We have developed and are refining a detailed three-dimensional computational model of the human cochlea. The model uses the immersed boundary method to calculate the fluid-structure interactions produced in response to incoming sound waves. An accurate cochlear geometry obtained from physical measurements is incorporated. The model includes a detailed and realistic description of the various elastic structures present. Initially, a macro-mechanical computational model was developed for execution on a CRAY T90 at the San Diego Supercomputing Center. This code was ported to the latest generation of shared memory high performance servers from Hewlett Packard. Using compiler generated threads and OpenMP directives, we have achieved a high degree of parallelism in the executable, which has made possible to run several large scale numerical simulation experiments to study the interesting features of the cochlear system. In this paper, we outline the methods, algorithms and software tools that were used to implement and fine tune the code, and discuss some of the simulation results

Bunn, J.

Givelberg, E.

Rajan, M.

English

Caltech Authors - Main

Mailing Address:  CACR Technical Publications, California Institute of Technology,  
Mail Code 158-79, Pasadena, CA 91125. Phone: (626) 395-6953 Fax: (626) 584-5917 
 
 2001 California Institute of Technology, Center for Advanced Computing Research.  
All rights reserved. 
 
 
 
 
 
 
 
 
CACR Technical Report 
 
CACR-190                                  July 2001 
 
Detailed Simulation of the Cochlea: 
Recent Progress Using Large Shared Memory Parallel Computers 
E. Givelberg, J. Bunn, M. Rajan 
 
 
 
 
 
