Mock circulatory systems (MCS) are very important tools for designing and testing cardiovascular prostheses.1–7In-vitro tests provide a better control of the working conditions with respect to animal tests while performing experiments. However, it is hard to correctly reproduce input and output hydraulic loads for the prostheses under testing. This is mainly due to difficulties arising when hydraulic components must mimic mathematical models describing the cardiovascular system, since every hydraulic component is affected by spurious effects.8 Moreover, if different devices have to be tested by means of the same MCS, or significant parameters have to be changed in real-time, costs and time to set up the MCS will rise significantly. In order to enhance and exploit the advantages of numerical simulations, even in the presence of hydraulic components and prostheses, new test benches aimed at obtaining an elastance-based behavior have been conceived.9–15
The solution reported in this work presents a modified left ventricular (LV) elastance model to control in real-time a mock ventricle, consisting of a valved piston-cylinder mechanism, which is able to reproduce the correct ventricular pressure- volume relationship of a native ventricle. The adopted solution represents the development of a previous work11 and overcomes the intrinsic limitations arising when fluid actuators are used to mimic the native LV in MCSs. Indeed, the presence of water hammer effect makes the classical elastance model16,17 unsuitable unless using mechanical filters. Water hammer is a pressure surge resulting when a fluid in motion is forced to stop. It commonly occurs when a valve is closed or opened suddenly, and caused by the conversion of kinetic energy to compressed-fluid energy.18
In this study, both numerical simulations and experiments on a MCS were carried out. In the former case, the modified elastance model performance, in terms of physiological requirements, was compared to one of the classical ventricular models.16,17 Next, the modified elastance model was used to drive in real-time the mock ventricle connected to a closed-loop hydraulic circuit for in-vitro experiments.
This system, comprising the systemic circulation and the left heart, allows modification of working conditions in real-time by varying the numerical parameters of the heart and/or those belonging to the hydraulic vascular part. The mathematical model of the heart is implemented on a personal computer. Thus, a very realistic closed-loop cardiovascular system, characterized by the correct interactions and variations of the main cardiovascular parameters, was obtained.
Materials and Methods
The LV Modified Elastance Model
In native hearts, the ventricular pressure-volume relationship during the filling phase is extremely flat. If the classical elastance model is used to model the diastolic pressure-volume relationship, high volume oscillations occur even in the presence of very modest pressure disturbances. Instead, pressure oscillations inside the native ventricle are dampened by the viscoelastic properties of the ventricular tissue and by the inertial effects of walls and intraventricular blood.19–22 On the contrary, mock ventricles made up of rigid pumping chambers are not provided with the same damping properties, while pumps with spring-like chambers (i.e., Bellofram pump) or pneumatic ventricles do not provide the possibility to exactly control the ventricular volume. In order to overcome these difficulties, a modified version of the classical Suga-Sagawa elastance model,16 enriched with the resistive term from Campbell et al.,17 was conceived. The basic mathematical model is as follows:
where t is the time, PLV and VLV are respectively the left ventricular pressure and volume, P0 and V0 are respectively the ventricular pressure and volume at the pivotal point of the elastance function E(t), Ri is the ventricular internal resistance acting during the ejection phase.
Modifications introduced into Equation (1) are: first, a nonlinear time-varying elastance model25–27; second, resistive and inductive terms (Ri(t) and Li, respectively) throughout the whole cardiac cycle. The ventricular pressure-volume relationship is thus described by the following equation:
Here, the ϕ[VLV(t),t] function describes the nonlinear pressure-volume relationship. It is dependent on the normalized contraction function fiso(t), whose value varies between 0 and 1 depending on the cardiac muscle activation. More details can be found in a previous work from our group27 or in Appendix. All parameters are listed in Table 1.
