Control Method of a Rotary Blood Pump for a Left Ventricular Assist Device

One of the main requirements to the circulatory support system is supplying of the adequate cardiac output. Generally, this requirement is realized with the help of control algorithms or methods for the implanted part of circulatory support system, i.e. rotary blood pump (RBP).

Method of control using pressure difference across the pump for calculation of the pulsatility index is suggested in the work [1]. Depending on the aims of control, a certain value of the gradient of pulsatility index with respect to pump speed is assigned, providing either a highest feasible flow or an average flow with a controlled opening of the aortic valve (AV) without any negative effect on the cardiovascular system. Wang et al. [2] have developed a control algorithm for preventing ventricular collapse by maintaining differential pump speed above user-defined threshold value and providing a sufficient flow by maintaining a reference pressure difference between the left ventricle (LV) and aorta. In the work [3] a method of establishing a balance between a cardiac output of the right ventricle and a combined flow rate of the LV and a pump is suggested. The value of pump flow pulsatility is used as feedback parameter in order to adjust the RBP flow.

The aim of the investigation is to develop a control method of a rotary blood pump to solve the following problems: estimation of the pump flow rate, achievement and maintaining of the desired flow rate through the continuous adjustment of the pump speed and prevention of adverse effects on the cardiovascular system.

Methods. Pump flow rate estimation is performed using mathematical model of RBP, which takes into account inertial and viscous properties of blood. Adverse effect on the cardiovascular system is prevented by controlling pumping states (backflow of blood through the pump P_BF, partial assist of the ventricle with a periodically opening aortic valve P_PA, full assist of the ventricle with a constantly closed aortic valve P_FA, partial and full ventricular collapse during cardiac cycle P_PVC and P_FVC).

A functional chart of the proposed control method is presented on Figure 1.

---

Figure 1. The functional chart of rotary blood pump (RBP) control

---

The main part of the functional chart is a RBP unit. The principal component of this unit is a mathematical model of the RBP, described by the following equation:

where L is the parameter, characterizing blood inertia in the given pump, which equals to 0.2 mm Hg·min²·L^–1; Q is pump flow rate (L/min); ω is pump speed (min^–1); H is pressure difference across the pump (mm Hg); a–g are coefficients, obtained by optimization on the basis of Levenberg–Marquardt method (their values are given in Table 1), each coefficient being related to the blood viscosity μ (сP) by the following linear relationship: y(μ)=k·μ+x.

---

Table 1. Coefficients of rotary blood pump model

---

Thus, the pump flow rate at a given time Q(t) is calculated on the basis of speed value ω, pressure difference H and blood viscosity μ, which value is set on the external control console.

The estimation unit of the functional chart is designed for storing the calculated value of Q(t), evaluation of the approximate and actual flow rate and identification of RBP states. Approximate flow Q_A is counted as a blood volume, pumped over by the pump during the time equal to nine cardiac cycles (6.75 s at a heart rate (HR) 80 beats per minute (bpm)); the obtained value is converted in liters per minute. The number of cardiac cycles necessary for estimation of the approximate flow may be arbitrary; in our case it was chosen to be nine in order to approximately evaluate a minute pump flow and to adjust it quickly if physiological conditions change. The actual flow Q_P is a blood volume pumped over by the pump per minute.

Pumping states are identified by the analysis of changes in the dynamics of derivatives, obtained from the pump mathematical model. To simplify the description of these changes some indices were introduced: e.g. S_BF uses to identify backflow of blood through the pump, S_AV for identification of partial and full assist of LV, S_PVC and S_FVC for partial and full collapse of the ventricle during cardiac cycle. A list of used indices and their values is given in Table 2. A detailed description of the RBP mathematical model and method of pumping states identification is presented in work [4].

---

Table 2. Indices for identification of rotary blood pump states

---

A new speed value ω(t+1) is formed in the speed adjustment unit. It depends on the difference between the approximate and desired flow Q_D and on the current pumping state. If Q_A and Q_D do not correspond, the pump speed will change with an increment of 100 rpm until their matching is set. When some undesired pumping state is identified (P_BF or P_FVC), which adverse effects on the cardiovascular system, the speed is forcibly increased or decreased independently of the pump flow at this moment.

