// MODEL NUMBER: 0360
// MODEL NAME: Rideout_PressureFlowNP
// SHORT DESCRIPTION: CV loop nonpulsatile model ported from Rideout
// (ACSL program PF-NP).
// Also in MATLAB.
import nsrunit; unit conversion on;
math Rideout_PressureFlowNP
{
// Choice Variable
choice Mode("Infusion",
"Left Ventricular Stiffness Change") = 1;
real TMAX = if (Mode = 1) 12 else 9;
// Independent Variable
realDomain t s; t.min = 0; t.max = TMAX; t.delta = .1;
// Parameters
real G1 = 24 ml/mmHg/s,// L. Vent. Preload Conductance
G2IC = .821 ml/mmHg/s,// L. Vent. Afterload Init. Conductance
G2NEW = .4105 ml/mmHg/s,// L. Vent. Afterload New Conductance
G3 = 40 ml/mmHg/s,// R. Vent. Preload Conductance
G4 = 3.889 ml/mmHg/s,// R. Vent. Afterload Conductance
RS = 1.0111 mmHg*s/ml, // Systemic Arterial Resistance
RP = .12222 mmHg*s/ml, // Pulmonary Arterial Resistance
CS = 2.6316 ml/mmHg, // Systemic Arterial Compliance
CR = 225 ml/mmHg, // Systemic Venous Compliance
CP = 6.9444 ml/mmHg, // Pulmonary Arterial Compliance
CL = 42.857 ml/mmHg, // Pulmonary Venous Compliance
QSIC = 1000 ml, // Systemic Arterial Initial Volume
QRIC = 5400 ml, // Systemic Venous Initial Volume
QPIC = 500 ml, // Pulmonary Arterial Initial Volume
QLIC = 1800 ml, // Pulmonary Venous Initial Volume
QSU = 750 ml, // Systemic Arterial Unstressed Volume
QRU = 4500 ml, // Systemic Venous Unstressed Volume
QPU = 375 ml, // Pulmonary Arterial Unstressed Volume
QLU = 1500 ml, // Pulmonary Venous Unstressed Volume
A = 25 ml/s, // Infusion Flow Amplitude
TSTT = 1 s, // Infusion flow at t > TSTT
WID = 2 s, // Infusion flow pulse width
TCH = 1 s; // Stiffness change at t > TCH
// Dependent Variables
real G2(t) ml/mmHg/s, // L. Vent. Afterload Conductance
FS(t) ml/s, // Systemic Arterial Flow
FR(t) ml/s, // Systemic Venous Flow
FP(t) ml/s, // Pulmonary Arterial Flow
FL(t) ml/s, // Pulmonary Venous Flow
FI(t) ml/s, // Infusion flow TSTT < t < TSTT + WID
PS(t) mmHg, // Systemic Arterial Pressure
PR(t) mmHg, // Systemic Venous Pressure
PP(t) mmHg, // Pulmonary Arterial Pressure
PL(t) mmHg, // Pulmonary Venous Pressure
QS(t) liter, // Systemic Arterial Volume
QR(t) liter, // Systemic Venous Volume
QP(t) liter, // Pulmonary Arterial Volume
QL(t) liter, // Pulmonary Venous Volume
QT(t) liter; // Total Volume
when (t = t.min) { QS = QSIC; QR = QRIC; QP = QPIC; QL = QLIC; }
// Equations
FI = if (t >= TSTT and t < TSTT + WID and Mode = 1) A else 0;
G2 = if (t >= TCH and Mode = 2) G2NEW else G2IC;
FL = G1 * PL - G2 * PS;
FR = G3 * PR - G4 * PP;
FS = (PS - PR) / RS;
FP = (PP - PL) / RP;
QS:t = FL - FS;
QR:t = FS - FR + FI;
QP:t = FR - FP;
QL:t = FP - FL;
QT = QS + QR + QP + QL;
PS = (QS - QSU) / CS;
PR = (QR - QRU) / CR;
PP = (QP - QPU) / CP;
PL = (QL - QLU) / CL;
}
/*
FIGURE:
Refer to Rideout, Figs. 4.6.1, page 118 and 4.6.2, page 120
Legend
---------------------------
F Flow [ml/s]
P Pressure [mmHg]
Q Volume [liter]
R Resistance [mmHg*s/ml]
G Conductance (G=1/R)
C Compliance [ml/mmHg]
A suffix is used to represent various segments in the CV loop. For
example, FS represents the flow (F) in the systemic arteries (S). Table
I shows suffixes used and anatomy represented. It also shows the values
of R, C and QU (unstressed volume) used in this model in the units shown
above. Conductances G1, G2 are used in place of RL; Conductances G3, G4
are used in place of RR. These conductances define the preload (prior to
contraction) and afterload (during ejection) conductance of the left and
right heart respectively. They are used to calculate the average outflow
from each ventricle as shown inthe detailed desciption. Table II gives
conductance values.
