- Email: [email protected]
An Adaptive Model of Ventricular Function Applied to the Study of Assist Device Pumping
AN ADAPTIVE MODEL OF VENTRICULAR FUNCTION APPLIED TO THE STUDY OF ASSIST DEVICE PUMPING J.W. Clark, M.E. Beasley and D.L. Baldridge Bioengineering Group Electrical Engineering Department Rice University Houston, Texas
ABSTRACT This paper is concerned with a digital computer simulation of the effects of series ventricular assist pumping (intra-aortic balloon pumping) on left ventricular mechanics and arterial hemodynamics. The human ventricle is, in general, characterized by a ventricular elastance function (E (Ev) v ) that is a function of end-diastolic volume (V. d ) and time, and the arterial load presented the ventricle is represented by a modified Windkessel model. A model for the intra-aortic balloon pump is also incorporated in arterial load model. The paper demonstrates (1) the characterization of the ventricular mechanics of the normal and failing (coronary artery disease, hypertrophied) heart in terms termS of elastance functions and (2) the utility of such an overall model in assessing the effects of ventricular assist device pumping on ventricular mechanics and arterial hemodynamics. INTRODUCTION Numerous models of ventricular function have been reported in the literature, many differing considerably abl y in complexity complexit y and in their abilitl to describe the mechanics of ventricular function( function ( ). ) . Ventricular elastance (Ev), (E v ) , defined as E.(t)
= p.(t) / Vv(t)
(1)
where Pvv and V Vvv are the instantaneous ventricy , has been used ular pressure and volume respectivel respectively, by several investigators investi ga tors as an adequate functional characterization of the ventricular v entricular mechanics of the denervated heart(2_0). heart( Z_o). The model of Greene, et al.(o) al. (o) has a distinct advantage advanta ge over other models of ventricular function in that E Evv is characterized by a family of elastance curves with enddiastolic volume (V.~) (V.!) as a controlling parameter [E, lfuen this model of ventricular [E . = E,(V.~,t)J. E.( Ve! , t)] . l-!hen mechanics is connected to suitable models of the arterial load, load , heart valves and atrium, the resulting model is capable capa b le of reflecting changes in both ventricular mechanics and circulatory circulator y hemodynamics in response to various preload and afterload conditions imposed upon the heart. This capability capabilit y makes this model very useful in the assessment of the ventricular performance of both the normal and the failing failin g heart under various load conditions.
Note:
The general purposes of this paper are to demonstrate (1) the elastance characterization of the ventricular mechanics of the normal and failing heart and (2) the utility of such a model in assessing the effects of assist device pumping (intraaortic balloom pumping) on ventricular mechanics and the hemodynamics hemodynarnics of the systemic arterial system. system . THE MODEL To characterize the circulatory effects of leftventricular pumping, adequate models of both the left heart and the systemic circulatory system s ystem must be considered. considered . Furthermore, to study the effects of assist device pumping, a model of the particular series or parallel* assist device must be also incorporated in the general model. A composite model for the evaluation of the effects of intra-aortic balloon pumping is shown in Figure 1 and consists of three parts: individual models of the left ventricle, the systemic arterial system s ystem and the balloon pump. pump . The ventricular model functionally functionall y char~c characterizes the mechanical properties of the ventricle in terms of a time-varying elastance (E that is a (Ev) v ) function of a controlling parameter, end-diastolic volume (V. d ) . The pressure-operated mitral and aortic valves are characterized by rectifiers and the left atrium is represented as a linear timevarying elastance (EA(t)). (E . (t)). Pulmonary Pulmonar y venous ve nous pressure and pulmonary venous resistance are represented by a constant pressure source (Pp,) (Ppv ) and a resistance (Rp v) v ) respectively. The rectifiers are assumed ideal (no backflow and with a constant resistance (~,RA) (~,R,) in the forward direction). The systemic s ystemic arterial model is represented by a lumped ~-type ~ -type network model analogous to that of Spencer and Denison(?) The lumped inertance L of Denison(7 ) the blood (primarily (primaril y in the descending aorta) and the series resistance R Rcb separate the two lumped representations of the proximal and distal portions of the systemic s y stemic arterial tree. tree . The proximal portion contains a lumped "coronary" and "cerebral" resistance (Rot.), (Rt.), and a series resistive-compliant *}\echanical circulatory assist devices are generally *}Iechanical classified as "series" or "parallel" according to the manner of their connection relative to the ventricle; i.e., series devices such as the intraaortic balloon pump are connected in series with the ventricle, while parallel devices such as the left ventricular by-pass pump are connected in parallel.