In Equation (2) the novelty resides in the last two terms on the right-hand side, both describing the viscoelastic behavior of the cardiac muscle. Their physiological meaning can be elucidated by means of stress/speed measurements of the cardiac sarcomere as well as by means of energetic considerations.19,20,28 In the classical elastance model the internal resistance Ri(t) is equal to zero throughout the whole cardiac cycle except during the ejection phase,17 while it has now the following mathematical law:
being RiMIN, RiMAX, and KRi appropriate coefficients (Table 1). Ri(t) varies between a minimum value (RiMIN) when fiso(t) = 0, and a maximum value when fiso(t) = 1. The maximum value is set to either (RiMIN + KRi) or RiMAX, being this latter the saturation value. Furthermore, the inertial term Li, not present in the classical model, was chosen equal to 0.0011 mm Hg × s2/ml.
Both the Ri(t) law and the Li value were empirically selected so that the R/L ratio resulted smaller during the filling phase (during which the sensitivity to disturbances is higher and the filtering action must be stronger) with respect to the ejection phase. Additional details regarding the modified elastance model filtering action can be found in Appendix.
The rationale for describing the ventricular pressure-volume relationship in the form of Equation (2) comes from what we experienced in our laboratory: the filling phase of the cardiac cycle produced abrupt vibrations of the mock ventricle. Indeed, the diastolic pressure-volume relationship is described by a low-slope mathematical entity (that is EMIN–Appendix), thus it was inferred that a very small pressure disturbance produced very large volume oscillations. This concept can be better explicated in the time-domain, by looking at the waveforms in Figure 1 obtained numerically. This Figure shows a comparison between the VLV calculated from both the classical elastance model and the modified elastance model. The two models behave alike when there is no disturbance (dotted lines), instead they behave in a different manner (solid lines) when the same sinusoidal pressure disturbance (amplitude 1 mm Hg and frequency 15 Hz) is applied. Indeed, it is shown that the VLV calculated from the classical model is very sensitive to the disturbance, especially during the filling phase, while the modified model filters out its effects.
Vascular Numerical Model and Simulations
The modified elastance model was numerically tested with the aim of elucidating its correct preload-afterload sensitivity. The results were thus compared to those obtained with the classical elastance model.
The two lumped parameter LV models were alternatively connected to an open-loop circulation, whose electric analog is shown in Figure 2. (Electric analogy was used to implement and solve the differential equations of the circuit).
The active atrium is described by a time-varying elastance model as follows:
The same equations as those reported in Appendix hold for the atrium as well, but with different contraction function27 and parameters as reported in Table 1 (the subscript A stands for atrium).
The arterial systemic afterload is modeled by a five-component Noordergraaf model.29 The five components for this model are: LCS, RCS, C1, CAS, RAS. Except the latter parameter, they were all tuned in physiologic conditions to obtain pressure and flow rate waveforms in agreement with Guyton’s study.30 The venous systemic compartment is modeled according to the Guyton’s model,31 but the venous compliance was replaced by a pressure source, the mean systemic pressure (PMS), which is assumed as the main determinant of ventricular preload. Both the systemic arterial resistance (RAS) and the systemic venous resistance (RVS) undergo a simple baroreflex control mechanism27: the mean aortic pressure (PAO_ave) and the mean atrial pressure (PLA_ave) are constantly compared to set point values properly adjusted to determine specific working conditions.
Ventricular inlet and outlet valves are modeled as inductors (Lv) in series connection with diodes characterized by low direct resistance (RVDIR) and high inverse resistance (RVINV).26,32 Several numerical simulations were carried out and four representative working conditions are reported for comparing the two models. These are: physiologic contractility, afterload increase (i.e., PAO_ave increase), preload increase (i.e., PLA_ave increase), and pathologic contractility. A systolic left heart failure was reproduced by setting: increased heart rate (HR); higher filling pressure (PMS) of the cardiovascular system; lower mean aortic pressure (PAO_ave); lower maximum isovolumic pressure (P*) with respect to the basal physiological value (Table 2 and Table 3). It is worth noting that the parameter P* was used to locate the end-systolic pressure-volume relationship (ESPVR) on the PV plane, thus mimicking different LV contractility conditions.
