Developing a physiological controller responding to the blood demand of the patients is a great challenge for the permanent implantation of left ventricular assist devices (LVADs).^{1} Therefore, several control strategies to adapt the LVAD pumping to the changing patient physiologic demand have been developed. Earlier, the blood pressure was used as the control variable of the control system. Giridharan and Skliar^{2} proposed a mathematical model of the circulation system and a control strategy in 2003, which allows to maintain the physiological perfusion by maintaining a constant mean pressure head (75 mm Hg) between the left ventricle and aorta. In the following year, he reported a control strategy,^{3} which maintained a constant average pressure difference between the pulmonary vein and the aorta (Pa). It is found that the Pa approach adapted to changing exercise and clinical conditions better than the constant rotational speed and constant different pressure control strategies. Subsequently, other physiological signals are researched to design control system by investigators. In 2005, Vollkron et al .^{4} suggested a control strategy, in which the flow rate was set to be proportional to the heart rate (HR). Chen^{5} proposed a control strategy, in which the blood demand of the circulatory system was indicated by HR and peripheral resistance of the vessels. In this approach, the desired flow rate of the pump is calculated according to the measurement of HR and peripheral resistance. Choi et al .^{6} reported a hemodynamic controller based on pulsatility ratio. In this work, a pulsatility ratio is defined to classify the status of the heart. The controller aims to maintain the pulsatility ratio in an optimal range to avoid suction. Moreover, Schima et al .^{7} developed a control system, which used the pump speed, pump power as inputs to estimate the blood flow ejected from the pump. The control strategy uses optimal algorithm to calculate the rotational speed of the pump. In 2009, Boston and coworkers^{8} designed a suction detector to indicate whether the suction was presented. They used the index of pulsatility as a suction detector. Karantonis et al .^{9} reported a novel control strategy, which regulates the pump speed in response to the active level index (ALI). The ALI is designed as a function of the HR and the output of a triaxial accelerometer. Pump speed is then varied linearly according to the ALI within a defined range. The above-mentioned control strategies have achieved good performance in numerical and in vitro experiments. As these control strategies need to use the simplified mathematic model, such as the relationship between the ALI and pump speed, to regulate the pump speed, it has some difficulties for clinical application.

An intraaorta pump is designed by the artificial heart research group of Beijing University of Technology.^{10} A global sliding mode control algorithm for the intraaorta pump was proposed in 2010.^{11} The global sliding mode controller has achieved a good performance. As the input of the controller is the rotational speed of the pump, the desired speed directly affects the performance of the pump. Hence, an algorithm to calculate the desired speed according to the blood demand of the circulatory system is necessary. In addition, because of the special structure of the pump, the pressure and flow sensors cannot be implanted on the blood pump to measure differential pressure and blood flow, i.e ., the control strategy needs feedback noninvasive signal to estimate the blood demand of circulatory system.

The relationship between the HR and arterial pressure (AP) has been researched by several groups. From these literatures, it is known that the HR responses to the change of AP by using the arterial baroreceptors.^{12} The baroreceptor is a pressure sensor located in the carotid sinus and aorta, 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.^{5} For example, if the pressure is lower than the set point, the systemic resistance and HR will increase to reduce the error between the current pressure and the set-point pressure. Therefore, the arterial baroreflex system can be considered as the pressure detection system of the LVADs. The HR is the output of the system. In addition, as the HR can be derived noninvasively and conveniently, it is fit to be used as the controlled variable of the intraaorta pump. If we can maintain it in a normal range, the intraaorta pump can be regulated in response to the demand of the circulatory system. Moreover, because the relationship between the HR and the rotational speed of the pump is too complex to derive an appropriate model for control algorithm, the conventional control algorithm cannot achieve a good performance.

The model-free adaptive control (MFAC) only depends on information from the input and output (IO) data of the controlled plant. The MFAC algorithm was proposed by Hou and Huang^{13} in 1997. In this article, a pseudo-partial derivative (PPD) is proposed to linearize iteratively the controlled plant, and an adaptive control law is designed to force the error between the desired output and the actual output to progressively converge on zero. This algorithm achieves good performances for single input single output (SISO) system. In 1998, Hou et al .^{14} extended this theory to the multiple input single output (MISO) system by substituting the pseudo-gradient (PG) for PPD. Because the design of the controller only depends on the IO data of the controlled plant, the MFAC algorithm has strong robustness and is used widely to control the complex system, such as the permanent magnet linear motor,^{15} the glass furnace control system,^{16} and so on. The cardiovascular system is a nonlinear time-vary system, in which the parameters' value and structure of the model in diastole are different from the one in systole. Therefore, the MFAC algorithm is fit for the control strategy based on the HR of the intraaorta pump.