Detailed Simulation of the Cochlea:
Recent Progress Using Large Shared Memory Parallel
Computers
E. Givelberg
Department of Mathematics,
University of Michigan,
Ann Arbor, MI
givelbrg@cims.nyu.edu
J. Bunn
Center for Advanced
Computing Research,
Caltech, Pasadena, CA
julian@cacr.caltech.edu
M. Rajan
Hewlett Packard Corporation,
Anaheim, CA
mrajan@cacr.caltech.edu
We have developed and are refining a detailed three-
dimensional computational model of the human cochlea. The
model uses the immersed boundary method to calculate the
fluid-structure interactions produced in response to incoming
sound waves. An accurate cochlear geometry obtained from
physical measurements is incorporated. The model includes a
detailed and realistic description of the various elastic
structures present. Initially, a macro-mechanical computational
model was developed for execution on a CRAY T90 at the San
Diego Supercomputing Center. This code was ported to the
latest generation of shared memory high performance servers
from Hewlett Packard. Using compiler generated threads and
OpenMP directives, we have achieved a high degree of
parallelism in the executable, which has made possible to run
several large scale numerical simulation experiments to study
the interesting features of the cochlear system. In this paper, we
outline the methods, algorithms and software tools that were
used to implement and fine tune the code, and discuss some of
the simulation results.
Background
The cochlea is the part of the inner ear where acoustic signals
are transformed into neural pulses that are then signaled to the
brain. It is a small snail-shell-like cavity in the temporal bone,
which has two openings, the oval window and the round
window. The cavity is filled with fluid and is sealed by two
elastic membranes that cover the windows. The spiral canal of
the cochlea is divided lengthwise into two passages by the so-
called basilar membrane. These passages are the scala vestibuli
and scala tympani and they connect with each other at the apex,
called the helicotrema. External sounds set the ear drum in
motion, which is conveyed to the inner ear by the ossicles,
three small bones of the middle ear, the malleus, incus and
stapes. The ossicles function as an impedance matching device,
focusing the energy of the ear drum on the oval window of the
cochlea. The piston-like motion of the stapes against the oval
window displaces the fluid of the cochlea, so generating
traveling waves that propagate along the basilar membrane.
Practically everything we know about the waves in the cochlea
was discovered in the 1940s by Georg von Békésy1 who carried
out experiments in cochleae extracted from human cadavers.
Despite its name the basilar membrane is in fact an elastic shell
whose stiffness varies exponentially with length. (Unlike an
elastic membrane, an elastic shell is not under inner tension, i.e.
when it is cut the edges do not pull apart.) The basilar
membrane’s elastic properties have an important role in the
wave mechanics of the cochlea. It is observed that a pure tone
input generates a traveling wave that reaches its peak at a
specific location along the basilar membrane: the position of
the peak depends on the frequency of the tone. Complex sounds
composed of several pure tones evoke a basilar membrane
response that is similar to a superposition of the membrane's
responses to the constituent frequencies. Resting on the basilar
membrane is the microscopic organ of Corti, a complicated
structure containing sensory receptors called hair cells. The hair
cells detect fluid motion and provide input to the afferent nerve
fibers that send action potentials to the brain. Thus, a pure tone
input activates only a narrow band of hair cells depending on
its frequency. These characteristic frequencies are
monotonically decreasing along the basilar membrane from the
stapes to the helicotrema.
Mathematically, the macro-mechanical system of the cochlea
can be described by the Navier-Stokes equations of
incompressible fluid mechanics coupled with equations
describing the elastic properties of the basilar membrane and
the membranes of the oval and the round windows. The
mathematical problem of solving this system of partial
differential equations on a three-dimensional domain with
intricate curved geometry is very difficult. Since the
displacements of the basilar membrane are extremely small (on
a nanometer scale), the system operates in a linear regime.
Analysis shows that the waves in the cochlea resemble shallow
water waves2
While the macro-mechanics of the cochlea break up the
incoming sound into its frequency components, it was
suggested as early as 1948 that a passive mechanical system
cannot explain the extreme sensitivity and frequency selectivity
of the cochlea; some kind of amplification is necessary3.Indeed,
analysis of cochlear macro-mechanics suggests that the
traveling wave focusing is not sufficiently sharp, with some
estimates suggesting that, at its threshold, human hearing is
about a hundred times more sensitive than expected from a
passive macro-mechanical filter of the cochlea.
The study of the active mechanism of the cochlea is the
ultimate goal of this project. This paper describes the
preliminary step towards that goal, namely the construction of a
comprehensive three-dimensional computational model of the
cochlear passive macro-mechanics.
Extensive research in cochlear modeling has been carried out
over the years. Because of mathematical difficulty mostly
simplified one or two-dimensional models that sought to
incorporate some of cochlear features have been studied.
Several three-dimensional models have been reported, such as
Kolston's model4, intended to simulate the micro-mechanics of
the cochlear partition in the linear regime (i.e. near the
threshold of hearing), and Parthasarati, Grosh and Nuttal's5
hybrid analytical-computational model using WKB
approximations and finite-element methods. The model we
have constructed is the only model that incorporates the full
curved three-dimensional geometry of the cochlea, uses the
viscous non-linear Navier-Stokes fluid equations, uses a shell
theory to model the basilar membrane and incorporates detailed
elastic models of the other materials.
The Cochlea Model
The small size and the complex geometry of the cochlea make
it very difficult to measure the vibrations of the basilar
membrane in vivo. We have constructed a three-dimensional