All parameters were chosen from literature data and then fine-tuned to reproduce correct cardiac output and venous return curves as reported in Guyton et al.30 and Sun et al.32
Experimental Setup and Experiments
A detailed description of the experimental setup as well as the characterization of its hydraulic components, were already reported elsewhere.11,33Figure 3 shows a top view of the MCS. Briefly, the mock ventricle is made up of a screw-driven piston-cylinder mechanism. The volume controlled pumping system, with bandwidth equal to 40 Hz, is able to provide a stroke volume up to 200 ml, and is driven by a direct current (DC) motor (Maxon RE 35) servo controlled by a four quadrant driver (Maxon ADS 50/5). The piston position is the control variable and is monitored by a built-in encoder. The hydraulic afterload and the venous return were reproduced by a modified Windkessel8,34 and Guyton’s model,31 respectively. The ventricular afterload (i.e., PAO_ave) is varied by increasing or decreasing the systemic arterial resistance, RAS. This latter is made up of a screw-clamp. The ventricular preload (i.e., PLA_ave, and thus the end-diastolic volume VED) is varied by changing the fluid content of the closed loop circuit, which in turn determines the mean systemic pressure, PMS. This closed loop setup represents a hydraulic simulator of the systemic circulation with left atrium. Passive spring-plate inlet and outlet valves, different from the rubber type previously used,11 are lodged in an appropriate support. Three pressure transducers (Honeywell International Inc.) measure pressure in the ventricular chamber, as well as in the arterial systemic and atrial compliances. A Sensoray DAQ board (mod.626) was used for all the signals with sampling period equal to 1 millisecond. The real time application was implemented by using Matlab-Simulink (The Mathworks©) and the Real Time Windows Target platform.
A total of four continuous run experiments based on variations of afterload (obtained by means of Ras variations) and preload (obtained by means of PMS variations) in presence of physiologic as well as pathologic ventricular contractility are shown. In presence of physiologic contractility, once a control condition was reached (HR = 60 bpm), the two following experiments were carried out 1) Afterload increase at constant preload: the PAO_ave was increased by clamping the RAS, while the PLA_ave (and therefore the VED) was kept constant by temporarily depleting fluid from the circuit (i.e., PMS decrease). 2) Preload increase at constant afterload: the PLA_ave (and therefore the VED) was increased by inoculating fluid into the circuit (i.e., PMS increase), while the PAO_ave was kept constant by temporarily loosening the RAS.
In presence of pathologic contractility, after a control condition was reached (HR = 100 bpm), the two following experiments were carried out 1) Afterload decrease: the PAO_ave was decreased by completely loosening the RAS clamp, thus lowering the RAS to its lowest value. 2) Preload decrease: the PLA_ave (and therefore the VED) was decreased by continuously depleting fluid from the circuit (i.e., PMS decrease).
This MCS is made up of a LV mathematical model implemented on a PC which drives in real time a volume controlled piston pump connected to a closed loop hydraulic circuit.11
The experimental setup mimics the hybrid test bench philosophy as reported in our previous work.11 However, in this work, two important aspects were improved: a more suitable LV mathematical model (see previous section), and a simpler pump controller.
Figure 4 shows a block diagram describing the present adopted control strategy. The light-dashed zone contains the numerical part of the setup. It communicates with the hydraulic part by means of A/D-D/A converters (blocks placed on one border of the light-dashed zone). The pressure measured into the mock ventricle is fed back into the LV mathematical model and the calculated reference volume (VLVREF) is then used to drive the piston position by means of a PI controller, whose parameters’ values are shown in the same Figure (thick-dashed box). Values of HR and contractility (i.e., value of P*) are inputs to the mathematical model, while the two parameters RAS and PMS are determined by the actions taken on the hydraulic circuit. Unit conversions are described by the Ks blocks; all the others blocks are self-explanatory.
