79'
TRACER STUDIES IN THE MATHEMATICAL MODELLING OF THE PULMONARY CIRCULATION* ANDREW J. LOVINGER Department of Chemical Engineering and Applied Chemistry Columbia University New York. N.Y.
T HERE are two types of ideal continuous chemical reactors. The first is the Continuous Flow Stirred Tank Reactor (CFSTR), a vessel characterized by perfect mixing so that composition and temperature are uniform in every part of the reactor. The entering material is mixed instantaneously with the fluid in the reactor, and the outflow stream is identical in both concentration and temperature to the contents of the reactor. The second type is called Plug Flow Tubular Reactor (PFTR), and is usually a tube into which material enters and leaves -at a steady state, flowing with a flat velocity profile. There is no mixing in the direction of flow (axial mixing), and all material at any cross section of the reactor is uniform (perfect radial mixing). The necessary and sufficient condition for ideal plug flow is that all flow elements have the same residence time in the reactor. While neither perfect mixing nor true plug flow can exist in actual practice, ideal reactors constitute a valid description of the processes in most real reactors.? A number of investigators~8 have described the behavior of circulatory segments in terms of chemical reactors by using the compartmental approach, i.e., representing the circulation as a series of chambers which are interconnected in various ways. There has been general agreement that the two heart chambers can be represented by stirred tank reactors, mixing being accomplished by the systolic and diastolic movements of the heart and by the eddies caused by the inflowing bloodstream. The degree to which such mixing approaches the ideal has been questioned by a number of investigators,7-9 but ideal CFSTRs have been adopted as a description of the right and left heart in the formulation of indicator-dilution theory, used in the determination of cardiac output and cardiopulmonary volumes as well as in the detection of intracardiac shunts and other circulatory anomalies. Vol. 52, No. 7, September 1976
A. J. LOVINGER
792
LOVINGER
792
800
RH
C/)
600
1-400
~~~~~~~LH
-
200
0 0
5
10
T I ME
15
20
(S E C)
Fig. 1. Typical precordial dilution curve. RH = right heart peak; LH = left heart peak.
There has not, however, been any investigation into the behavior of pulmonary circulation as a chemical reactor, although a number of authors have used lung models in their studies. Newman et al.4 used a single CFSTR, 5oo cc. in volume, as their model, albeit questioning this representation. Love and co-workers10 modelled pulmonary circulation by use of four stirred tanks connected in parallel, mixing being improved by the addition of glass beads. This paper presents a chemical reactor model of pulmonary circulation after examining a number of possible lung models consisting of combinations of ideal chemical reactors. METHOD
OF
INVESTIGATION
The indicator dilution technique, first proposed by Stewart,"1 is Bull. N. Y. Acad. Med.
MODELLING OF PULMONARY CIRCULATION
793
employed extensively in radiocardiographic determinations of cardiac output, ventricular volumes, and circulatory shunts. This technique involves injection of a dye or, most commonly, a radioactive tracer (99Tc, 1311) into the right ventricle, followed by subsequent monitoring of gamma radiation activity over the precordium. A precordial dilution curve such as the one shown in Figure I is obtained. The first peak represents the practically instantaneous injection of indicator into the right heart. As blood flows into the pulmonary circulation, activity decreases. However, a second peak is observed when the radioactive particles arrive at the left heart, followed by a further decrease in activity as the tracer enters systemic circulation. A final, much lower peak may be observed after return of the greatly diluted radioactive particles into the right heart. Determination of cardiopulmonary parameters then can be made by use of the precordial dilution data in the Fick equation and its corollaries.'2 Before this technique was used to simulate cardiopulmonary configuration, a study into its accuracy, applicability, and limitations was undertaken. An actual mechanical model of the circulation was constructed and used to obtain indicator dilution data under controlled conditions. A special computer program was written14 that automatically performs a complete radiocardiographic analysis of radioistope-dilution data. Use of the above apparatus and computer program indicated that the precordial dilution technique produces cardiac output results that are accurate to within 3 % of the actual value and heart volumes to within 3%0 and io%, depending upon the method used.l3, '5 These results also define the limits of confidence in the use of the precordial dilution technique for the simulation of the cardiopulmonary circulation. The effect of pulmonary configuration on the shape of the radioisotope dilution curves is very pronounced and characteristic. The greater the extent of mixing in the lungs, the more uniform will be the distribution of tracer in the bloodstream, and, thus, the peak in the left heart curve will tend to be less pronounced. Consequently, the shape of the curve of the left heart can be related directly to pulmonary configuration since, as was noted, there is general agreement that the other two circulatory segments through which the tracer has flowed (i.e., the heart chambers) can be regarded as valid approximations of ideal CFSTRs. Figure 2 shows the variation of pulmonary transit time (defined Vol. 52, No. 7, September 1976
794
A. J. LOVINGER
13 12
11
V10\ 9
-_
7
_0X
I-
c_
0
5
0
4_ Cl-
3 2 1
.'
0
0
1
.I I. I1.1 2
3
4
5
6
1
1
7
8
9
10
I
I
11
12
CARDIAC OUTPUT (LIT/MIN) Fig. 2. VariAbion of pulmoiary transit time (time-distance between right and left heart peaks) with flow rate.15
as the time distance between the right and left heart peaks) with cardiac output, obtained from the mechanical model mentioned previously.15 Peak separation is indicative of the delay in flow from the right to the left heart. Since, in the absence of chemical reactions, an ideal PFTR constitutes a pure delay, peak separation can be used to determine the relative size of a plug-flow reactor required to account for the time delay during passage of the indicator through the pulmonary
circulation. On the basis of the above, a mathematical model of the circulatory Bull. N. Y. Acad. Med.
MODELLING OF PULMONARY CIRCULATION
- - - -- PULMONARY CIRCULATION
r
lC1 | 7
- -
RIGHT HEART Vi = 130 cc
nLEFT HEART
7
IC4
V15
7 9 5
IV4 = L-
130 cc
SYSTEMIC CIRCULATION -
l
--7
I1000I
IccI
IJ C5
I |
1 ~~~~~~500cc
3000 cc
|
Fig. 3. Flow diagram of the mathematical model of the circulatory system. (See text for further description.)
system was constructed by use of tubular and stirred tank reactors. The system of simultaneous differential equations describing the flow of the tracer after a pulse injection into the right heart was solved by Vol. 52, No. 7, September 1976
7
796
A. J. LOVINGER LOVINGER A.
J.
use of a digital computer program, utilizing the Continuous System Modelling Program (CSMP) of International Business Machines. Precordial dilution curves were then plotted by computer for different pulmonary configurations and compared to actual curves obtained under corresponding conditions of cardiac output.
