Ventricular assist devices (VADs) have been used as a bridge to transplantation since 1980s,^{1} and recently, the rotary VADs (rVADs) are increasingly used as a permanent treatment for the end-stage heart failure patients.^{2} An important engineering challenge impeding the expand use of left ventricular assist device (LVADs) is the development of an appropriate controller that can regulate the pump to meet the circulatory demand of the patient.^{3} Besides robustness and reliability, such a controller must satisfy the blood demand for cardiac output (CO) and mean arterial pressure (MAP) to compensate the physiological parameters change of patients. Researchers proposed some control strategies to settle these difficulties. Oshikawa *et al.*^{4} proposed a control strategy in 2000, which used the index of current amplitude (ICA) to evaluate the assistance level of the rVADs. They find that the ICA has a peak point that corresponded closely with the turning point from partial to total assistance and a trough that corresponded with the beginning point of ventricular collapse. The rVADs work at the point that is before the beginning point of ventricular collapse. Although this control strategy regulates the rotational speed of the pump according to the state of the native heart, it cannot prevent the ventricular collapse. Because this method can only detect ventricular collapse but not predict it. Giridharan and Skliar^{5} proposed a mathematical model of the circulatory system and a control strategy in 2003, which maintained the physiological perfusion by maintaining a constant mean pressure head (75 mm Hg) between the left ventricle and aorta. This control system achieves good performance *in vitro* experiments. However, this method needs two pressure sensors to measure the different pressure. Hence, it is not fit for long-term assistance. Vollkron *et al.*^{6} suggested a control strategy in 2005, in which the flow rate was adjusted depending on the heart rate (HR). In this approach, the pump flow rate is designed as linear function of the HR. In actual circulatory system, the relationship between pump flow rate and HR is very complex. Hence, this approach cannot meet the requirement of circulatory very well. In 2009, Boston and coworkers^{7} designed a suction detector to indicate whether the suction was presented. In this approach, they use the index of pulsatility as a suction detector. However, this control strategy needs the implantation of a flow sensor.

An intraaorta pump is developed autonomously by the artificial heart research group of Beijing University of Technology.^{8} The special structures of intraaorta pump bring some difficulties to the design of the control system (Chang and Gao. A global sliding mode controller design for an intraaorta pump, accepted).^{20} For instance, as there are no percutaneous wires, the pressure and flow sensors cannot be implanted on the blood pump to measure arterial pressure and blood flow. Therefore, the control strategy needs to use noninvasive signal to estimate the blood demands of circulatory system. Also, conventional control algorithms, which used arterial pressure or blood flow as the controlled variable, are not fit for the intraaorta pump.

The relationship between the HR and arterial pressure has been investigated by some groups.^{9} It is well known from these literatures that HR can indicate the status of the arterial pressure. The native heart is controlled by the baroreflex system and nervous system, which can monitor the status of the hemodynamic properties of the cardiovascular system. When the blood demands of the cardiovascular system changes, the HR will be adjusted by sympathetic nerve and vagal nerve.^{10} For instance, when a patient takes exercises, his blood demand increases, and in consequence, the HR increases to satisfy this demand. That is, the native heart can be considered as a “sensor” to monitor of the blood demand, and the HR is the signal of the sensor output. If we choose the HR as the controlled variable and maintain the HR in a normal range, the controller can regulate the pump in response to the blood demand of blood. Unfortunately, the mechanism of the HR regulation is too complex to derive an accuracy model for designing controller. Hence, the conventional control algorithm, such as proportional-integral-derivative (PID) and optimal control, cannot achieve good performances.

The fuzzy logic control (FLC) does not directly use a model of the cardiovascular pump system. It has an ability to summarize information and to extricate from the collections of masses of data impinging on the human brain and only those facts that are relevant to performance of the task at hand. ^{11} Therefore, the FLC is fit for the control strategy based on the HR of the intraaorta pump.

In this article, a lumped parameter model of the cardiovascular pump system is established to simulate the status of the circulatory system, in which the HR is designed as a nonlinear function of the MAP. The controller of the intraaorta pump based on the feedback fuzzy logic control (FFC) is designed. The HR is used as the controlled variable. The aim of the controller is to maintain the HR in a normal range. The computer simulations have been performed to verify the robustness and the dynamic characters of the fuzzy feedback control.

