different regions of
the cochlea, variation between the calculated impedances were expected.
During
the experiments it was difficult to determine the exact thickness of the TM between
the cantilever and coverslide because obtaining images directly below the cantilever
was not possible. Therefore, initial comparisons between preparations were made for
measurements taken at the first point for which the cantilever was believed to touch
59
10-2
E
10
-3
C4
1O010
10 180
a
'0
I
90-
0
7
--
-90
-180
10
100
1000
10000
Frequency(Hz)
Figure 4-3: Typical TM impedance. The solid black line shows the measured mechanical impedance of the TM (Table 4.1, preparation 1), when the distnace between
the TM and coverslide was 40 pm. The dotted and dashed lines show the fluid
"impedance" with the coverslide 200 pm below the cantilever and at 50 pm, just
below the point where the TM touches the cantilever. The gray solid line represents
the cantilever impedance, which has a resonant frequency in fluid of approximately
27 kHz. The shaded gray region shows where the TM impedance measurements lie
within 6 dB of the fluid "impedance".
60
the TM. Subsequent comparisons were made for equivalent TM thicknesses to the
nearest 10 Pm.
At the point of first contact, the frequency response of the impedances decreased
with increasing frequency with a slope between -4.5 and -6.6 dB/octave, with a median slope of -4.8 dB/octave (Figure 4-4). The resonances observed between 300 and
500 Hz in some of the TM impedance measurements are identical to those observed in
fluid measurements (Figure 4-3) and result from a side-resonance of the chamber that
sets the fluid in motion. For frequencies below the resonances, the TM impedances
run roughly parallel to one another, and those impedances that are unaffected by
the resonances continue in the same trajectory. However, in other measurements,
the calculated TM impedances cross each other at the resonances and then proceed
to decrease with similar slopes in other frequency regions. The phase response was
consistent across all preparations. It remained fairly constant between -45 and -900
throughout the entire frequency range except at the chamber resonant frequencies.
The overall variation of the magnitude response of the TM impedance for all
indentations in the eight preparations was between -3.6 and -6.6 dB/octave (Figure 45) with a median slope of -4.8 dB/octave, while the phase varied between -45 and
-1000 (Figure 4-6). Most of the deviations in slope and phase were from preparation
5. The data presented show that the experimental method to measure the TM's
transverse impedance generated reliable and repeatable results.
4.5
Effect of Increasing TM Indentation
Impedance trends as the TM was compressed within each preparation were also studied. As the microscope stage was raised in 10 pm increments, the amount of indentation on the TM generated by the cantilever increased as did the corresponding
impedance as shown in Figure 4-7. For each preparation, the magnitudes of the
frequency response remained roughly parallel to one another with little variation in
slope except for preparation 5 (Figure 4-5). In general, the phases also did not change
much with increasing indentation (Figure 4-5).
61
.1....................I
*~'
10
A
~\
\
~
\,\
.~
-
\,~
z
A
"A 1 '
_0
~
-
4
*
\~
'tlJ
~
'
II
I
ii'
k
~"
5)10-
41
~
i
i
U11
0U)
C
n)
90 -
10
100
1000
Frequency (Hz)
Figure 4-4: Calculated TM impedance of all the preparations in Table 4.1 at the
point when the cantilever and TM first come in contact. The magnitude responses are
separated by shade and line style such that each individual trace can be distinguished.
For TM impedances unaffected by the chamber resonances between 300 and 500 Hz,
the impedances run parallel to one another. The remainder of the traces cross one
another at the resonant frequencies.
62
-3
-o
-7
1
2
3
4
5
6
7
8
all
TM Preparation
Figure 4-5: Estimated slopes of the magnitude of the frequency response of the TM
impedance for each preparation in Table 4.1. Each point represents the value for a
different indentation within each preparation. The last column pools the data from
all experiments. The box shows the interquartile range, and the bar gives the median
slope.
Estimates of the TM thickness between the cantilever and glass coverslide to the
nearest 10 p-m were used to calculate the point stiffness of the TM at various indentations (Figure 4-8). For each preparation, the measurement with the largest impedance
was assigned a thickness of 10 pm. The next level of compression completely squeezed
the TM (thickness of 0 p-m) and provided an estimate of the velocity of the chamber.