THE MATHEMATICAL MODEL A flow diagram of the mathematical model of the circulatory system is shown in Figure 3. A closed system containing four major segments was used to describe the circulation. These segments correspond to the two heart chambers and the pulmonary and systemic circulation. CFSTRs with volumes equal to 130 cc. were used to model the right and left heart chambers. Systemic circulation is modelled by a combination of two CFSTRs and one PFTR in series. The first stirred tank has a volume V6 = 5oo cc. and simulates the vigorous mixing that occurs in the aorta and the other large channels. The second CFSTR of volume V7 3,000 cc. models the dilution of tracer through the systemic blood pool. The plug-flow reactor of volume V5 = 1,ooo cc. is added to account for the delay of blood flow in the body before return to the right heart. Pulmonary circulation is modelled in general by a stirred tank in series with a plug-flow reactor; in all cases their combined volume (V2 + V3) is equal to 400 cc. The individual reactor capacities can vary anywhere from o cc. to 400 cc., so that in the extreme cases pulmonary circulation will be represented by either a single ideal CFSTR or a pure delay, with combinations of these constituting the intermediate cases. The selection of a series combination of a CFSTR and a PFTR for the pulmonary model is substantiated by the nature of blood flow through the circulation. Mixing effects. occur in the pulmonary artery and veins, as well as in the other major channels, and at all junctions. In a model of pulmonary circulation by chemical reactors these effects can be incorporated into a single CFSTR. The need for a tubular reactor arises from the fact that blood flow between the right and left heart is not instantaneous; rather, a time interval must elapse, representing the various delays encountered during flow through the lungs. The tubular reactor, therefore, is added to account for these delays. Finally, the series connection of a PFTR and a CFSTR instead of Bull. N. Y. Acad. Med.
MODELLING OF PULMONARY
CIRCULATION
79 7
a parallel one is based on the requirement that all blood particles must encounter the same conditions while passing through the circulation. Otherwise, it would be possible for some of them to move instantaneously from one point to another without having to pass through a PFTR, whereas others might travel through a pure delay without being subjected to any dilution. Thus, the proposed model of pulmonary circulation is in principle a valid representation of this circulatory segment, and exhibits a strong resemblance to the actual physiological system. The precise composition of the model, i.e., the relative volumes of the stirred tank and tubular reactors, now can be determined by solving the mathematical model equations. These equations, whose derivation is found in the literatures1' 7 are the typical mass balances for a CFSTR and a PFTR in the absence of chemical reaction. If Vi is the volume of the jth reactor and CQ the concentration of tracer in the exit stream from that reactor; if F represents cardiac output, t the time, and Tj the time constant for the jth reactor (defined by equation 4); and if, finally, the time-dependent flow rate and concentration of the injected tracer are designated by F. and C., respectively, then the mass balances for the circulatory model of Figure 3 become:
dC, V1-d= F.(t) * C. + F * (C7-C1) dt TJ
(I)
dCQ d j = Cj-1- Ci
j=,2416,7.
(2)
C>(t)~ CJ-1 (t-ril)
j 3, 5. jI, . . . ,7.
(3) (4) (5)
where Tr = Vj / F Initial conditions: C1(o-)
o
j=I, . . . ,7.
THE COMPUTER PROGRAM
The above set of simultaneous differential equations was solved by use of the CSMP. This programming language, a detailed description of which is given elsewhere,16' 18 combines the best characteristics of analog computation in engineering simulation with the extensive mathematical, logical, and control capabilities of FORTRAN IV. CSMP was selected for this study because of its provision of advanced mathVol. 52, No. 7, September 1976
798 798
A. A. J. J. LOVINGER
ematical functions (integrators, differentiators, delays), signal sources (step and impulse functions), and extraordinary output facilities. Numerical integrations were performed by use of the Runge Kutta fourth order routine with a variable step size. The injection of radioactive tracer was described by use of a FORTRAN assignment statement, e.g., a unit-pulse injection of 0.2 sec. duration, occurring at t = I.o sec. and representing F. (t) in equation i, is coded as the difference between two step functions: FX = STEP (i.o) - STEP (I.2) The exceptional output facilities provided by CSMP include plots of variables with their limits automatically adjusted, printing of output in either tabular or equation format, and use of titles and labels without necessary resort to FORTRAN format statements. The investigation of different lung models was made by use of successive runs of the program with different values for the volumes of the CFSTR and PFTR, ranging from o cc. to 400 cc.
RE~sULTS
AND
DISCUSSION
Of the curves obtained from the execution of the above program, four are shown in Figures 4 to 7. The first column in each figure contains the independent variable (time, in seconds), while corresponding values of tracer concentration (in mc./liter) are printed in the second column and plotted alongside. In the case of Figure 4, the entire pulmonary circulation was represented by an ideal stirred tank reactor. No left heart peak is visible; it has become part of the initial downslope. This occurs because the vigorous mixing in the lungs drastically reduces the pulmonary mean transit time, thus transposing the left heart curve toward the origin. In addition, the perfect mixing that occurs in the pulmonary segment further dilutes the tracer, so that the contribution of the left heart curve to the total dilution curve is substantially decreased. In this case, the entire cardiopulmonary circulation effectively becomes a series of three ideal stirred tanks, with no specific delay function added. The precordial dilution curve of Figure 5 is for an ideal CFSTR, 200 CC. in volume, attached in series to an ideal PFTR of equal capacity. The typical shape of precordial dilution is evident. However, this configuration still causes too much mixing, as is seen by the shape of Bull. N. Y. Acad. Med.
MODELLING OF PULMONARY CIRCULATION
CFSTtno400.C
PFTR O.0 CC
CC,
MINIM'1UM coo
TIME 0.0
5.0000E-01 1.0000E 00
1.5000E 00 2.00OO1 00 2 .SO00 3 0OOOO
00
co CO 3.5000E 00
4.00O0F 00
4.5000E 5.00000 5.SOOOE 6.0000E 6.5COOE 7.0000E 7.5000E 8.0O00E 8.50001 9.0COOE 9.5000E
00 00 00 GO CO 00 00 00 00 Co 00
I.OCOOE 01 1.0500E 01 1.COOZE 01 1.1500E 01 1.20COE 01 1.2 5001E 01 1.3000E 01 1.3500E 01 1.40O0E 01 1.4500E Cl 1.5OOO 01 1.5500E 01 1.6000F 01 1.6500E 01 1.7OOOF 01 1.7500E 01 1.8000E 01 1 .8500E 01 1.9000E 01 1.500E 01 2.OOOOE Cl 2.0500F 01 2.1000E Cl 2.1500E 01 2.200OE 0t 2.2500F Cl 2.30001 Cl 2.3'o3(E Cl 2?.0Oof 01 2.45001 Ct 2 .5000 Cl
7 99
PRECOR 0.0 0.0
0.0 1. 3411tE-0-3 O.4 E-0 7.5051E-04 *1045E-O4 S.14051E-0 4.4935E-04 4.0117E-04 3.65C2E-0 3.3542E-10 3e.0976E--0 2.8656E-04 2.6S06E-04
2.4671E-04
2.2585E-0 2.0795E-04 1.91 18-04
PRECOR VERSUS TIME
I +
+
- --------- ----4 --------------------------------------
-
-+
------------------------+ -------+
------------
1.7555E-0'4 1.6109E-0'4
1.4778F-04' 1.3563F-.-1.24591-04 1.1463'-04 1.056@E-04 ---4 9.7727E-05 --4 9.0667E-O5 --0 8.4451E-OS 7.9014E-05 --. 7.4293E-05 7.02241E-05 ---4 6.67 51-5 6*.3816E-05 -6*.1366E-5 -+ -* 5.93521E-0 '5.728E-05 -* 5. 6451 E-0-O 5.5482E-05 -, -+ 5.4764F-05 -, 5.4325E-05 S. 4071 E-055 5.'006E-35 5.4092S-05-5 5.43131-05 -, S.'.6 .ir-05-9 -, 5.507S5-05 5.5562F-053 -
S.61 51E-05-5 S. 6 ?t E-05 -+ -, 5.74343-05
Fig. 4. Computer output from the simulation of the precordial dilution technique on the mathematical model of Figure 3. CFSTR = 400 cc., PFTR = 0 cc.