For both numerical and experimental validation, the following parameters were monitored: the end-diastolic and the end-systolic LV volume (VED and VES, respectively); the mean LV volume (VLV_ave) and stroke volume (SV); the ejection fraction (EF); the LV flow rate (Q_ave); the mean aortic pressure (Pao_ave) and the mean atrial pressure (PLA_ave); the first derivative of the LV pressure (dP/dt). Values of the monitored quantities were compared to those reported in in vivo studies.16,31,35Tables 2 and Table 3 list the corresponding values. Input parameters’ values, such as HR, P*, PMS, and RAS are also provided. Numerical results are plotted shrunk in the PV loops only (Figure 5). Instead, for the experimental results PV loops as well as pressure and volume waveforms are shown (Figure 6 and Figure 7).
For each experiment, the RAS value was calculated as the ratio of the pressure difference (PAO_ave − PMS) over Q_ave. The dP/dt values reported in Table 3 were calculated as the ratio of the pressure rise over time during the systolic isovolumic phase.
The aim of the simulations was to prove the suitability of the modified elastance model in terms of physiological requirements, i.e., its respect of the Starling law of the heart, ahead of its use on the mock circulation.
Figure 5A (control condition) shows a comparison between PV loops obtained from both the classical elastance and the modified elastance model in presence of physiologic contractility. In virtue of the presence of the two dissipative terms Ri(t) and Li, the two parameters P* and P0 were properly set (Table 2) in order to match the working conditions of the two models. Figure 5B shows the PV loops obtained from the modified elastance model before and after an afterload increase in presence of physiologic contractility. The PAO_ave set point value was increased from 100 to 120 mm Hg (Table 2), thus producing what follows: a decrease of Q_ave, SV, and EF; an increase of dP/dt, VES, and VLV_ave. Due to the open loop configuration, the preload PLA_ave was controlled, and thus the VED remained almost constant.
Figure 5C shows the PV loops obtained from the modified elastance model before and after a preload increase in presence of physiologic contractility. The PLA_ave set point value was increased from 7 to 9 mm Hg (Table 2), thus producing what follows: an increase of Q_ave, dP/dt, VED, VLV_ave, SV, and EF. Due to the open loop configuration, the afterload PAO_ave was controlled and thus the VES remained almost constant.
Figure 5D shows the PV loops obtained from the modified elastance model in presence of pathologic contractility. In order to reproduce such a condition, the input parameters were varied as reported in Table 2, and the following was obtained: dP/dt, SV, and EF all decrease; VES, VED, and VLV_ave all increase. Q_ave resulted from all the autoregulation mechanisms.
Figure 6 shows the PV loops recorded during four experiments carried out on the mock circulation. For each experiment (from A to D), the starting control condition as well as the steady state reached after the working condition change are indicated by the arrows. All the PV loops describing the evolution from one condition to another as well as the end-diastolic pressure-volume relationship (EDPVR) and the ESPVR curves are plotted. Parameter values and results coupled with each plot are reported in Table 3. For the same four experiments, in Figure 7, pressure and volume waveforms versus time are plotted before and after each working condition change. In the same Figure, this instant is indicated by means of black vertical lines together with a tag describing the corresponding action taken. On the left hand side of Figure 7, LV pressure, arterial systemic pressure, and atrial pressure are shown together, while the ventricular volume is on a separate plot. A zoom in of the same waveforms is also plotted on the right hand side.
Figure 6A shows the afterload increase experiment in presence of a ventricle characterized by physiologic contractility. Starting from a control condition (arrows on the plot), the RAS was clamped while keeping about constant VED (i.e., constant PLA_ave). Results show that the system behaves according to the Starling mechanism: due to the PAO_ave increased, the VES increased and thus the SV decreased. As a consequence, the Q_ave and the EF both decreased. Figure 7A (left) shows how the waveforms evolve after the RAS increase was taken. In Figure 7A (right), both rough and filtered LV pressure signal are superimposed (respectively, gray and black lines pointed by the Plv arrow). A low-pass filter with cutoff frequency equal to 32 Hz was used to obtain the filtered waveform. For all the other cases, only the filtered signal is plotted.
