Lutchen et al. (1982) is referred to as Lutchen.
Figure 1: Flow Q1 (Case 1).
The flow in the dead space airway is plotted as a
function of time( solid black line). This duplicates
the upper panel in Lutchen, Figure 2, with the caveat
that the x-axis annotation in Lutchen is incorrect.
The period of each breath is 4 seconds. Six breaths
should end at 24 seconds instead of 28 seconds.
Figure 2: Volume V2 (Case 1).
The volume of either alveolar compartment is plotted
as a function of time. This plot corresponds to the
middle panel in Lutchen, Figure 2, with the same caveat
given in Figure 1.
Figure 3: Concentrations (Case 1).
The solid line is the concentration of nitrogen gas
at the mouth normalized to 1 (maximum nitrogen gas
concentration). During the first breath, the subject
is breathing air. On the subsequent breaths, the
patient is breathing oxygen and we can see the nitrogen
gas washout curve. At the beginning of the second
breath, the concentration in either alveolar space
(dotted line) does not change until the dead space is
cleared (approximately 0.15 seconds). Then the
concentration declines in the alveolar spaces because of
the inflow of oxygen which expands the alveolar volumes.
On exhalation, the concentration of nitrogen at the mouth
rises from zero to the alveolar concentration as the
dead space is cleared. This plot corresponds to the
bottom panel in Lutchen, Figure 2, with the same caveat
given in Figure 1.
Statistics for Case 1 corresponding to Lutchen, Table
II, can be found under the Stat plot and displaying
text. The are given here:
Case 1:
End-Expiratory Volumes JSim Lutchen
V2(E) 1.50 L 1.50 L
V3(E) 1.50 L 1.50 L
Tidal Volumes
VT 0.78 L 0.75 L
VT2 0.39 L 0.38 L
VT3 0.39 L 0.38 L
End Inspiration Volume
Ratio 1.00 1.00
Pendelluft Fraction 0.00 0.00
Figure 4: Flow Q1 (Case 6):
The flow in the dead space airway is plotted as a
function of time. This duplicates The upper panel in
Lutchen, Figure 4, with the same caveat as Figure 1.
Although the driving Pressure is the same, the flow is
different. Notice that it does not approach zero at
either the end of inspiration or expiration. Notice also
the asymmetry in the maximum and minimum flows.
Figure 5: Volumes V2 and V3 (Case 6):
The obstructed lung, alveolar compartment 2, has a greater volume,
but a smaller tidal range (dotted line) than the unobstructed tidal
volume in alveolar compartment 3 (solid line).
Figure 6: Concentrations (Case 6):
This figure is different from figure 3. Note that the concentration
at the mouth is rising during exhalation because of outflow from the
obstructed which is not synchronized with the normal lung. Note also
that the clearance from the obstructed lags the clearance from the
normal lung.
Figure 7: Flows (Case6Sine):
The inflows and outflows are not synchronous. Compartments 2 and 3 take
turns emptying and filing each other and their flows are approximately one
second out of phase with each other. The pendelluft fraction is 20% (see
plotpage Stat in text-mode).
Figure 8: Volumes V2 and V3 (Case6Sine):
V3 (blue dotted line), the volume of the obstructed lung as a function
of time increases to a higher mean value compared to Case 6. The
volume fluctuations of the unobstructed are approximately the same as
in case 6.
Figure 9: Concentrations (Case6Sine):
In this plot of concentrations of normalized nitrogen gas concentrations
in the two lung lobs and at the airway opening, the washout of nitrogen
gas is substantially slower than in Case 6 because of the increased
pendelluft fraction (increased from 4.3% to 20%. The washout is slower
because the two lungs are not synchronized for inhalation and
exhalation, and consequently, they refill each other.
Figure 10: Display as Text; Compare Lutchen Table II (Case1 & Case6)
This is not a Figure (all lines displayed in near invisible yellow).
Displayed as text, it yields values which can be compared with
Lutchen. The values are in close, but not exact agreement because
the non-exchanging airway of Lutchen (relative dispersion = 0%)
has been replaced with 100 serially connected well mixed compartments
which have a relative dispersion of 10%.