the left heart curve, which is considerably lower and flatter than actual radioisotope dilution curves. The closest similarity to actual physiological data is seen in the curve of Figure 6, which depicts the results for a CFSTR of IOO cc. in volume, in series with a PFTR of volume equal to 300 cc. The posiVol. 52, No. 7, September 1976
8oo
A. J. LOVINGER
CFSTPu2oo.o CC.
PFTku2O0.0 CC PRECOR VERSUS TIME
IIlJ4
mr N
C.o PRECCR
TIME 0.0
s.oc4oE-ol 1hoooOE CO
1.5000S 2.OOOOE 2.5000E 3.OOOOE 3 .500O 4.OlOOF 4.5000E 5.OOOOE 5.5000E
00
c0 CO 00
ao
0O 00 00 00 00 '6.0000E &.5000E CO 7.0000 00 7.500OF 00 .OOOOE 00 S.5000E 00 q.oOooE 00 9.5000E 00 1.O000OE 01
1.0500E
01
1.100Uf
01
I.!500E 01 1.20COE 01 1.2 00E 1.3000E 1.3500E 1.4000E 1.4500E 1.5000E 1.5500E 1.6000F 1.6500E 1.700OE 1.7500E
I.SOOOE 1.0500F 1.9000E 1 .9 SOO' 2.0COOF 2.0500E 2.1000E 2.1500E
01
+
0.0 0.0 0.0
+
76 cF03 1.327 9.2844S-J34
6.492PE-04 4.5406E-4 3.20!Q-O U4 2.722Pf-04 2.8902'-04 3.2727c-04
-------------------
-----------
3.6340 -04 3. PT3?r-04 ?.*9664-0 4
-----------
3.9225'-04
-
---------
3.5377E-34 3.256bE-04 2.95C4E-04
----------
2.63q!E'04
2.3335 -04 2. 0 6! E-04 1.4119 - '4
1.543',E-04
-------+ +
----.
1.3333-O4
----+
1.1497E-04
---+
9.9192E-05 58 IO0-05
---+
01
8.
01 Ol 01 01
--+ 6.535SE-05 -5 5. 783AE-05
01
01. 01 01 01
0l 01 01 01
Cl 01 2.2COOE 01 2.2SOOE Cl 2.3000k 01 2. 3 500 E Cl ?.4000E 01 2.45(CF 01 2.95COE 01
-
3. Th94 c-O 4
7.46C9E-05
--
--+
5.1925F-05 -' -+ 4.7122E-05 4.354eF-05 -* 4,.0928E-O5 -+ 3.9124E-05 -+ 3. 8004F-0 -+ 3. 7TS4E-0 5 -+ 3.7377E-05 -+ '.7687F-05 -, 3.8312E-05 -+ ..9189'-0 5
4.0264F-05 4.1492E-05 4.
,.4257E-05 4.573?E-3 5 1.7240EO-05 -.o375?5-'oi 5.
5.
-*
Z634f-0 S
(266E-05 '7421--35
Fig. 5. Same
as
-, -, -
-* -
Figure 4. CFSTR
200
cc.,
PFTR =200
cc.
don of all inflection points, the time interval between the right and left heart maxima, and the individual concentration values exhibit the best fit to actual data. Finally, the curve of Figure 7, obtained when only an ideal PFTR, representing a pure delay, was used as the pulmonary model, typifies Bull. N. Y. Acad. Med.
8o i
MODELLING OF PULMONARY CIRCULATION
PrTr430)0.O CC
crSTR10.o.o CC,
TI ME 0.0 5 .OCOOF- 01 00 I .0000 1.5000E 00 2 .OCOUE GO 2 .5000F I00 3 .OOOOE 00 3.5 000 400 1
4
.0000r
Co
4.5000E 00 S.OOCO'F 100 5.5004W 400 6.0004WC 00 6.5000F
00
T OOOOOE 1(00 7.0004W) 7.50001?
CO
8.OOOO0E
00
8.5000c 9.00300E 00 9.5000E 00 01 1.0 00
000w 0l
1.!00;uE 0O1
1.1500E I01 1.2OOOE (01 1.2500£ I01 I1.3C00E 0' 1.3504W 01 1.4000E 0! 1.4500F 101 1.5000E 01 1.5500E 01 1.6000E 01 1*.6500E4 D1 1.7000E 01 I.7500E CD1 1.80COE 01 1.115C'OFC01 1.9000! 01 1. 5005! Hl 2.00001E4 01 2.050O0 Cl 2.IOOOE 4C1 2.1500fF 01 2.2000E C 2.2 S(40' Ol 2.3OOO9f Cl 2.35000h 4Cl ?.40(()'~ 2 *4(iC r t1I ;!.S(COL I 01
MP1INIMUt.4 C.0 I
PRECOR 0.0
PRCCOR VERSUS TIME
+ +
0.0 0.0
3274E-0 3 '0.2844F-04 6.4928E-04 4.54C6E-04 3.1754F-)4 2. 220CF-) 4 l.553S'e-04. 1.6193E-O 2.5850F-04 3.6596E-04 4*.4318E-04 4. 809LE-04
-_
* _
1.
----------------------+ -------------4
---------------
4.8385F--04
4.6lC7F-04 4.2203E-34 3.7459F-U4 3.2455F-04 2.7590E-04 2.