The point stiffnesses varied as a function inversely proportional to a value between
the thickness, t, and thickness squared, P.
4.6
Other Dependences
Results taken from different mice did not vary as greatly as those from one set of
preparations from the same mouse. Preparations 2 and 3 were taken from the cochlea
of one mouse, as were preparations 5 and 6, yet the difference between the data from
preparation 5 and preparation 6 was larger than that between other preparations.
This showed that the results were not mice dependent. The median values of the
slopes and phases for most of the preparations were within the interquartile range
63
M-45 -
-90
BCD -45
D-90 -
(n
CZ
-450,
C
N
-90 -
a1
2
3
4
5
6
7
8
all
TM Preparation
Figure 4-6: Phase of the frequency response of the TM impedance for each preparation in Table 4.1) at 10, 100, and 1000 Hz. Each point represents the value for a
different indentation within each preparation. The last column pools the data from all
experiments. The box shows the interquartile range and the horizontal bar provides
the median phase.
64
-1
E
C,)
z
10
1
-2
3
10
C,
180
o)
90
-70
0~
-180
10
102
103
Frequency(Hz)
Figure 4-7: Effect of increasing the indentation of the cantilever into the TM for
preparation 3. The dotted lines represent fluid measurements, while the solid lines
represent the calculated impedances of the TM with approximately 0, 10, and 20 ,um
of static compression with increasing impedance corresponding to increasing indentation. The dashed line shows the measurement taken when the experimental chamber
was firmly pressed against the cantilever. These are the calculated impedances from
the velocity data plotted in Figure 4-2.
65
10
N
1
t
ZN
Cn-
0
510-(N
10
20
30
TM Thickness (gm)
40
Figure 4-8: TM stiffness dependence on final thickness after static compression. The
point stiffness at 1 kHz is plotted as a function of the estimated thickness of the
TM between the TM and coverslide. To distinguish the trends between each of
the 8 preparations, connecting lines between points are of different line styles and
shade. For comparison, dotted-dashed lines show trends for functions that vary as t
and 1/t 2 . The boxes show the interquartile range for the point stiffnesses, with the
longer horizontal bar representing the median point stiffness for each thickness and
the shorter horizontal bars showing the extremes.
66
across all the preparations.
The length of time from sacrifice to first measurement ranged from 65 and 460
minutes across the preparations. Nevertheless, the amount of scatter in the data
did not appear to depend on time, but rather on the quality of the preparation
and how much of the TM was being compressed by the cantilever. The quality of
the preparations were judged by the images of the TM taken at the beginning of
each experiment.
As the TM's quality degraded, it also became less transparent.
In preparation 5, in which the scatter in the slopes between the indentations was
relatively large, only a small fraction of the cantilever area was on the TM.
67
68
Chapter 5
Discussion
This chapter explains the issues surrounding the quantitative measurements obtained
in this study. It will also discuss how they support previous data and provide new
insight into the gel model of the TM.
5.1
Preparation limitations
As mentioned earlier, there are advantages to an isolated TM preparation. The ability
to decouple the TM from other cochlear structures permitted the measurement of the
mechanical impedance of the TM without the confounding effects of adjacent cochlear
structures. However, this preparation also has its disadvantages. The results here
cannot be easily compared to in situ experiments in which the TM is not detached
from other structures on the organ of Corti. Additionally, to understand the TM's
role in cochlear micromechanics, its interactions with the rest of the cochlea must be
considered. Another limitation to this preparation comes from the TM being glued to
a glass coverslide while its motions are being measured. This unnatural attachment
differs greatly from the TM's natural state of suspension above hair cells in the scala
media.
Nevertheless, the isolated TM provides a means to describe the material
properties of the TM without interference from other structures.
The quantitative description of the TM's mechanical impedance is also limited
by the accuracy to which the TM's thickness was determined. Since the microscope
69
stage brought the tissue towards the cantilever in 10 pim increments, the estimated
thickness, which was between 31 and 42 ptm, can vary by as much as 33%.