#### Materials and Methods

##### Modeling of the Cardiovascular Pump System

According to the previous researches of our group (Chang and Gao. Modeling and identification of an intra-aorta pump, accepted),^{21} the mathematic model of the intraaorta pump is described as a function of the flow rate, pressure head, and rotational speed of the pump, which is denoted by Equation 1.

where Q_{PO} represented the flow rate of the pump (L/min), P_{P} is the pressure head of the pump (mm Hg), ω is the rotational speed (R/s), ω_{limit} denotes the threshold speed, L_{P} is the inertia of blood in intraaorta pump, *g*(·) represents the step function that is denoted by Equation 2.

The fuzzy feedback control itself does not directly use a model of cardiovascular pump system. However, a model is required to evaluate the behavior of the controller. The model used here, shown in Figure 1, is a fifth-order nonlinear time-varying lumped parameter model,^{12} which is described by a set of differential equations as follows:

where γ(x) represents the ramp function

Equation (Uncited) Image Tools |
Equation (Uncited) Image Tools |
Equation (Uncited) Image Tools |

Equation (Uncited) Image Tools |

In this model, C_{R} represents the compliance of preload and pulmonary circulations; R_{M} and R_{A} model, the resistance properties of the mitral valve and aortic valve; D_{M} and D_{A} mimic the properties of one-directional valve; the left ventricle is represented by the time-varying compliance C(t) and C_{A} represents the aortic compliance; and afterload is represented by the four-element Windkessel model R_{C}, L_{S}, C_{S}, and R_{S}. Resistor R_{S} represents the peripheral resistance of the circulatory system. The behavior of the left ventricular is modeled by a time-varying elastance function E(t)=1/C(t). In this work, the elastance function has been defined as follows:

where E_{max} and E_{min} denote the maximum and minimum values of E(t); E_{n}(t) represents the normalized time-varying elastance function.^{13}

where t_{n}=t/t_{max} and t_{max}=0.15×t_{c}+0.2; t_{c} represent the cardiac cycle interval, that is, t_{c}=60/HR, where HR represent the HR of the native heart. The ability of the model to emulate the hemodynamic of the left ventricular is demonstrated by simulating the model for normal heart condition. Figure 2 shows the simulation waveforms of the hemodynamics for an adult with HR of 75 bpm. From Figure 3, we can derive that the systolic and diastolic aortic pressure is 104 and 72 mm Hg, respectively. The mean CO is 5.54 L/min. These numbers and waveforms are consistent with data that is derived from clinical measurement. That is, this model can be used to emulate the hemodynamic status of the left ventricular.

Equation (Uncited) Image Tools |
Figure 2 Image Tools |
Figure 3 Image Tools |

We note that the HR in the model described previously is considered as a constant, and this is not consistent with the clinical experience. It is well known from the literatures^{9,14} that the HR is a nonlinear function of the arterial pressure with a negative slope. The HR response can be written as follows:

where HR_{0} and HR_{M} are the threshold values of the HR, PASN represents the normalized arterial pressure (PASN = MAP/100), and μ_{HR} is a constant, which is set to 8 in this work.

Finally, the cardiovascular pump model, in which the HR is adjusted by MAP automatically, has been established. The HR is calculated from Equation 10, and then it was used as the actual HR in the next time interval. In this model, it has been designed as a variable that can be used as the control objective.

##### Designing of the Fuzzy Feedback Control

A fuzzy logic feedback controller for the intraaorta pump is designed following Mamdani approach.^{11} The block diagram is shown in Figure 3. The error between the desired HR and the actual HR is used as the input of the controller. The aim of the control algorithm is to maintain the actual HR tracking the desired one. The output of the controller is the gain of the rotational speed increment. The rotational speed of the pump is updated according to Equation 11.

where ω_{k} is the rotational speed of the pump in kth time interval; K_{k} is the gain of the rotational speed increment; Δ*HR*_{k} is the error between desired HR and actual HR. The input membership sets are defined on the range spanned by the error of HR with one crosspoint between consecutive sets at μ=0.5. The output membership sets for K_{k}, the input, and output membership functions are shown in Figure 4. Note that the range of the output variable is limited from 0 to 1. The height defuzzification method is used to calculate the crisp value of the output in this work. It is described as Equation 12.

Equation 12 Image Tools |
Figure 4 Image Tools |

where *m* is the number of fired rules; *c*_{k} are the centers of the membership functions corresponding to these rules, and f_{k} are their heights.

#### Results

Computer simulations of the fuzzy logic feedback controlled system were performed to verify the robustness and the dynamic characters of the controller. The main parameters of the model for heart failure patients are listed in Table 1. Because of the slow change of the HR, the control frequency is set at 20 Hz.

