Secondary Logo

Journal Logo

Special Reports

Modeling Ventricular Function during Cardiac Assist: Does Time-Varying Elastance Work?

Vandenberghe, Stijn; Segers, Patrick; Steendijk, Paul; Meyns, Bart; Dion, Robert A.E.; Antaki, James F.; Verdonck, Pascal

Author Information
doi: 10.1097/01.mat.0000196525.56523.b8
  • Free


New left ventricular assist devices (LVADs) are slowly infiltrating hospitals for the treatment of patients with end-stage heart failure. Two basic categories of devices are in use: displacement-type pumps and turbine pumps. The former operate similar to the heart in that they contain a chamber that undergoes a filling and an ejecting phase. The filling is usually passive as blood drains through a valved conduit from the left ventricle (LV) or atrium into the pump. Next, the blood is forcefully ejected in the aorta through a valved outlet conduit. Turbine pumps or “rotary blood pumps” contain an impeller that spins at a constant high speed, sucks blood out of the ventricle, and expels it into the aorta. Therefore, they do not operate in a phasic manner and contain no valves.

Prior to clinical use, the hemodynamic impact of LVADs on the native cardiovascular system is often investigated by creating a model of the device and the human circulation. The advantage of this modeling is that it is reproducible, and it can be used to simulate a broad field of different patient conditions and device status. The most popular modeling methods are mathematical lumped parameter models and in vitro models. The interaction between the LV and an LVAD is usually of great concern, and consequently, most models are limited to the LV and left atrium and an afterload. An important component of these models is the simulation of left ventricular function, where the time-varying elastance concept developed by Suga et al. is the “gold standard”.1

This theory states that the relation between pressure and volume variation in the LV is described by an elastance curve according to the following formulation:

where p(t) is the time-varying pressure (mm Hg), V(t) is the time-varying volume (ml), E(t) is the time-varying elastance (mm Hg/ml), and V0 is the volume at which the LV can no longer generate pressure (ml).

The LV is thus considered as a cardiac chamber with its stiffness changing in time, and with the intracavity pressure solely determined by instantaneous LV volume. According to early investigations, the elastance curve is independent of ventricular preload and afterload within a physiological range, and its maximum (Ees, end-systolic elastance) has been used widely as a marker of the contractility of the LV.2–7 Because it is independent of the cardiac load, each individual case has its own elastance curve that only changes when the basic contractile function of the heart changes, e.g., under the influence of inotropic drugs or disease or healing processes.8 In fact, normalization of this curve, with respect to time and amplitude, results in a general curve that is typical for the species, which is an appealing feature for modeling.9 This concept, however, was developed in the early 1970s and validated with isolated animal hearts under normal physiological conditions. During cardiac assist with displacement-type blood pumps, the heart is subjected to extreme loading conditions that vary quickly. When such a device is used in a counterpulsation manner, for instance, severe unloading occurs during LV systole, because all the blood can be “dumped” in the filling LVAD. Consequently, it should be questioned whether the original elastance theory is still valid for mechanically supported hearts. This will be illustrated using data obtained in two different settings, i.e., mechanical support using a displacement-type pump and a rotary blood pump.

Materials and Methods

Interaction with a Displacement-Type Pump

Left ventricular pressure and volume and pump data were acquired from a healthy calf that underwent an implantation of a Novacor (World Heart Corporation, Ottawa, Canada) LV assist device as part of a surgical training program in the Leiden University Medical Center (The Netherlands). The Novacor was implanted with apical to ascending aorta cannulation, and several pump modes were tried and combined with altered filling status of the calf (obtained by transferring blood to the cardiopulmonary bypass circuit) as part of the training. Apart from the routine surgical instrumentation, the animal was additionally instrumented with a hi-fi catheter tip pressure transducer (Millar Instruments Inc., Houston, TX) in the aorta and a five-segment pressure-conductance catheter (CardioDynamics BV, Leiden, The Netherlands) in the LV that provides volume measurement. The parallel conductance of tissue surrounding the LV is picked up by the catheter as a volume offset error, and its value was determined from injections with hypertonic saline (10 ml bolus of 10% NaCl solution) following recognized standard procedures.10 The V0 value of the ventricle (25 ml) was determined by extrapolation of end-systolic LV pressure and volume data at different filling status of the calf, with the pump turned off.