I 1 02 E-0
3.58165-05 3.2390E-05 3.0147E-0 S 2. q897E-3 5
2.8t436E-05 2.8597E-05 2.')244E-0 3.0 68E-05
3.1578E-05
3.3104E-95
3.47CbF-05 3.6576-O 3o.fl437E-05 4.224P r-0
as
-
-+ -+ -4 -4
-4
+
+ + + +
+ -4
-4 -
-4
4.415L2;-O5 'i - 0 5 0' . 7 ! 6" l- 5 4. 91, 5 -4
Fig. 6. Same
------
----
-
1.9116E-04 1.5680E-04 1.2785E-04 1 .03 96 -04 8. 45 95 S-0 5 6.h 163F-0 5 5.70S7E-05 4.7839E-05 4.0906E-05
rp. L391E-;
--------------4
-,
+ -
5
Figure 4. CFSTR
=
100
cc.,
PFTR = 300
cc.
the other end of the spectrum. Again, the curve is not representative of human precordial dilution curves, this time because, in the absence of any mixing, the input to the left heart is simply the delayed output from the right heart. Thus, the right heart curve is allowed to decay almost completely before the beginning of the left heart curve; this Vol. 52, No. 7, September 1976
8o 2
A. J. LOVINGER
CFSTRmO.O CC9
PFTA*400.0 CC
'41rNIMIUM
T1 ME 0.0
5.0C00E-01
10000OF 1.5000E 2.0000E 2.5000E
00 00
00 00
3.0000E 00 3.5CO0! CO 4.00001 00 4.5000E 00 5.0000E 00 5.5000E 00 6.00000 00 6.5000w 00 7.0000E 00 7.5000 CO 8.)000E 00 8.5000E 00 9.00001 00 9.5000E 00 I.OOOOE 01 1.0500E 01 01 S.tG1C0 114500E1 Cl I .20001e 01 1.25001 C1 1.3000E 01 1.3500E 01 1.400O1 01 1.4500E 01 1.5000E 01 1.5500E 01 1-.6000E 01 1.6 50' 01 1.7000E 01 1.7500E 01 1 60001E 01
PRECOR 0.0
coo
PRECOR VERSUS tINE
I
0.00
0.0 1.3276C-03 9.2844E-04 6.4928F-04
------
-----------
4.5406F-04
----------
3.l75't-04 2.2206-04
-------
1.5530E-04 1. 08601-
--_
7* 544O1-5
--4
1.4055E-04 4.1012E-Ol 5.3190F-04
---------
5.5237E-04 5.190CF-04 4.6C8?E-34 3.S439E-34
?.2QC&F-0 4
2*6957F-04 2.1812E-04 1. 7522C-04
1392ZE-O4
l1035E-04 1-
----------
------------------------------+ ----------------------+
-------
------+ ---+
1.8500E 01 1.9000E 01
A.7'32E-05 6.95361E-0 5. 5767F-OS-5 4.5344E-095 -+ 3.7611--05 -+ 3.2022F-05 + 2.8135F-39 + 2*56COF-Os 2.4135F-05 + * 2.3472E-05 + 2.3463E-05 2.3960h-0 5 + 4 2.o485OF-05 + 2. 6039E-0 5
1.9500F 01
2 . 7454'-C) 5
2.0000 1 21z.'O3 r-n3.0734!-;i5 2.0500E C1 3. 25 1 E-0 2*10O0E Ct .43 36F-05 2.1 500Cl 01 3. 61 93 -0 ' 2.2000E C1 . 80 30E-05 2.25001 01 3.98i56r-05 2.3000E 0 4.165.3 F-0 2.3500E 01 4.34 C%1-05 2 *4000E 01 4.511.4F-05 2.4500F Cl
-
+
+ +
-+
-4
-+ -e
-e -+
-e
2.5000E 01
4. 743F-05 Fig. 7. Same as Figure 4. CFSTR = 0 cc., PFTR = 400 cc.
results in an abnormally low local minimum and a visible distortion of the expected total dilution curve. From these data it is seen that an ideal CFSTR in series to an ideal PFTR, with a corresponding volume ratio of one to three would give the best model of pulmonary circulation by use of ideal chemical reBull. N. Y. Acad. Med.
MODELLING OF PULMONARY CIRCULATION
803
actors. Use of a time-varying flow rate (to account for the pulsatile motion of blood in the circulation), as well as of real reactors, such as PFTRs with axial mixing,17 would yield a more complicated solution. This, however, would not be expected to improve the validity of the model, since precise data regarding, for example, the extent of axial mixing in the various segments of the circulation, are not available. Further, the limiting factor in this investigation is not the ideal assumption for the reactor models used, but rather the inherent difficulties associated with the actual application of the precordial dilution technique,15 e.g., imperfect alignment of the detector above the heart, inclusion of stray radiation, noninstantaneous injections, and pulsatile flow. SUMMARY A mathematical model of the circulatory system, composed of ideal plug flow and stirred tank reactors, was constructed in order to determine a valid reactor representation of the pulmonary circulation. Physiological considerations led to a series model of a stirred tank and a tubular reactor, the first to represent the extent of mixing in the lungs while the second accounts for the delay in passage through the pulmonary circulation. Solution of the mathematical model equations was obtained by use of CSMP, which is particularly suited for engineering simulation. The indicator-dilution technique was tested by use of a controlled mechanical model to ensure its accuracy and applicability to this study, and was subsequently used to produce precordial dilution curves in the simulation of pulmonary circulation. Such curves, obtained from the mathematical model for various volume ratios of the two reactors and compared with actual data, indicate that pulmonary circulation can be represented by an ideal stirred tank in series with a plug-flow reactor, with a corresponding volume ratio of one to three.
Vol. 52, No. 7, September 1976
804
A. J. LOVINGER
REFERENCES 1. Levenspiel, O.: Chemical Reaction Enigineering. New York, Wiley, 1972. 2. Bassingthwaighte, J. B., Ackerman, F. H.) and Wood, E. H.: Applications of the lagged normal density curve as a model for arterial dilution curves. Circ. Res. 18:398-415, 1966. 3. Harris, T. R. and Newman, E. V.: An analysis of mathematical models of circulatory indicator dilution curves. J. Appl. Physiol. 28:840-50, 1970. 4. Newman, E. V., Merrell, M., Genecin, A., Monge, C., Milonso, WV. R., and McKeeves, 'V. P.: 'rhe dye dilution method for describing the central circulation. Circulation 4:735-46, 1951. 5. Schlossmiacher, E. J., Weinstein, H., Lochaya, S., and Schaffer, A. B.: Perfect mixers in series model for fitting venoarterial indicator-dilution curves. J. Appl. Physiol. 22:327-32, 1967. 6. Shames, D. M. and Weber, P. M.: A general logical structure for quantitative analysis of radiocardiographic data. Cliin. Res. 232:210-14, 1972. 7. Irisawa, H., Wilson, M. F., and Rushmer, R. F.: Left ventricle as a mixing chamber. Circ. Res. 8:183-87, 1960. 8. Pavek, E., Pavek, K., and Boska, D.: Mixing and observation errors in indicator dilution studies. .J. Appi. Physiol. 298:753-57, 1970. 9. Vliers, A. C. A. P. and Ziljstra, W. G.: Zum Problem der Mischung von Indikator und Blut. Z. Kreislaufforsch. 58: 79, 1969. 10. Love, WV. D., O'Meallie, L. P., and Burch, G. E.: Clinical estimation of the
volumes of blood in the right heart, left heartand lungs by use of 1131 albumin. Amer. Heart J. 61:397-407, 1961. 11. Stewart, G. N.: Researches on the circulation time in organs and on the influences which affect it. J. Physiol. 15:31, 1894. 12. Fick, A.: Ueber die Messung des Blutquantums in den Herzenventrikeln. Translated by Hoff, H. E. and Scott, H. J. New Eng. J. Med. 239:120, 1948. 1:3. Lovinger, A. J., Castellana, F. S., and Pierson, R. N., Jr.: A study of the accuracy of precordial radiocardiography by use of a mechanical model. Medicine (Greece) 52:281-87, 1974. 14. Lovinger, A.: Computer programming for automatic radiocardiographic analysis. Acad. Med. (Greece) 38:58-69, 1974. 15. Lovinger, A. J.: Analysis of the applicability and Limitations of the Precordial Dilution Technique for the Calculation of Cardiopulmonary Parameters. M. S. thesis. New York, Columbia University, 1971. 16. System/360 Continuous System Modellihg Program., Application Description. Publication H 20-0240-1. White Plains, N. Y., IBM Tech. Pub. Dept. 17. Aris, R.: Introduction to the Analysis of Chemical Reactors. Englewood Cliffs, N. J., Prentice-Hall, 1969. 18. System/360 Continuous System Modelling Program, User's Manual. Publication H-20-0367-3. White Plains, N. Y., IBM Tech. Pub. Dept.