The first simulation is to verify the dynamic characters of the controller. The desired HR is set to 100 bpm at the beginning of the simulation, and then it changes to 80 bpm at 32 seconds. The results of this simulation are shown in Figure 5. It is seen that the settling time is 8 seconds without steady-state error. Because of the initial values of aortic pressure, blood flow and arterial pressure are set according to the data of heart failure patients; the HR is up to 150 bpm at the beginning of the simulation. From Figure 6, it is also seen that due to the increase in rotational speed of the pump, the blood flow increases significantly. When the desired HR decreases from 100 to 80 bpm, the blood flow increases accordingly. The phenomenon is consistent with the clinical data.^{14} We also note that the pulsatile of the instantaneous arterial pressure decreases compared with the instantaneous arterial pressure without intraaorta pump. This is because the HR is regulated by the MAP but not by the instantaneous arterial pressure. Figure 6 illustrates the pressure waveform of the cardiovascular pump model. In Figure 6, the LVP represents left ventricular pressure; the LAP denotes left atrial pressure; and the AOP is aortic pressure. It can be seen that at 32 seconds, the HR of the model decreases from 100 to 80 bpm, and to achieve this aim, the rotational speed of the pump increases to deliver more blood flow from left ventricle. Hence, the amplitude of the pressure waveform decreases.

Figure 5 Image Tools |
Figure 6 Image Tools |

To further investigate the robustness of the controller, another computer simulation is performed. The desired HR is set to 80 bpm throughout entire process. The peripheral resistance in the model changes from 1.0 to 0.7 mm Hg/ml at 32 seconds for simulating the peripheral resistance under exercise. The simulation results are shown in Figure 7. It is seen that the settling time of HR (80 bpm) is 8 seconds without steady-state error when the peripheral resistance increases from 1.0 to 0.7 mm Hg/ml, and the amplitude of the blood flow increases correspondingly to ensure that the arterial pressure is constant.

#### Discussion

In the cardiovascular pump models proposed by other groups, the HR is generally considered as a constant. It is not consistent with the clinical data. As the matter of fact, on one hand, the HR is affected by both pressure and CO. When the aortic pressure and CO is changed by the rVADs, the HR accordingly changed, owing to the HR is controlled by the hormonal and nervous system. On the other hand, the change of HR can adjust the CO and mean aortic pressure.^{14} Therefore, the HR should be considered as a state variable identical with aortic pressure and blood flow.

According to the Ref. ^{15}, the HR responses to the change of arterial pressure by using the arterial baroreceptor. The baroreceptor is a pressure sensor located in the carotid sinus and arterial vessel, which converts pressure into afferent firing frequency. Then, the afferent firing frequency is translated into efferent signals by the nervous system: sympathetic firing frequency and vagal firing frequency. These efferent signals are the inputs of the regulation effectors, which change the HR. Therefore, the control strategy used here is to use the nervous system and baroreflex system to monitor the change of blood demand of patients for us. In response to the Refs. ^{16 and 17}, it is seen that even if the patients suffered from the end-stage heart failure, the regulation system of HR also regulate the HR according to the change of arterial pressure. However, if the patients suffer from severe arrhythmia, the mechanism of HR regulation is abnormal^{18}; the active of sympathetic nerves is different from the normal persons. Therefore, the control algorithm reported in this article is not fit for the patients who suffer from severe arrhythmia, and a HR analysis system should be added to the controller to decide whether the HR measured from patients is normal. If the HR cannot indicate the blood demand accurately, other algorithm will be adopted. This work will be investigated in the future.

The HR of the model is a nonlinear function of MAP according to the Ref. ^{15}. The HR is considered as a sensor of MAP; consequently, the controller based on HR can regulate the MAP. For instance, in Figure 6, when the desired HR changes from 100 to 80 bpm at 32 seconds, the arterial pressure increases accordingly from 100 to 120 mm Hg. Because the desired HR of patients is different from each other, the increment of aortic pressure leads to the decrement of the HR. To maintain the actual HR tracking the desired one, the blood flow of the pump is increased by the controller. When the patients take exercises, the peripheral resistance of their aortic vessels will decrease to admit more blood flow into aorta.^{10} That is, when the blood flow demand of the circulatory system increases, the peripheral resistance of the aortic vessel will decrease. As shown in Figure 7, when the peripheral resistance decreases from 1.0 to 0.7 mm Hg/ml, the blood flow through the pump increases correspondingly to ensure that the arterial pressure is constant. That means the HR can indicate the blood demands of patients, and the fuzzy logic feedback controller can regulate the intraaorta pump to response to the demand of the circulatory system.