Interaction with a Rotary Blood Pump

Left ventricular pressure and volume and pump flow were acquired in an acute study in seven healthy sheep that underwent acute implantation of a rotary blood pump at the Center for Experimental Surgery and Anesthesiology (Leuven, Belgium). Details of the procedure were described previously.11 The particular device was a Medos microdiagonal pump (Medos Medizintechnik GmbH, Stolberg, Germany) which was used in each sheep subsequently with left atrial and left ventricular (through the mitral valve) inflow cannulation. For each cannulation, the pump was run at different mean pump flows (0–3 l/min in 0.5-l/min increments) for approximately 2-minute periods, ending with 10–20 seconds of data acquisition. Before the acquisition, it was verified that the transient effects of the speed increase had dissipated, and stable signals were present for several ventilations. Data were also acquired at baseline for each cannulation, where the device was shut off and the cannulas clamped. Each animal was instrumented with a conductance catheter (CardioDynamics BV) and a hi-fi catheter tip pressure transducer (Millar Instruments Inc.) in the LV. The parallel conductance was determined for each sheep from hypertonic saline injections, and the V0 value of the LV was determined by extrapolation of end-systolic pressure and volume data from LV filling alterations by progressively occluding the inferior vena cava at baseline.


Interaction with a Displacement-Type Pump

Figure 1 shows an LV data sequence where the Novacor had been pumping in a fixed rate mode (85 BPM) for several minutes and was subsequently switched off. The “Novacor on” part demonstrates the interaction between the native heart and the device; the asynchronous support is obvious from the repeated sequence of five beats in the LV pressure plot. These varying LV pressures indicate that the native heart is subject to quickly varying loading conditions.

Figure 1.
Figure 1.:
Aortic pressure (Ao) and left ventricular (LV) pressure, volume, and elastance measured in a calf with a Novacor assist device. The device was pumping in fixed-rate mode (85 BPM) and then switched off. The graphs show one continuous data sampling interval of 16 seconds.

The bottom panel of Figure 1 displays the elastance calculated from the acquired pressures and volumes (with V0 = 25 ml). The right hand side (“Novacor off”) shows stable elastance curves, whereas the “Novacor on” side shows sequential beats with very different maxima (Ees). A similar phenomenon was previously reported by Yoshizawa et al.7 According to the time-varying elastance theory, this would mean that the contractility of the heart changes dramatically from beat to beat. Because contractility is considered an intrinsic property of the heart muscle (its “strength”), it is expected to change over longer periods. We acknowledge that rapid contractility increase or decrease is possible as a result of pharmaceutical interventions or impaired coronary perfusion. Also, atrial fibrillation or electrical pacing may induce sudden changes in contractility as a result of postextrasystolic potentiation, where premature electrical activation results in altered mechanical contraction due to distorted cellular calcium release.12–14 In our study, however, no inotropic pharmaceuticals were used, and heart rate was maintained without external pacing. It is unlikely that the here observed magnitude and speed of elastance variation is solely the result of coronary flow variation and its impact on contractility. Moreover, there is no such variation observed in the aortic pressure (Figure 1), which indicates that the coronary flow was unaltered by the cardiac assist. A more acceptable explanation is that the observed rapid variations in elastance result from the heart–device interaction, and that the relation between elastance and contractility is no longer applicable when a second pump is present in the systemic circulation. This finding makes Ees an unreliable indicator for LV contractility with cardiac assist.

Normalization of all the separate beats with respect to time and amplitude, as shown in Figure 2, results in a series of coinciding graphs for the “Novacor off” section, whereas the “Novacor on” normalized graphs differ considerably.

Figure 2.
Figure 2.:
Left ventricular pressure–volume loops (A, top) and time-normalized elastance curves (B, bottom) derived from the data displayed in Figure 1.