In this article, the controller of the intraaorta pump is designed based on the MFAC algorithm. The HR is chosen as the controlled variable. The aim of the controller is to maintain the HR in a normal range. Computer simulations are conducted to verify the robustness and the dynamic characters of the MFAC controller in the presence of a pathological left ventricle and slightly physical active.

Materials and Methods
Modeling of the Cardiovascular-Baroreflex System
Because the MFAC algorithm does not need to know the mathematic model of the controlled plant, the model of the cardiovascular-baroreflex system is only used to generate the IO data (HR) and to verify the performances of the controller. The model used here is a fifth-order nonlinear time-varying lumped parameter model reported by Chen et al.^{17} The behavior of the left ventricle is modeled by a time-varying elastance function E (t ) = 1/C (t ), which is the function of HR. In this work, the HR is automatically regulated by nervous system^{18} to respond to the change of blood demand of the circulatory system (Figure 1 ).

Figure 1.:
The block of the cardiovascular-baroreflex system.

According to Ursino's^{19} report, the HR is regulated by sympathetic nerves and vagal fibers. The relationship between them and the HR are described in Equations 1–4 :

Where f _{es} (P ) is the frequency of spikes in the afferent fibers. f _{min} and f _{max} are the upper and lower saturation of the frequency discharge. P is the carotid sinus pressure, which is considered as the same with the AP to simplify the model.

Where f _{es} is the frequency of spikes in the efferent sympathetic nerves. f _{es∞} , f _{es0} are constant parameters, which satisfy f _{es0} > f _{es∞} .

Where f _{ev} is the frequency of spikes in the efferent vagal fibers. f _{ev0} and f _{ev∞} are constant parameters (with f _{ev∞} > f _{ev0} ).

According to the frequency of spikes in the efferent sympathetic and vagal fibers, the change of the cardiac cycle due to the sympathetic and vagal stimulation can be calculated, respectively. The details of the calculation are reported by Ursino.^{19} ΔT _{s} (t ) is used to represent the change due to the sympathetic stimulation, ΔT _{v} (t ) is used to represent the change due to the vagal stimulation, and T _{0} is the basic cardiac cycle without any stimulation. The overall cardiac cycle is described in Equation 4 .

Where T (t ) is the overall altered cardiac cycle.

Modeling of the IntraAorta Pump
According to the previous researches of our group, the mathematic model of the intraaorta pump is described as a function of the flow rate, pressure head, and rotational speed of the pump,^{20} which is denoted in Equation 5

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, and g (·) represents the step function that is denoted in Equation 6 .

Designing of the Model-Free Adaptive Controller
The relationship between the HR and the rotational speed of the intraaorta pump is described in Equation 7 .

Where HR(k ) is the HR that is regulated by the baroreflex system, u (k ) is the rotational speed of the intraaorta pump, and n _{HR} and n_{μ} are the order of the output HR(k ) and the input u (k ), respectively. f (···) represents the unknown nonlinear function.

According to Ref. ^{13} , for the nonlinear system, (Equation 7 ) there must exist φ (k ), called PPD. When u (k ) − u (k − 1) ≠ 0, we have:

The aim of the controller is to force the output of the system to track the desired value; therefore, a weighted one-step-ahead control input criterion function and a parameter estimation criterion function are designed as Equations 9 and 10 to calculate the control law.

Where HR_{d} (k ) is the desired HR, and ë and μ are the constant weight parameters.

Substituting Equation 8 into Equations 9 and 10 , the minimization of above two criterion functions gives control law algorithm.

And estimation algorithm:

Where ñ_{k} and ç_{k} in the Equations 11 and 12 are step-size constants series, which are added to make them general.

To make the condition u (k ) − u (k − 1) ≠ 0 satisfy, and meanwhile to make parameter estimation algorithm (Equation 12 ) have stronger ability to track time-varying parameter, a reset measurement of estimation algorithm should be taken.

Where å is a small positive number, and φ (1) is the initial estimation value of φ (k ).

