Steady-State Model of Pressure-Flow Characteristics Modulated by Occluders in Cardiopulmonary Bypass Systems

Cardiopulmonary bypass is a complex procedure that involves the maintenance of heart and lung functions using an external system during cardiac surgery. It is prone to significant human errors, which are mainly caused by inappropriate occluder operation that controls the perfusion balance by adjusting flow rates. Hence, there is a requirement for automatic occluder control; however, the relationship between occluder operation and flow rates remains unclear. The aim of this study is to use a steady-state model to evaluate the influence of occluder control on the flow and pressure of a cardiopulmonary bypass system. Perfusion experiments were performed using Newtonian (glycerin solution) and non-Newtonian (erythrocyte turbid solution) fluids to investigate the pressure-flow characteristics modulated by the occluder. We also visualized the fluid to verify the validity of the measurement data. Based on these experimental data, an exponential occlusion-pressure model is derived to express the relationships between the opening ratio of the occluder and flow rate. Estimation is then achieved by combining the occlusion-pressure model with the linear pressure-flow model. Results reveal that the combined model fitted the perfusion experiment data well for the venous and arterial line side ( $R^{2} = 0.946$ and 0.985, respectively; $p < 0.01$ ). Further, leave-one-out cross-validation and Bland–Altman analysis confirmed that the combined model could predict flow rates accurately with minimal proportional and bias errors. Therefore, the proposed model can serve as a basis for the further development of cardiopulmonary manipulation systems.


I. INTRODUCTION
Cardiopulmonary bypass (CPB) is a procedure that involves the temporary replacement of heart and lung functions for the maintenance of systemic circulation during cardiac surgery. CPB drains blood from the right atrium and bypasses the circulation of the heart and lungs. This bypass perfuses blood into the systemic circulation from the aorta after the gas exchange in the oxygenator. In the perfusion process, sufficient preservation of systemic blood pressure is critical for organ impairment prevention, especially with respect to their functions [1]. Therefore, a CPB operator is required to carry The associate editor coordinating the review of this manuscript and approving it for publication was Jenny Mahoney. out delicate operations for the prevention of significant variations in perfusion pressure and to ensure that the arterial-and venous-line side flow rates are equal.
To prevent a decrease in blood pressure owing to abrupt blood removal at the start of CPB [2], in addition to air embolisms owing to low blood retention levels [3], it is necessary to minimize human error and meticulously carry out the CPB procedure. During the first CPB procedure by Gibbon [4], a safety sensor was installed in the reservoir to monitor the perfusion balance between the arterial-and venous-line sides for human operation support, as is currently common practice. Nonetheless, human error is the main cause of CPB accidents, not equipment failure [5], [6]. Thus, complex CPB operations require simplification, automation, or both for the prevention of human error [7], [8]. Therefore, a system that can monitor and automatically control the blood reservoir level by measuring the blood reservoir static pressure was developed [9]. However, it was not practically implemented owing to its requirement of a specialized device and the development of improved disposable blood reservoirs. Moreover, a scheme that can control perfusion on the arterial-and venous-line sides by measuring blood flow on the venous-line side was developed and implemented in clinical applications [10]. However, this scheme can only be applied to CPB systems that employ roller pumps as the supply pump and not centrifugal pumps. Given that the roller pump may lead to abnormal increases in pressure, thus leading to the transfer of a large amount of air [11] to the arterial-line side, gas microemboli are present [12]- [16]. Nevertheless, many hospitals typically use roller pumps as blood pumps, owing to operation complexities when the centrifugal pump is used [17], [18]. In the centrifugal pump, the perfusion balance is adjusted by operating the occluders attached to the venous and arterial lines. The blood flow rate should be estimated from the occluder closure to achieve automatic maintenance of the blood reservoir level.
As reported, the relationship between the differential pressure and Reynolds number can be applied to viscous fluids in general differential pressure flow meters, from which the flow rate from the fluid differential pressure can be obtained using an orifice plate based on Bernoulli's theorem [19]- [21]. Based on these reports, a system that can continuously monitor blood viscosity by measuring the differential pressure and flow rate of the oxygenator was proposed [22]- [24]. Additionally, a mathematical model that can express the relationship between the blood flow rate and differential pressure for perfused viscous fluids was also proposed [25]- [27]. However, the effects of the occluder on the flow rate have not been revealed. If flow rate can be estimated from the operation amount of the occluder, it can serve as a basis for the development of a system that links the venous and arterial line occluder operations, thus simplifying the CPB operation and decreasing the probability of human error [25]. The ultimate goal of this study is to construct an assisted cardiopulmonary manipulation control system that can reduce the burden on the operator. Thus, in this study, we model the relationship between the occluder operation amount and flow rate, which can provide a basis for the control system. Such a model needs to include an occluder operation amount term explicitly and be simple enough for real-time control. Therefore, the model only focused on steady-state flow.
We propose an exponential model to address the relationships between the opening ratio of the occluder and the differential pressure. Combining this model with the previous model [22]- [24], [26], which links differential pressure and flow rate, allows the estimation of the flow rate from the opening ratio of the occluder. The following experiments were carried out to determine the model parameters. First, the occluders on the arterial-and venous-line sides were modulated to investigate the differential pressure-flow rate characteristics of Newtonian (glycerin solution) and non-Newtonian fluids (erythrocyte turbid solution), which were perfused using a CPB system. Additionally, the influence of the fluid viscosity on frictional losses was experimentally investigated with respect to occluder perfusion and viscosity. Finally, we investigated the characteristics of the flow in the blood circuit by visualizing perfusion near the occluder to confirm the validity of the measurement data.

