# Biomedical Engineering Theory And Practice/Physiological Modeling and Simulation

 Biomedical Engineering Theory And Practice Physiological System Physiological Modeling and Simulation Bioelectric phenomena

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 softwares 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.The single compartment model
Figure 4.2.Graphical illustration of the quantity of tracer, q versus time for relatively high and low values of the fractional turnover, k.

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:

q: the quantity of tracer in the compartment at time, t, and
F: the outflow.

If we define the fractional turnover, k, as the ratio of these two parameters, i.e.

${\displaystyle k={\frac {-dq/dt}{q}}}$

which can be rewritten as:

${\displaystyle {\frac {dq}{dt}}=-kq}$

And the solution to this equation is:

${\displaystyle q=q_{0}\ exp(-kt)\,\!}$

where qo is the quantity of tracer present at time, t = 0. This equation is plotted in Figure 4.2.

### Two Compartment Model-a closed system

Figure 4.3 Closed two compartment model
Figure 4.4 The change in the quantity of tracer in Compartments #1 and #2 versus time in a closed compartment model.

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,

${\displaystyle {\frac {dq_{1}}{dt}}=k_{21}q_{2}-k_{12}q_{1}}$ and ${\displaystyle {\frac {dq_{2}}{dt}}=k_{12}q_{1}-k_{21}q_{2}}$.

Since there is no loss of tracer from the system,

${\displaystyle q_{1}+q_{2}={\text{constant}}=q_{0}\,\!}$

Therefore,

${\displaystyle {\frac {dq_{1}}{dt}}=-{\frac {dq_{2}}{dt}}\,,}$

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,

 ${\displaystyle q_{1}=q_{0}\,\!}$ and ${\displaystyle q_{2}=0\,,\,\!}$

So, at the initial stage

 ${\displaystyle {\frac {dq_{1}}{dt}}=-k_{12}q_{0}}$ and ${\displaystyle {\frac {dq_{2}}{dt}}=k_{12}q_{0}}$

The solutions to this system are:

${\displaystyle q_{1}=q_{0}\left\lbrack 1-{\frac {k_{12}}{k_{12}+k_{21}}}\left\lbrace 1-\ {\text{exp}}\ -(k_{12}+k_{21})t\right\rbrace \right\rbrack }$

and

${\displaystyle q_{2}=q_{0}\left\lbrack {\frac {k_{12}}{k_{12}+k_{21}}}\left\lbrace 1-\ {\text{exp}}\ -(k_{12}+k_{21})t\right\rbrace \right\rbrack }$

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.

Figure 4.5 Open catenary two compartment model.
Figure 4.6 The quantity of tracer versus time in the open catenary two compartment model.

In this model,

${\displaystyle {\frac {dq_{1}}{dt}}=-k_{12}q_{1}}$ and ${\displaystyle {\frac {dq_{2}}{dt}}=k_{12}q_{1}-k_{20}q_{2}}$

The solutions to these equations are:

${\displaystyle q_{1}=q_{0}\ {\text{exp}}(-k_{12}t)}$

and

${\displaystyle q_{2}=q_{0}{\frac {k_{12}}{k_{12}-k_{20}}}\lbrack {\text{exp}}(-k_{20}t)-{\text{exp}}(-k_{12}t)\rbrack \,,}$

and the behaviour of q1 and q2 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"
Figure 4.7 Open mamillary two compartment model.
Figure 4.8 The quantity of tracer versus time in the open mamillary two compartment model.

In this case,

${\displaystyle {\frac {dq_{1}}{dt}}=-k_{10}q_{1}-k_{12}q_{1}+k_{21}q_{2}}$ and ${\displaystyle {\frac {dq_{2}}{dt}}=k_{12}q_{1}-k_{21}q_{2}}$

At t = 0:

${\displaystyle q_{1}=q_{0}\,\!}$ and ${\displaystyle q_{2}=0\,\!}$

So,the solutions are:

${\displaystyle q_{1}=q_{0}\left\lbrack {\frac {k_{21}-a_{1}}{a_{2}-a_{1}}}{\text{exp}}(-a_{1}t)+{\frac {k_{21}-a_{2}}{a_{1}-a_{2}}}{\text{exp}}(-a_{2}t)\right\rbrack }$ and

${\displaystyle q_{2}={\frac {q_{0}k_{12}}{a_{2}-a_{1}}}\lbrack {\text{exp}}(-a_{1}t)-{\text{exp}}(-a_{2}t)\rbrack }$

## Cardiovascular Model and Control

Modeling the Cardiovascular System

## Respiratory Model and Control

Modeling Respiratory system

## Neural Networks for Physiological Control

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.