// MODEL NUMBER: 0005
/* MODEL NAME: Lutchen
SHORT DESCRIPTION:"A nonlinear model combining Pulmonary Mechanics
and Gas Concentration Dynamics." IEEE Trans.,BME-29, 1982, p. 629-641.
Lutchen, F.P. Primiano Jr., G.M. Saidel
*/
import nsrunit; unit conversion on;
math Lutchen {
realDomain t sec;t.min = 0;t.max = 24; t.delta = 0.1;
real L = 10 cm, Ndiv=101;
realDomain x cm; x.min=0; x.max=L; x.ct=Ndiv;
// PARAMETERS:
real Rtaw = 1.6 cmH2O*s/L, // A total airway resistance
Rtp = 1.9 cmH2O*s/L, // Total pulmonary resistance
Rc = 0.85*Rtaw, // Conducting airway resistance
Rp = Rtp-Rc, // Total peripheral resistance
R2 = 1.08 cmH2O*s/L, // Compartment 2 resistance
R3 = 1.08 cmH2O*s/L, // Compartment 3 resistance
R1 = 1.36 cmH2O*s/L, // Conducting airway resistance coefficient
b2 = 1.62 cmH2O*sec, // Compartment 2 specific resistance
b3 = 1.62 cmH2O*sec, // Compartment 3 specific resistance
eps2 = 0.68052 L, // Compartment 2 volume cofficient
eps3 = 0.68052 L, // Compartment 2 volume cofficient
h2 = 0.55169 cmH2O, // Compartment 2 pressure coefficient
h3 = 0.55169 cmH2O, // Compartment 2 pressure coefficient
VD = 0.15 L, // Dead space volume
V20 = 1.5 L, // Compartment 2 initial volume
V30 = 1.5 L, // Compartment 3 initial volume
ComL = 0.2 L/cmH2O, // Lung Compliance
Com2 = ComL/2, // Compartment 2 compliance
Com3 = ComL/2, // Compartment 3 compliance
cycle = 4 sec, // Breathing cycle
inspiration = cycle/3, // Length of inspiration
expiration = cycle*2/3, // Length of expiration
tauF = 0.22 sec, // Fall time of expiration
tauR = tauF/2, // Rise time of inspiration
P0 = 5 cmH2O, // Reference Pressure
A = 4 cmH2O; // Amplitude of Pressure fluctuation
// VARIABLES
real VL(t) L, // Lung volume
V2(t) L, // Compartment 2 volume
V3(t) L, // Compartment 3 volume
Q1(t) L/sec, // Compartment 1 (dead space) flow
Q2(t) L/sec, // Compartment 2 flow
Q3(t) L/sec, // Compartment 3 flow
QL(t) L/sec; // Flow
// PRESSURE FORCING FUNCTION
real Ptp(t) cmH2O, // Periodic transpulmonary pressure
tmod(t) sec; // tmod will repeat every cycle
tmod = rem(t, cycle);
choice choose("exp breath","sine breath")=1;
real chosen=choose;
real f = 1/cycle;
Ptp=if(chosen<1.5) (
if(tmod<=inspiration) // Transpulmonary pressure forcing function
P0+A*(1-exp((-tmod)/tauR))
else
P0+A*exp((inspiration-tmod)/tauF)
)
else P0 + A*(1-cos(2*PI*f*t));
// INITIAL CONDITIONS AND GOVERNING EQUATIONS
when (t=t.min) { V2 = V20; V3 = V30;}
V2:t= V2/(V2*R1+b2)*(-R1*V3:t-h2*exp(V2/eps2)+Ptp);
V3:t= V3/(V3*R1+b3)*(-R1*V2:t-h3*exp(V3/eps3)+Ptp);
VL= VD+V2+V3;
Q1=Q2+Q3;
Q1=QL;
Q2=V2/(V2*R1+b2)*(-R1*V3:t-h2*exp(V2/eps2)+Ptp);
Q3= V3/(V3*R1+b3)*(-R1*V2:t-h3*exp(V3/eps3)+Ptp);
// GAS TRANSPORT
extern real Cin(t) mmol/ml; // Input concentration
real copyCin(t) = Cin(t);
real C0 = 1 mmol/ml, // Initial Lung concentration