Superior numbers refer to similarly similarl y numbered
~rences ~rences at the end of this paper.
325
calculate volume from the cinefilm. A cubic spline function fit to the computed elastance function (1) was obtained in each of the three cases (Normal, (Normal , subsequentl y CAD, Hypertrophy) and this function was subsequently Evy in the model equations. equations . The charutilized as E acterization of ventricular elastance as a function of V. d and time requires multiple determinations of instantaneous left ventricular pressure and volume at different filling volumes. Since the accumulative effects of successive injections of contrast material are possibly possibl y harmful to the patient ) , only one injection of contrast material was (1 (1 4 ), studies . As a result, E Evy is deutilized in these studies. termined at only one value of V. d • Results from lab orator y and prelimiboth dog experiments in our laboratory nary results from data on paced human subjects, however, lead uS to believe belie ve that ventricular elastance is a function of both end-diastolic volume and time state . With adequate within a defined inotropic state. (i . e., multiple determinations of ~ data available (i.e., ~v ), ), methods have been developed(6) developed (6 ) for characand V terizing the resultin resultingg family famil y of elastance curves Ev(V. d , t).
pair (Rl,CL) (R1,CL) representing the viscoelastic properties of the ascending aorta and aortic arch region. region . The distal element contains the lumped systemic peripheral res is tance RR and a series R-C pair characterizing the lumped viscoelastic properties of the descending and abdominal aorta. The intra-aortic balloon ventricular assist technique utilizes a polyurethane balloon inserted through the femoral artery and advanced under fluoroscopy to a suitable position in the descending aorta . The balloon is pneumatically driven and conaorta. trolled so as to raise central aortic pressure soon after left ventricular ejection and aortic valve closure. Aortic pressure remains elevated throughout diastole and is lowered via balloon deflation before the succeeding ventricular ejection. Timing information for operation of the balloon "out-ofphas~1 with the heart is typically t ypicall y provided by the phas~1 ECG and balloon pumping is controlled in either an open_loop(8_10) open_loop (B_1 0) or closed-Ioop(ll closed-loop (ll ,12) ,l a ) fashion. fashion . The "balloon" itself is basically baSically a non-elastic cigarshaped, collapsible, polyurethane bag connected via a stiff, wide-bore catheter to a penumatic driv driving ing vacuum) . The model of source (under pressure or vacuum). the balloon pump utilized in this st study udy is shown in Fi gure 1I and consists of an equivalent balloon presFigure sure source P6 Ps S(t); s (t); a switch S6 Ss which is closed during balloon b alloon filling and emptying empt y ing and open otherwise [for example, when the balloon b alloon is fully full y inflated, it has no further f urther "circulatory effect" other than offering resistance to blood b lood flow, i.e., i.e . , under this condition blood is no longer being displaced and stored in the proximal and distal compliances (CL and CR ) ] ; a balloon resistance Re(t) which has a small nominal value when the balloon is fully full y deflated and rises to a larger lar ger value of resistance in an approximately exponential fashion during balloon filling.
Model runs were made for the various disease states in both the unassisted and assisted (balloon operational) cases; in general, these studies indicate b oth arterial blood good diastolic augmentation of both pressure through press ure and flow thr ough resistor res is tor ~ (coronary(coronar y cerebral cer ebral pathway) pathway ) during durin g diastole with appropriately appropriat e l y controlled assist ass is t pumping. By" appropriate ly By "appropriatel y controlled" pumping we mean that the balloon was inflated during du rin g diastole with an appropriate delay dela y (D) after aortic valve closure and remained deflated for an appropriate duration (B); one that is long enough to bring about a b out good g ood augmentation of diastolic pressure, but short enough so as not to "load" the failing ventricle on the succeeding succe e ding cycle. cy cle . Instantaneous pressure-volume pressure -vol ume loops constructed for the unassisted and assisted cases indicate a general reduction circulatory r e duction in heart work with mechanical circulat or y assistance. This is consistent c onsist e nt with the findings findin gs of Mullins, (15) Hullins, et al. (l S) who used an arterial countercount e rpulsation technique techniq ue in dogs. dogs .
