A physiological controller using long-term reliable signals is essential for the permanent implantation of rotary blood pumps (RBPs). Previously, we proposed a state-space-based adaptive controller for RBPs.1,2 Incorporating an adaptive estimation algorithm, the controller was able to estimate the varying aortic pressure in different activity levels and different levels of left heart failure. As the estimated aortic pressure follows a reference signal, the blood pump responds automatically to the varying demands of physiological perfusion. In the previous work, the controller used the actual values (without noise) of pump pressure rise (ΔP) and pump speed (ω) as feedback signals. Both feedback signals were obtained through long-term reliable sensors. Specifically, the pump pressure rise (ΔP) was derived from the axial position of impeller.1 This approach is applicable for magnetically levitated axial flow pumps, but not for other type of blood pumps, such as pumps with contact bearings.
In this article, we use computer simulation to study the feasibility of using other intrinsic pump parameters such as pump current (I) as feedback signals. Measurement noises were modeled as random variables and were added to the feedback signals. The adaptive controller was implemented for two pumps, MY2 pump and Nimbus pump. MY2 pump simulates the magnetically levitated axial flow pump. It uses pump pressure rise (ΔP) and pump speed (ω) as feedback signals, which include measurement noise up to ±10% and ±5% of their actual values, respectively. Nimbus is an axial flow pump with contact bearing. It uses pump current (I) and pump speed (ω) as feedback signals with measurement noise up to ±5% of their actual values. The level of measurement noise is conservative compared with commercially available current and speed sensors, which have a lower level of measurement noises.3 By comparing the controller’s performance for these two pumps, we can verify if the adaptive controller is implementable using pump intrinsic parameters and if its performance is affected by measurement noise.
Numerical models of the human cardiovascular system have been widely used to study the interaction between the human circulation system and blood pumps.4–6 A similar model has been developed by the author, which innovatively includes the muscle pump function.7 In this model, the pressure and volume relation of the ventricle was described by a scaled quadratic-type equation in systole and an exponential-type equation in diastole. The variation of heart function in rest, exercise, or heart failure can be simulated by adjusting the parameters of these two equations. The muscle pump is modeled as a collapsible tube8 compressed by an intramuscular pressure resource.8–10 This numerical model is used in this article to study the performance of physiological controller. The blood pump is placed between the left ventricle and the aorta, exactly where it will be placed in implantation surgery as a left ventricular assist device.
Modeling of Rotary Blood Pump
Rotary blood pumps are hydraulic pumps driven by electrical direct current (DC) motors. The characteristics of DC motors can be described as:
where J is the mass moment of inertia of the impeller; ω is the motor/pump speed; K is the torque coefficient of motor; I is the motor/pump current; B is the damping coefficient (e.g., coming from the bearing); T load is the load torque on the motor, which is also the hydraulic power to generate the pump flow Q, and pump pressure rise ΔP. Because of their small sizes, RBPs usually have small inertia J. T load can be further described by the following equation:
where η is the pump efficiency, which can be found through a lookup table of ΔP, ω, and Q. In the motor electrical equation of MY2 pump, the inertia J is neglected. B is also negligible because MY2 pump simulates a magnetically levitated pump. The load torque of Nimbus pump was found through experiment to be a function of pump speed ω and pump flow Q, as in Equation 3.11
The hydrodynamic characteristics of RBPs (i.e., the relationship between the pump pressure rise ΔP, the pump flow rate Q, and the pump speed ω) can be described by either dynamic equation (including differential of some variables),11
or static equation (second or third order polynomial equations).1
Here, b i (i = 0, 1, 2) and αi (i = 0, 1, 2) are some constants for specific pumps. Static equation is more like the steady-state version of dynamic equation, which ignore the effect of hydraulic loss when the pump speed is varied. As shown in Table 1, MY2 is a pump modeled by the static hydrodynamic equation and Nimbus pump is depicted by a dynamic hydrodynamic equation.
One feature (advantage) of the Nimbus pump is that only one hydrodynamic variable (i.e., pump flow Q) is present in the motor characteristic equation. So, Q can be estimated from the measurement of the current I and the speed ω. Q can be further employed in the hydrodynamic characteristic equation to derive the other hydrodynamic variable ΔP. This provides a method for controlling the blood pump using the measurement of the current and speed.3 However, this method is not applicable for the MY2 pump as both hydrodynamic variables (ΔP and Q) are coupled together in the motor characteristic equation. Only when one hydrodynamic variable is measured can the other hydrodynamic variable be derived. Combining the hydrodynamic equation with motor electrical equation, we can find that the physical variables adjusted by the controller for MY2 and Nimbus pumps are the voltage V and the current I, respectively.