Interaction with a Rotary Blood Pump

The data in Figure 3 all originate from one animal (sheep #7) with atrial inflow cannulation. For each pump flow level, several sequential beats were averaged to eliminate respiration effects and obtain the displayed curves. The elastance curves of panels B and C are normalized in time to filter out the effect of heart rate variation (range: 98–102 BPM) for better comparison. From the PV-loops in Figure 3A, it can be derived that with increasing pump flow (from baseline to 3 l/min), the end-systolic volume stays constant while the end-diastolic volume decreases and the maximum LV pressure increases.

Figure 3.
Figure 3.:
Left ventricular pressure-volume loops (A) and the according elastance curves (B) of a sheep (#7) with a Medos microdiagonal pump with left atrial inflow cannulation. Panel C displays the elastance curves after scaling to the baseline amplitude by correcting V0 for each pump flow setting (see text). The black curves show data acquired at baseline (BL) and pump flows of 0, 0.5, 1, 1.5, 2, 2.5, and 3 l/min. The curves are arranged between “BL” and “3 l/min” in that respective order. The gray dashed line shows similar data at a pump flow of 3.5 l/min in the same sheep.

The elastance curves in Figure 3B demonstrate how this results in a gradual increase in Ees. Figure 4 displays a similar effect, albeit to different extents, in all seven sheep for both atrial and ventricular cannulation. The displayed deformations of the PV-loops are obviously the result of the heart-pump interaction and the unloading of the LV by the pump. Because of the continuous nature of the rotary blood pump, there were no rapid beat-to-beat variations observed in Ees. There was, however, an increase in mean arterial pressure that was directly related to the pump flow: 61 mm Hg at baseline to 74 mm Hg at 3 l/min for this specific animal. Therefore, it is possible that increased coronary flow was responsible for the increase in contractility. However, the increase in Ees (37% for this animal) is large enough to suspect dissociation between the actual contractility and the Ees due to the presence of a second pump in the circuit. Consequently, quantification of LV contractility during cardiac assist with a rotary blood pumps should not be performed with Ees as the only measure.

Figure 4.
Figure 4.:
Summary of the effect of pump flow of the rotary blood pump on end-systolic elastance (Ees) for all studied sheep, with both left atrial and left ventricular inlet cannulation. Ventricular cannulation was not performed in sheep #3.

In the analysis of these data, V0 of each sheep at baseline was also used for the data processing of the various support levels, because controlled preload variations (vena cava occlusions, necessary for V0 calculation) could not be performed repeatedly in between the acquisitions. Therefore, one could argue that the reported Ees changes are due to shifts in V0 as a result of the assist. To investigate this, we scaled the Ees of the different curves to the baseline Ees by iteratively changing V0 (“new V0”). This meant that for higher pump flows, new V0 values below the baseline value were needed. For the data of sheep #7 (displayed in Figure 3), with a baseline V0 of 10.7 ml, a new V0 of 0.4 ml was needed at a pump flow of 3 l/min at baseline to yield the same Ees (2.03 mm Hg/ml). A linear relationship was found between the new V0 values at intermediate pump flow levels. This scaling technique revealed fundamentally different elastance curve morphologies for each support level, as depicted in Figure 3C: a flatter rising (systolic) leg for increasing pump flows. The same was concluded for all sheep, and also after normalization of the original elastance curves (all derived with baseline V0) with respect to amplitude. This indicates that higher pump flows would slow down intraventricular pressure rise.

For the particular sheep data shown in Figure 3, it was also found that increasing the pump flow beyond 3 l/min resulted in a gradual decrease of Ees and an alteration of the waveform (see dashed line). The maximum elastance point shifted to an earlier time in the cycle thus resulting in a longer time-normalized curve. This means that the cardiac assist also influences the systolic/diastolic time ratio, if Ees is assumed to indicate the end of systole. A similar phenomenon was observed in all seven sheep, for both left atrial and left ventricular cannulation, but the pump flow level, at which the waveform changes between subjects.


In most existing mathematical and in vitro models that simulate LV function, one general elastance curve is used to calculate the pressure and volume variations that have to be generated according to the original time-varying elastance theory of Suga et al.15–18 Before a simulation, that curve is usually scaled in amplitude and time to simulate different contractilities and heart rates, but during a simulation, the resulting elastance curve is fixed. The data presented above, however, indicate that the elastance curve can vary from beat to beat as a result of the interaction between the heart and a left ventricular assist device. Consequently, the original time-varying elastance theory seems not appropriate for this kind of modeling, even though it has been used extensively up to now. A more complicated left ventricular model is deemed indispensable for simulations that focus on the heart–device interaction.

