# Estimation Methods for Viscosity, Flow Rate and Pressure from Pump-Motor Assembly Parameters

^{1}

^{2}

^{*}

## Abstract

**:**

^{2}= 0.98, root mean squared error (RMSE) = 46 mL

^{.}min

^{−1}; pressure difference r

^{2}= 0.98, RMSE = 8.7 mmHg; and viscosity r

^{2}= 0.98, RMSE = 0.049 mPa

^{.}s. The results suggest that the presented methods can be used to accurately predict blood flow rate, pressure, and viscosity online.

## 1. Introduction

_{2}R) [2] devices are combined with protective ventilation strategies to let the lungs recover. However, ECCO

_{2}R is a highly invasive procedure that is associated with elevated risks of infection, thrombosis and—due to the high priming volumes—high load on the heart [3]. An intracorporeal membrane oxygenator could be used as a supplementary mode of oxygenation along with mechanical ventilation and thus reduce the mechanical load on the lungs due to the high ventilator pressures. Additionally, an intracorporeal membrane oxygenator could help patients with sufficient lung capacity recover from various lung diseases and injuries by supplementing their spontaneous breathing.

^{.}min

^{−1}for the static and 444 mL

^{.}min

^{−1}for the dynamic estimation of flow rate in water–glycerol mixtures. Most of the approaches employed a parametric polynomial model that was acquired by fitting the parameters to the measured characteristics of the device. The problem with that method is that an assumption about the underlying function of the process must be made. Furthermore, during operation often the characteristics of the system change—e.g., hematocrit levels and consequently viscosity, thus the estimation degrades with time. To combat that some groups have developed an online estimation of the viscosity of the blood from the pump-motor assembly parameters by either inducing vibrations in the radial direction [20,21], by occluding the pump exit [17] or by forcing the pump with a random signal for 10s every 2 min to get a frequency rich signal [18]. Hijikata et al. report their viscosity estimation methods with water–glycerol mixtures [21] (RMSE = 0.12 mPa

^{.}s in a range 1.18–5.12 mPa

^{.}s) and in vitro with blood [20] (RMSE = 0.12 mPa

^{.}s in a range 2.32–2.75 mPa

^{.}s). The inclusion of the viscosity compensation helped them improve the average error from 1830 mL

^{.}min

^{−1}to 360 mL

^{.}min

^{−1}in a range of 3000–5000 mL

^{.}min

^{−1}.

## 2. Materials and Methods

#### 2.1. Data Collection and Data Processing

#### 2.2. Viscosity and Blood Modeling

^{−1}[23]. However, rotary blood pumps usually develop shear rates in the order of 10

^{4}s

^{−1}[24], thus blood can be approximated as a Newtonian fluid with constant viscosity of 3.4 mPa

^{.}s in those shear rate ranges. The viscosity of different water–glycerol mixtures was measured and modeled by various groups [25]. It was shown by Cheng [26] that his formula has an average predicting error of 1.3% in a range of 0–100 °C and 0–100% water in the water–glycerol mixture. The analytic kinematic viscosity was thus calculated by:

#### 2.3. Measurement Protocol

_{max}to no flow rate Q

_{0}and then monotonously rising flow rates Q

_{0}to Q

_{max}and their corresponding pressure drops △p. Additionally, training data consisting of combinations of throttle setting and rotations speed in random steps was acquired, in order to capture the dynamical behavior of the system. The I, ω, Q, △p quadruplets distributed over the whole operation range of the pump were recorded for 3 different test liquids (85/15, 65/35 and 50/50 water–glycerol volume percentage) with different viscosity (Table 1.). The test data set consisted of random combinations of ω and throttle setting and due to its randomness was different for each liquid mixture.

#### 2.4. Gaussian Process Models

#### 2.5. Gaussian Process Models Optimization

- Mean Function: {none, constant, linear, quadratic}
- Covariance Function: {automatic relevance determination (ARD) exponential, ARD Matern kernel 3/2, ARD Matern kernel 5/2, ARD rational quadratic, ARD squared exponential, exponential, Matern kernel 3/2, Matern kernel 5/2, rational quadratic, squared exponential}

#### 2.6. Viscosity Estimation GP

#### 2.7. Estimation Algorithm

## 3. Results

#### 3.1. System Characterization

#### 3.2. Estimating the Flow Rate and Pressure Difference

^{2}= 0.98, while the RMSE = 46 mL

^{.}min

^{−1}and the maximum error (definition see Appendix A below) ERR

_{max}= 391 mL

^{.}min

^{−1}(Figure 5) and for pressure difference (Figure 6) r

^{2}= 0.98, RMSE = 8.7 mmHg and ERR

_{max}= 67 mmHg. In comparison these values for the predictions ${E}_{ij}$, with i ≠ j, were: r

^{2}= 0.94, RMSE = 236 mL

^{.}min

^{−1}, ERR

_{max}= 555 mL

^{.}min

^{−1}for blood flow rate and r

^{2}= 95, RMSE = 29 mmHg, ERR

_{max}= 109 mmHg for pressure difference.