Feedback Signal (Signal Processing of ΔP)
Previous study showed that the developed adaptive controller can estimate the mean aortic pressure using the feedback signals of the pump speed ω and the maximum pump pressure rise in each heartbeat ΔP max.1 This control algorithm is based on the findings that the trajectory of ΔP max follows the response of a linear state-space system, whose states are linearly related to the mean aortic pressure. The estimated aortic pressure can then be used by the adaptive controller to adjust the pump speed/flow so that the aortic pressure will follow a reference signal. It is worth noting that maximum pump pressure rise ΔP max occurs in diastole in each heartbeat. The benefit of using ΔP max instead of the whole trajectory of pump pressure rise ΔP max as feedback signal is the former approach eliminates the unknown effect of heart contraction (i.e., systole). However, the measurement of ΔP max can be distorted by measurement noise. So in this article, we propose to use the pump pressure rise in the diastole portion of each heart beat ΔP diastole instead. This processing will improve the controller’s noise rejection capability while maintaining the validity of the developed linear state-space model (i.e., eliminating the effect of heart contraction in systole). An upper limit of ±100 mm Hg/s for the first order differential of ΔP along with a lower limit of 80 mm Hg for ΔP was found to be reasonable criteria to extract the diastolic portion. The resultant signal may miss a small portion of diastole or include a small portion of systole depending on the heart condition. Nonetheless, these small portions can be ignored because of their negligible effect on the general trend of ΔP diastole. The resultant ΔP diastole was processed through a 10-point moving-average filter to obtain a continuous signal, which is used as the feedback signal of the adaptive controller. Because of the delay effect of moving-average filter, a lag is expected in this feedback signal.
Adaptive Control Algorithm
The structure of the developed adaptive speed/flow controller is illustrated in Figure 1. Previous studies show that the response of human circulation system can be approximated by a linear state-space model. The states of this state-space model are defined as: LVPdiastole − AoP, VP − AoP, CO, where LVPdiastole is the left ventricular pressure in diastole, AoP is the aortic pressure, VP is the venous pressure, CO is the cardiac output. Given the states of the state-space model, the aortic pressure can be derived.1 Based on this state-space model, the adaptive estimator can estimate total peripheral resistance (TPR), states of the linear state-space model, and the mean aortic pressure. The estimated aortic pressure is adjusted by the adaptive controller to follow a reference signal. The feature of this adaptive controller is that it is not a fixed value feedback controller. The linear state-space model and the controller gain are updated as TPR changes. Because of this feature, the estimation error of aortic pressure and control effort can be minimized. The error between the estimated aortic pressure and the reference signal, together with the estimated states and the updated controller gain, is given to the controller to regulate the control variable (V for MY2 and I for Nimbus).
The reference signal in Figure 1 is not a constant value. Higher reference signal corresponds to higher pump flow if other conditions remain the same. The reference signal is updated by a nonlinear function of the feedback signal. This nonlinear function is selected in such a way that when the volume of left ventricle increases the reference signal will increase. The rationale of this function lies in the positive relation between the preload (i.e., the end-diastolic pressure) and the body’s need for blood flow. Here, we will explain how the reference signal is updated for MY2 pump using the feedback signal ΔP. The same mechanism applies for Nimbus pump because ΔP can be derived from its feedback signals using the pump model. The variation of left ventricular volume can be detected from the pump pressure rise in systole ΔP sys. The detection is based on the fact that the increase of left ventricular volume will cause the increase of left ventricular systolic pressure, which is larger than the increase of aortic pressure in systole. As a result, the value of ΔP sys (=aortic pressure − left ventricular pressure) decreases. A lower limit of ±100 mm Hg/s for the first order differential of ΔP along with an upper limit of 40 mm Hg for ΔP was found to be reasonable criteria to extract the systolic portion. The update of reference signal has two advantages: 1) it can increase blood flow when metabolic needs increase, as in exercise and 2) it can compensate for the estimation error of aortic pressure. Both of these two scenarios will cause the variation of venous return, resulting in the variation of blood volume inside left ventricle.