Several models are readily available but have never been used or validated for simulations with LVADs. Suga et al. extended their own time-varying elastance model in 1980 to counteract for inaccuracies in pressure estimation of differently loaded heart beats.19 They, therefore, included terms that are related to instantaneous volume variation, peak ejection velocity, and the total ejected volume. To our knowledge, this model has only been used in the heart–device interaction studies by Ferrari et al.20,21 Several models have been developed that mimic left ventricular function not only by elastance, but also with additional resistive and inertial components,22–24 or by isovolumic characteristics.25 Such models yield a more accurate representation of natural cardiac pressure and volume variations, but so far they have not been evaluated for assisted hearts or used in cardiac assist models.

To determine the elastance curve of a specific heart according to the original theory, one needs pressure and volume curves under several loading conditions to determine V0 and the elastance curve. This is, however, not sufficient for a more complicated model. The elastance-resistance model, for instance, additionally requires data sampled during an isovolumic beat to determine the ventricular resistance, and the isovolumic description model requires multiple isovolumic beats at different volumes. Consequently, it is not straightforward to acquire the necessary data for accurate cardiac modeling from experiments that focus on cardiac assist. Moreover, the presence of the device and the associated mutilation of the heart (e.g., by apical cannulation) have an impact on the remaining heart function, and should be incorporated in the cardiac model. It is, therefore, recommended not only to implement more accurate LV function models in simulations, but also to base these models on data that were acquired with the device in place, optionally with clamped cannulas.

Study Limitations

It is acknowledged that our observations are based on data measured with the conductance catheter, which is an indirect technique that derives volume from measurement of conductance of the blood within the cavity, and which requires offset and amplitude calibration.10,26,27 In the presence of an assist device, the measured data may be influenced by the nonoptimal working conditions for the conductance catheter. The electric field may be affected by the presence and functioning of the device, parallel conductance may be more difficult to assess, and LV shapes may deviate from the expected geometry due to severe LV unloading.


The currently used time-varying elastance model fails to simulate left ventricular function accurately when there is interaction between the LV and a mechanical cardiac assist device. This was demonstrated by two sets of animal data for a displacement-type pump and a rotary blood pump, respectively. Further investigation of the heart–device interaction and the modeling thereof should preferably be performed with different types of assist devices and measuring equipment to assure that the here discussed results (although clear and repeatedly observed) are not due to instrumentation and calibration errors.


The authors are most grateful to Takahiro Nishida for acquiring the in vivo sheep data. This research was performed with the financial support of the Institute for the Promotion of Innovation by Science and Technology in Flanders (IWT-993171, granted to the first author).


