We present results from a series of compute-intensive simulation experiments employing a realistic and detailed three-dimensional model of the human cochlear macro-mechanics. Our model uses the immersed boundary method to compute the fluid-structure interactions within the cochlea. It is a three-dimensional model based on an accurate cochlear geometry obtained from physical measurements. It includes detailed descriptions of the elastic material components immersed in the fluid, and is based on the previously developed immersed boundary method for elastic shells. The basilar membrane is modeled by a fourth-order partial differential equation of shell theory. The results reproduce the basic well known characteristics of cochlear mechanics and constitute a successful initial step in model validation

Bunn, Julian

Givelberg, Edward

Caltech Authors - Main

  1 
Computational experiments with a three-
dimensional model of the Cochlea 
 
Edward Givelberg 
 
Division of Computer Science, University of California, Berkeley 
givelber@cs.berkeley.edu 
 
Julian Bunn 
 
Center for Advanced Computing Research, California Institute of Technology, Pasadena 
Julian.Bunn@caltech.edu 
 
Abstract: We present results from a series of compute-intensive simulation 
experiments employing a realistic and detailed three-dimensional model of the 
human cochlear macro-mechanics. Our model1 uses the immersed boundary 
method2,3 to compute the fluid-structure interactions within the cochlea. It is a 
three-dimensional model based on an accurate cochlear geometry obtained 
from physical measurements. It includes detailed descriptions of the elastic 
material components immersed in the fluid, and is based on the previously 
developed immersed boundary method for elastic shells 4. The basilar 
membrane is modeled by a fourth-order partial differential equation of shell 
theory. The results reproduce the basic well known characteristics of cochlear 
mechanics and constitute a successful initial step in model validation. 
 
1. Introduction 
 
The development of powerful computers increases the significance of computational modeling 
as a tool of research in cochlear mechanics. The construction of cochlear models using a 
combination of asymptotic analysis and computing methods has been extensively explored 
over five decades. Three-dimensional models were considered by Steele and Taber5, de Boer6, 
Kolston7,8 and Parthasarati, Grosh and Nuttal9. 
     We present results of a series of computer simulation experiments with a realistic and 
detailed three-dimensional model of the human cochlear macro-mechanics. By cochlear macro-
mechanics we mean the mechanics of a passive physical system consisting of elastic material 
components interacting with a viscous incompressible fluid; the system does not include 
details of the organ of Corti nor a model of the active mechanism of the cochlea. More 
precisely, this system is a model consisting of the bony walls of the scala tympani and the 
scala vestibuli, the cochlear partition (comprising the basilar membrane and the bony shelf), 
and the oval and the round windows, each covered with elastic membranes. The study of the 
mechanics of this system amounts to the study of the motion of the basilar membrane in 
response to excitation of the oval window. This system can be completely described by a set 
of partial differential equations defined on a domain of three-dimensional space. Such a 
description is not unique. While the viscous, incompressible fluid is canonically described by 
the Navier-Stokes equations, there is no unique and obvious choice for the description of the 
elastic components of the cochlea. 
     We use the immersed boundary method2,3 to compute the fluid-structure interactions. The 
immersed boundary method is a general method for modeling systems with elastic tissue 
immersed in a viscous, incompressible fluid. It was originally developed by Peskin and 
McQueen to study blood flow in the heart. The method has been applied to a variety of bio-
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fluid and engineering problems. Our implementation of the immersed boundary method allows 
great flexibility in the choice of the models for the elastic components. The present model1 is 
based on the immersed boundary method for elastic shells 4. In particular, the basilar membrane 
is modeled as an isotropic elastic shell by a fourth-order partial differential equationi.  
 
2. The Model 
 
While the immersed boundary method enables construction of realistic models, it often leads 
to very intensive computing requirements, which can only be satisfied by the development of 
specialized algorithms for execution on systems with multiple CPUs. This is certainly true in the 
case of the cochlea. We have used a 32 processor shared-memory HP “Superdome” computer 
to construct and execute a cochlea model code that is based on a 2563-point computational 
grid for the fluid, and several grids that model the elastic materials within the cochlea, a total of 
approximately 750,000 points. 
     The geometric model of the cochlear anatomy is based on measurements that include the 
position of the center line 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 the scala vestibuli and the scala tympani and the semi-elliptical walls sealing 
the cochlear canal and containing the oval and the round window membranes (see Fig1). 
 