Results
Simulation of the Model-Free Adaptive Controller
Computer simulations have been performed to verify the dynamic characters and the robustness of the controller. The main parameters of the cardiovascular-baroreflex system used in this work are listed in the Tables 1 and 2 . In the simulation, the peripheral resistance is changed from 1.5 to 0.8 mm Hg · s/ml at 19 seconds to mimic physical active. In addition, the desired HR is increased from 65 to 85 bpm at 38 seconds to maintain the control system stable and the response time <8 seconds, the weight parameter λ = 10, μ = 1, and the initial estimation value φ (1) = −0.5.

Table 1: The Parameters of the Cardiovascular Model

Table 2: The Parameters of the Baroreflex System

The simulation results of the MFAC are shown in Figures 2–5 . Because the initial values of the aortic pressure (AOP), blood flow rate, and AP are set in response to the heart failure patients, the HR increases up to 150 bpm (Figure 2 ), at the beginning of the simulation. From Figure 3 , it is seen that the settling time of tracking the desired HR (65 bpm) is 5 seconds without static error or overshoot. At 19 seconds, the peripheral resistance changes from 1.5 to 0.8 mm Hg/ml, the HR accordingly increases from 65 to 87 bpm because the AP decreases (Figure 3 ). To compensate the decrease of the AP caused by the change of peripheral resistance, the rotational speed of the pump increases, and the settling time is <5 seconds (Figure 4 ). Because the rotational speed increases at 18 seconds, the left ventricular pressure (LVP) and AOP decreases obviously (Figure 5 ). When the desired HR changes from 65 to 85 bpm at 38 seconds, the rotational speed decreases to reduce the AP, and the settling time is 4 seconds without overshoot or static error.

Figure 2.:
The response of the heart rate.

Figure 3.:
The response of the arterial pressure.

Figure 4.:
The response of the rotational speed of the pump.

Figure 5.:
The response of the LVP, LAP, and AOP. LVP, left ventricular pressure; LAP, left atrial pressure; AOP, aortic pressure.

Discussion
The HR in the cardiovascular-baroreflex system is regulated by the baroreflex system. When the AP increases, the frequency of spikes in the afferent fibers will increase. The sympathetic nerves and vagal fibers choose the frequency of the spikes in the afferent fibers as the input. When the frequency increases, the sympathetic will be inhibition and on the contrary, the vagal will be activity. This phenomenon results in the decrease of the HR. According to Ref. ^{18} , the baroreflex can not only affect the HR but also the peripheral resistance and the contractility of the left ventricular. They work together to regulate the AP. Among the three variables, the HR can be measured conveniently, accurately, and noninvasively. Hence, in this work, the HR is chosen as the controlled variable.

The control algorithm proposed in this article is used the HR to estimate the blood demand of circulatory system. However, if the patients suffer from severe arrhythmia, the mechanism of HR regulation will be abnormal.^{18} 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.

From Refs. ^{9 and 19} , it is seen that the reaction of the HR to the AP exists a pure delay, which changes according to different people, time, and location. Hence, it is difficult to derive an appropriate model, which is fit for control algorithm, to describe the relationship between the HR and rotational speed of the pump. All conventional control algorithms need to know the model of the controlled plant; hence, they are not fit for the control strategy based on the HR. The MFAC algorithm does not need the model of controlled plant. It can iteratively establish the model of control system, according to the IO data derived from the sensors. Then, it calculates the control law depending on model established by itself. Therefore, the MFAC algorithm is fit to control the complex system, whose parameters' value, structure, and mathematic model are unknown or change with time.

From Figure 5 , it is seen that the LVP decreases with the desired HR. When the peripheral resistance decreases, the AP will decrease correspondingly; this will lead to increased HR. This phenomenon will force the rotational speed of the pump to increase to compensate the error of the HR. It indicates that if the desired HR is not appropriate or the peripheral resistance change acutely, the phenomenon of the ventricular collapse will occur. Therefore, the suction detection algorithm is needed in this control system. In addition, the controller regulates the intraaorta pump, in response to the HR. That means, if the receptors suffer from arrhythmia or the native heart loses its pump function totally, the controller reported in this study cannot regulate the pump to respond to the change of the circulatory system demands. Hence, the detection algorithm, which can analyze and identify whether the HR is physiological significance, is necessary for this control system.

Conclusion
The mathematic model of the cardiovascular-baroreflex system combined the intraaorta pump is established. The HR is chosen as the controlled variable. The controller aims to maintain the HR in a normal range. The model-free adaptive controller of the intraaorta pump, which only needs IO data to calculate the control output, is designed. As a key feature, the proposed controller provides a defined and adjustable HR and automatically regulates the pump speed according to the status of circulatory system. To verify the dynamic characters and robustness of the controller, couples of computer simulations are performed. The simulation results demonstrate that the proposed control algorithm can keep the HR stable. Furthermore, no matter the change of desired HR or the disturbance of the peripheral resistance, the settling time of the control system is <5 seconds without overshoot or static error.