Table 2 shows the simulation results in different physiological scenarios. The normal left heart and the failed left heart scenarios were simulated using the same setting of numerical circulatory model as those in the previously published article.7 Recovering left heart scenarios were simulated by setting the value of left ventricular contractility between its normal value and failed value. For the benefit of comparing with pump-assisted scenarios, the unassisted left heart failure scenarios neglected the compensatory increase of heart rate and vascular resistance, both of which were observed in heart failure patients.12 As a result, AoP and left ventricular end-diastolic pressure (LVEDP) in the unassisted failed left heart (LH) scenario were lower than their actual values for heart failure patients. The moderate exercises were simulated by increasing heart rate and heart contractility, and lowering TPR and pulmonary resistance.7 The intramuscular pressure generated by the working muscle (e.g., muscle pump effect) is also included in moderate exercise simulation. Mild exercises were simulated by setting the values of those parameters between their values in rest and moderate exercise.
Table 2 indicates that there is no significant difference in controller performance when ΔP is measured (as in MY2) or when ΔP is derived from I and ω (as in Nimbus). With the assistance of blood pump, the cardiac output and the mean aortic pressure were restored back to the normal range. Additionally, suction was not detected for any of the simulated scenarios as negative values of LVEDP were not observed.
To further demonstrate the transition of hemodynamic variables and controller variables from rest to mild exercise, the simulated results of Nimbus-assisted failed LH scenario are shown in Figure 2. The mild exercise was introduced at 60 seconds. In the course from rest to mild exercise, the estimated values of TPR and AoP vary, after the trajectory of their actual values with a little delay, about two heartbeats. The controller gain changes too, after a similar trajectory as that of TPR.
In the developed adaptive controller, no flow sensor was used. The pump flow, which is derived from feedback signals using the pump model, was assumed to be the total flow to the human circulation system. This assumption is not true when the left ventricle ejects directly to the aorta. As a result, the estimation error of aortic pressure increases when the outflow of native left ventricle increases. When the left ventricle outflow accounts for <10% of the cardiac output, the estimation error is very small (<1 mm Hg).1 As the flow contribution from left ventricle increases, the estimation error increases too. The estimation error may be higher than 10 mm Hg, as observed in the recovering LH in moderate exercise scenario when native left ventricle accounts for 45% of the cardiac output. The estimation error is further enlarged in exercise scenarios, because the muscle pump effect is not accessible thus ignored in the adaptive controller. The higher is the exercise level, the larger is the estimation error due to muscle pump effect. However, the estimation error of aortic pressure because of the ignorance of left ventricle outflow and muscle pump effect will not deteriorate controller performance. This error is compensated by the update of reference signal. As the estimation error increases, the discrepancy between the actual pump flow and the body need is larger, resulting in the variation of left ventricular volume and pump pressure rise ΔP. Thus, the reference signal is updated by the nonlinear function of ΔP until the discrepancy is resolved. It is also worth noting that the function to update the reference signal in the adaptive controller is carefully selected to actually allow small outflow of left ventricle in rest condition. This selection is to make sure aortic valve opens frequently to prevent aortic valve fusion.
As shown in Figure 2, there is a lag between the estimated aortic pressure and the actual aortic pressure, about two heartbeats. This lag is caused by the delayed effect of the moving average filter in the processing of feedback signal. Because most of the variation of physiological scenarios (e.g., the variation of heart failure level, the variation of physical exercise) occurs in a much longer time scale, this delay is not expected to affect the performance of physiological controller.
It is well known that as the blood viscosity changes, the hydrodynamic characteristics of the pump change.13,14 In this article, the blood pump model used in the adaptive controller is fixed; therefore, the actual pump flow may be different from the pump flow detected by the controller when the blood viscosity changes. This discrepancy, similar to the ignorance of left ventricular outflow, will cause the mismatch between the pump flow and the body need. The function to update the reference signal, if selected right, is also expected to compensate for this discrepancy. The performance of the controller at different blood viscosity will be studied in the future and addressed in another article.
With the designed adaptive controller, the abnormal hemodynamic values indicating congestive heart failure in a left ventricular assist device patient, including total blood flow, mean aortic pressure, and LVEDP, are all successfully restored to normal ranges. This good performance is consistent for both MY2 and Nimbus pump in the variation of activities and left ventricular failure levels. This consistency also demonstrates that the designed controller can be applicable when pump pressure rise (ΔP) is either measurable or derivable through the pump’s intrinsic parameters (such as current and speed). In addition, the performance of controller is not affected by measurement noise. The results in this article are based on computer simulation study for a set of physiological conditions, which are representative of different heart failure and activity levels. Many factors were ignored, such as neurohumoral responses and the variation of blood volume. A series of in vitro tests is necessary to verify the performance of the designed controller and will be the next step for future research.
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