II. MATERIALS AND METHODS
This section describes the in vitro experiments using the CPB system, which is shown in Fig. 1. Co., Ltd., (Tokyo, Japan)). There are two types of tubes commonly used in cardiopulmonary blood circuits: 10 mm and 12 mm. In both cases, they are reported to behave in such a way that the same curve is drawn. A three-way stopcock was connected to the occluder inlet and outlet, and the pressure was measured from the stopcock using pressure sensors (40PC015, Pressure range 0-15psi, Total accuracy ±0.4%; Honeywell Japan, Inc. Tokyo, Japan). Moreover, an electromagnetic blood flow meter (FD-M5AT, Flow range 0-10 L/min, Repetition accuracy 10s: ±1%; Keyence Corp, Tokyo, Japan) was installed at the oxygenator outlet.

B. EXPERIMENTAL FLUIDS
A glycerin solution and a human erythrocyte turbid solution were employed as working fluids. Glycerol solution was substituted for human blood as a more readily available working fluid, based on our previous reports [22]- [24]. Each fluid was perfused in the CPB system, and the viscosities were measured using a vibrating viscometer (SV-10; A & D. Co., Ltd., Tokyo, Japan). The glycerin solution was prepared by adding glycerin to water, and then stored in a venous reservoir. Glycerin solution samples with viscosities of 1.88 mPas (20.7 • C), 2.84 mPas (18.8 • C), and 3.60 mPas (17.7 • C) were prepared by dilution with water. Glycerin solution viscosity corresponds to a viscosity of diluted blood, which is generally used in CPB cases [27]. Human erythrocyte solution (Cosmo Bio. Co., Ltd., Tokyo, Japan) was used with a physiological saline solution. In particular, a sample with a viscosity of  2.66 mPas (24.3 • C) and hematocrit level of 31.0% was prepared.