Bull. N. Y. Acad. Med.
TRACER STUDIES IN THE MATHEMATICAL MODELLING OF THE PULMONARY CIRCULATION* ANDREW J. LOVINGER Department of Chemical Engineering and Applied Chemistry Columbia University New York. N.Y.
T HERE are two types of ideal continuous chemical reactors. The first is the Continuous Flow Stirred Tank Reactor (CFSTR), a vessel characterized by perfect mixing so that composition and temperature are uniform in every part of the reactor. The entering material is mixed instantaneously with the fluid in the reactor, and the outflow stream is identical in both concentration and temperature to the contents of the reactor. The second type is called Plug Flow Tubular Reactor (PFTR), and is usually a tube into which material enters and leaves -at a steady state, flowing with a flat velocity profile. There is no mixing in the direction of flow (axial mixing), and all material at any cross section of the reactor is uniform (perfect radial mixing). The necessary and sufficient condition for ideal plug flow is that all flow elements have the same residence time in the reactor. While neither perfect mixing nor true plug flow can exist in actual practice, ideal reactors constitute a valid description of the processes in most real reactors.? A number of investigators~8 have described the behavior of circulatory segments in terms of chemical reactors by using the compartmental approach, i.e., representing the circulation as a series of chambers which are interconnected in various ways. There has been general agreement that the two heart chambers can be represented by stirred tank reactors, mixing being accomplished by the systolic and diastolic movements of the heart and by the eddies caused by the inflowing bloodstream. The degree to which such mixing approaches the ideal has been questioned by a number of investigators,7-9 but ideal CFSTRs have been adopted as a description of the right and left heart in the formulation of indicator-dilution theory, used in the determination of cardiac output and cardiopulmonary volumes as well as in the detection of intracardiac shunts and other circulatory anomalies. Vol. 52, No. 7, September 1976
A. J. LOVINGER
792
LOVINGER
792
800
RH
C/)
600
1-400
~~~~~~~LH
-
200
0 0
5
10
T I ME
15
20
(S E C)
Fig. 1. Typical precordial dilution curve. RH = right heart peak; LH = left heart peak.
There has not, however, been any investigation into the behavior of pulmonary circulation as a chemical reactor, although a number of authors have used lung models in their studies. Newman et al.4 used a single CFSTR, 5oo cc. in volume, as their model, albeit questioning this representation. Love and co-workers10 modelled pulmonary circulation by use of four stirred tanks connected in parallel, mixing being improved by the addition of glass beads. This paper presents a chemical reactor model of pulmonary circulation after examining a number of possible lung models consisting of combinations of ideal chemical reactors. METHOD
OF
INVESTIGATION
The indicator dilution technique, first proposed by Stewart,"1 is Bull. N. Y. Acad. Med.
MODELLING OF PULMONARY CIRCULATION
793
employed extensively in radiocardiographic determinations of cardiac output, ventricular volumes, and circulatory shunts. This technique involves injection of a dye or, most commonly, a radioactive tracer (99Tc, 1311) into the right ventricle, followed by subsequent monitoring of gamma radiation activity over the precordium. A precordial dilution curve such as the one shown in Figure I is obtained. The first peak represents the practically instantaneous injection of indicator into the right heart. As blood flows into the pulmonary circulation, activity decreases. However, a second peak is observed when the radioactive particles arrive at the left heart, followed by a further decrease in activity as the tracer enters systemic circulation. A final, much lower peak may be observed after return of the greatly diluted radioactive particles into the right heart. Determination of cardiopulmonary parameters then can be made by use of the precordial dilution data in the Fick equation and its corollaries.'2 Before this technique was used to simulate cardiopulmonary configuration, a study into its accuracy, applicability, and limitations was undertaken. An actual mechanical model of the circulation was constructed and used to obtain indicator dilution data under controlled conditions. A special computer program was written14 that automatically performs a complete radiocardiographic analysis of radioistope-dilution data. Use of the above apparatus and computer program indicated that the precordial dilution technique produces cardiac output results that are accurate to within 3 % of the actual value and heart volumes to within 3%0 and io%, depending upon the method used.l3, '5 These results also define the limits of confidence in the use of the precordial dilution technique for the simulation of the cardiopulmonary circulation. The effect of pulmonary configuration on the shape of the radioisotope dilution curves is very pronounced and characteristic. The greater the extent of mixing in the lungs, the more uniform will be the distribution of tracer in the bloodstream, and, thus, the peak in the left heart curve will tend to be less pronounced. Consequently, the shape of the curve of the left heart can be related directly to pulmonary configuration since, as was noted, there is general agreement that the other two circulatory segments through which the tracer has flowed (i.e., the heart chambers) can be regarded as valid approximations of ideal CFSTRs. Figure 2 shows the variation of pulmonary transit time (defined Vol. 52, No. 7, September 1976
794
A. J. LOVINGER
13 12
11
V10\ 9
-_
7
_0X
I-
c_
0
5
0
4_ Cl-
3 2 1
.'
0
0
1
.I I. I1.1 2
3
4
5
6
1
1
7
8
9
10
I
I
11
12
CARDIAC OUTPUT (LIT/MIN) Fig. 2. VariAbion of pulmoiary transit time (time-distance between right and left heart peaks) with flow rate.15
as the time distance between the right and left heart peaks) with cardiac output, obtained from the mechanical model mentioned previously.15 Peak separation is indicative of the delay in flow from the right to the left heart. Since, in the absence of chemical reactions, an ideal PFTR constitutes a pure delay, peak separation can be used to determine the relative size of a plug-flow reactor required to account for the time delay during passage of the indicator through the pulmonary
circulation. On the basis of the above, a mathematical model of the circulatory Bull. N. Y. Acad. Med.
MODELLING OF PULMONARY CIRCULATION
- - - -- PULMONARY CIRCULATION
r
lC1 | 7
- -
RIGHT HEART Vi = 130 cc
nLEFT HEART
7
IC4
V15
7 9 5
IV4 = L-
130 cc
SYSTEMIC CIRCULATION -
l
--7
I1000I
IccI
IJ C5
I |
1 ~~~~~~500cc
3000 cc
|
Fig. 3. Flow diagram of the mathematical model of the circulatory system. (See text for further description.)
system was constructed by use of tubular and stirred tank reactors. The system of simultaneous differential equations describing the flow of the tracer after a pulse injection into the right heart was solved by Vol. 52, No. 7, September 1976
7
796
A. J. LOVINGER LOVINGER A.
J.
use of a digital computer program, utilizing the Continuous System Modelling Program (CSMP) of International Business Machines. Precordial dilution curves were then plotted by computer for different pulmonary configurations and compared to actual curves obtained under corresponding conditions of cardiac output.