5.2
5.2.1
Implication of TM Measurements
Point Stiffness
The median value of the point stiffness in this study varied between 0.11 N/m and
0.84 N/m over a frequency range of 10 Hz to 4 kHz (Figure 5-1). The point stiffnesses
increased very gradually with frequency such that the overall variation is less than
one order of magnitude. At 10 Hz, the median point stiffness was 0.11 N/m with
an interquartile range of 0.059 N/m-0.22N/m; at 100 Hz, the median was 0.20 N/m
with an interquartile range of 0.11-0.29 N/m; at 1000 Hz, the median was 0.35 N/m
with an interquartile range of 0.27-0.38 N/m. This is the first set of measurements
for which the stiffness of the TM was determined at such a high frequency. For the
lower frequencies, the values obtained for the point stiffness are in agreement with
those of previous measurements as demonstrated in Figure 5-1.
The transverse impedance of the TM is similar to the measured longitidinal stiffness and smaller than the radial stiffness (Abnet and Freeman, 2000). Since the TM
does not have collagen fibrils running in the transverse direction, it was expected
that its stiffness would be comparable to that of the longitudinal direction. Because
collagen has been shown to stiffen tissues such as tendon and articular cartilage (Zhu
et al., 1996; Kempson et al., 1973), it should have the same effect on the TM's fibrils,
which run radially (Lim, 1972). The findings in this study are consistent with the
TM's anatomy.
5.2.2
Compression dependence
The indentation of the cantilever into the TM can potentially have two different effects (Figure 5-2). One possibility is static compression, in which the solid component
directly below the cantilever is compressed into a much smaller volume as fluid leaves
70
Mr-
- ---.
ao
~...
1
.
I
I
I
I
l
i
I
I
I
I
I
Iaa Ia
I
Iaa
a
-- . -
((e)
0.01- 4
10
0
C .
.
1
. 1
I
I
.
.
I
1000
100
Frequency (Hz)
Figure 5-1: Calculated TM point stiffness values from various experiments. Solid
line is the median point stiffness of the TM when it first comes in contact with
the cantilever and the gray region represents the interquartile range for the data
in this study. (a) Dotted line represents the median radial stiffness (Abnet and
Freeman, 2000). (b) Dotted line represents the median longitudinal stiffness (Abnet
and Freeman, 2000). (c) Radial stiffness at 20 Hz (Zwislocki and Cefaratti, 1989). (c)
Transverse stiffness at 20 Hz (Zwislocki and Cefaratti, 1989). (e) Static depression
stiffness (Bekesy, 1960). (f) Stiffness at 200 Hz (Bekesy, 1960). Bekesy found the TM
to be rigid when he probed it with a needle vibrating at 200 Hz.
71
the area. This increases the bulk modulus and overall stiffness of the TM proportionally to the increase in density. As a result, to double the indentation into the
TM, the force required would be larger than twice its original value because additional force would be needed to compress a material with a higher bulk modulus.
Alternatively, pushing an object into the TM could force both the elastic matrix and
interstitial fluid out to the sides. The overall volume and density of the tissue remain
constant; hence, the bulk modulus would not change. Therefore, the force to create
an additional indentation of a similar size should be identical to that needed for the
first indentation. However, as the cantilever pressed into the TM, its thickness decreases, thereby increasing the overall stiffness, which is inversely proportional to the
thickness. Both proposed theories would explain an increased impedance observed in
figure 4-7 as the indentations into the TM increased.
The data in Figure 4-8 show that the stiffness increased with decreasing thickness with a factor between 1/t and 1/t 2 , where t is the thickness of the TM at each
indentation. If the stiffness varied inversely proportional to t, then the effect of the
indentation was pure deformation. Static compression increases the density of the
tissue at a rate inversely proportional to the thickness since the area under compression was constant. Combined with the 1/t factor from decreased thickness, the
overall increase in stiffness due to static compression would be proportional to 1/t 2 .
Since the TM's stiffness was inversely proportional to a value between t and
t2,
the
effect of indentation on the TM was a combination of pure deformation and static
compression.