C. PERFUSION EXPERIMENTS
In the in vitro experiments, the two types of fluids described in section II.B (EXPERIMENTAL FLUIDS), namely, the glycerin solutions and human erythrocyte solution, were perfused through the oxygenator using a centrifugal pump. The rotation speed of the centrifugal pump was 1700-2200 rpm. The Hoffman clamp was placed in the circuit tube on the venous line side. The occluders in the arterial and venous line sides were operated so that the venous reservoir level was always maintained at a steady level. In this study, the venous reservoir level stabilization indicates that the flow rates in the occluders on the arterial and venous line sides are equal. When the occluder is fully opened, the flow rate is 3 L/min. In the experiment, the flow rate was reduced by closing the occluder on the venous line in 5% decrements. At the same time, the occluder on the arterial line side was manipulated to equalize the flow rates of both sides and reservoir level. In the same manner, the occluder on the arterial line was then closed in 5% decrements, and the occluder on the venous line was adjusted to equalize the reservoir level. The occluder inlet and outlet pressures and average flow rate were recorded for each adjustment of the opening ratio of the occluder. These experimental procedures were carried out to evaluate dispersion in the glycerin solutions with three different viscosities and human erythrocyte solutions. The measurements were repeated five times for each glycerin and human erythrocyte solution. Fig. 4 shows an additional CPB system for visualizing the separation flow generated by the occluder closure. Ltd., (Tokyo, Japan)). The vertical axis represents the opening ratio of occluder, and the horizontal axis represents the manipulator value. Image (b) shows the manipulator value controlled by the knob handle, which is shown on display. The occluder has an exponential relationship between the opening ratio of occluder and the manipulator value, as shown in the blue line in (a), to facilitate flow rate control by an operator.

D. VORTEX FORMATION VISUALIZATION
The perfusion circuit was composed of a centrifugal pump (CX-SL45X; Terumo Cardiovascular Systems Corp, Tokyo, Japan) and electromagnetic blood flow meter (FD-M5AT, Flow range 0-10 L/min, Repetition accuracy 10s: ±1%; Keyence Corp, Tokyo, Japan), which were connected using polyvinyl chloride tubes. The glycerin solution with added polystyrene beads (Daiyaion TM ; Mitsubishi Chemical Holdings Corp, Tokyo, Japan) was used as perfusate. The perfusate in the system was illuminated using an infrared laser (Z-LASER; Edmund Optics R Headquarters, New Jersey, USA) and imaged using a Universal Serial Bus (USB) camera (LU105M-IO; Lumenera Corp, Canada). To visualize the polystyrene beads flowing through the blood circuit, a transparent occluder was crafted. The material of the crafted occluder is acrylic resin. The size and shape of the occluder are the same as those of occluder HAS-RH200, manufactured by Senko Inc. The crafted acrylic occluder was attached to the tube on the venous side, as shown in Fig. 4. A water level tube was installed between the acrylic occluder and HAS-RH200 to confirm that the acrylic occluder had the same opening ratio of occlusion as that of HAS-RH200. If the water level of the cylinder is stable, the flow rate to the acrylic occluder is the same as that of HAS-RH200. Therefore, the opening ratio of the occluder can be determined by HAS-RH200, and the flow can be visualized by the acrylic occluder. Moreover, the occluder was closed in steps of 5%, and the same measurement procedure as described in the previous section was repeated. The captured images were analyzed using fluid measurement software (Flow-PVI v6.01; Library Co., Ltd., Tokyo, Japan). As a measurement method, the sequential dismissal method for the accumulation of luminance differences was chosen. The tracking parameters were set to 15 × 15 pixel grid spacing, 11-pixel tracking interval, and 11-pixel reference size for image analysis.

