// MODEL NUMBER: 0318
/* MODEL NAME: MultiCircSystem
SHORT DESCRIPTION: The MultiCompartment Circulatory System is based on
pages 450-452 from J. Keener and J. Sneyd, Mathematical Physiology, Vol. 8
in series Interdisciplinary Applied Mathematics, Springer-Verlag New York Inc.,
1998, p. 766.
*/
import nsrunit; unit conversion on;
math MultiCircSystem {
// PARAMETERS AND VARIABLES
real Q L/min; // Flow rate
real F min^(-1); // heart beats per minute
real Psa mmHg; // Pressure, systemic arteries
real Ps1 mmHg; // Pressure between systemic arteries
// and systemic capillaries
real Ps2 mmHg; // Pressure between systemic capillaries
// and systemic veins
real Psv mmHg; // Pressure, systemic veins
real Ppa mmHg; // Pressure, pulmonary arteries
real Pp1 mmHg; // Pressure between pulmonary arteries
// and pulmonary capillaries
real Pp2 mmHg; // Pressure between pulmonary capillaries
// and pulmonary veins
real Ppv mmHg; // Pressure, pulmonary veins
real Rsa mmHg*L^(-1)*min; // Resistance, systemic arteries
real Rsv mmHg*L^(-1)*min; // Resistance, systemic veins
real Rpa mmHg*L^(-1)*min; // Resistance, pulmonary arteries
real Rpv mmHg*L^(-1)*min; // Resistance, pulmonary veins
real Rs mmHg*L^(-1)*min; // Resistance, systemic capillaries
real Rp mmHg*L^(-1)*min; // Resistance, pulmonary capillaries
real Csa L/mmHg; // Compliance, systemic arteries
real Csv L/mmHg; // Compliance, systemic veins
real Cpa L/mmHg; // Compliance, pulmonary arteries
real Cpv L/mmHg; // Compliance, pulmonary veins
real Cld L/mmHg; // Compliance, left heart, diastolic
real Cls L/mmHg; // Compliance, left heart, systolic
real Crd L/mmHg; // Compliance, right heart, diastolic
real Crs L/mmHg; // Compliance, right heart, systolic
real Vsa L; // Volume, systemic arteries
real Vsv L; // Volume, systemic veins
real Vpa L; // Volume, pulmonary arteries
real Vpv L; // Volume, pulmonary veins
real V0s L; // Initial volume of systemic arteries
real V0p L; // Initial volume of pulmonary arteries
real V0 L; // Total Volume
// VALUES
Q = 6.0;
F = 80;
Psa = 100;
Ps1 = 20;
Ps2 = 10;
Psv = 5;
Ppa = 30;
Pp1 =20;
Pp2 = 10;
Ppv = 2;
// ASSUMED THAT SYSTEMIC VOLUME = 11*PULMONARY VOLUME
// ASSUMED VENOUS VOLUMES = 60% TOTAL VOLUME
Vsa = 2.2;
Vsv = 3.3;
Vpa = 0.2;
Vpv = 0.3;
/* Compliances must be positive. We used a,b,c, and d and adjusted
them so that the compliances and initial volumes are all positive.
*/
real a = 60.0 dimensionless;
real b = 10.0 dimensionless;
real c = 0.08 dimensionless;
real d = 0.10 dimensionless;
Cld=a*Cls;
Crd=b*Crs;
Csa=c*Csv;
Cpa=d*Cpv;
/*
Because Q must be positive, the following relations must hold:
(Cld/Cls)Ppv>Psa>Ps1>Ps2>Psv
60*2>100>20>10>5
(Crd/Crs)*Psv>Ppa>Pp1>Pp2>Ppv
10*5>30>20>2
*/
// Governing equations
// Systemic arteries
Q = (Psa-Ps1)/Rsa;
Q = F*(Cld*Ppv-Cls*Psa);
Vsa = V0s+Csa/2*(Psa+Ps1);
// Systemic veins
Q = (Ps2-Psv)/Rsv;
Q = (Ps1-Ps2)/Rs;
Vsv = Csv/2*(Psv+Ps2);
// Pulmonary arteries
Q = (Ppa-Pp1)/Rpa;
Q = F*(Crd*Psv-Crs*Ppa);
Vpa = V0p+Cpa/2*(Ppa+Pp1);
// Pulmonary veins
Q = (Pp2-Ppv)/Rpv;
Q = (Pp1-Pp2)/Rp;
Vpv = Cpv/2*(Ppv+Pp2);
// Conservation of blood volume
V0=Vsa+Vsv+Vpa+Vpv;
}
/*
FIGURE:
PULMONARY CIRCULATION
Rp
Pp1--------------------/\/\/\/\--------------------Pp2
| |
... ...
. . . .
. Rpa . ^ | . Rpv .
. Cpa . | Q Q | . Cpv .
. Vpa . | v . Vpv .
. . . .
... ...
| |
Ppa------................. .................------Ppv
. . . .
. Right . Left .
. Heart . Heart .
. Crd,Crs . Cld,Cls .
. . .
Psv------................. .................-----Psa
| |
| |
... ...
. . . .
. Rsv . ^ | . Rsa .
. Csv . | Q Q | . Csa .
. Vsv . | v . Vsa .
. . . .
... ...
| |
Ps2--------------------/\/\/\/\--------------------Ps1
Rs
SYSTEMIC CIRCULATION
Diagram based on Figure 15.8 of reference.
DETAILED DESCRIPTION:
This is an algebraic model for a Multicompartment Circulatory System based
on the reference, pages 450-452. The values for the various parameters and
the ratios of the compliances were chosen so that the compliances and initial
volumes would all be positive. The volumes were distributed so that the systemic
volume was eleven times greater than the pulmonary volume, and the venous volumes
were 60% of the total volume. The flow, pressures, beats per minute, volumes and ratio
of compliances were set, and the resistances, compliances, and initial volumes
were solved for.
The second model in this project, MultiCircSystem2, sets the compliances
and resistances, and solves for the pressures, volumes, and ratios of
the compliances.
The subscript conventions are given as
a arterial
v venous
s systemic
p pulmonary
r right
l left
d diastolic
s systolic
e.g.
Csa Compliance of the systemic arteries.
Cld Diastolic compliance of left heart.
Ps1 Pressure at the ill-defined border
between systemic arteries and systemic
capillaries.
KEY WORDS:
Multicompartment, circulatory, linear, systemic, pulmonary,
resistance, compliance, circulation, heart, two chambers
REFERENCES:
J. Keener and J. Sneyd, Mathematical Physiology, Vol. 8 in series
Interdisciplinary Applied Mathematics, Springer-Verlag New York Inc., 1998.
REVISION HISTORY:
06/01/2011 Model written from equations in reference by GMR.
02/14/2012 Added rtml interface and display text plot pages.
JSim SOFTWARE COPYRIGHT AND REQUEST FOR ACKNOWLEDGMENT OF USE:
JSim software was developed with support from NIH grants HL088516,
and HL073598. Please cite these grants in any publication for which
this software is used and send one reprint of published abstracts or
articles to the address given below. Academic use is unrestricted.
Software may be copied so long as this copyright notice is included.
Copyright (C) 1999-2011 University of Washington.
Contact Information:
The National Simulation Resource,
Director J. B. Bassingthwaighte,
Department of Bioengineering,
University of Washington, Seattle, WA
98195-5061
*/