In clinical practice the balloon is always operated in a non-occlusive fashion. fashion . The model, therefore, assumes that the major circ circulatory ulat or y effect of the fullyy inflated balloon is resistive. full resisti v e . Consequently, Conseq uentl y, any inertial effects produced by the narrowed annular orifice for blood flow are neglected. neg lected . This appears to be a reasonable assumption when one wishes to characterize characteriz e the general aspects asp e cts of balloon inflation(10) flation (l O) .
b alloon pumping The effectiveness of intra-aortic balloon was also judged jud ged by utilizing utili z ing a performance index ind ex dede (11, 12) veloped in previous studies (11,12) This scalar performance index reflects refl ects the general ge neral purposes of balloon pumping (namely, (namel y , diastolic diast olic augmentation au gmentati on of blood b lood flow to the coronary coronar y and cerebral circulations and reduction defined red uction of heart work) defi ned as:
RESULTS RESULT S The Th e system s y stem equations for the network model of Figure Fi gu re 1 were programmed on an IIBM BH 370/155 3 70 / 15 5 computer. c omputer. Typical scal scaled e d values v alues for many ma ny of the "arterial" and "heart" mode]. parameters were obtained previob tained from prev ious studies(o) Values studies (o ) Val ues for balloon resistance as a function radi us (volume) ( v ol ume) were obtained f unction of balloon radius from Brown(lo). Brown (l o ) . "Typical" "T ypical" elastance elastanc e curves for the normal human h uman heart and hearts with coronary artery disease (CAD) and hypertrophy hypertroph y were constructed (Figure (Fi gure 2) based on left ventricular ventric ular pressure-volume data* obtained at a single end-diastolic volume (V. ob ( V. d ). The left-ventricular volume data was obtained via single-plane cineangiography cineangiograph y at 60 fps and a standard ellipsoidal formula(13) formula (1 3 ) was used to *This data was supplied by Dr. Dr . James S. Cole, Cardiac Catheterization Laboratory, Lab oratory , The Methodist Hethodist Hospital, Houston, Texas.
where = scalar performance functional whe re J = f unctional which is a function (D) and f unction of two variables, v ariables , balloon delay dela y CD) duration (B). (B) . (Here, ( Here, delay dela y (D) is defined as the t he time ti me between b etween aortic valve val ve closure clos ure (T,) (T 5 ) and the onset of balloon inflation in the th e diastolic period. Balth e time interval inter val between the loon duration (B) is the actuator Signals signals for balloon inflation and deflation. deflation . p.(t) PaCt) AMDP ='
AEDP
326 32 6
= aortic (T~ ( T~
root pressure (mm Hg). Hg ). T 1I ; p. p.(( t)dt where T~ -T, -T , ) eT, T;
P.(T P. ( T)J ))
r l~
=
time at enddiastole. diastole .
[aortic end-diastolic pressure]
AMSP
= 1-
K33 K
desired arterial end-diastolic pressure (AEDP*)
Tss T
REFERENCES
jIsP.(t)dt where time t=O refers to the 0 time of aortic valve opening.
1 i f (Ks -AEDP) otherwise
o
<
1. Noordergraaf, A. (1969) "Hemodynamics" in Biomedical Engineering. H.P. Schwan, ed., pp. 391545. 2. Beneken, J.E.W. (1965) "A mathematical apph . D. Thesis, proach to cardiovascular function," ph.D. Institute of Medical Physics, Utrecht, The Netherlands. Biomech.l: 3. Snyder, M.F., et a!. (1968) J. Biomech.1: 341-353. 10: 509-515. 4. Suga, H. (1969) Jap. Heart J. !Q: 5. Suga, H. (1971) IEEE Trans. BME-18:47-55 BME 18:47-55.. M. E., J.W. Clark, D.N. Mohr, H.M. 6. Greene, M.E., 126-134 . Bourland (1973) Med. & BioI. Engr. 11: ll: 126-134. 7. Spencer, M.P. M.P . and A.B. A. B. Denison (1963) "Pulsatile Blood Flow in the Vascular Sys tern," Handbook Vol . 11, 839-864. 839-864 . of Physiology, Sect. 2, Vol. a1. (1970) Trans. Amer. Soc. 8. Jaron, D., et a!. Artif . Int. Organs 16: 466-471. Artif. 9. Buckley, M.J., et al. (1970) Circulation 41, 130-134 . Suppl. 11: 130-134. 10 . Brown, B.G. (1969) "Cardiac assistance by 10. intra-aortic balloon," Ph.D. Thesis, Johns Hopkins University . University. G. R., J.W. Clark, H.M. Bourland (1971) 11 . Kane, G.R., 11. Trans . Amer. Soc. Soc . Artif. Int. Organs 12: 148-152. Trans. 12. Clark, J.W., G.R. Kane, H.M. Bourland (1973) "On the feasibility of closed-loop control of intraaortic balloon pumping," IEEE Trans. BME (Sept. issue). J . W. Kennedy (1969) ~. 13. Kasser, E.S. and J.W. ~. Radiol. 4: 83-90. al . (1967) Invest. Invest . Radiol. 14. Kloster, F.E., et al. 353-359 . 2: 353-359. IS. Mullins, C.B., et al. (1971) Amer. J. 15. Physiol. 1lQ: 694-698.