5.2.3
Frequency Dependence
Calculated impedances are frequency dependent (Figure 4-4). Because the phase lies
between 0' and -90', the TM's mechanical properties are between that of a viscous
damper and a spring, both of which are important for the entire frequency range of
10 Hz to 4 kHz. This is consistent with previous data (Abnet and Freeman, 2000),
and supports studies which show that the TM is a polyelectrolyte gel (KronesterFrei, 1978; Steel, 1983; Thalmann et al., 1987; Thalmann et al., 1993; Hasko and
72
+ indentation
S cantilever
0
woo
static compression
(increase in solid/fluid ratio)
Scantilever
pure deformation
(constant solid/fluid ratio)
Figure 5-2: Possible effects of indentation on a solid-fluid matrix. When the cantilever
pushes into the gel-like structure, two things can happen. In static compression, the
probe compresses the solid matrix, increasing the solid-to-fluid ratio in the volume
directly beneath. The second possibility is pure deformation in which the applied
force of the probe pushes out the elastic network along with interstitial fluid, keeping
the density of the tissue constant.
73
Richardson, 1988; Shah et al., 1995; Tsuprun and Santi, 1996; Weiss and Freeman,
1997). Similar frequency dependencies have been observed in cartilage and other soft
connective tissues, which have been characterized as polyelectrolyte gels (Frank and
Grodzinsky, 1987; Grodzinsky, 2000).
The frequency response of TM mechanics can be modeled by the simplest poroelastic theory with some slight modifications. In the uniaxial confined compression
scenario, the dynamic stiffness is expressed by (Frank and Grodzinsky, 1987)
AC = 6HAY coth y6,
(5.1)
where 6 is the thickness of the tissue. y2 = jw/HAk, where HA and k are the same
constants as in Equation 3.16.
Dividing equation 5.1 by jw gives the impedance
of the material. The magnitude of the resulting frequency response has a slope of
-3 dB/octave and a phase of -450 at frequencies above 10 Hz. In such cases, the
elastic and viscous properties of the tissue are equally important. Because in confined compression, a larger fraction of the applied force must be used to drive out
the fluid through the specimen, the viscous properties would be more important than
that of unconfined compression, in which a portion of the applied forces would be
used to deform the elastic matrix. As a result, it is to be expected that in unconfined compression, the TM would exhibit stronger elastic than viscous properties, as
demonstrated by median phase of -700 (Figure 4-5). The resistive effects of the fluid
flow still affect the mechanical properties of the TM, nonetheless.
Since the TM is inherently more elastic than viscous, the slope of the magnitude
of the frequency response should lie between -3 dB/octave and -6 dB/octave, the
slope for a purely elastic material. One can use the slope of the magnitude response
to predict the tissue's phase response. For a slope of 0 dB/octave, the material is
purely viscous with 0' phase, while a slope of -6 dB/octave corresponds to a phase of
-90'. The phase corresponding to the median slope of -4.6 dB/octave was calculated
to be -691, which is consistent with the phase data from this study. This shows that
the TM impedance data obtained through the cantilever method can be explained
74
with reasonable modifications to the uniaxial confined compression behavior.
5.3
Summary
The TM is believed to play an important role in cochlear micromechanics, yet little is
understood about its mechanical behavior. A new in vitro technique using an atomic
force cantilever to determine the transverse mechanical impedance of the TM was developed. A broadband stimulus was presented to measure the frequency response of
the TM under various static indentations. This complements previous measurements
of the longitudinal and radial stiffnesses from isolated TMs using forces of physiological magnitude (Abnet and Freeman, 2000). Additionally, this is the first investigation
that has quantified the TM's mechanical properties over a large frequency range.
Impedance measurements from eight TM preparations uniformly decreased with
increasing frequency with a median slope of -4.6 dB/decade while the phase response
remained relatively constant with a median phase of approximately -70'. This data
supports the poroelastic theory for gels, thereby adding another dimension to the
current polyelectrolyte gel model for the TM (Weiss and Freeman, 1996; Weiss and
Freeman, 1997). Results also show that the transverse impedance of the TM is similar
to the longitudinal impedance and smaller than the radial impedance, agreeing with
the anatomical structure of the TM. Lastly, this study also shows that applied forces
both compress and deform the TM.
75
76
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