THE MATHEMATICAL MODEL A flow diagram of the mathematical model of the circulatory system is shown in Figure 3. A closed system containing four major segments was used to describe the circulation. These segments correspond to the two heart chambers and the pulmonary and systemic circulation. CFSTRs with volumes equal to 130 cc. were used to model the right and left heart chambers. Systemic circulation is modelled by a combination of two CFSTRs and one PFTR in series. The first stirred tank has a volume V6 = 5oo cc. and simulates the vigorous mixing that occurs in the aorta and the other large channels. The second CFSTR of volume V7 3,000 cc. models the dilution of tracer through the systemic blood pool. The plug-flow reactor of volume V5 = 1,ooo cc. is added to account for the delay of blood flow in the body before return to the right heart. Pulmonary circulation is modelled in general by a stirred tank in series with a plug-flow reactor; in all cases their combined volume (V2 + V3) is equal to 400 cc. The individual reactor capacities can vary anywhere from o cc. to 400 cc., so that in the extreme cases pulmonary circulation will be represented by either a single ideal CFSTR or a pure delay, with combinations of these constituting the intermediate cases. The selection of a series combination of a CFSTR and a PFTR for the pulmonary model is substantiated by the nature of blood flow through the circulation. Mixing effects. occur in the pulmonary artery and veins, as well as in the other major channels, and at all junctions. In a model of pulmonary circulation by chemical reactors these effects can be incorporated into a single CFSTR. The need for a tubular reactor arises from the fact that blood flow between the right and left heart is not instantaneous; rather, a time interval must elapse, representing the various delays encountered during flow through the lungs. The tubular reactor, therefore, is added to account for these delays. Finally, the series connection of a PFTR and a CFSTR instead of Bull. N. Y. Acad. Med.
MODELLING OF PULMONARY
CIRCULATION
79 7
a parallel one is based on the requirement that all blood particles must encounter the same conditions while passing through the circulation. Otherwise, it would be possible for some of them to move instantaneously from one point to another without having to pass through a PFTR, whereas others might travel through a pure delay without being subjected to any dilution. Thus, the proposed model of pulmonary circulation is in principle a valid representation of this circulatory segment, and exhibits a strong resemblance to the actual physiological system. The precise composition of the model, i.e., the relative volumes of the stirred tank and tubular reactors, now can be determined by solving the mathematical model equations. These equations, whose derivation is found in the literatures1' 7 are the typical mass balances for a CFSTR and a PFTR in the absence of chemical reaction. If Vi is the volume of the jth reactor and CQ the concentration of tracer in the exit stream from that reactor; if F represents cardiac output, t the time, and Tj the time constant for the jth reactor (defined by equation 4); and if, finally, the time-dependent flow rate and concentration of the injected tracer are designated by F. and C., respectively, then the mass balances for the circulatory model of Figure 3 become:
dC, V1-d= F.(t) * C. + F * (C7-C1) dt TJ
(I)
dCQ d j = Cj-1- Ci
j=,2416,7.
(2)
C>(t)~ CJ-1 (t-ril)
j 3, 5. jI, . . . ,7.
(3) (4) (5)
where Tr = Vj / F Initial conditions: C1(o-)
o
j=I, . . . ,7.
THE COMPUTER PROGRAM
The above set of simultaneous differential equations was solved by use of the CSMP. This programming language, a detailed description of which is given elsewhere,16' 18 combines the best characteristics of analog computation in engineering simulation with the extensive mathematical, logical, and control capabilities of FORTRAN IV. CSMP was selected for this study because of its provision of advanced mathVol. 52, No. 7, September 1976
798 798
A. A. J. J. LOVINGER
ematical functions (integrators, differentiators, delays), signal sources (step and impulse functions), and extraordinary output facilities. Numerical integrations were performed by use of the Runge Kutta fourth order routine with a variable step size. The injection of radioactive tracer was described by use of a FORTRAN assignment statement, e.g., a unit-pulse injection of 0.2 sec. duration, occurring at t = I.o sec. and representing F. (t) in equation i, is coded as the difference between two step functions: FX = STEP (i.o) - STEP (I.2) The exceptional output facilities provided by CSMP include plots of variables with their limits automatically adjusted, printing of output in either tabular or equation format, and use of titles and labels without necessary resort to FORTRAN format statements. The investigation of different lung models was made by use of successive runs of the program with different values for the volumes of the CFSTR and PFTR, ranging from o cc. to 400 cc.
RE~sULTS
AND
DISCUSSION
Of the curves obtained from the execution of the above program, four are shown in Figures 4 to 7. The first column in each figure contains the independent variable (time, in seconds), while corresponding values of tracer concentration (in mc./liter) are printed in the second column and plotted alongside. In the case of Figure 4, the entire pulmonary circulation was represented by an ideal stirred tank reactor. No left heart peak is visible; it has become part of the initial downslope. This occurs because the vigorous mixing in the lungs drastically reduces the pulmonary mean transit time, thus transposing the left heart curve toward the origin. In addition, the perfect mixing that occurs in the pulmonary segment further dilutes the tracer, so that the contribution of the left heart curve to the total dilution curve is substantially decreased. In this case, the entire cardiopulmonary circulation effectively becomes a series of three ideal stirred tanks, with no specific delay function added. The precordial dilution curve of Figure 5 is for an ideal CFSTR, 200 CC. in volume, attached in series to an ideal PFTR of equal capacity. The typical shape of precordial dilution is evident. However, this configuration still causes too much mixing, as is seen by the shape of Bull. N. Y. Acad. Med.