0
Kz, and the constants K1 , Ks > 0 and K 2 , ~ < O. In equation (2) the term K1AMDP relates to the "circulatory" objectives of balloon pumping; particularly to enhanced diastolic coronary and carotid flow brought about by a maximization of this term. The KzAMSP ~o(Ks-AEDP)z term K 2AMSP and the penalty function ~o(Ks-AEDP)2 reflect the attempt to minimize heart work. With constants K ~ assigned appropriate signed K1,K 1 ,K 2z , and K4 values, maximization of J implies optimal satisfaction of the stated objectives of balloon pumping in some sense. Figure 3 indicates the model generated wave shape of aortic pressure with and without balloon pumping. In this case, the "normal" ventricular elastance curve of Figure 2 is used and the resulting diastolic pressure augmentation achieved during balloon pumping (D = 33.6 and B = 361.2 msec) is evident. In general, the performance index J(D,B) yields a unimodal isocontour surface when plotted versus delay D and duration B as seen in Figure 4. The isocontours (locus of equal values of J) indicated 151 of on the performance surface are located every 151 performance index maximum and the delay-duration times are expressed in normalized time (T) units (see Figure 4). Of course, the optimal operating point J.ax(Dopt,Bo.t) for the balloon varies with the particular elastance characteristics assumed for the failing ventricular myocardium. ACKNOWLEDGMENT The authors wish to express their gratitude to Dr J ames S. Cole, James Cole. Dept. of Medicine, Baylor College of Medicine for supplying the pressure-volume data stud y. This research was supported utilized in this study. in part by PHS Research Grant #HE-09251 from the Institute, National InstiNational Heart and Lung Institute. tutes of Health.
I
I
I
RA
I I I I
I" I I
r-----,
-t';"L1 IS: Pao
I
Re( t)
I
v\I'
1
I
Ppv PpV
Ev(Ved , t)
I RG
CLI
I
RI 1 I
I
B
I L
Pab
~,-------,
~"--I
I I I
P. (t) I PBS(t): BS I
:-----I'-----L,---_-.....L.._-_-_....,....---------' :----~------~~-----~_--_--_--J~----------~ ~
L_ L __
J
L L
BALLOON MODEL
HEART MODEL FIG. 1.
ARTERIAL LOAD
load . The arterial load also contains balVentricular model with "modified Windkessel" arterial load. loon pump model as indicated. indicated . Left ventricular (~), (~), aortic (P ao ), and abdominal aortic (P ab ab) pressures are given in units of mmHg, mmH g, resistances in units of mmHg/liter/sec, mmH g/ liter / sec, compliances in units of liter liter/mmHg, / mmH g, and inertance (L) in units of mmHg·sec mmH g ·sec 22 /1iter. / 1iter. See text for further explanation. 327
I
., . ,
2.5 2.5
:'"""
I I
I'
~.,. E
,
E
Z
,
t- 1.5 ....
,
1.0
er
t.... '"zZ
o
o
'"er !,!
\
\
~
0 .2 0.2
'"o
\
.~' --~-------_. 0.4
0 .6
0 .8 0.8
1.0 10
TI ME -SECONDS
FIG. 2.
~
J-
2
" -4' "
CS
Isocontours on the performance index surface; T = ~ t'HR'60, ~ heart rate, t ' HR'60, where HR = T = ~ normalized time in seconds; contours every e ver y 15 4 of performance index maximum;
Doopt pt
268 .8 268.B
4032
806.4
,( t) for the FIG. 3. Computed aortic root pressure p. p.,(t) normal patient with and without ventricular assistance. assistance . (D ~= 33.6 msec, B = 361.2 361 . 2 msec)
c c
FIG. 4.
o
TIME- msec T1ME-
Computed ventricular elastance curves for normal, CAD, and hypertrophy patients. Experimental elastance points were obtained ~y and V Vvy data using uSing (1) and then fit from P by a cubic spline interpolating function. function .
c: ::::: c::
50
t.... er
\
Or--f--------==~=e:~::=;:~~o
8ALLOON
'"er..."'I
\
".
BALLOON
....ot-
\
(!
..J -'
W/O
vi V> W w
I
l
>
0 .5 .......wt- 0.5
~IOO E
I I
,
If
W w
f 100
II
,1--\
'"er
:5
,"
,
w W
.,.