 
Fig 1 A rendering of the geometric model of the cochlea with several parts of 
the outer shell removed in order to expose the cochlear partition, consisting 
of the narrow basilar membrane and the bony shelf. The round window is 
located directly below the oval window and in this picture it is partially 
obscured by the cochlear partition. 
 
     The basilar membrane is modeled as an elastic shell following the prototype tested earlier10. 
It is 3.5 cm long and we assume its width grows linearly from 0.0015 cm at the stapes to 0.0052 
cm at the helicotrema. The elastic force that the membrane exerts on the surrounding fluid is 
given by a fourth order partial differential operator acting on the vector field that measures the 
displacement of the membrane from its elastic equilibrium. We have assumed that the Lamé 
coefficients of the shell are constant, and the variable compliance of the basilar membrane is 
                                                                 
i In the future this model will be replaced by a more accurate orthotropic shell model. 
 
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due to its changing width and thickness. We have used this shell equation to estimate the 
thickness necessary to achieve the increasing compliance measured by von Bekesy. Indeed, 
with only a slight variation in the thickness the estimated compliance per unit length increases 
as xe ? , where 7.01 ??? cm. 
     The oval window and the round window membranes are also modeled as elastic shells, but 
unlike in the case of the basilar membrane, the compliance of each of these shells is constant 
throughout. The windows are geometrically identical: each is modeled by a rectangular grid, all 
of whose points outside a given circle are fixed. Hence each such grid represents a circular 
plate within a rectangular piece of bone. 
3. Numerical Experiments 
 
Our experiments typically consisted of applying a pure tone input of given frequency at the 
stapes and a study of the subsequent motion of the basilar membrane. The input at the stapes 
was simulated by specifying an external periodic force vector field on the oval window grid. A 
very small time step of 30 nano-seconds was chosen to guarantee numerical stability and good 
detail. The choice of the time step was made as a result of the convergence study of the 
system4. Our initial aim in the numerical experiments was to reproduce the qualitative 
characteristics of cochlear mechanics predicted by asymptotic analysis and previously 
reported computational models. Consequently, we have attempted to capture the steady state 
response of the basilar membrane to pure tones and in each of our experiments we have run the 
system for the duration equivalent to several input cycles. For example, for a 10 kHz input tone 
we have run the system for more than 30,000 time steps, equivalent to 0.9 milli-seconds of total 
simulated time. On 16 processors of the HP SuperDome this computation required more than 20 
hours to complete. Every 10th time step of the simulation the program generated an output file 
containing the instantaneous position of the computational grids. The total amount of storage 
required for a single experiment is measured in tens of Gigabytes of disk space. 
Correspondingly, the 2 kHz experiment required five times as much computational time and 
storage.      
4. Results 
 
Our experiments reproduced the basic characteristic features of cochlear mechanics. Initially, 
we verified that all of our experiments remained in the system's linear regime, i.e. the input force 
at the stapes was kept very small, resulting in basilar membrane displacements on the order of 
nano-meters. Indeed, increasing the force by a factor of 10 resulted in basilar membrane 
displacements almost exactly ten times bigger. Since in each experiment the system was started 
from rest, we have observed initial oscillatory transient response, which was followed by the 
smoothing of the traveling wave. 
     In each instance, in response to a pure tone input frequency, we have observed a traveling 
wave propagating from the stapes in the direction of the helicotrema. The amplitude of the 
wave gradually increases until it reaches a peak at a characteristic location along the basilar 
membrane, and this location depends on the input frequency. The speed of the wave is sharply 
reduced as it propagates further along the basilar membrane. These features can be observed 
in the following animation (Fig 2) of the results of a simulation experiment with an 8 kHz 
driving force. 
  4 
 
Fig 2 Animation of the 8 kHz pure tone experimental results 
Mm. 1.  Fig 2 (41MBytes)  [http://pcbunn.cacr.caltech.edu/cochlea/tall.mpeg]  
 
     The wave propagation and its abrupt shutdown are even more clearly illustrated by the 
motion of the points of the middle line of the basilar membrane in the animation shown in Fig 3 . 
The animation also demonstrates the transient response of the model. The calculation of the 
wave envelope begins after the transient response has subsided. 
 