Table I:
Suffix Anatomy Represented R C QU
-----------------------------------------------------------
S Systemic Arteries 1.0111 2.6316 0.75
R Systemic Veins 225 4.5
P Pulmonary Arteries 0.12222 6.9444 0.375
L Pulmonary Veins 42.857 1.5
-----------------------------------------------------------
Total Unstressed Blood Volume: 7.125
Table II:
No Anatomy Represented G
-----------------------------------------------------------
1 Left Ventricle Preload 24
2 Left Ventricle Afterload 0.821
3 Right Ventricle Preload 40
4 Right Ventricle Afterload 3.889
-----------------------------------------------------------
DETAILED DESCRIPTION:
Nonpulsatile models are useful for pharmacokinetic studies which
typically have time constants in minutes. Ignoring nonlinearities, the
ventricular pressure-volume locus is counterclockwise between two
straight lines whose slopes are SD=1/CD and SS=1/CS. CD and CS are the
diastolic and maximum systolic compliances of the myocardium. If we
denote the end-diastolic pressure PED and end-systolic pressure PES,
then the stroke volume for the ventricle is:
QSV = PED * CD - PES * CS
Multiplying by the heart rate H, gives the average outflow:
F = (CD * H) * PED - (CS * H) * PES
Assuming a direct relationship between average atrial (Pat) pressure
and PED, and between average arterial (Part) pressure and PES, we get:
F = Gpre * Pat - Gafter * Part
Where Gpre and Gafter are the preload (prior to contraction) and
afterload (during ejection) conductances, and are inversely proportional
to the diastolic and systolic stiffness, respectively.
Denoting the left ventricle conductances G1 and G2, and the right
ventricle conductances G3 and G4, the following equations describe the
nonpulsatile ventricular flow:
FL = G1 * PL - G2 * PS
FR = G3 * PR - G4 * PP (Eqs. 1)
The pressure drops over the systemic and pulmonary peripheral
resistances are given by Ohm's law:
PS - PR = RS * FS
PP - PL = RP * FP (Eqs. 2)
Assuming fixed compliances and unstressed volume (under zero pressure)
for each of the four components:
PS = (QS - QSU) / CS
PR = (QR - QRU) / CR
PP = (QP - QPU) / CP
PL = (QL - QLU) / CL (Eqs. 3)
Finally, integrating the flow through each component give the volume.
QS:t = FL - FS
QR:t = FS - FR + FI
QP:t = FR - FP
QL:t = FP - FL (Eqs. 4)
Since this is a nonpulsatile model, the steady state is a unified
constant flow through the loop. It is interesting to model a sudden
change and the system response until a new steady state is obtained.
Two such conditions are modeled with a JSim choice variable:
(1) Infusion: Infusion flow pulse FI is added to systemic arteries
(2) Left Ventricular Stiffness Change: G2 is halved at time TCH
In infusion mode, a flow pulse FI is added to the systemic arteries
flow (second equation in Eqs. 4). The infusion pulse is rectangular,
starting at time TSTT = 1 sec and lasting WID = 2 seconds. In this
mode all conductances are fixed.
In Left Ventricular Stiffness Change mode, there is no infusion and
total blood volume is constant. Left ventricular afterload
conductance G2 is halved (left systolic stiffness is doubled) at
TCH = 1 sec.
Figure 1 shows the flow pulse FI in infusion mode. Figure 2 shows an
increase of 50 ml in total blood volume (integral of flow). Figure 3
shows the flows FR, FP, FL and FS in infusion mode. Figures 4 through 11
show the corresponding volumes and pressures.
Figure 12 shows the flows FR, FP, FL and FS in Left Ventricular
Stiffness Change mode. Figures 13 through 16 show the corresponding
pressures. The increase in systolic stiffness results in increased PS
and QS (as evident from Eqs. 3). PR and QR also increase, but somewhat
more slowly. As a result of these increases, the variables PP, QP, PL,
and QL decrease (total blood volume must remain constant). All flows
settle to the same increased value because of the stronger left heart.
KEY WORDS:
Cardiovascular, CV, Left, Right, Ventricle, Ventricular, Non Pulsatile,
Myocardial Infarction, Blood Infusion, Heart Rate, Systolic, Diastolic,
Cardiac Output, Stiffness, Muscle, Atrial, Arterial, Venous, Systemic,
Pulmonary, Unstressed, Locus, Pressure-Flow, Ohm's Law, Resistance,
Resistive, Conductance, Preload, Afterload, Compliance, Compliant,
Stroke Volume, Rideout
REFERENCES:
Rideout VC. Mathematical computer modeling of physiological systems.
Prentice Hall, Englewood Cliffs, NJ, 1991, Section 4.6, pp. 117-125
Rideout VC. Linear analysis of the cardiovascular system. Ch. 11, pp.
156-157
REVISION HISTORY:
Ported from ACSL by DH 5/1/12
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) 1999-2009 University of Washington.
Contact Information:
The National Simulation Resource,
Director J. B. Bassingthwaighte,
Department of Bioengineering,
University of Washington, Seattle, WA
98195-5061
*/