W. Robert J. Funnell
Dept. BioMedical Engineering, McGill University
Type of differential equations  Example  

Lumped systems  Ordinary  RLC circuits 
Distributed systems  Partial  Electromagnetic fields 
In a ‘lumped’ model, the system characteristics are lumped into idealized discrete components with no (or negligible) spatial extent.
The only differentiation is with respect to time.
There is a welldeveloped theory for lumpedcircuit analysis, originally developed for electrical circuits.
The foundations are
The basic components are linear and timeinvariant:
There are also voltage sources, current sources and transformers.
The components can also be nonlinear and/or timevarying.
Analogies among electrical, mechanical & acoustical circuits:
Electrical  Mechanical  Acoustical  

v  voltage  f  force  p  pressure  
i  current  u  velocity  U  volume velocity  
R  resistance  R  resistance  R  resistance  
L  inductance  m  mass  M  mass  
C  capacitance  1/k  compliance (spring)  C  compliance (volume)  
v = iR  f = Ru  p = RU  
v = L di/dt  f = m du/dt  p = M dU/dt  



Transformers are required to convert between different domains in a circuit model.
Electrical  Mechanical  Acoustical 

voltage  force  pressure 
current  velocity  vol. velocity 
resistor  dashpot  mesh 
inductor  mass  tube 
capacitor  spring  volume 
Why does one circuit seem to be in parallel while the other two are in series?
In an electrical circuit, which is easier to measure: voltage or current?
In a mechanical circuit, which is easier to measure: force or velocity?
The electrical/mechanical analogy is sometimes made the other way around, by associating voltage with velocity rather than with force, and current with force rather than with velocity. The electrical/acoustical analogy may also be inverted.
There are advantages and disadvantages to both methods, and in fact the whole issue is more complicated than it first appears.
References:
In real life, circuit components are not ideal, e.g.,
The middle ear lies between the external ear canal and the cochlea.
The middle ear includes
The middle ear also contains
Block diagram of middleear model
This block diagram, and the circuit model that we shall develop from it, apply equally well to human, cat and guineapig middle ears.
How to model the air cavities?
Represent air cavities by C's.
Represent passage between cavities by R & L.
How to model ear canal?
Represent volume by capacitor.
This assumes that the input pressure and volume velocity are measured close to the eardrum.
How to model the malleus and incus?
Assume that they're fixed together.
Represent malleus/incus complex by RLC.
How to model the eardrum?
Conceptually divide it into 2 regions, à la Békésy.
Use one RLC branch for part of eardrum tightly coupled to malleus,
and a second branch for the part which shunts energy directly to
the cavities.
How to model the stapes and cochlea?
Represent stapes and cochlea each by RLC.
Ignore incudostapedial joint and many other things.
If experimental data consist of input impedance measurements, which components can be distinguished?
Combine
components which cannot be distinguished.
Variable elements.
There are now 11 model parameters.
Independently determine as many model parameters as possible.
This is a siliconerubber casting of the air cavities of a guineapig middle
ear. Measuring the
cavity volumes gives us C_{e}, C_{b1} and C_{b2}.
Estimate some model parameters by comparison with impedance
measured with eardrum removed.
Estimate remaining model parameters by comparison with impedance
of intact ear.
Lack of direct connection between parameter values and anatomical or physiological properties. For example:
R. Funnell Last modified: Sat, 2005 Feb 5 18:03:57
Equations converted from T_{E}X to HTML by T_{T}H. I have seen the equations show up properly under Windows using Mozilla 1.7, Firefox 1.0, Netscape 4 & 7, and IE 5 & 6. They do not show up properly for me under Windows using Netscape 6.2, nor under GNU/Linux using Mozilla 1.0.