From Figure 1, it is seen that the coronary ostia locate at the inlet of the pump. That means the blood to the coronary vessel will be sucked by the pump, when the rotational speed of the pump is higher than the appropriate speed. To overcome this problem, coronary artery bypass will be operated to increase coronary perfusion. This will be investigated in the future.

From Figure 6, we find that the minimal value of the left ventricular pressure decreases, when the HR decreases. That means if the desired HR is not appropriate for the patients, the phenomenon of suction maybe occur. Therefore, a suction detection algorithm should be added in the controller to detect suction. Because the desired HR is different from each other, the algorithm can decide that whether the desired HR is appropriate for the patients is needed. This work is investigated by other person in our group.

Figure 6 shows the statuses of HR, blood flow, and arterial pressure. It is seen that, from 0 to 2 seconds, the HR is up to 150 bpm, because the rotational speed of the pump is very low (approximate to 0 rpm). Also, the blood flow and arterial pressure in this period represent the situation of the heart failure without supported. The mean flow rate in this period is approximately 2 L/min, and the MAP is approximately 60 mm Hg, which are consistent with the clinical data proposed by literatures. From 2 to 8 seconds, because of the increase of rotational speed of the pump regulated by the controller, the mean flow rate and MAP increase significantly. When the HR has tracked the desired HR (after 8 seconds), the mean flow rate and MAP increases to 5.5 L/min and 100 mm Hg, respectively. It is seen that the controller based on HR can regulate the pump according to the blood demands of patients and maintain the arterial pressure and blood flow in a normal range.

The theoretical source of the physiological control by maintaining HR is from the fact that the natural regulatory mechanism of the body is able to effectively represent the physiological demands. Unfortunately, when the regulatory mechanism does not perform normally, such as hypertension, there is a decrease in the performance of the HR control algorithm if the control algorithm is based on the mathematic model of the cardiovascular pump system, because the relationship between the HR and mean aortic pressure has changed. In this case, Equation 10 has to be modified to adapt to the status of patients. However, the fuzzy logic feedback control does not directly use the model of the system, and it is insensitive to the change of cardiovascular pump system. Hence, the fuzzy logic feedback control is an appropriate control algorithm for the intraaorta pump controller.

The fuzzy logic controller monitors the error of the HR to regulate the gain of the system. The rotational speed is actually controlled by the feedback controller. The structure of the controller used in this work has advantages. On one hand, the fuzzy logic feedback control is a combination of fuzzy logic supervisory and classical feedback control. The feedback control algorithm can always achieve good performance.^{19} However, because the cardiovascular pump model is a complex nonlinear model with much internal uncertainty and external disturbance, it is difficult to obtain an appropriate gain of the feedback for all status of the model. To overcome this problem, the fuzzy logic supervisory has been derived into this control strategy to adjusted gain of the feedback control intelligently. The fuzzy logic supervisory does not use the mathematical model but the knowledge and experience to adjust the gain; hence, the fuzzy logic controller has more robustness than other control algorithms based on model. Despite the accuracy and dynamic characters, which are shortages of the fuzzy logic control,^{11} from Figures 6 and 7, we note that when the desired HR decrease from 100 to 80 bpm, or the peripheral resistance decreases from 1.0 to 0.7 mm Hg/ml, the controller can maintain the actual HR tracking the desired one without steady-state error, that means the fuzzy logic feedback control obtains the predominance of feedback control and fuzzy logic control. On the other hand, because the cardiovascular pump system is a complex nonlinear system, for a single-objective control algorithm, it is difficult to achieve good performance for all situation of the system.^{3} The fuzzy logic supervisory algorithm provides a method to solve this problem by being determined by the experience and *a priori* knowledge of the designers, and it is easy to be modified to a multiobjective control algorithm, which maybe include such hemodynamic parameters as HR, mean flow rate, pulsatile of the pressure waveform, and blood flow waveform.

#### Conclusion

The mathematical model of the cardiovascular pump system has been established. For the reason that the HR could reflect the status of the hemodynamic of the system, the HR in the model is designed as a variable that is a nonlinear function of MAP. The fuzzy logic feedback controller of the intraaorta pump is designed, in which HR is considered as the controlled variable. The aim of the controller is to maintain the actual HR tracking the desired one. Couples of simulations are performed to verify the robustness and the dynamic characters of the controller. The simulation results demonstrate that the settling time of the controller is <10 seconds without steady-state error.

#### Acknowledgment

Supported partly by the National Natural Science Foundation of China (Grant No. 11072012), the Furtherance Program of 10 Beijing University of Technology on Strengthening Talents Education (31500054R5001), and the National Academic Working Group-Talents Training Program (01500054R8001).