MODELLING OF PULMONARY CIRCULATION
CFSTtno400.C
PFTR O.0 CC
CC,
MINIM'1UM coo
TIME 0.0
5.0000E-01 1.0000E 00
1.5000E 00 2.00OO1 00 2 .SO00 3 0OOOO
00
co CO 3.5000E 00
4.00O0F 00
4.5000E 5.00000 5.SOOOE 6.0000E 6.5COOE 7.0000E 7.5000E 8.0O00E 8.50001 9.0COOE 9.5000E
00 00 00 GO CO 00 00 00 00 Co 00
I.OCOOE 01 1.0500E 01 1.COOZE 01 1.1500E 01 1.20COE 01 1.2 5001E 01 1.3000E 01 1.3500E 01 1.40O0E 01 1.4500E Cl 1.5OOO 01 1.5500E 01 1.6000F 01 1.6500E 01 1.7OOOF 01 1.7500E 01 1.8000E 01 1 .8500E 01 1.9000E 01 1.500E 01 2.OOOOE Cl 2.0500F 01 2.1000E Cl 2.1500E 01 2.200OE 0t 2.2500F Cl 2.30001 Cl 2.3'o3(E Cl 2?.0Oof 01 2.45001 Ct 2 .5000 Cl
7 99
PRECOR 0.0 0.0
0.0 1. 3411tE-0-3 O.4 E-0 7.5051E-04 *1045E-O4 S.14051E-0 4.4935E-04 4.0117E-04 3.65C2E-0 3.3542E-10 3e.0976E--0 2.8656E-04 2.6S06E-04
2.4671E-04
2.2585E-0 2.0795E-04 1.91 18-04
PRECOR VERSUS TIME
I +
+
- --------- ----4 --------------------------------------
-
-+
------------------------+ -------+
------------
1.7555E-0'4 1.6109E-0'4
1.4778F-04' 1.3563F-.-1.24591-04 1.1463'-04 1.056@E-04 ---4 9.7727E-05 --4 9.0667E-O5 --0 8.4451E-OS 7.9014E-05 --. 7.4293E-05 7.02241E-05 ---4 6.67 51-5 6*.3816E-05 -6*.1366E-5 -+ -* 5.93521E-0 '5.728E-05 -* 5. 6451 E-0-O 5.5482E-05 -, -+ 5.4764F-05 -, 5.4325E-05 S. 4071 E-055 5.'006E-35 5.4092S-05-5 5.43131-05 -, S.'.6 .ir-05-9 -, 5.507S5-05 5.5562F-053 -
S.61 51E-05-5 S. 6 ?t E-05 -+ -, 5.74343-05
Fig. 4. Computer output from the simulation of the precordial dilution technique on the mathematical model of Figure 3. CFSTR = 400 cc., PFTR = 0 cc.
the left heart curve, which is considerably lower and flatter than actual radioisotope dilution curves. The closest similarity to actual physiological data is seen in the curve of Figure 6, which depicts the results for a CFSTR of IOO cc. in volume, in series with a PFTR of volume equal to 300 cc. The posiVol. 52, No. 7, September 1976
8oo
A. J. LOVINGER
CFSTPu2oo.o CC.
PFTku2O0.0 CC PRECOR VERSUS TIME
IIlJ4
mr N
C.o PRECCR
TIME 0.0
s.oc4oE-ol 1hoooOE CO
1.5000S 2.OOOOE 2.5000E 3.OOOOE 3 .500O 4.OlOOF 4.5000E 5.OOOOE 5.5000E
00
c0 CO 00
ao
0O 00 00 00 00 '6.0000E &.5000E CO 7.0000 00 7.500OF 00 .OOOOE 00 S.5000E 00 q.oOooE 00 9.5000E 00 1.O000OE 01
1.0500E
01
1.100Uf
01
I.!500E 01 1.20COE 01 1.2 00E 1.3000E 1.3500E 1.4000E 1.4500E 1.5000E 1.5500E 1.6000F 1.6500E 1.700OE 1.7500E
I.SOOOE 1.0500F 1.9000E 1 .9 SOO' 2.0COOF 2.0500E 2.1000E 2.1500E
01
+
0.0 0.0 0.0
+
76 cF03 1.327 9.2844S-J34
6.492PE-04 4.5406E-4 3.20!Q-O U4 2.722Pf-04 2.8902'-04 3.2727c-04
-------------------
-----------
3.6340 -04 3. PT3?r-04 ?.*9664-0 4
-----------
3.9225'-04
-
---------
3.5377E-34 3.256bE-04 2.95C4E-04
----------
2.63q!E'04
2.3335 -04 2. 0 6! E-04 1.4119 - '4
1.543',E-04
-------+ +
----.
1.3333-O4
----+
1.1497E-04
---+
9.9192E-05 58 IO0-05
---+
01
8.
01 Ol 01 01
--+ 6.535SE-05 -5 5. 783AE-05
01
01. 01 01 01
0l 01 01 01
Cl 01 2.2COOE 01 2.2SOOE Cl 2.3000k 01 2. 3 500 E Cl ?.4000E 01 2.45(CF 01 2.95COE 01
-
3. Th94 c-O 4
7.46C9E-05
--
--+
5.1925F-05 -' -+ 4.7122E-05 4.354eF-05 -* 4,.0928E-O5 -+ 3.9124E-05 -+ 3. 8004F-0 -+ 3. 7TS4E-0 5 -+ 3.7377E-05 -+ '.7687F-05 -, 3.8312E-05 -+ ..9189'-0 5
4.0264F-05 4.1492E-05 4.
,.4257E-05 4.573?E-3 5 1.7240EO-05 -.o375?5-'oi 5.
5.
-*
Z634f-0 S
(266E-05 '7421--35
Fig. 5. Same
as
-, -, -
-* -
Figure 4. CFSTR
200
cc.,
PFTR =200
cc.
don of all inflection points, the time interval between the right and left heart maxima, and the individual concentration values exhibit the best fit to actual data. Finally, the curve of Figure 7, obtained when only an ideal PFTR, representing a pure delay, was used as the pulmonary model, typifies Bull. N. Y. Acad. Med.
8o i
MODELLING OF PULMONARY CIRCULATION
PrTr430)0.O CC
crSTR10.o.o CC,
TI ME 0.0 5 .OCOOF- 01 00 I .0000 1.5000E 00 2 .OCOUE GO 2 .5000F I00 3 .OOOOE 00 3.5 000 400 1
4
.0000r
Co
4.5000E 00 S.OOCO'F 100 5.5004W 400 6.0004WC 00 6.5000F
00
T OOOOOE 1(00 7.0004W) 7.50001?
CO
8.OOOO0E
00
8.5000c 9.00300E 00 9.5000E 00 01 1.0 00
000w 0l
1.!00;uE 0O1
1.1500E I01 1.2OOOE (01 1.2500£ I01 I1.3C00E 0' 1.3504W 01 1.4000E 0! 1.4500F 101 1.5000E 01 1.5500E 01 1.6000E 01 1*.6500E4 D1 1.7000E 01 I.7500E CD1 1.80COE 01 1.115C'OFC01 1.9000! 01 1. 5005! Hl 2.00001E4 01 2.050O0 Cl 2.IOOOE 4C1 2.1500fF 01 2.2000E C 2.2 S(40' Ol 2.3OOO9f Cl 2.35000h 4Cl ?.40(()'~ 2 *4(iC r t1I ;!.S(COL I 01
MP1INIMUt.4 C.0 I
PRECOR 0.0
PRCCOR VERSUS TIME
+ +
0.0 0.0
3274E-0 3 '0.2844F-04 6.4928E-04 4.54C6E-04 3.1754F-)4 2. 220CF-) 4 l.553S'e-04. 1.6193E-O 2.5850F-04 3.6596E-04 4*.4318E-04 4. 809LE-04
-_
* _
1.
----------------------+ -------------4
---------------
4.8385F--04
4.6lC7F-04 4.2203E-34 3.7459F-U4 3.2455F-04 2.7590E-04 2.