C2(t) mmol/ml, // Compartment 2 concentration
C3(t) mmol/ml; // Compartment 3 concentration
realState Cm(t) mmol/ml; // Mixing node concentration
when(t=t.min) Cm=C0;
real Cao(t) mmol/ml, // Airway opening concentration
uQ1(t) dimensionless, // Compartment 1 flow switch
umQ1(t) dimensionless, // Compartment 1 flow switch
uQ2(t) dimensionless, // Compartment 2 flow switch
umQ2(t) dimensionless, // Compartment 2 flow switch
uQ3(t) dimensionless, // Compartment 3 flow switch
umQ3(t) dimensionless; // Compartment 3 flow switch
// SWITCHES FOR FLOW IN COMPARTMENTS
uQ1 = if ( Q1>0) 1 else 0;
umQ1 = if (-Q1>0) 1 else 0;
uQ2 = if ( Q2>0) 1 else 0;
umQ2 = if (-Q2>0) 1 else 0;
uQ3 = if ( Q3>0) 1 else 0;
umQ3 = if (-Q3>0) 1 else 0;
when (t=t.min) { C2=C0; C3=C0;}
// VARIOUS STATISTICS
realState V2max(t) L, // Compartment 2 maximum volume
V2min(t) L, // Compartment 2 minimum volume
V3max(t) L, // Compartment 3 maximum volume
V3min(t) L; // Compartment 3 minimum volume
real V2T L, // Compartment 2 tidal volume
V3T L, // Compartment 3 tidal volume
VT(t) L; // Pulmonary Tidal volume
when(t=t.min) {V2max=-100;
V2min=100;
V3max=-100;
V3min=100;
VT=0;}
event (t>=24 and V2max<V2) V2max=V2; // Calculated over last breath
event (t>=24 and V2min>V2) V2min=V2; // Calculated over last breath
event (t>=24 and V3max<V3) V3max=V3; // Calculated over last breath
event (t>=24 and V3min>V3) V3min=V3; // Calculated over last breath
VT:t=if(t<24 or Q1<0) 0 else Q1; // Integrated over last inspiration
V2T=V2max(t.max)-V2min(t.max);
V3T=V3max(t.max)-V3min(t.max);
real EndInspireVolRatio, // End-inspiratory volume ratio
PendelluftFraction ; // Pendelluft Fraction
EndInspireVolRatio=V2max(t.max)/V3max(t.max);
PendelluftFraction=(V2T+V3T)/VT(t.max)-1;
// MIXING CHAMBER
real Cmouth(t) mmol/ml, Cend(t) mmol/ml;
event(t>t.min)
Cm = (uQ1*Q1*Cend+umQ2*(-Q2)*C2+umQ3*(-Q3)*C3)/(uQ1*Q1+umQ2*(-Q2)+umQ3*(-Q3));
// BTEX10 PDE FOR NON-DISPERSIVE
// BI-DIRECTIONAL DEAD SPACE
real Dds = 0 cm^2/sec; // zero diffusion for no mixing in dead space
real Cnn(t,x) mmol/ml;
when(t=t.min) Cnn=C0;
real FlowFromMouth(t);
real FlowFromLungs(t);
FlowFromMouth=if(Q1>0) Q1 else 0;
FlowFromLungs=if(Q1<0) Q1 else 0;
when(x=x.min) {-FlowFromMouth*L/VD*(Cnn-Cin)+Dds*Cnn:x=0; }
when(x=x.max) {-FlowFromLungs*L/VD*(Cnn-Cm) +Dds*Cnn:x=0; }
Cnn:t=-Q1*L/VD*Cnn:x + Dds*Cnn:x:x;
Cmouth=if(t=t.min) C0 else Cnn(t,0);
Cend =if(t=t.min) C0 else Cnn(t,L);
// Equations
Cao = Cmouth;
// THE LUNGS
V2*(C2:t)=Q2*(Cm-C2)*uQ2;
V3*(C3:t)=Q3*(Cm-C3)*uQ3;
} // END OF MML CODE
/* Figure
| |
| Qao |
| | ao |
| v | Non-dispersive
| 1 | Airway
| | Q1 |
.....| v |.....
..... | m | .....
.... ......| / \ |...... ....
... ..... .| / \ |. ..... ...
... ... v v ... ...
... .. Q2 | Q3 .. ...
.. .. | .. ..