I I
I,
I
V>
~
I ,,
-Hypertrophy
i
II
u
-
I~
II
w
• 'I\
I
I
,,
'"E 2.0
:I:
---Normal - -- --Coronary Artery Arter'1 Disease Oisease -----Coronary
\\
~ 4; Bopt = ~ 43. =
328
ABSTRACT This paper is concerned with a digital computer simulation of the effects of series ventricular assist pumping (intra-aortic balloon pumping) on left ventricular mechanics and arterial hemodynamics. The human ventricle is, in general, characterized by a ventricular elastance function (E (Ev) v ) that is a function of end-diastolic volume (V. d ) and time, and the arterial load presented the ventricle is represented by a modified Windkessel model. A model for the intra-aortic balloon pump is also incorporated in arterial load model. The paper demonstrates (1) the characterization of the ventricular mechanics of the normal and failing (coronary artery disease, hypertrophied) heart in terms termS of elastance functions and (2) the utility of such an overall model in assessing the effects of ventricular assist device pumping on ventricular mechanics and arterial hemodynamics. INTRODUCTION Numerous models of ventricular function have been reported in the literature, many differing considerably abl y in complexity complexit y and in their abilitl to describe the mechanics of ventricular function( function ( ). ) . Ventricular elastance (Ev), (E v ) , defined as E.(t)
= p.(t) / Vv(t)
(1)
where Pvv and V Vvv are the instantaneous ventricy , has been used ular pressure and volume respectivel respectively, by several investigators investi ga tors as an adequate functional characterization of the ventricular v entricular mechanics of the denervated heart(2_0). heart( Z_o). The model of Greene, et al.(o) al. (o) has a distinct advantage advanta ge over other models of ventricular function in that E Evv is characterized by a family of elastance curves with enddiastolic volume (V.~) (V.!) as a controlling parameter [E, lfuen this model of ventricular [E . = E,(V.~,t)J. E.( Ve! , t)] . l-!hen mechanics is connected to suitable models of the arterial load, load , heart valves and atrium, the resulting model is capable capa b le of reflecting changes in both ventricular mechanics and circulatory circulator y hemodynamics in response to various preload and afterload conditions imposed upon the heart. This capability capabilit y makes this model very useful in the assessment of the ventricular performance of both the normal and the failing failin g heart under various load conditions.
Note:
The general purposes of this paper are to demonstrate (1) the elastance characterization of the ventricular mechanics of the normal and failing heart and (2) the utility of such a model in assessing the effects of assist device pumping (intraaortic balloom pumping) on ventricular mechanics and the hemodynamics hemodynarnics of the systemic arterial system. system . THE MODEL To characterize the circulatory effects of leftventricular pumping, adequate models of both the left heart and the systemic circulatory system s ystem must be considered. considered . Furthermore, to study the effects of assist device pumping, a model of the particular series or parallel* assist device must be also incorporated in the general model. A composite model for the evaluation of the effects of intra-aortic balloon pumping is shown in Figure 1 and consists of three parts: individual models of the left ventricle, the systemic arterial system s ystem and the balloon pump. pump . The ventricular model functionally functionall y char~c characterizes the mechanical properties of the ventricle in terms of a time-varying elastance (E that is a (Ev) v ) function of a controlling parameter, end-diastolic volume (V. d ) . The pressure-operated mitral and aortic valves are characterized by rectifiers and the left atrium is represented as a linear timevarying elastance (EA(t)). (E . (t)). Pulmonary Pulmonar y venous ve nous pressure and pulmonary venous resistance are represented by a constant pressure source (Pp,) (Ppv ) and a resistance (Rp v) v ) respectively. The rectifiers are assumed ideal (no backflow and with a constant resistance (~,RA) (~,R,) in the forward direction). The systemic s ystemic arterial model is represented by a lumped ~-type ~ -type network model analogous to that of Spencer and Denison(?) The lumped inertance L of Denison(7 ) the blood (primarily (primaril y in the descending aorta) and the series resistance R Rcb separate the two lumped representations of the proximal and distal portions of the systemic s y stemic arterial tree. tree . The proximal portion contains a lumped "coronary" and "cerebral" resistance (Rot.), (Rt.), and a series resistive-compliant *}\echanical circulatory assist devices are generally *}Iechanical classified as "series" or "parallel" according to the manner of their connection relative to the ventricle; i.e., series devices such as the intraaortic balloon pump are connected in series with the ventricle, while parallel devices such as the left ventricular by-pass pump are connected in parallel.
Superior numbers refer to similarly similarl y numbered
~rences ~rences at the end of this paper.