#### 3.3. Estimating the Viscosity of the Test Liquid

^{2}= 0.98, while RMSE = 0.11 mPa

^{.}s and ERR

_{max}= 0.77 mPa

^{.}s. If the 33% and 36.5% water volume fraction liquids are also included in the training process, the estimation error improves to RMSE = 0.049 mPa

^{.}s. The results of the improved estimation can be seen in Figure 8.

#### 3.4. Estimation of Uncertainty

^{.}min

^{−1}for Q and 5.5 mmHg for △p for ${E}_{ij}$, with i = j, and 31 mL

^{.}min

^{−1}for Q and 6.2 mmHg for △p for ${E}_{ij}$, with i ≠ j. Figure 9 summarizes and visualizes those results and compares them to the RMSE. In all cases ${E}_{ij}$, with i = j the SD is lower compared to the ${E}_{ij}$, with i ≠ j, thus SD could be used as a signal that gives information about the error of the prediction model.

## 4. Discussion

_{50/50}= 700 mL

^{.}min

^{−1}, Q

_{85/15}= 950 mL

^{.}min

^{−1}, Q

_{65/35}= 1050 mL

^{.}min

^{−1}, i.e., the mapping is not unique for the pump-motor assembly, but for the pump-motor-liquid system. Therefore, a separate GPR model was built for each liquid with different viscosity. Another possible solution is to train a single GPR model with an additional input parameter—the viscosity of the liquid. However, this approach is much more computationally expensive as the matrix inversion during the optimization/estimation process scales with ~O(N

^{3}). The non-uniqueness of the mapping also makes it necessary to include the additional signal measurement for the estimation algorithm in Figure 2b. Without it, the estimation could be completely wrong without any indication, since the motor current and rotation speed ranges are almost identical for all test liquids. At low rotation speeds, it is hard to distinguish between the characteristic curves of different test liquids, which is not a problem for high rotation speeds. Consequently, at a low rotation speed, even a wrong model would predict correctly. However, in this case, the controller would not know that the model is wrong since the SD of the prediction would also be low.

^{.}s—even though the identification signal itself is new, and unknown to the model. However, the estimation of the viscosity of the new liquids is not as good—RMSE = 0.6 mPa

^{.}s—because the model has not been trained with data in that region. The viscosity estimation was improved by including more training points (Figure 9). This implies that the estimation accuracy can be improved further simply by introducing more data in a region of interest. For example, one might be interested in high estimation accuracy of blood viscosity with around 40% hematocrit. Then it would be sufficient if data is collected at several distinct hematocrit levels around the 40% mark and subsequently used to re-train the GPR model. The hydraulic resistance that the pump is exposed to does not change in our system, therefore the identification signal was always acquired in the same throttle position. This might not be the case for heart pumps.

^{.}min

^{−1}and 8.7 mmHg, respectively. The accuracy of the sensors is reported to be ~40 mL

^{.}min

^{−1}and 7.5 mmHg for flow rate and pressure, respectively. Thus, it can be concluded that the presented models cover almost all the variability in the data and the uncertainty in the prediction is mostly of an aleatoric character. Additionally, there might be some bias in the test sets presented in this work, because the throttle was operated by hand, but the bias was in no way intended.

^{2}= 0.98, RMSE = 46 mL

^{.}min

^{−1}; pressure difference r

^{2}= 0.98, RMSE = 8.7 mmHg) and it was shown that it could be easily implemented for online use by including a viscosity measurement (r

^{2}= 0.98, RMSE = 0. 0.049 mPa

^{.}s). In future work, the performance of the method will be tested in vitro with blood as a test liquid.