.. .. | .. ..
.. . | . ..
. .. Alveolar | Alveolar .. .
.. . Compartment 2 . Compartment 3 . ..
. . . . .
.. . . . ..
. .. . .. .
.. . . . . ..
.. .. .. .. .. ..
.. .. .. .. .. ..
... .. .. .. .. ...
... ... ... ... ... ...
... ..... .... Ppl .... ..... ...
.... ...... ...... ....
...... ......
.......... ..........
.........
DETAILED DESCRIPTION:
Model Structure with conducting airway {1} and alveolar compartments
{2} and {3}. Qao, Q1, Q2, and Q3 are the flows. {m} is the mixing node.
The conducting airway in Lutchen is non-dispersive. Here it is modeled
as BTEX10 with no axial diffusion.
DESCRIPTION:
This model simulates "A Nonlinear model combining pulmonary
mechanics and gas concentration dynamics" (Lutchen, et al., 1982,
hereafter referred to as Lutchen).
A non-dispersive airway is connected to a bifurcating alveolar
compartment. The naming of variables and parameters follows the
nomenclature of the paper. The primary difference between the
model in Lutchen and the model presented here is that the non-
dispersive airway has been simulated by a non-dispersive BTEX10
model (axial diffusion set to zero).
The experiment being simulated is as follows: the human subject
is breathing air. The first breath (inspiration and expiration)
is also air. At the beginning of the second breath, the air is
replaced by pure oxygen for breaths two through seven. It is noted
that Figures 2 and 4 in Lutchen have an incorrect abcissa axis
annotation. The axis is for 7 breaths (7 breaths*4 seconds/breath
equals 28 seconds. Six breaths (shown) would end at 24 seconds.
(There is also a minor error in Equation 19.) The normalized
nitrogen concentration is tracked in both alveolar compartments
and the dead space as washout curves.
The model is run for Lutchen's cases 1 and 6 (Parameter sets Case1
and Case6). Figures 1, 2, and 3 correspond to Lutchen's Figure 2.
Figures 4, 5, and 6 correspond to Lutchen's Figure 4. Text Table
10 (Stat_ plot displayed as text) gives the results of
numerical calculations which may be compared with Lutchen's TABLE
II.
Case 1 is for normal alveolar compartments which have synchronized
flows, concentration changes, and volume changes. The exponential
pressure forcing ranges from 5 mmHg to 9 mmHg.
Case 6 is for obstructive lung disease affecting just the alveolar
compartment labled 3. Using the periodic exponential pressuring forcing,
The obstructed lung has a larger volume with reduced tidal range.
Clearance in the obstructed lung is slower than in the unobstructed
lung.
Case 6 Sine for sinusoidal pressure forcing. The pressure fluctuation
at the mouth has a range of 5 mmHg to 9 mmHg.
The pendelluft fraction is the sum of the two alveolar tidal volumes
minus the tidal volume of the pulmonary system (integrated flow
at the airway opening over an inspiration), all divided by the
tidal volume of the pulmonary system indicates the percentage of
inspired air that is being recirculated in the lungs. In Case 6 Sine,
it accounts for 46% of the ventilation of the obstructed lung.
Pulmonary Tidal Volume * Pendelluft Fraction .434 * 0.200
-------------------------------------------- = -------------- =~0.46.
V3T (Tidal Volume) .188
KEY WORDS: lung, compliance, resistance, RC, circuit, mechanics, airflow, trachea,
tidal volume, positive pressure, ventilation, pendelluft, publication
REFERENCES:
K.R. Lutchen, F.P. Primiano Jr., G.M. Saidel, "A nonlinear model
combining Pulmonary Mechanics and Gas Concentration Dynamics."
IEEE Trans.,BME-29, 1982, p. 629-641.
REVISION HISTORY:
Original Author: GR Date: 11/06/08
Revised: GR Date: 01/27/11
The bi-directional btex10 modeling the trachea replaced
the large number of compartmental models originally used in
this program.
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 to 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) 1999-2008 University of Washington.
Contact Information:
The National Simulation Resource,
Director J. B. Bassingthwaighte,
Department of Bioengineering,
University of Washington, Seattle WA
98195-5061.
Academic use is unrestricted. Software may be copied so long as this copyright
notice is included.
*/