Acknowledgment
Supported, in part, by the National Natural Science Foundation of China (Grant numbers: 11072012, 10872013, and 31070754), the Furtherance Program of 10 Beijing University of Technology on Strengthening Talents Education (31500054R5001), and the National Academic Working Group-Talents Training Program (01500054R8001).

References
1. Boston JR, Antaki JF, Simaan MA: Hierarchical control of heart-assist devices. IEEE robot.

Autom Mag 10: 54–64, 2003.

2. Giridharan GA, Skliar M: Control strategy for maintaining physiological perfusion with rotary blood pumps.

Artif Organs 27: 639–648, 2003.

3. Giridharan GA, Pantalos GM, Gillars KJ,

et al : Physiologic control of rotary blood pumps: An in vitro study.

ASAIO J 50: 403–409, 2004.

4. Vollkron M, Schima H, Huber L,

et al : Development of a reliable automatic speed control system for rotary blood pumps.

J Heart Lung Transplant 24: 1878–1885, 2005.

5. Chen SH:

Baroreflex-Based Physiological Control of A Left Ventricular Assist Device . Pittsburgh, PA, University of Pittsburgh, 2006.

6. Choi S, Boston JR, Antaki JF: Hemodynamic controller for left ventricular assist device based on pulsatility ratio.

Artif Organs 31: 114–125, 2007.

7. Schima H, Vollkron M, Quittan M,

et al : Clinical evaluation of an automatic physiologically responsive control system for rotary blood pumps.

IFMBE Proceedings, World Congress on Medical Physics and Biomedical Engineering . Munich, Germany, 2009, pp. 707–710.

8. Ferreira A, Boston JR, Antaki JF: A control system for rotary blood pumps based on suction detection.

IEEE Trans Biomed Eng 56: 656–665, 2009.

9. Karantonis DM, Lim E, Mason DG,

et al : Noninvasive activity-based control of an implantable rotary blood pump: Comparative software simulation study.

Artif Organs 34: E34–E45, 2010.

10. Gao B, Chen NN, Chang Y: A hemodynamic predict of an intra-aorta pump application in vitro using numerical analysis.

International Conference, WISM 2009 . Shanghai, China, 2009, pp. 177–185.

11. Chang Y, Gao B: A global sliding mode controller design for an intra-aorta pump.

ASAIO J 56: 510–516, 2010.

12. Chen F:

Baroreceptor Modeling with Its Applications to Biosignal Processing . Hong Kong, The Chinese university of Hong Kong, 2004.

13. Hou ZS, Huang WH: The model-free learning adaptive control of a class of siso nonlinear systems.

Proceedings of the American Control Conference. Albuquerque, New Mexico, 1997, pp. 343–344.

14. Hou ZH, Han CW, Huang WH: The Model Free Learning Adaptive Control of a Class of MISO Nonlinear Discrete Time Systems. IFAC Low Cost Automation. Shanghai, People's Republic of China, Elsevier Science: 1998, pp. 227–232.

15. Cao RM, Bai LP, Hou ZH: Study on model-free learning adaptive control in permanent magnet linear motor.

2008 Chinese Control and Decision Conference (

CCDC 2008 ). Yantai, Shandong, China, 2008, pp. 2946–2949.

16. Qi JL, Ma G: Design of glass furnace control system based on model-free adaptive controller.

2010 Second International Conference on Computer Modeling and Simulation . Sanya, Hainan, China, 2010, pp. 130–133.

17. Chen SH, Ferreira A, Simaan MA,

et al : A mathematic model of a cardiovascular system regulated by the baroreflex.

Proceedings of the 2006 American Control Conference . Minneapolis, Minnesota, 2006, pp. 701–706.

18. Borchard U: The role of the sympathetic nervous system in cardiovascular disease.

J Clin Basic Cardiol 4: 175–177, 2001.

19. Ursino M: Interaction between carotid baroregulation and the pulsating heart: A mathematical model.

Am J Physiol 275: H1733–H1747, 1998.

20. Chang Y, Gao B: Modeling and identification of an intra-aorta pump.

ASAIO J 56: 504–509, 2010.