I 1 02 E-0
3.58165-05 3.2390E-05 3.0147E-0 S 2. q897E-3 5
2.8t436E-05 2.8597E-05 2.')244E-0 3.0 68E-05
3.1578E-05
3.3104E-95
3.47CbF-05 3.6576-O 3o.fl437E-05 4.224P r-0
as
-
-+ -+ -4 -4
-4
+
+ + + +
+ -4
-4 -
-4
4.415L2;-O5 'i - 0 5 0' . 7 ! 6" l- 5 4. 91, 5 -4
Fig. 6. Same
------
----
-
1.9116E-04 1.5680E-04 1.2785E-04 1 .03 96 -04 8. 45 95 S-0 5 6.h 163F-0 5 5.70S7E-05 4.7839E-05 4.0906E-05
rp. L391E-;
--------------4
-,
+ -
5
Figure 4. CFSTR
=
100
cc.,
PFTR = 300
cc.
the other end of the spectrum. Again, the curve is not representative of human precordial dilution curves, this time because, in the absence of any mixing, the input to the left heart is simply the delayed output from the right heart. Thus, the right heart curve is allowed to decay almost completely before the beginning of the left heart curve; this Vol. 52, No. 7, September 1976
8o 2
A. J. LOVINGER
CFSTRmO.O CC9
PFTA*400.0 CC
'41rNIMIUM
T1 ME 0.0
5.0C00E-01
10000OF 1.5000E 2.0000E 2.5000E
00 00
00 00
3.0000E 00 3.5CO0! CO 4.00001 00 4.5000E 00 5.0000E 00 5.5000E 00 6.00000 00 6.5000w 00 7.0000E 00 7.5000 CO 8.)000E 00 8.5000E 00 9.00001 00 9.5000E 00 I.OOOOE 01 1.0500E 01 01 S.tG1C0 114500E1 Cl I .20001e 01 1.25001 C1 1.3000E 01 1.3500E 01 1.400O1 01 1.4500E 01 1.5000E 01 1.5500E 01 1-.6000E 01 1.6 50' 01 1.7000E 01 1.7500E 01 1 60001E 01
PRECOR 0.0
coo
PRECOR VERSUS tINE
I
0.00
0.0 1.3276C-03 9.2844E-04 6.4928F-04
------
-----------
4.5406F-04
----------
3.l75't-04 2.2206-04
-------
1.5530E-04 1. 08601-
--_
7* 544O1-5
--4
1.4055E-04 4.1012E-Ol 5.3190F-04
---------
5.5237E-04 5.190CF-04 4.6C8?E-34 3.S439E-34
?.2QC&F-0 4
2*6957F-04 2.1812E-04 1. 7522C-04
1392ZE-O4
l1035E-04 1-
----------
------------------------------+ ----------------------+
-------
------+ ---+
1.8500E 01 1.9000E 01
A.7'32E-05 6.95361E-0 5. 5767F-OS-5 4.5344E-095 -+ 3.7611--05 -+ 3.2022F-05 + 2.8135F-39 + 2*56COF-Os 2.4135F-05 + * 2.3472E-05 + 2.3463E-05 2.3960h-0 5 + 4 2.o485OF-05 + 2. 6039E-0 5
1.9500F 01
2 . 7454'-C) 5
2.0000 1 21z.'O3 r-n3.0734!-;i5 2.0500E C1 3. 25 1 E-0 2*10O0E Ct .43 36F-05 2.1 500Cl 01 3. 61 93 -0 ' 2.2000E C1 . 80 30E-05 2.25001 01 3.98i56r-05 2.3000E 0 4.165.3 F-0 2.3500E 01 4.34 C%1-05 2 *4000E 01 4.511.4F-05 2.4500F Cl
-
+
+ +
-+
-4
-+ -e
-e -+
-e
2.5000E 01
4. 743F-05 Fig. 7. Same as Figure 4. CFSTR = 0 cc., PFTR = 400 cc.
results in an abnormally low local minimum and a visible distortion of the expected total dilution curve. From these data it is seen that an ideal CFSTR in series to an ideal PFTR, with a corresponding volume ratio of one to three would give the best model of pulmonary circulation by use of ideal chemical reBull. N. Y. Acad. Med.
MODELLING OF PULMONARY CIRCULATION
803
actors. Use of a time-varying flow rate (to account for the pulsatile motion of blood in the circulation), as well as of real reactors, such as PFTRs with axial mixing,17 would yield a more complicated solution. This, however, would not be expected to improve the validity of the model, since precise data regarding, for example, the extent of axial mixing in the various segments of the circulation, are not available. Further, the limiting factor in this investigation is not the ideal assumption for the reactor models used, but rather the inherent difficulties associated with the actual application of the precordial dilution technique,15 e.g., imperfect alignment of the detector above the heart, inclusion of stray radiation, noninstantaneous injections, and pulsatile flow. SUMMARY A mathematical model of the circulatory system, composed of ideal plug flow and stirred tank reactors, was constructed in order to determine a valid reactor representation of the pulmonary circulation. Physiological considerations led to a series model of a stirred tank and a tubular reactor, the first to represent the extent of mixing in the lungs while the second accounts for the delay in passage through the pulmonary circulation. Solution of the mathematical model equations was obtained by use of CSMP, which is particularly suited for engineering simulation. The indicator-dilution technique was tested by use of a controlled mechanical model to ensure its accuracy and applicability to this study, and was subsequently used to produce precordial dilution curves in the simulation of pulmonary circulation. Such curves, obtained from the mathematical model for various volume ratios of the two reactors and compared with actual data, indicate that pulmonary circulation can be represented by an ideal stirred tank in series with a plug-flow reactor, with a corresponding volume ratio of one to three.
Vol. 52, No. 7, September 1976
804
A. J. LOVINGER
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volumes of blood in the right heart, left heartand lungs by use of 1131 albumin. Amer. Heart J. 61:397-407, 1961. 11. Stewart, G. N.: Researches on the circulation time in organs and on the influences which affect it. J. Physiol. 15:31, 1894. 12. Fick, A.: Ueber die Messung des Blutquantums in den Herzenventrikeln. Translated by Hoff, H. E. and Scott, H. J. New Eng. J. Med. 239:120, 1948. 1:3. Lovinger, A. J., Castellana, F. S., and Pierson, R. N., Jr.: A study of the accuracy of precordial radiocardiography by use of a mechanical model. Medicine (Greece) 52:281-87, 1974. 14. Lovinger, A.: Computer programming for automatic radiocardiographic analysis. Acad. Med. (Greece) 38:58-69, 1974. 15. Lovinger, A. J.: Analysis of the applicability and Limitations of the Precordial Dilution Technique for the Calculation of Cardiopulmonary Parameters. M. S. thesis. New York, Columbia University, 1971. 16. System/360 Continuous System Modellihg Program., Application Description. Publication H 20-0240-1. White Plains, N. Y., IBM Tech. Pub. Dept. 17. Aris, R.: Introduction to the Analysis of Chemical Reactors. Englewood Cliffs, N. J., Prentice-Hall, 1969. 18. System/360 Continuous System Modelling Program, User's Manual. Publication H-20-0367-3. White Plains, N. Y., IBM Tech. Pub. Dept.
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