The development and testing of the RBP control method was carried out using the mathematical model of the cardiovascular system in which RBP was connected between LV and aorta [4]. All results were obtained at blood viscosity value μ=3.6 cP.

Results and Discussion. A waveform of flow rate Q_P and Q_A (L/min), pump speed ω_p (rpm/1,000) and indices for identification of pumping states at desired flow rate Q_D=4.5 L/min is presented in Figure 2. After estimation of the approximate flow Q_A it is compared with the desired Q_D and, if necessary, the RBP speed is changed; in this case the speed increases by 100 rpm every time.

---

Figure 2. A waveform of pump flow QP and QA (L/min), pump speed ωp (rpm/1,000) and indices SBF, SAV, SPVC and SFVC for QD=4.5 L/min

---

As seen on the figure, the increase of RBP speed results in a certain dynamics of every index. Thus, partial assist of the ventricle P_PA corresponds to the decrease of S_BF index and increase of S_AV, while partial ventricular collapse during a cardiac cycle corresponds to the decrease of S_PVC and S_FVC indices. Transitions from one state to another, which characterized by the changes in index dynamics, are pointed by color markers: a blue diamond-shaped marker on S_BF(t) designates the moment of transition from P_BF state to the state of partial LV unloading P_PA. A red square marker on S_AV(t) marks the moment of constant AV closure, which corresponds to the transition from partial assist state to the full assist state P_FA.

A red round marker on S_PVC(t) corresponds to the transition in P_PVC state, denoting a partial collapse of the ventricle during systole. A violet round marker on S_FVC(t) corresponds to the transition in P_FVC state, in this case the pump speed reduces by 500 rpm. As the desired flow level was not achieved, pump speed continues to increase.

A waveform of pump flow Q_P and Q_A (L/min), pump speed ω_p (rpm/1,000), flow through the aortic valve Q_AV (L/min), S_AV index for the selected value Q_D=3.8 L/min at LV contractility changes C_LV (%) is shown on Figure 3.

---

Figure 3. A waveform of pump flow QP and QA (L/min), pump speed ωp (rpm/1,000), flow through the aortic valve QAV (L/min), SAV and ΔSAV indices for QD=3.8 L/min at left ventricle contractility changes CLV (%)

---

In this case the reduction of C_LV by 10% does not alter the pump speed, which does not allow to detecting AV closure and transition to the P_FA state. To trace the influence of such physiological changes, a differential index ΔS_AV was introduced, which is described by the following equation:

where i is the time step corresponded to estimation of current approximate flow Q_A; i–1 is the estimation of previous Q_A value.

The increase of contractility by 10% is resulted in increase of pump speed due to the augmentation of AV flow, which is shown on Q_AV(t). The increase of S_AV index with the consecutive increase of speed by 200 rpm corresponds to the partial assist pumping state P_PA and AV opening, which is indicated by empty green square markers. At the same time, change of ΔS_AV is not considered — it is pointed by an empty black triangle marker on ΔS_AV(t).

The next characteristic change of ΔS_AV is related to the reduction of LV contractility to the initial level. Such alteration simultaneously with the decrease of S_AV index corresponds to the AV closure and transition to P_FA state; it is pointed by a red triangular marker. Following decrease of speed by 100 rpm corresponds to the state of full assist state also due to the increase of S_AV index at the decrease of the pump speed. However following reduction of the speed by 100 rpm and decrease of S_AV index denotes the change in index dynamics and corresponds to the transition from the P_FA state to the state of partial LV unloading (pointed by a green square marker).

The decrease of C_LV by 10% does not lead to the speed change, therefore ΔS_AV index is used to trace the effect of this physiological change on the pumping state. In this case the characteristic change of ΔS_AV from the negative to positive value at a decreased S_AV corresponds to the constant AV closure and to the transition in P_FA state; it is pointed by a red triangular marker.