325
calculate volume from the cinefilm. A cubic spline function fit to the computed elastance function (1) was obtained in each of the three cases (Normal, (Normal , subsequentl y CAD, Hypertrophy) and this function was subsequently Evy in the model equations. equations . The charutilized as E acterization of ventricular elastance as a function of V. d and time requires multiple determinations of instantaneous left ventricular pressure and volume at different filling volumes. Since the accumulative effects of successive injections of contrast material are possibly possibl y harmful to the patient ) , only one injection of contrast material was (1 (1 4 ), studies . As a result, E Evy is deutilized in these studies. termined at only one value of V. d • Results from lab orator y and prelimiboth dog experiments in our laboratory nary results from data on paced human subjects, however, lead uS to believe belie ve that ventricular elastance is a function of both end-diastolic volume and time state . With adequate within a defined inotropic state. (i . e., multiple determinations of ~ data available (i.e., ~v ), ), methods have been developed(6) developed (6 ) for characand V terizing the resultin resultingg family famil y of elastance curves Ev(V. d , t).
pair (Rl,CL) (R1,CL) representing the viscoelastic properties of the ascending aorta and aortic arch region. region . The distal element contains the lumped systemic peripheral res is tance RR and a series R-C pair characterizing the lumped viscoelastic properties of the descending and abdominal aorta. The intra-aortic balloon ventricular assist technique utilizes a polyurethane balloon inserted through the femoral artery and advanced under fluoroscopy to a suitable position in the descending aorta . The balloon is pneumatically driven and conaorta. trolled so as to raise central aortic pressure soon after left ventricular ejection and aortic valve closure. Aortic pressure remains elevated throughout diastole and is lowered via balloon deflation before the succeeding ventricular ejection. Timing information for operation of the balloon "out-ofphas~1 with the heart is typically t ypicall y provided by the phas~1 ECG and balloon pumping is controlled in either an open_loop(8_10) open_loop (B_1 0) or closed-Ioop(ll closed-loop (ll ,12) ,l a ) fashion. fashion . The "balloon" itself is basically baSically a non-elastic cigarshaped, collapsible, polyurethane bag connected via a stiff, wide-bore catheter to a penumatic driv driving ing vacuum) . The model of source (under pressure or vacuum). the balloon pump utilized in this st study udy is shown in Fi gure 1I and consists of an equivalent balloon presFigure sure source P6 Ps S(t); s (t); a switch S6 Ss which is closed during balloon b alloon filling and emptying empt y ing and open otherwise [for example, when the balloon b alloon is fully full y inflated, it has no further f urther "circulatory effect" other than offering resistance to blood b lood flow, i.e., i.e . , under this condition blood is no longer being displaced and stored in the proximal and distal compliances (CL and CR ) ] ; a balloon resistance Re(t) which has a small nominal value when the balloon is fully full y deflated and rises to a larger lar ger value of resistance in an approximately exponential fashion during balloon filling.
Model runs were made for the various disease states in both the unassisted and assisted (balloon operational) cases; in general, these studies indicate b oth arterial blood good diastolic augmentation of both pressure through press ure and flow thr ough resistor res is tor ~ (coronary(coronar y cerebral cer ebral pathway) pathway ) during durin g diastole with appropriately appropriat e l y controlled assist ass is t pumping. By" appropriate ly By "appropriatel y controlled" pumping we mean that the balloon was inflated during du rin g diastole with an appropriate delay dela y (D) after aortic valve closure and remained deflated for an appropriate duration (B); one that is long enough to bring about a b out good g ood augmentation of diastolic pressure, but short enough so as not to "load" the failing ventricle on the succeeding succe e ding cycle. cy cle . Instantaneous pressure-volume pressure -vol ume loops constructed for the unassisted and assisted cases indicate a general reduction circulatory r e duction in heart work with mechanical circulat or y assistance. This is consistent c onsist e nt with the findings findin gs of Mullins, (15) Hullins, et al. (l S) who used an arterial countercount e rpulsation technique techniq ue in dogs. dogs .
In clinical practice the balloon is always operated in a non-occlusive fashion. fashion . The model, therefore, assumes that the major circ circulatory ulat or y effect of the fullyy inflated balloon is resistive. full resisti v e . Consequently, Conseq uentl y, any inertial effects produced by the narrowed annular orifice for blood flow are neglected. neg lected . This appears to be a reasonable assumption when one wishes to characterize characteriz e the general aspects asp e cts of balloon inflation(10) flation (l O) .