## Author Contributions

## Funding

## Conflicts of Interest

## Appendix A

#### Measures of Fitness

^{2}), the RMSE and the maximum error (ERR

_{max}) were used to evaluate the performance of the estimation. The r

^{2}represents the extent to which the variance of the data is explained by the model. An r

^{2}= 1, would suggest that all variance of the system is captured by the model, while r

^{2}= 0, would suggest that the model is not explaining the underlying phenomena at all. The correlation coefficient r

^{2}, is defined as:

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**Figure 2.**Flowchart of the two proposed algorithms for online estimation of hydrodynamic parameters. First the viscosity is automatically detected and the most suitable model for flow rate estimation and/or pressure becomes active. (

**a**) The pressure and flow rate are continuously estimated with the same model as long as the time elapsed since the last viscosity identification is below a threshold. When the threshold time T elapses, the viscosity is re-identified; (

**b**) The second algorithm needs an additional measurement signal to automatically detect when the viscosity should be re-identified. Here the pressure is measured along with the motor current and rotation speed and the flow rate is estimated. If the uncertainty of the estimation is above a threshold, the viscosity is re-identified. Otherwise, the flow rate is continuously estimated until the uncertainty becomes high.

**Figure 3.**(

**a**) Q–△p characteristics of the motor-pump assembly used in the experiment. Three clusters of curves can be seen corresponding to three different motor speeds, while the color coding refers to different viscosities of the test liquid; (

**b**) ω–I characteristics of the motor - pump assembly. The motor current is plotted in a boxplot versus motor speed and viscosity of the test liquid.

**Figure 4.**(

**a**) The I–Q characteristics of the system; (

**b**) The I–△p characteristics of the system. The three clusters of curves in both plots correspond to different motor speeds while the color coding refers to different viscosities of the test liquid.

**Figure 5.**Results of the flow rate prediction. The estimation is done with a model that is trained on the same viscosity as the test signal. (

**a**) Time series of the true and predicted flow rate with confidence intervals; (

**b**) A scatter plot with fitness measures of the estimated vs. measured flow rate.

**Figure 6.**Results of the pressure difference prediction. The estimation is done with a model that is trained on the same viscosity as the test signal. (

**a**) Time series of the true and predicted flow rate with confidence intervals; (

**b**) A scatter plot with fitness measures of the estimated vs. measured pressure difference.

**Figure 7.**(

**a**) Results of the viscosity estimation plotted versus volume fraction; (

**b**) Fitness metrics of the estimation.

**Figure 8.**(

**a**) Results of the viscosity estimation plotted versus volume fraction after inclusion of the 0.33 and 0.365 volume fraction points in the training process; (

**b**) Fitness metrics of the estimation.

**Figure 9.**Illustrating the correlation between standard deviation and root mean squared error of the prediction. The left column of matrices shows the results for flow rate and the right column for pressure difference. The top row shows the matrices for standard deviation and the bottom row for RMSE. The matrices themselves show results for the 3 presented liquids thus far the 506,585 volume percentage water in the water–glycerol mixture. The diagonal values correspond to predictions ${E}_{ij}$, with i = j, while off-diagonal values correspond to ${E}_{ij}$, with i ≠ j. It can be seen that in both RMSE and standard deviation (SD) the diagonal values are much smaller than the off-diagonal values.

**Table 1.**Kinematic and dynamic viscosities of different water–glycerol mixtures calculated analytically using Equations (1)–(4).

Water/Glycerol Vol% | 100/0 | 95/5 | 90/10 | 85/15 | 80/20 | 75/25 | 70/30 | 65/35 | 60/40 | 55/45 | 50/50 |
---|---|---|---|---|---|---|---|---|---|---|---|

Kinematic Viscosity [µm^{2.}s^{−1}] @23 °C | 0.934 | 1.08 | 1.26 | 1.49 | 1.78 | 2.15 | 2.64 | 3.28 | 4.16 | 5.38 | 7.11 |

Dynamic Viscosity [µPa^{.}s^{−1}] @23 °C | 1006 | 1156 | 1341 | 1571 | 1861 | 2231 | 2711 | 3344 | 4194 | 5359 | 6995 |

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**MDPI and ACS Style**

Elenkov, M.; Ecker, P.; Lukitsch, B.; Janeczek, C.; Harasek, M.; Gföhler, M.
Estimation Methods for Viscosity, Flow Rate and Pressure from Pump-Motor Assembly Parameters. *Sensors* **2020**, *20*, 1451.
https://doi.org/10.3390/s20051451

**AMA Style**

Elenkov M, Ecker P, Lukitsch B, Janeczek C, Harasek M, Gföhler M.
Estimation Methods for Viscosity, Flow Rate and Pressure from Pump-Motor Assembly Parameters. *Sensors*. 2020; 20(5):1451.
https://doi.org/10.3390/s20051451

**Chicago/Turabian Style**

Elenkov, Martin, Paul Ecker, Benjamin Lukitsch, Christoph Janeczek, Michael Harasek, and Margit Gföhler.
2020. "Estimation Methods for Viscosity, Flow Rate and Pressure from Pump-Motor Assembly Parameters" *Sensors* 20, no. 5: 1451.
https://doi.org/10.3390/s20051451