computational model of the cochlea using the immersed
boundary method in order to study cochlear mechanics.
The immersed boundary method of Peskin and McQueen6 is a
general numerical method for modeling an elastic material
immersed in a viscous incompressible fluid. It has proved to be
particularly useful for computer simulation of various biofluid
dynamic systems. In this framework the elastic (and possibly
active) biological tissue is treated as a collection of elastic
fibers immersed in a viscous incompressible fluid. For details
of this formulation of the method together with references to
many applications see [6].
The immersed boundary framework is suitable for modeling
not only elastic fibers, but also different elastic materials
having complicated structure. The cochlea model incorporates
a shell theory into the immersed boundary framework,
obtaining a practical computational method for modeling an
elastic shell immersed in fluid7. The main advantage of the
method is its conceptual simplicity: the viscous incompressible
fluid is described by the Navier-Stokes equations, the geometry
of the model mirrors the real-life curved three-dimensional
cochlear geometry and models for elastic and active material
components can be naturally integrated.
The geometric model of the cochlear anatomy is based on
measurements that include the position of the basilar
membrane, its width and the cross-sectional area of the scalae.
There are six surfaces in this model: the basilar membrane, the
spiral bony shelf, the tubular walls of scala vestibuli and scala
tympani and the membranes covering the oval window and the
round window. The basilar membrane is modeled as an elastic
shell and the oval window and the round window membranes
are modeled by computational grids, whose points are
interconnected by elastic springs. Figure 1 shows a partially
transparent cochlea model as it is seen from the outside.
Figure 1 Partially transparent cochlea model as seen from the
outside.
In Figure 2 the two external walls forming the outer tube have
been removed to expose the two membranes representing the
oval and the round windows and the cochlear partition
consisting of a bony shelf and a narrow basilar membrane. This
is a snapshot of the system taken after several time steps of the
simulation algorithm had been executed. One can see the input
force applied to the oval window and the resulting traveling
wave beginning to propagate along the basilar membrane.
Figure 2 Snapshot of the internal parts of the system taken after
several time steps of the simulation algorithm.
We use a first-order immersed boundary numerical scheme,
which is the easiest to implement and has the important
advantage of modularity: integration of various models of
immersed elastic material is straightforward. The bulk of the
computation is related to the fluid equations and to the fluid-
material interaction. The solution of the fluid equations uses the
Fast Fourier Transform algorithm. (see [7]).
Software Implementation
We have developed a suite of software codes to support our
studies of the cochlea using the immersed boundary method.
These include codes for the generation of simulation input
models (implemented in C++), the main immersed boundary
numerical solver, and those for the analysis and visualization of
results.
The main workhorse in this suite is the general purpose
immersed boundary code, which is written in C and requires
extensive computing resources, in terms of CPU power,
memory, and disk space for storage of the results. There are
several C++ programs used to generate the input models for the
immersed boundary code. The principal program generates an
approximate cochlear geometry (as seen in Figures 1 and 2) and
the material properties of various surfaces. To achieve
approximately uniform mesh density the six surfaces of
immersed material in the cochlea model are partitioned into 25
computational grids comprising approximately 750,000 points
in total. They include an elastic shell modeling the basilar
membrane, two elastic membranes modeling the oval and the
round windows and 22 grids describing the bony walls and the
shelf.
The immersed boundary code has been optimized to run on
several platforms: the Silicon Graphics Origin 2000 parallel
architecture, the Cray T90 vector-parallel machine and the HP
9000 V-class and Superdome systems. There is also a version
for a PC running under Windows, suitable for testing small
models. The code is very well instrumented with calls to
system timing routines. This information proves invaluable
during porting and tuning of the software on different
architectures. The largest and most recent numerical
experiments have been carried out on the 32 processor HP
Superdome installed at the Center for Advanced Computing
Research (CACR) at Caltech.
The codes that are used to analyze and visualize the simulated
data include a C++/OpenGL tool that runs on SGI and a
Java/Java3D tool that runs on most platforms. These tools take
as input the vertex coordinates for all computational grids in the
cochlea model from the result files for each time step in the
simulation. The tools generate a full 3D rendering of the model
geometry. Since the displacements of the basilar membrane are
very small (measured by a fraction of a nano-meter at the
hearing threshold), they are color-coded to reveal the dynamics
of the system. An essential insight into the basilar membrane
dynamics is provided by the plot of the normal displacement of
its center line (see Figure 3). Other Java tools display various
important characteristics of the system dynamics, such as the
response of individual points on the basilar membrane as a
function of time. All of the graphics tools have built-in
facilities for generating animation.
The Computational Challenge
The principal practical difficulty in the implementation of the
macro-mechanical model of the cochlea is due to the fact that
the basilar membrane is very narrow relative to the size of the
whole cochlea. In humans it is approximately 3.5 cm long and
its width grows from about 150µm near the stapes to
approximately 560 µm near the helicotrema, while the size of
the whole cochlea is on the order of 1cm x 1cm x 1cm.
Consequently, a fluid grid of at least 2563 points is necessary to
adequately resolve the small scale. In order to reduce the
computing time several small changes were made to the