b alloon pumping The effectiveness of intra-aortic balloon was also judged jud ged by utilizing utili z ing a performance index ind ex dede (11, 12) veloped in previous studies (11,12) This scalar performance index reflects refl ects the general ge neral purposes of balloon pumping (namely, (namel y , diastolic diast olic augmentation au gmentati on of blood b lood flow to the coronary coronar y and cerebral circulations and reduction defined red uction of heart work) defi ned as:
RESULTS RESULT S The Th e system s y stem equations for the network model of Figure Fi gu re 1 were programmed on an IIBM BH 370/155 3 70 / 15 5 computer. c omputer. Typical scal scaled e d values v alues for many ma ny of the "arterial" and "heart" mode]. parameters were obtained previob tained from prev ious studies(o) Values studies (o ) Val ues for balloon resistance as a function radi us (volume) ( v ol ume) were obtained f unction of balloon radius from Brown(lo). Brown (l o ) . "Typical" "T ypical" elastance elastanc e curves for the normal human h uman heart and hearts with coronary artery disease (CAD) and hypertrophy hypertroph y were constructed (Figure (Fi gure 2) based on left ventricular ventric ular pressure-volume data* obtained at a single end-diastolic volume (V. ob ( V. d ). The left-ventricular volume data was obtained via single-plane cineangiography cineangiograph y at 60 fps and a standard ellipsoidal formula(13) formula (1 3 ) was used to *This data was supplied by Dr. Dr . James S. Cole, Cardiac Catheterization Laboratory, Lab oratory , The Methodist Hethodist Hospital, Houston, Texas.
where = scalar performance functional whe re J = f unctional which is a function (D) and f unction of two variables, v ariables , balloon delay dela y CD) duration (B). (B) . (Here, ( Here, delay dela y (D) is defined as the t he time ti me between b etween aortic valve val ve closure clos ure (T,) (T 5 ) and the onset of balloon inflation in the th e diastolic period. Balth e time interval inter val between the loon duration (B) is the actuator Signals signals for balloon inflation and deflation. deflation . p.(t) PaCt) AMDP ='
AEDP
326 32 6
= aortic (T~ ( T~
root pressure (mm Hg). Hg ). T 1I ; p. p.(( t)dt where T~ -T, -T , ) eT, T;
P.(T P. ( T)J ))
r l~
=
time at enddiastole. diastole .
[aortic end-diastolic pressure]
AMSP
= 1-
K33 K
desired arterial end-diastolic pressure (AEDP*)
Tss T
REFERENCES
jIsP.(t)dt where time t=O refers to the 0 time of aortic valve opening.
1 i f (Ks -AEDP) otherwise
o
<
1. Noordergraaf, A. (1969) "Hemodynamics" in Biomedical Engineering. H.P. Schwan, ed., pp. 391545. 2. Beneken, J.E.W. (1965) "A mathematical apph . D. Thesis, proach to cardiovascular function," ph.D. Institute of Medical Physics, Utrecht, The Netherlands. Biomech.l: 3. Snyder, M.F., et a!. (1968) J. Biomech.1: 341-353. 10: 509-515. 4. Suga, H. (1969) Jap. Heart J. !Q: 5. Suga, H. (1971) IEEE Trans. BME-18:47-55 BME 18:47-55.. M. E., J.W. Clark, D.N. Mohr, H.M. 6. Greene, M.E., 126-134 . Bourland (1973) Med. & BioI. Engr. 11: ll: 126-134. 7. Spencer, M.P. M.P . and A.B. A. B. Denison (1963) "Pulsatile Blood Flow in the Vascular Sys tern," Handbook Vol . 11, 839-864. 839-864 . of Physiology, Sect. 2, Vol. a1. (1970) Trans. Amer. Soc. 8. Jaron, D., et a!. Artif . Int. Organs 16: 466-471. Artif. 9. Buckley, M.J., et al. (1970) Circulation 41, 130-134 . Suppl. 11: 130-134. 10 . Brown, B.G. (1969) "Cardiac assistance by 10. intra-aortic balloon," Ph.D. Thesis, Johns Hopkins University . University. G. R., J.W. Clark, H.M. Bourland (1971) 11 . Kane, G.R., 11. Trans . Amer. Soc. Soc . Artif. Int. Organs 12: 148-152. Trans. 12. Clark, J.W., G.R. Kane, H.M. Bourland (1973) "On the feasibility of closed-loop control of intraaortic balloon pumping," IEEE Trans. BME (Sept. issue). J . W. Kennedy (1969) ~. 13. Kasser, E.S. and J.W. ~. Radiol. 4: 83-90. al . (1967) Invest. Invest . Radiol. 14. Kloster, F.E., et al. 353-359 . 2: 353-359. IS. Mullins, C.B., et al. (1971) Amer. J. 15. Physiol. 1lQ: 694-698.
