MODELING OF THE CARDIOVASCULAR SYSTEM WITH

VARIABLE VENTRICLE COMPLIANCES

 

Prepared by Alp MANYAS, 2004

Instructor: Prof. Ziya IDER

 

To obtain the time courses of the Cardiovascular System variables, a simple model can be used that assumes the ventricles as variable compliances. This model can be summarized by the circuitry below;

 

 

 

 

CLICK HERE FOR THE APPLET SIMULATING THE ABOVE MODEL

 

 

The system parameters of this model are as follows (inputs);

 

TH         : Heart Beat Period (sec)

TS         : Systolic Period (sec)

 

CLV     : Compliance of the left ventricle (cc/mmHg)

CRV     : Compliance of the right ventricle (cc/mmHg)

 

RS         : Systematic resistance (mmHg.sec/cc)

RP         : Pulmonary resistance (mmHg.sec/cc)

 

CAS     : Systematic arterial compliance (cc/mmHg)

CVS     : Systematic venous compliance (cc/mmHg)

CAP     : Pulmonary arterial compliance (cc/mmHg)

CVP     : Pulmonary venous compliance (cc/mmHg)

 

RMV    : Mitral valve resistance

RAV   : Aortic valve resistance

RTV    : Triscupid valve resistance

RPV      : Pulmonary valve resistance

 

          B       : Total blood volume except the ventricles (cc)

 

The system variables of this model are as follows (outputs);

       

PAS       : System arterial pressure (mmHg)

PAP      : Pulmonary arterial pressure (mmHg)

PVP       : Pulmonary venous pressure (mmHg)

PVP       : Systematic venous pressure (mmHg)

 

QAS      : System arterial volume (cc)

QAP      : Pulmonary arterial volume (cc)

QVP      : Pulmonary venous volume (cc)

QVS      : System venous volume (cc)

 

QLV       : Left ventricle volume (cc)

PLV       : Left ventricle pressure(mmHg)

 

QRV      : Right ventricle volume (cc)

PRV       : Right ventricle pressure(mmHg)

 

FMV       : Flow through the mitral valve (cc/sec)

FAV       : Flow through the aortic valve (cc/sec)

 

FMV       : Flow through the triscupid valve (cc/sec)

FAV       : Flow through the pulmonary valve (cc/sec)

 

In the above model, there is no source of pressure (voltage) or flow (current) therefore, it may seem like there is no flow in this model at a first glance. However time dependence of the system parameters (especially the time varying compliances of the ventricles) drives the circuit and a circulation occurs.

 

 

 

 

Time dependence of the ventricles:

 

The time dependence of ventricle compliance is usually defined through its stiffness which is 1/compliance. The stiffness of a ventricle is usually given in an expression, which depends on two variables that are SLD and SLS for left the ventricle and SRD and SRS for right the ventricle. The expression is same for the both ventricles except these two parameters. The time dependence of the left ventricle compliance (CLV) and the left ventricle stiffness (SLV) is given below;

 

 

 

The Solution of the model:

 

Since the analytic solution of the above circuit is impossible due to the time dependence of the system variables, numeric computation is carried out to solve the differential equations derived from the circuit above. Computation is done by Euler integration with small time increment and in a finite time interval. The time increment (dt) and the time interval are parameters that can both be specified in the Applet by the user.

 

 

 

 

How to use the Applet:

 

In the Applet there are four plots, each having a choice above them. User can select any system variable or system parameter in these choices therefore can see the time dependence of any four outputs or input at the same time.

 

When a system parameter is selected in a choice a button labeled “change” appears below this choice. This button allows the user to specify the time dependence of the input parameter. Note that this button only appears when an input is selected. When “Change” button is pressed a dialog that allows the user to change the selected parameter is shown on the screen. To specify the time dependence of a parameter, the user doesn’t have to enter the value of that parameter at all computation points. The user can specify the time variance just by specifying enough number of points that can roughly represent the time dependence. These points are then linearly interpolated and the interpolated time variance of the parameter is shown to the user. If the user thinks that the time dependence is okay then he/she presses the “Apply” button. When the “Apply” button is pressed, the selected parameter is changed and all the variables are recomputed according to it. If the selected parameter is ventricle compliance, then a different dialog appears in which there are two parameters that the user can change. These are SLD and SLS for left the ventricle and SRD and SRS for right the ventricle. The relation between a ventricle compliance and these two parameters is stated above.