The increase of contractility to the initial value results in increase of S_AV and the characteristic change of ΔS_AV, which corresponds to the transition in partial assist state and pointed by a green triangular marker on S_AV(t).

A waveform of pump flow Q_P and Q_A (L/min), pump speed ω_p (rpm/1000), flow through the aortic valve Q_AV (L/min), S_AV and ΔS_AV indices for Q_D=3.8 L/min HR changes demonstrates, that decrease of HR to 70 bpm does not alter the pump speed (Figure 4). In this case ΔS_AV index is also used to determine effect of physiological changes on the pumping state — its characteristic change in case of S_AV decrease allows to detect AV closure (pointed by a red triangular marker).

---

Figure 4. A waveform of pump flow QP and QA (L/min), pump speed ωp (rpm/1,000), flow through the aortic valve QAV (L/min), SAV and ΔSAV indices for QD=3.8 L/min at heart rate (HR) changes (bpm)

---

Increase of S_AV at a characteristic change of ΔS_AV, which is the opposite of the previous one, corresponds to the transition in P_PA state and AV opening (marked by a green triangle). With further HR changes pump speed at first increases and then reduces by 100 rpm. In the first case the speed increase is accompanied by the increase of S_AV index, in the second one speed decrease results in the decrease of the index. Both of these changes corresponds to the P_PA state and, therefore, are pointed by the empty square green markers.

A waveform of pump flow Q_P and Q_A (L/min), pump speed ω_p (rpm/1,000), minimum volume of LV during a cardiac cycle V_LV[min] (ml) and S_FVC and ΔS_FVC indices for Q_D=4.4 L/min at HR changes illustrates, that decrease of HR to 70 bpm leads to the increase of S_FVC index (Figure 5). In spite of the fact that increase of S_FVC with the increase of pump speed corresponds to the transition in P_FVC state, this change is not associated with the transition to P_FVC due to the characteristic change of ΔS_FVC. The situation like this is pointed by empty violet markers over the whole time range (by the round markers on S_FVC(t), and triangular markers on ΔS_FVC(t)).

---

Figure 5. A waveform of pump flow QP and QA (L/min), pump speed ωp (rpm/1,000), minimum volume of left ventricle during a cardiac cycle VLV[min] (ml) and SFVC and ΔSFVC indices for QD=4.4 L/min at heart rate (HR) changes (bpm)

---

The increase of S_FVC with the increase of speed in other cases corresponds to the transition in full ventricular collapse (P_FVC) state, i.e. to the decrease of volume lower reference value (120 ml), that corresponds to the zero pressure in the ventricle — in this situation pressure in the ventricular chamber during systolic phase is constantly negative. The moment of transition to this state is pointed by round violet markers on ΔS_FVC(t), the pump speed also decreasing by 500 rpm.

It should be noted, that increase of HR to 100 bpm allows reaching the desired flow level without any LV collapse, which is seen at V_LV[min](t).

Thus, the proposed control method of a RBP allows to achieve the desired flow rate under various physiological conditions by continuous adjustment of pump speed. The estimation of the RBP performance is carried out by means of the pump mathematical model, which uses values of blood viscosity, pressure difference across the pump and pump speed. The control of RBP states allows to avoid adverse states in the cardiovascular system associated with backflow of blood through the pump or full LV collapse. A long-term RBP operation in the full assist state (P_FA) results in the valve leaflet fusion and thrombus formation, therefore assessment of the AV condition is also necessary [5].

The proposed control method has been tested under changing LV contractility and HR. The results demonstrate feasibility of achievement and maintaining of RBP flow rate on the required level and control adverse states in the cardiovascular system under varying physiological conditions. Identification of RBP states is possible in all considered cases, including a case of constant pump speed.

Conclusion. The presented control method of a rotary blood pump provides the desired flow rate and prevents adverse effects on the cardiovascular system under different physiological conditions. This method is supposed to be used in the development of adaptive control system for the left ventricular assist devices.

Study Funding. The study was supported by the grant of the Russian Scientific Fund (project No.14-39-00044).

Conflicts of Interest. The authors declare no conflicts of interest related to this study.