0
Kz, and the constants K1 , Ks > 0 and K 2 , ~ < O. In equation (2) the term K1AMDP relates to the "circulatory" objectives of balloon pumping; particularly to enhanced diastolic coronary and carotid flow brought about by a maximization of this term. The KzAMSP ~o(Ks-AEDP)z term K 2AMSP and the penalty function ~o(Ks-AEDP)2 reflect the attempt to minimize heart work. With constants K ~ assigned appropriate signed K1,K 1 ,K 2z , and K4 values, maximization of J implies optimal satisfaction of the stated objectives of balloon pumping in some sense. Figure 3 indicates the model generated wave shape of aortic pressure with and without balloon pumping. In this case, the "normal" ventricular elastance curve of Figure 2 is used and the resulting diastolic pressure augmentation achieved during balloon pumping (D = 33.6 and B = 361.2 msec) is evident. In general, the performance index J(D,B) yields a unimodal isocontour surface when plotted versus delay D and duration B as seen in Figure 4. The isocontours (locus of equal values of J) indicated 151 of on the performance surface are located every 151 performance index maximum and the delay-duration times are expressed in normalized time (T) units (see Figure 4). Of course, the optimal operating point J.ax(Dopt,Bo.t) for the balloon varies with the particular elastance characteristics assumed for the failing ventricular myocardium. ACKNOWLEDGMENT The authors wish to express their gratitude to Dr J ames S. Cole, James Cole. Dept. of Medicine, Baylor College of Medicine for supplying the pressure-volume data stud y. This research was supported utilized in this study. in part by PHS Research Grant #HE-09251 from the Institute, National InstiNational Heart and Lung Institute. tutes of Health.
I
I
I
RA
I I I I
I" I I
r-----,
-t';"L1 IS: Pao
I
Re( t)
I
v\I'
1
I
Ppv PpV
Ev(Ved , t)
I RG
CLI
I
RI 1 I
I
B
I L
Pab
~,-------,
~"--I
I I I
P. (t) I PBS(t): BS I
:-----I'-----L,---_-.....L.._-_-_....,....---------' :----~------~~-----~_--_--_--J~----------~ ~
L_ L __
J
L L
BALLOON MODEL
HEART MODEL FIG. 1.
ARTERIAL LOAD
load . The arterial load also contains balVentricular model with "modified Windkessel" arterial load. loon pump model as indicated. indicated . Left ventricular (~), (~), aortic (P ao ), and abdominal aortic (P ab ab) pressures are given in units of mmHg, mmH g, resistances in units of mmHg/liter/sec, mmH g/ liter / sec, compliances in units of liter liter/mmHg, / mmH g, and inertance (L) in units of mmHg·sec mmH g ·sec 22 /1iter. / 1iter. See text for further explanation. 327
I
., . ,
2.5 2.5
:'"""
I I
I'
~.,. E
,
E
Z
,
t- 1.5 ....
,
1.0
er
t.... '"zZ
o
o
'"er !,!
\
\
~
0 .2 0.2
'"o
\
.~' --~-------_. 0.4
0 .6
0 .8 0.8
1.0 10
TI ME -SECONDS
FIG. 2.
~
J-
2
" -4' "
CS
Isocontours on the performance index surface; T = ~ t'HR'60, ~ heart rate, t ' HR'60, where HR = T = ~ normalized time in seconds; contours every e ver y 15 4 of performance index maximum;
Doopt pt
268 .8 268.B
4032
806.4
,( t) for the FIG. 3. Computed aortic root pressure p. p.,(t) normal patient with and without ventricular assistance. assistance . (D ~= 33.6 msec, B = 361.2 361 . 2 msec)
c c
FIG. 4.
o
TIME- msec T1ME-
Computed ventricular elastance curves for normal, CAD, and hypertrophy patients. Experimental elastance points were obtained ~y and V Vvy data using uSing (1) and then fit from P by a cubic spline interpolating function. function .
c: ::::: c::
50
t.... er
\
Or--f--------==~=e:~::=;:~~o
8ALLOON
'"er..."'I
\
".
BALLOON
....ot-
\
(!
..J -'
W/O
vi V> W w
I
l
>
0 .5 .......wt- 0.5
~IOO E
I I
,
If
W w
f 100
II
,1--\
'"er
:5
,"
,
w W
.,.
I I
I,
I
V>
~
I ,,
-Hypertrophy
i
II
u
-
I~
II
w
• 'I\
I
I
,,
'"E 2.0
:I:
---Normal - -- --Coronary Artery Arter'1 Disease Oisease -----Coronary
\\
~ 4; Bopt = ~ 43. =
328