III. MATHEMATICAL MODEL
In this section, we derive a simple model from the Navier-Stokes equation that can be applied to automatically control VOLUME 8, 2020 of the occluder in the future. To this end, we assume that the blood flow in the tube of the CPB circuit is laminar. The Navier-Stokes equation is given by the following: where t is time; x is the axial position of the tube; r is the radial distance; u and v are the flow velocity in the x and r directions, respectively; F x and F r are volume forces; ρ is the density of the fluid; µ is the viscosity, and p is the pressure. For simplification, the blood flow in the circuit is assumed to be a steady flow through a tube with a constant cross-sectional area, and the volume force is ignored (F x = F r = 0). From the steady-state flow assumption, the time derivative terms become 0, v = 0, and ∂u/∂x = (∂ 2 u)/(∂ 2 x) = 0; therefore, (1) and (2) are rewritten as the following equations: By integrating both sides of the Equation (3) twice about r, we obtain the following equation: Because u is finite, C 1 = 0. C 2 can be obtained using the condition that u = 0 for the tube wall (r = r 0 ). Then, the velocity in this flow is given by The flow rate Q is obtained by multiplying the flow velocity u by the small cross-sectional area, 2πrdr, and integrating with r as the following: Therefore, the relationship between the pressure loss P and Q between any two points is found by where l is the length of the tube, and this equation is known as the equation of Hagen-Poiseuille flow. We then formulated the relationship between the manipulator value (see Fig. 2) and the flow rate based on Equation (8). The variables are henceforth subscripted by i (arterial, i = a; and venous, i = v) to distinguish the arterial and venous lines of the CBP circuit. Here, Q 0,i = −P i /(R i + R 0,i ) is the flow rate when the occluder is fully opened (O cc,i = 100%), where R i = 8µl i /πr 4 0,i is the tube friction, and P i is the applied pressure. The tube length l i does not include the channel length of the occluder, and R 0,i is the channel friction of the occluder when O cc,i = 100%. Because the occluder channel is sufficiently short compared to the tube length l i , we can approximate Q 0,i ≈ −P i /R i . When the occluder is closed, flow rate Q i can be obtained by the following equation: where R i is the increased tube friction caused by closing the occluder. By denoting the measured pressures at the inlet and outlet of the occluder as P in,i and P out,i , the pressure loss owing to the occluder can be expressed by P i = P out,i − P in,i . Using the relationship P i = Q i R i , (9) can be summarized in the following form This equation allows us to determine unknown R i and Q 0,i from Q i , P in,i and P out,i measured by the CPB system.
We then introduced a model that can estimate the pressure loss from the manipulator value of the occluder. Fig. 2 shows a schematic diagram of a tube pressed by an occluder. When the blood flows through the pressed tube, separating flow occurs, causing a pressure loss, as shown in the figure. It is difficult to theoretically derive the pressure loss P i caused by closing the occluder because the cross-section of the tube compressed by the occluder is complexly deformed, as shown on the right side of Fig. 2. In this study, for simplicity, we assumed that P i is inversely proportional to the opening ratio of occluder d c /d (see Fig. 2 and Fig. 3). Figure 3 shows the relationship between the manipulator value and the opening ratio of occluder. Let us define a function that links the manipulator value of occluder, O cc,i , with opening ratio of occluder, d c /d, as the following: P i can then be expressed by the following equation: where R i is the loss factor. R i can then be defined by the following equation: Finally, substituting (13) into (10) and solving for Q i gives the following equation representing the relationship between manipulator value O cc,i and flow rate Q i .
In this study, we employed occluder manufactured by Senko Inc. Fig. 3 shows the relationship between the manipulator value and the opening ratio of the occluder employed. According to this figure, the manipulator value and opening ratio of the occluder d c /d can be expressed by an exponential function using the occluder opening O cc,i as follows.
where A i and K i are calibration coefficients. R i and Q i thus can be estimated by substituting right-hand side of (15) for f O cc,i in Equations (13) and (14). Here, the calibration coefficients in (13) and (15) can be determined by fitting Equation (13) to the measured data ( The model can be applied to other occluders by replacing f O cc,i with an occluder-dependent function that can express the relationship between the manipulator value, O cc,i , and the opening ratio of occluder, d c /d. A two-step estimation allows us to separate the fluid mechanical parameters and occluder-dependent calibration parameters, as shown in (10) and (13). This separation provides an advantage to generalize the model to other occluders.

A. PARAMETER DETERMINATION AND VERIFICATION
The calibration coefficients Q 0 and R i were determined by the least square method. The parameters A i and K i were determined using a non-linear regression method (Levenberg-Marquardt algorithm). The fitting accuracy of the proposed model (10)- (13) and (14) was evaluated using the coefficient of determination.
Leave-one-out cross-validation was then carried out to examine the validity of the determined parameters and the prediction accuracy of the proposed model. The flow rate prediction accuracy and its systematic errors were evaluated using the coefficient of determination and Bland-Altman analysis, respectively.
Here, regression and Bland-Altman plot analyses were conducted using statistical software R and the graphical user interface Easy R (Saitama Medical Center, Jichi Medical University, Saitama, Japan). A significance level of 0.01 was assumed in all the statistical tests (p < 0.01).