 
Fig 3 Animation of the motion of the centre line of the basilar membrane, in 
the 8kHz tone experiment  
Mm.2.(7MBytes) 
[http://pcbunn.cacr.caltech.edu/cochlea/Animated_8kHz_LineX10.avi] 
  5 
     As expected, we observe that the higher the input frequency, the closer the peak of the 
wave is to the stapes: the wave envelopes of the steady state solutions for experiments with 
four different frequencies are shown in Fig 4. 
 
 
Fig 4 Steady state envelopes of the basilar membrane displacement, for 
experiments with 16 kHz, 8 kHz, 5 kHz and 4 kHz tones. 
 
     Since experiments at low frequencies require much longer compute times, our results are 
currently limited to frequencies = 2kHz. At the other extreme, model simulations of the 
response of the high frequency tuned region of the basilar membrane appear to suffer from 
insufficient resolution. By frequency-selecting the middle region of the basilar membrane we 
are able to generate the cochlear tuning curves shown in Fig 5. 
 
 
Fig 5 Tuning curves for the human cochlea, derived from our experiments 
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5. Summary and Conclusions      
 
We have presented results of several large-scale computer simulation experiments using a 
detailed three-dimensional model of cochlear macro-mechanics. Our results demonstrate the 
basic characteristic features of cochlear mechanics. Extensive testing of the model reveals 
some limitations of the method, which requires significant computing resources.  We are 
currently developing a distributed algorithm to efficiently deploy a large numb er of processors 
concurrently to tackle this simulation. We believe that the results presented here justify this 
approach, and we expect the algorithms together with a numerical method of higher order 
accuracy will remove the current limitations, and will make it possible to enrich the model with 
more detail, for instance by allowing the incorporation of a model of the organ of Corti. 
 
Acknowledgements  
 
We thank Santiago Lombeyda for creating the three-dimensional animation presented in this 
paper, and Sarah Emery Bunn for assistance in preparation of the manuscript. 
 
 
References 
 
1 E. Givelberg and J.Bunn, “A comprehensive three-dimensional model of the cochlea”, To be published in 
J.Comp.Phys.  
2 S. Peskin and D.M. McQueen, “A general method for the computer simulation of biological systems 
interacting with fluids,” in Proc. Of SEB Symposium on Biological Fluid Dynamics, Leeds, England(1994). 
3 C.S. Peskin, “The immersed boundary method,” Acta Numerica, 11, 479--517 (2002). 
4 E. Givelberg, “Modeling elastic shells immersed in fluid,” Submitted for publication 
5 C.R. Steele and L.A. Taber, “Comparison of wkb calculations and experimental results for three-
dimensional cochlear models.” J. Acoust. Soc. Am., 65, 1007--1018 (1979). 
6 B.P. de Boer, “Solving cochlear mechanics problems with higher order differential equations.” J. 
Acoust.Soc.Am., 72, 1427--1434 (1982). 
7 P.J. Kolston, “Finite element micromechanical modeling of the cochlea in three dimensions,” 
J.Acoust.Soc.Am., 99, 455--467 (1996). 
8 P.J. Kolston, “Comparing in vitro, in situ  and in vivo experimental data in a three dimensional model of 
mammalian cochlear mechanics,” Proc. Natl..Acad.Sci. USA,  96, 3676--3681 (March 1999). 
9 A.A. Parthasarati, K. Grosh, and A.L. Nuttall, “Three-dimensional numerical modeling for global cochlear 
dynamics,” J. Acoust. Soc. Am., 107, 474--485 (2000). 
10 E. Givelberg, Modeling Elastic Shells Immersed in Fluid , Ph.D. thesis, New York University (1997). 
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Computational experiments with a three-dimensional model of the Cochlea

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Computational experiments with a three-dimensional model of the Cochlea

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