Figure 6B shows the preload increase experiment in presence of a ventricle characterized by physiologic contractility. Starting from a control condition (arrows on the plot), fluid was introduced into the circuit, thus causing a PMS increase and as a consequence a PLA_ave increase, while the RAS clamp was temporarily loosened for keeping the VES constant. Results show that the system behaves according to the Starling mechanism for this case as well: the SV increased because the VED increased as a consequence the PLA_ave increased. In turn, the Q_ave and the EF both increased. Figure 7B (left) shows the waveform trend during the same experiment, and some zoomed in cycles once the new steady state was reached (right).
The control condition for the two experiments characterized by pathologic contractility (Figure 6C and D–Figure 7C and D) was obtained by properly adjusting, with respect to the physiologic condition, the experiment control parameters: via software, HR and EMIN were increased, while P* was reduced; by acting on the mock circulation, PLA_ave was increased and PAO_ave reduced, by means of variations of PMS and RAS, respectively (Table 3).
In Figure 6C the afterload decrease experiment is shown. Starting from a control condition (arrows on the plot), the RAS was loosened in order to determine its lowest value. The effect of this variation can be compared with the one resulting from an afterload reduction26,27,36–38: the PAO_ave decreased, and so did the VES; the PLA_ave and VES both decreased; the SV increased, and as a consequence the Q_ave increased as well; the EF improved. Figure 7C (left) shows the waveforms’ trend during the same experiment, and some zoomed in cycles once the new steady state was reached (right).
In Figure 6D, the preload decrease experiment is shown. Starting from a control condition (arrows on the plot), water was depleted from the hydraulic circuit in order to obtain a PMS decrease. The following can be observed: the PAO_ave, the PLA_ave, the VES, the VES, the SV, and the Q_ave all decreased, while the EF increased. Figure 7D (left) shows the waveforms’ trend during the experiment, and some zoomed in cycles once the new steady state was reached (right).
Since the aortic pressure, as well as the atrial pressure, was measured at the top of the correspondent compliances, then their oscillation-free waveforms (Figure 7) were due to the filtering action of the air.
Numerical and experimental tests were carried out with the aim of showing that the proposed modified elastance model, as well as the classical elastance model,16,17 behaves according to the Starling mechanism.
Numerical as well as experimental results show that the modified elastance model is able to mimic the native LV preload and afterload sensitivity as observed in vivo. In particular, in case of an afterload increase (Figure 5B and Figure 6A), the SV and thus the cardiac output both decreased due to the VES increase.16,39–41 In case of a preload increase (Figure 5C and Figure 6B), the SV and thus the cardiac output both increased due to the VED increase.16,42–44 Moreover, pathologic conditions (Figure 5D and Figure 6C and D) showed typical hemodynamic consequences.16 Values in Table 2 and Table 3 further quantify these phenomena.16,30,31
It must be said that the use of the classical elastance model when numerical simulations are to be performed could be much preferable, since the modified model introduces the additional constant parameter Li, while Ri(t) is not zero throughout the whole cardiac cycle. Nevertheless, in this study the focus was put on proving the feasibility of an alternative architecture for the classical elastance model to be used on MCSs and not on the exact identification of the parameters Ri(t) and Li. The need for an alternative model comes from the evidence, experienced in our laboratory,45 that the classical elastance model is unsuited for translating mathematical models into hydraulic applications in virtue of the presence of water-hammer effect, not naturally dampened as it may happen in biological tissues.