IV. RESULTS
The individual analyses conducted on the three glycerin solutions at different viscosities (1.88, 2.84, and 3.60 mPas) indicate that the linear flow-pressure model was well fitted with the measured data (R 2 > 0.977 (venous line) and R 2 > 0.983 (arterial line); p < 0.01), as shown in Fig. 5 a and b. VOLUME 8, 2020 FIGURE 5. Relationship between differential pressure and flow rate for glycerin solutions. Circular markers indicate measured data, whereas filled color distinguishes the viscosity level. Solid and dotted lines indicate the fit obtained using the proposed model described in (10). Here, (a) and (b) depict measured data and model fit to (10) observed on venous-and arterial-line sides, respectively. Images (c) and (d) depict fitting to equation (10) for all measured data with respect to viscosity on venous-and arterial-line sides, respectively. FIGURE 6. Relationship between differential pressure and flow rate for glycerin solutions. Circular markers indicate measured data, whereas filled color indicates viscosity. Solid and dotted lines indicate the fit obtained using the proposed model described in (13). Images (a) and (b) depict measured data and fits of the model (13) observed on venous-and arterial-line sides, respectively. In these figures, the models were respectively fitted to the data measured from glycerin solutions with different viscosities. Images (c) and (d) depict the fit of the model to (13) for all measured data on venous-and arterial-line sides, respectively.
A combined analysis also showed that the model was well fitted to the measured data (R 2 > 0.942 (venous line) and R 2 > 0.958 (arterial line); p < 0.01), as shown in Fig. 5 c  and d. Fig. 6 shows measured R a and R v modified by the manipulator values of the occluders (O cc,v and O cc,a ) and the regression lines of the exponential occlusion-pressure model (13). The individual analyses of the three glycerin solutions viscosities show that the exponential occlusionpressure model was well fitted to the measured data (R 2 > 0.994 (venous line) and R 2 > 0.995 (arterial line); p < 0.01), as shown in Fig. 6a and b. The combined analysis is also well fitted to the measured data (R 2 > 0.997 (venous line)   and R 2 > 0.999 (arterial line); p < 0.01), as shown in Fig. 6 c and d. The experimental results and regression lines are shown in Figs. 7 and 8, respectively. Fig. 7 shows that the linear flow-pressure model regression was well fitted to the measured data on the venous-and arterial-line sides (R 2 = 0.985 and R 2 = 0.997, respectively; both produced p < 0.01). Moreover, the exponential occlusion-pressure model was well fitted to the measured data on the venous-and arterial-line sides (R 2 = 0.989 and R 2 = 0.988, respectively; both with p < 0.01), as shown in Fig. 8. Table 1 lists the proposed model calibration parameters determined by fitting the estimated data to the measurements (see Materials and Methods; Mathematical Model). Based on these parameters, the flow rate from the opening ratio of the occluder can be estimated using (14). Fig. 9 shows the relationship between the opening ratio of the occluder (O cc,v , O cc,a ) and flow rates (Q v , Q a ), in addition to the fitted lines based on (14). These results show significant high coefficients of determination between the estimated and measured values for the venous-and arterial-line sides (R 2 = 0.956 and R 2 = 0.993, respectively; both with p < 0.01).
Additionally, leave-one-out cross-validation was carried out to verify the proposed method prediction accuracy. Fig. 10 shows the predicted Q i plotted against the measured  flow rate. The coefficient of determination obtained from the venous-line side using cross-validation is 0.979 (p < 0.01) (Fig. 10a), and that of the arterial-line side is 0.996 (p < 0.01) (Fig. 10c). Bland-Altman analysis was then used to evaluate the prediction accuracy of (14), as shown in Figs. 10b and 10d. There are no constant errors on the arterial-and venous-line sides given that zero was included in the 95% confidence interval. Further, proportional error was not found given that the R 2 p-values on the venous and arterial sides are 0.890 and 0.677, respectively. These results indicate that systematic errors are minimal.
The relationship between the flow rate and manipulator values of the occluder was dichotomized in the venous side (see Fig. 9). We thus visualized the flow in the venous side to investigate the cause of this dichotomy. Fig. 11 shows that occluder closure resulted in the perfusate separation. The separated flows were situated at the occluder restrictor upper rear part, and further closure promoted the spread of the vortex throughout the blood circuit behind the occluder. The onset of exfoliation flow occurred at 95% closure in the first, second, and fifth trials and 90% in the third and fourth trials. The spread of the vortex throughout the blood circuit occurred at 80% closure in the second trial; 75% closure in the first, fourth, and fifth trials; and 70% closure in the third trial.