geometric model of the cochlear anatomy. In the altered model
the cochlea is packed more tightly and the whole structure fits
in a half-cube, making it possible to use a fluid grid of 256 x
256 x 128 points. This configuration requires approximately 1.5
Gigabytes of main memory. Each time step of the immersed
boundary computation consists of computing the forces the
immersed material applied to the fluid, spreading these forces
from the computational grid of the material to the
computational grid of the fluid, solving the discretized Navier-
Stokes equations and advancing the position of the immersed
material using the newly obtained fluid velocity. The
computation of the material forces is relatively inexpensive in
time. The fluid solver relies primarily on the efficient parallel
Fast Fourier Transform routines. The other two demanding
parts of the time step required development of a specialized
algorithm carefully partitioning the fluid and the material grids
into portions distributed to different processors in such a way as
to prevent different processors from operating simultaneously
on the same portion of the data.
Extensive optimization of the immersed boundary numerical
solver was necessary to make numerical experiments with the
cochlea model practical. We have achieved a “wall-clock”
performance of approximately 1.3 seconds per time step for the
cochlea model on the HP Superdome running in a dedicated
mode. The HP 9000 Superdome at CACR contains 32 RISC
processors arranged in a cell-based hierarchical crossbar
architecture, with each cell consisting of 4 cpus with 8Gb of
memory and an I/O sub-system. This architecture supports the
shared memory programming model and the code efficiency
was achieved primarily through the use of OpenMP
parallelization directives.
In addition to the code instrumentation several software tools
such as the HP CXperf and the KAI Guideview were used to
display parallel efficiency of every section in the code and to
examine data cache and TLB misses. As a result, the cochlea
code achieves excellent scaling on the Superdome, as depicted
in Figure 3.
Figure 3: Wall clock time in seconds per one time-step vs. the
number of system processors.
Simulation Results
The achieved efficiency of the immersed boundary solver
allowed several large numerical experiments to be successfully
completed. Each experiment consisted of providing an input of
a pure tone of a given frequency at the stapes and the study of
the subsequent motion of the basilar membrane. A very small
time step of 40 nano-seconds was chosen to guarantee
numerical stability. A typical experiment was run for 50,000
time steps, equivalent to 2 milliseconds of simulated time, and
took approximately 18 hours of dedicated computation on the
Superdome.
There have been many experimental and modeling studies of
cochlear response to a pure tone input and our simulations
reproduce the characteristic features of the cochlear macro-
mechanics. We have run experiments with varying input sound
frequencies of 20, 10, 5 and 2 KHz. In each instance we
observe a traveling wave propagating from the stapes in the
direction of the helicotrema. The amplitude of the wave is
gradually increasing until it reaches a peak at a characteristic
location along the basilar membrane depending on the input
frequency. The higher the frequency the closer the peak is to
the stapes. Another characteristic feature observed was that
after reaching the peak the wave drops off very sharply,
essentially shutting down. Figure 4 shows a snapshot of a
single time step of a 10Khz experiment, as displayed by the
Java analysis tool. The wave pattern on the basilar membrane
(black line) can be clearly seen. The red lines indicate the
envelope of the wave computed over a series of previous time
steps.
Figure 4 Normal displacement of the center line of the basilar
membrane after 23850 time steps. The peak displacement is
approximately 0.5 nanometers.
The interested reader is invited to view several animations of
our results by visiting the Web site:
http://pcbunn.cacr.caltech.edu/Cochlea/default.html
Conclusion
We have constructed a comprehensive three-dimensional
computational model of the cochlea using the immersed
boundary method. Extensive optimization and parallelization
made it possible to complete several large-scale numerical
simulations on a 32-processor shared memory HP Superdome
computer. The pure tone experiments capture the most
important properties of the cochlear macro-mechanics. We
intend to continue testing the model and compare the results
with the available experimental and modeling data. For
example, we would like to construct a cochlear map for this
model, i.e. to determine the characteristic frequency of each
location on the basilar membrane. Subsequently we will extend
the present model to capture the micro-mechanics of the organ
of Corti in order to study the microscopic mechanism of
amplification in the cochlea. We anticipate such a model to be
even more computationally demanding than the present model.
We would like to thank Sarah Emery Bunn for her assistance in
the preparation of the manuscript.
1 G. von Békésy, Experiments in Hearing (E.G. Weaver trans.),
McGraw Hill, New York, 1960
2 Randall J. Leveque, Charles S. Peskin, Peter D. Lax, Solution
of a Two-dimensional Cochlea Model with Fluid Viscosity.
SIAM J. Appl. Math., 48, No.1, (1988), 191-213.
3 Gold, T. Hearing II. The physical basis of the action of the
cochlea. Proc R. Soc Lond [Biol] 1948, 135:492-498
4 Kolston, P.J., Finite element micromechanical modeling of
the cochlea in three dimensions. J. Acoust. Soc. Am. Vol.99, (1)
455-467
5 Parthasarati, A. A., Grosh, Kl, Nuttall, A.L. Three-
dimensional numerical modeling for global cochlear dynamics.
J. Acoust. Soc. Am., 107(1), Jan 2000, 474-485
6 Charles S. Peskin, David M. McQueen, A General Method for
the Computer Simulation of Biological Systems Interacting
with Fluids. Proc. Of SEB Symposium on Biological Fluid
Dynamics, Leeds, England, July 5-8, 1994
7 E. Givelberg, Modeling Elastic Shells Immersed in Fluid.
Ph.D. Thesis, NYU, 1997


Detailed Simulation of the Cochlea: Recent Progress Using Large Shared Memory Parallel Computers

http://authors.library.caltech.edu/28155/1/cacr.190.pdf

Detailed Simulation of the Cochlea: Recent Progress Using Large Shared Memory Parallel Computers

Abstract

Similar works

Full text

Available Versions

Caltech Authors - Main

Caltech Authors - Main