Real-time control of mock ventricles falls within the framework of hybrid systems conceived to study the correct ventricle- environment interaction.9–15 Different solutions were adopted. Baloa et al.9 conceived a pressure controlled bellows pump driven in real time by the time-varying elastance function (real time control of the pressure-volume relationship). They tested the mock ventricle responsiveness to changes in preload, afterload, and contractility by means of PV loops analysis. It must be noted that the presence of an elastic diaphragm inside the ventricular chamber filters the pressure disturbances, although makes difficult to exactly measure the instantaneous pump volume. Gwak et al.12 showed experimental preliminary studies on an elastance-based MCS equipped with gear pumps as fluid actuators. The gear pumps were controlled to mimic the principal active and passive elements of the circulatory system, thus reproducing the impedances of the cardiovascular system. This solution was successfully tested on a numerical basis, but the mock loop experiments (limited to only one working condition) showed the strong limitation of pronounced pressure ripples due to the type of pump used. Ferrari et al.10 showed the feasibility of the interaction between hydraulic and numerical environment. This interaction was obtained by electro-hydraulic interfaces driven by numerical simulations and mimicking appropriate flow and pressure signals at the connections with the hydraulic environment. The main limitation of their work is the absence of an explicit real time control of the elastance function: the LV pressure-volume relationship is controlled by adaption to a fixed volume reference signal which does not change in real time with the working conditions. Rutten et al.15 used a pressure controlled positive displacement pump driven in real time by a Hill-type heart contraction model (instead of an elastance-based model). A baroreflex heart rate control mechanism was included as well. In this work, the mock circulatory system interaction with different types of LVADs (pulsatile and continuous flow) was analyzed with appreciable results.
Results of the present work demonstrate that the modified elastance model can be exploited for controlling mock ventricles connected to MCSs. Indeed, the experiments carried out show its correct sensitivity to preload and afterload, and the capability to reproduce different scenarios and working conditions (Figures 6 and 7).16,30,31 This is helpful for investigating the LV interaction with the environment connected to it. Indeed, any passive or active cardiac assist device modifies the cardiovascular physiology. In order to quantify the effects of their insertion and/or eventual improvement to their design, in vitro experiments must mimic real scenarios. One example could be the development and test of a VAD control strategy: the correct sensitivity to preload and afterload of the mock ventricle is a key issue to approach reliable solutions. However, this is not achievable with present mock ventricles commonly used world wide since they are affected by limited preload sensitivity and unsatisfactory LV mechanics.46
The PI controller performance was tested by means of a sinusoidal volume reference signal with amplitude equal to 33.3 ml and frequency equal to 120 bpm (the MCS was fully loaded with water). The root mean square error (RMSE) between the reference signal and the measured position was equal to 7.98 ml, the normalized root mean square error (NRMSE–with respect to the peak-to-peak value of the reference volume signal) was equal to 0.12, the maximum error was equal to 13.1 ml, and the delay in correspondence of the maximum error was equal to 35 ms. The controller responsiveness completely fulfilled the system dynamic requirements.
Some aspects, all related to the valves used, have to be better investigated and solved. As shown in the zoomed-in cardiac cycles (Figure 6 and Figure 7–right), low frequency pressure oscillations exist during the ejection and the filling phases (i.e., the valve itself acts as a mass-spring oscillating system). These oscillations reflect the undampened motion of the passive spring-plate valve during the same phases. Presently, the pump outlet is connected to a three-way conduit which has passive spring-plate valves mounted in two lateral branches. The length of the connections together with their curved path respectively increase inertial and resistive pressure drop through the valves. Valves’ dynamical behavior as well as their fluid dynamics should be improved by using clinically available artificial valves and by a better design of their lodging conduits. The latter should be characterized by a direct connection between the chamber of the mock ventricle and the inlet and outlet valve.
Regarding the deviation from the end diastolic pressure-volume curve (Figure 6), this can be reasonably explained by considering the dynamic nature of the filling phase. In vivo measurements showed that the LV pressure-volume data during the filling phase does not strictly lie on the EDPVR.47–49 Furthermore, negative pressure values during the same phase are due to the considerably high direct resistance of the inlet valve. A similar effect was observed in vivo, to demonstrate diastolic suction in the human ventricle, during balloon inflation experiments aimed at occluding the mitral orifice.50 The present elastance-based control strategy, even in the presence of the mentioned setup limitations, behaves accordingly. By improving the valves’ performance, the mock ventricle will work in less critical conditions, thus better results should be obtained.