1. Suga H, Sagawa K: Instantaneous pressure-volume relationships and their ratio in the excised, supported canine left ventricle. Circ Res 35: 117–126, 1974.
2. Elzinga G, Westerhof N: The effect of an increase in inotropic state and end-diastolic volume on the pumping ability of the feline left heart. Circ Res 42: 620–628, 1978.
3. Gorcsan J, Gasior TA, Mandarino WA, et al: Assessment of the immediate effects of cardiopulmonary bypass on left ventricular performance by on-line pressure-area relations. Circulation 89: 180–190, 1994.
4. Ishihara H, Yokota M, Sobue T, Saito H: Relation between ventriculoarterial coupling and myocardial energetics in patients with idiopathic dilated cardiomyopathy. J Am Coll Cardiol 23: 406–416, 1994.
5. Sasayama S, Asanoi H: Coupling between the heart and arterial system in heart failure. Am J Med 90:14S-18S, 1991.
6. Schreuder JJ, BiervlietJD, van der VeldeET, et al: Systolic and diastolic pressure-volume relationships during cardiac surgery. J Cardiothorac Vasc Anesth 5: 539–545, 1991.
7. Yoshizawa M, Iemura S, Abe K, et al: Parameter optimization approach to estimation of emax under cardiac assistance. In: Koyanagi H (ed), Heart Replacement: Artificial Heart 6. Tokyo, Springer-Verlag, 1997, pp. 378–381.
8. Suga H, Sagawa K, Shoukas AA: Load independence of the instantaneous pressure-volume ratio of the canine left ventricle and effects of epinephrine and heart rate on the ratio. Circ Res 32: 314–322, 1973.
9. Senzaki H, Chen CH, Kass DA: Single-beat estimation of end-systolic pressure-volume relation in humans: A new method with the potential for noninvasive application. Circulation 94: 2497–2506, 1996.
10. Steendijk P, Staal E, Jukema JW, Baan J: Hypertonic saline method accurately determines parallel conductance for dual-field conductance catheter. Am J Physiol Heart Circ Physiol 281: H755–H763, 2001.
11. Vandenberghe S, Nishida T, Segers P, et al: The impact of pump speed and inlet cannulation site on left ventricular unloading with a rotary blood pump. Artif Organs 28: 660–667, 2004.
12. Cooper MW: Postextrasystolic potentiation. Do we really know what it means and how to use it? Circulation 88: 2962–2971, 1993.
13. Brookes CI, White PA, StaplesM, et al: Myocardial contractility is not constant during spontaneous atrial fibrillation in patients. Circulation 98: 1762–1768, 1998.
14. Yamada H, Martin DO, Mowrey KA, et al: Effects of coupled pacing on cardiac performance during acute atrial tachycardia and fibrillation: An old therapy revisited for a new reason. Am J Physiol Heart Circ Physiol 285: H2630–H2638, 2003.
15. Garcia D, Barenbrug PJ, Pibarot P, et al: A ventricular-vascular coupling model in presence of aortic stenosis. Am J Physiol Heart Circ Physiol 288: H1874–H1884, 2005.
16. Sun Y, Sjoberg BJ, Ask P, Loyd D, Wranne B: Mathematical model that characterizes transmitral and pulmonary venous flow velocity patterns. Am J Physiol 268: H476–489, 1995.
17. Stergiopulos N, Meister JJ, Westerhof N: Determinants of stroke volume and systolic and diastolic aortic pressure. Am J Physiol 270: H2050–H2059, 1996.
18. Vollkron M, Schima H, Huber L, Wieselthaler G: Interaction of the cardiovascular system with an implanted rotary assist device: Simulation study with a refined computer model. Artif Organs 26: 349–359, 2002.
19. Suga H, Sagawa K, Demer L: Determinants of instantaneous pressure in canine left ventricle. Time and volume specification. Circ Res 46: 256–263, 1980.
20. Ferrari G, Nicoletti A, De Lazzari C, et al: A physical model of the human systemic arterial tree. Int J Artif Organs 23: 647–657, 2000.
21. Ferrari G, De Lazzari C, Kozarski M, et al: A hybrid mock circulatory system: Testing a prototype under physiologic and pathological conditions. ASAIO J 48: 487–494, 2002.
22. Shroff SG, Janicki JS, Weber KT: Evidence and quantitation of left ventricular systolic resistance. Am J Physiol 249: H358–370, 1985.
23. Campbell KB, Kirkpatrick RD, Knowlen GG, Ringo JA: Late-systolic pumping properties of the left ventricle. Deviation from elastance-resistance behavior. Circ Res 66: 218–233, 1990.
24. Campbell KB, Ringo JA, Knowlen GG, et al: Validation of optional elastance-resistance left ventricle pump models. Am J Physiol 251: H382–H397, 1986.
25. Palladino JL, Rabbany SY, Mulier JP, Noordergraaf A: A perspective on myocardial contractility. Technol Health Care 5: 135–144, 1997.
26. van der Velde ET,van Dijk AD, Steendijk P, et al: Left ventricular segmental volume by conductance catheter and cine-ct. Eur Heart J 13 Suppl E:15-21, 1992.
27. Baan J, van der Velde ET, de Bruin HG, et al: Continuous measurement of left ventricular volume in animals and humans by conductance catheter. Circulation 70: 812–823, 1984.
Copyright © 2006 by the American Society for Artificial Internal Organs