# Biomedical Engineering Theory And Practice/Physiological Modeling and Simulation

The higher efficiency and lower cost of computational resources have an enormous impact on modeling and design. The easy availability of powerful computer workstations and software or programming languages (e.g., MATLAB, LabView, wxMaxima, R and so on) allow for the interactive design of high-performance, robust controllers. The risk of failure often precludes the design and evaluation of controllers in live humans, so that devices and treatments are often tested on computer and animal models first. In the past, animals have been the preferred, but this approach is continually reevaluated as animals are rarely perfect models of human diseases and conditions. Tissue cultures start to satisfy some of this need, and computer modeling and simulation may also do this need. Today, computer models is used to (1) demonstrate feasibility,(2) increase confidence in controller designs by complementing animal studies, (3) help design better animal and clinical experiments, and (4) reduce the number of required animal and human experiments. In the future, use of the computer model would be extended.

## Basic Models of Physiologic Systems-Compartment Models

This analysis involves dividing the physiological system into a number of interconnected compartments - where a compartment can be any anatomical, physiological, chemical or physical subdivision of a system. A basic assumption is that the tracer is uniformly distributed throughout a compartment. Among various compartment models, the simplest is the single compartment model.

### Single Compartment Model

Figure 4.1 shows a single compartment model that the flow of a tracer through a blood vessel follows an ideal bolus injection. The compartment in the system is closed except for the inflow and outflow of the trace, and the tracer is injected as illustrated. In this theory, the tracer will mix immediately and uniformly throughout the compartment according to its injection. And its quantity will reduce with time depending on the rate of outflow. The variables used in the system are:
If we define the fractional turnover, k, as the ratio of these two parameters, i.e.
$k={\frac {-dq/dt}{q}}$
which can be rewritten as:
${\frac {dq}{dt}}=-kq$
And the solution to this equation is:
$q=q_{0}\ exp(-kt)\,\!$
where q$_o$ is the quantity of tracer present at time, t = 0. This equation is plotted in Figure 4.2.

### Two Compartment Model-a closed system

In a closed system the tracer simply moves between the two compartments without any overall loss or gain as shown in Figure 4.3. Therefore,
${\frac {dq_{1}}{dt}}=k_{21}q_{2}-k_{12}q_{1}$ and ${\frac {dq_{2}}{dt}}=k_{12}q_{1}-k_{21}q_{2}$.
Since there is no loss of tracer from the system,
$q_{1}+q_{2}={\text{constant}}=q_{0}\,\!$
Therefore,
${\frac {dq_{1}}{dt}}=-{\frac {dq_{2}}{dt}}\,,$
As the quantity of tracer in Compartment #1 decreases, the quantity in Compartment #2 increases, and vice versa. And when the tracer is injected into Compartment #1 at time, t = 0,
| $q_{1}=q_{0}\,\!$ | and | $q_{2}=0\,,\,\!$ |
So, at the initial stage
| ${\frac {dq_{1}}{dt}}=-k_{12}q_{0}$ | and | ${\frac {dq_{2}}{dt}}=k_{12}q_{0}$ |
The solutions to this system are:
$q_{1}=q_{0}\left\lbrack 1-{\frac {k_{12}}{k_{12}+k_{21}}}\left\lbrace 1-\ {\text{exp}}\ -(k_{12}+k_{21})t\right\rbrace \right\rbrack$
and
$q_{2}=q_{0}\left\lbrack {\frac {k_{12}}{k_{12}+k_{21}}}\left\lbrace 1-\ {\text{exp}}\ -(k_{12}+k_{21})t\right\rbrace \right\rbrack$
Figure 4.4 shows one of these system if the volume of the compartment are the same.

### Two Compartment Model - Open Catenary System

Two compartments connected like chain and the last compartment have a sink as shown in Figure 4.5.
In this model,
${\frac {dq_{1}}{dt}}=-k_{12}q_{1}$ and ${\frac {dq_{2}}{dt}}=k_{12}q_{1}-k_{20}q_{2}$
The solutions to these equations are:
$q_{1}=q_{0}\ {\text{exp}}(-k_{12}t)$
and
$q_{2}=q_{0}{\frac {k_{12}}{k_{12}-k_{20}}}\lbrack {\text{exp}}(-k_{20}t)-{\text{exp}}(-k_{12}t)\rbrack \,,$
and the behaviour of q$_1$ and q$_2$ is shown in Figure 4.6.

### Two compartment Model- Open Mammillary System

The central compartment have a sink which is not related to the other compartment. Although these compartments do not necessarily have a physiological significance, common designations are:
* Comp 1 (central) - blood and well perfused organs, e.g. liver, kidney, etc.; "plasma"
* Comp 2 (peripheral) - poorly perfused tissues, e.g. muscle, lean tissue, fat; "tissue"
In this case,
${\frac {dq_{1}}{dt}}=-k_{10}q_{1}-k_{12}q_{1}+k_{21}q_{2}$ and ${\frac {dq_{2}}{dt}}=k_{12}q_{1}-k_{21}q_{2}$
At t = 0:
$q_{1}=q_{0}\,\!$ and $q_{2}=0\,\!$
So,the solutions are:
$q_{1}=q_{0}\left\lbrack {\frac {k_{21}-a_{1}}{a_{2}-a_{1}}}{\text{exp}}(-a_{1}t)+{\frac {k_{21}-a_{2}}{a_{1}-a_{2}}}{\text{exp}}(-a_{2}t)\right\rbrack$ and
$q_{2}={\frac {q_{0}k_{12}}{a_{2}-a_{1}}}\lbrack {\text{exp}}(-a_{1}t)-{\text{exp}}(-a_{2}t)\rbrack$

## Respiratory Model and Control

'See also A Mathematical Model of the Human Respiratory Control System

## Neural Networks for Physiological Control

'See also Distributed neural networks for controlling human locomotion: lessons from normal and SCI subjects.
The term neural networks usually refer to a class of computational algorithms that are loosely based on the computational structure of the nervous system.In other words,the design of a neural network includes the specification of the neuron, the architecture, and the learning algorithm.

## External Control of Movements

'See also External Control of Movements

## The Fast Eye Movement Control System

'See also Wikipedia,eye movement

## Further reading

* Bronzino, Joseph D. (April 2006). The Biomedical Engineering Handbook, Third Edition. [CRC Press]. ISBN 978-0-8493-2124-5.
* Villafane, Carlos, CBET. (June 2009). Biomed: From the Student's Perspective, First Edition. [Techniciansfriend.com]. ISBN 978-1-61539-663-4.{{cite book}}: CS1 maint: multiple names: authors list (link)
* Information related to biomedical engineering.

## References

* Compartmental Analysis in Biology and Medicine, 2nd ed., The University of Michigan Press, 1985.
* Evans, W. C., Linear Systems, Compartmental Modeling, and Estimability Issues in IAQ Studies, in Tichenor, B., Characterizing Sources of Indoor Air Pollution and Related Sink Effects, ASTM STP 1287, pp. 239–262, 1996 ISBN 0-8031-2030-3.
* Bronzino J. (ed.) The biomedical engineering handbook, 3 Vol. set. (3ed., CRC, 2006)ISBN 0849321212