The overall pump hydrodynamics has also a strong relationship with the best choice of the two parameters Ri(t) and Li (the latter can even be modeled as time dependent). Finally, the screw-clamp resistance for the RAS could be substituted with an electronically controlled resistance to improve the repeatability of the experiments. This could also be exploited to implement baroreflex control mechanisms acting on the systemic arterial resistance.
The presented modified elastance model was validated on a numerical basis and then used to control a mock ventricle in real-time, where sudden pressure disturbances represents a key issue and are not negligible. A physiological filtering action implied in the LV model19–22 was used so that, especially during the LV filling phase, the LV pressure disturbances did not cause large LV volume oscillations (typical if the classical elastance model is used): the resistive and the inertial terms added to the classical elastance model carry out this filtering action. Although the hydraulic components, and in particular the valves, must be improved to overcome the actual experimental setup limitations, this modified elastance model proved to be a very good candidate to control elastance-based mock ventricles in real-time in order to reproduce a correct LV-preload-afterload interaction. Therefore, such a system can be used for better designing and testing of cardiovascular prostheses.
Considerations on the Filtering Action of the LV Modified Elastance Model
In order to analyze the filtering action of the modified elastance model, the relationship between ventricular pressure and volume of the two models (classical and modified) are compared. The ϕ[VLV(t),t] pressure-volume relationship of Equation (2) is represented by the following nonlinear equation27:
Here, ϕa[VLV(t)] describes the end-systolic pressure-volume relationship (ESPVR) by means of a parabolic function, ϕp[VLV(t)] describes the end-diastolic pressure-volume relationship (EDPVR) by a combination of a linear relationship and an equilateral hyperbola, and fiso(t) represents the normalized contractility function. The ϕ a[VLV(t)] and ϕ p[VLV(t)] equations are:
K, VSAT, P*, and V* are, respectively: the EDPVR LV saturation coefficient, the LV saturation volume, the maximum pressure that can be developed by the LV (parameter used to modify the LV contractility), the volume-coordinate in correspondence of P*.
To simplify the analysis of the filtering action, let us linearize the pressure-volume relationship in (A 1) and consider only the filling and ejection phase (i.e., when fiso(t) = 0 or fiso(t) = 1, respectively). Indeed, when fiso(t) = 0 or fiso(t) = 1, ϕ[VLV(t),t] reduces to ϕp[VLV(t)] or ϕa[VLV(t)] only. These two latter relationships can be simplified by approximating them by secant lines connecting the (P0,V0) point and the end-diastolic/end-systolic pressure-volume coordinates, respectively. These lines have slopes equal to EMIN and EMAX.
During ventricular filling, the analysis is therefore simplified by considering that fiso(t) is equal to zero, the internal resistance is equal to RiMIN, the ϕ[VLV(t),t] function is approximated by the value EMIN, and P0 and V0 are set equal to zero. In the Laplace domain, Equations (1) and (2) can be respectively represented by the two following transfer-functions between ventricular pressure and volume:
Both the Equations (A 4) and (A 5) have the same DC-gain, which is equal to 1/EMIN (i.e., 66.7 ml/mm Hg). Being the classical model a static one, sensitivity to pressure disturbances results high due to this gain. On the contrary, the modified model is a dynamic one and it works as a second order low-pass filter with cutoff frequency of about 0.28 Hz.
The analysis becomes more complex during the ventricular ejection phase mainly because the fiso(t) function is not constant. This means that the elastance term in Equations (A 4) and (A 5) oscillates now between the minimum value, EMIN, and the maximum value, EMAX, and the same does the resistive term, which varies according to Equations (3) and (4). During the ejection phase, the DC-gain value is, therefore far lower (being EMAX about one hundred times bigger than EMIN) and the cutoff frequency, calculated when fiso(t) = 1, is equal to about 3.48 Hz. The modified elastance model can be thus seen as a low-pass filter whose DC-gain and cutoff frequency both vary during the heart cycle. However, by considering the fiso(t) values of 0 and 1 as bound conditions for the filling and ejection phases, the presented simplified analysis shows that disturbances with frequencies of about 50 Hz result attenuated of 42 dB for both the bound conditions.
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