V. DISCUSSION
This study evaluated the influence of the manipulator value of the occluder on the differential pressure. We assumed that the relationship between the manipulator value, O cc,i , and opening ratio of the occluder d c /d is exponential, and d c /d is inversely proportional to tube friction R i . This assumption is justified by the high accuracy of the fitting to the measured data of both glycerin solutions and blood (see Fig. 6). To facilitate flow control by an operator, CPB occluders are commonly configured in such a way that the relationship  (11) in this study, to an appropriate monotonic non-linear function.
The fitting results confirm that the relationship between the differential pressure and flow rate, in addition to that between the differential pressure and opening ratio of the occluder, is not significantly influenced by changes in the fluid viscosity (see Figs. 5 and 6 a.b.c.d). These results indicate that it is unnecessary to include viscosity as a parameter of the model.
The analysis result showed that the correlation on the venous side was lower than that on the arterial side. This is because the measured flow rates dichotomized (see Fig. 9a). To investigate the cause of this, we visualized the flow around the occluder. By visualizing the venous side, we found that the spread of the vortices in the entire blood circuit does not occur at a specific opening ratio of the occluder. This observation suggests that Hopf bifurcation is occurring in the blood circuit [28], which is not controllable and can cause a certain prediction error of the flow rate. This error is unavoidable because our model focuses on steady-state flow. Therefore, an assisted cardiopulmonary manipulation control system to be developed in the future requires combining feedforward control using the proposed model to shift between the steady states and feedback control to absorb error caused by the turbulent flow.
In this study, we assumed the working fluid to be a simple viscous fluid (Newtonian fluid); however, the viscosity of actual blood nonlinearly changes and affects the pressure drop. Nonetheless, the blood and glycerin approximated by the same model equation demonstrated high coefficients of determination, where R 2 > 0.998 (p < 0.01) for the venous line and R 2 > 0.988 (p < 0.01) for the arterial line. These results indicate that the proposed model can be applied to blood.
The parameters were determined using the experimental data obtained from the perfusion system connected by tubes with a diameter of 0.5 and 0.375 on the venous-and arterial-line sides, respectively. The system configuration and difference between the non-Newtonian fluid characteristics may have an influence on the calibration parameters. For example, the differences between the occluder positions and circuit lengths may alter the flow velocity distribution, which has an influence on the differential pressure [29]. Therefore, the three parameters in the proposed model require tuning to the specific CPB system configuration.
By building an inverse model of the proposed model, O cc,i can be determined from the desired flow rate. This inverse model can be easily derived owing to the simplicity of the proposed model. Therefore, the proposed model can be used for feedforward control to shift the steady-state, which serve as a basis for the development of control methods and the realization of assisted cardiopulmonary manipulation control systems. In the future, transient responses of physiological functions need to be considered. By combining feedback control and feedforward control to absorb the errors caused by such transient responses, a cardiopulmonary manipulation support control system can be realized. Furthermore, to realize a control system for various scenarios in cardiac surgery, the technology to monitor the reservoir level in real-time is required, which is the next stage of research. By building an inverse model of the proposed model, Occ can be determined from the desired flow rate, and the steady state can be shifted. This model is simple and it is easy to construct the inverse model. Therefore, the proposed model can serve as a basis for the development of control methods and the realization of assisted cardiopulmonary manipulation control systems.

VI. CONCLUSION
This study developed a model to enable estimation of flow rate from the manipulator value of the occluder used in the CPB circuit. The results demonstrate that the proposed model can accurately estimate the relationship between the differential pressure, flow rate, and opening ratio of the occluder for non-Newtonian fluids perfused in the circuit. The results of this study can serve as basis for further development of cardiopulmonary operation interlocking systems as well as simplification of occluder operation.