# Micro-Viscometer for Measuring Shear-Varying Blood Viscosity over a Wide-Ranging Shear Rate

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## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

#### 2.1. Numerical Simulation

^{−4}.

#### 2.2. Fabrication of the Micro-Viscometer

#### 2.3. Experimental Setup

#### 2.4. Viscosity Measurement of the Micro-Viscometer

_{r}and N

_{s}are the number of channels respectively filled with the reference and sample fluids in the counting section. Thus, because the viscosity of the reference fluid (μ

_{r}) is already a known value, it is possible to predict the viscosity of the sample fluid (μ

_{s}) by determining the input flow rates of each fluid and the number of fluid-filled channels.

#### 2.5. Determination of the Optimal Flow Rate Ratio

_{r}), a denominator in Equation (5), is too small, then a change in a single channel with the reference fluid will have a large influence on the viscosity measurement. In other words, as the number of reference fluid channels increases, a reliable viscosity measurement becomes possible. In the presented micro-viscometer, therefore, a higher PBS to blood flow rate ratio is chosen in order to increase the number of channels occupied by a reference fluid (PBS).

## 3. Results and Discussion

#### 3.1. Parametric Study for Design Optimization of the Micro-Viscometer

#### 3.2. Preliminary Test: Newtonian Viscosity

#### 3.3. Numerical Demonstration

#### 3.4. Experimental Demonstration

#### 3.5. Blood Sample Viscosity Measurements with Variation in Hematocrit Levels

## 4. Conclusions

## Supplementary Materials

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## References

- Letcher, R.L.; Chien, S.; Pickering, T.G.; Sealey, J.E.; Laragh, J.H. Direct relationship between blood pressure and blood viscosity in normal and hypertensive subjects. Am. J. Med.
**1981**, 70, 1195–1202. [Google Scholar] [CrossRef] - Lowe, G.; Rumley, A.; Norrie, J.; Ford, I.; Shepherd, J.; Cobbe, S.; Macfarlane, P.; Packard, C. Blood rheology, cardiovascular risk factors, and cardiovascular disease: The west of scotland coronary prevention study. Thromb. Haemost.
**2000**, 84, 553–558. [Google Scholar] [PubMed] - Danesh, J.; Collins, R.; Peto, R.; Lowe, G.D. Haematocrit, viscosity, erythrocyte sedimentation rate: Meta-analyses of prospective studies of coronary heart disease. Eur. Hear. J.
**2000**, 21, 515–520. [Google Scholar] [CrossRef] - Mayer, G.A. Hematocrit and coronary heart disease. Can. Med. Assoc. J.
**1965**, 93, 1151–1153. [Google Scholar] [PubMed] - Chabanel, A.; Flamm, M.; Sung, K.L.; Lee, M.M.; Schachter, D.; Chien, S. Influence of cholesterol content on red cell membrane viscoelasticity and fluidity. Biophys. J.
**1983**, 44, 171–176. [Google Scholar] [CrossRef] - Jeong, S.K.; Cho, Y.I.; Duey, M.; Rosenson, R.S. Cardiovascular risks of anemia correction with erythrocyte stimulating agents: Should blood viscosity be monitored for risk assessment? Cardiovasc. Drugs Ther.
**2010**, 24, 151–160. [Google Scholar] [CrossRef] [PubMed] - Schramm, G. A practical Approach to Rheology and Rheomtery, 2nd ed.; Thermo Haake Rheology: Karlsruhe, Germany, 1994. [Google Scholar]
- Lin, Y.-Y.; Lin, C.-W.; Yang, L.-J.; Wang, A.-B. Micro-viscometer based on electrowetting on dielectric. Electrochim. Acta
**2007**, 52, 2876–2883. [Google Scholar] [CrossRef] - Keen, S.; Yao, A.; Leach, J.; Leonardo, R.D.; Saunter, C.; Love, G.; Cooper, J.; Padgett, M. Multipoint viscosity measurements in microfluidic channels using optical tweezers. Lab Chip
**2009**, 9, 2059–2062. [Google Scholar] [CrossRef] [PubMed] - Li, Y.; Burke, D.T.; Kopelman, R.; Burns, M.A. Asynchronous Magnetic Bead Rotation (AMBR) microviscometer for label-free DNA analysis. Biosensors
**2014**, 4, 76–89. [Google Scholar] [CrossRef] [PubMed] - Srivastava, N.; Davenport, R.D.; Burns, M.A. Nanoliter viscometer for analyzing blood plasma and other liquid samples. Anal. Chem.
**2005**, 77, 382–392. [Google Scholar] [CrossRef] [PubMed] - Han, Z.; Tang, X.; Zheng, B. A PDMS viscometer for microliter Newtonian fluid. J. Micromech. Microeng.
**2007**, 17, 1828–1834. [Google Scholar] [CrossRef] - Srivastava, N.; Burns, M.A. Analysis of non-Newtonian liquids using a microfluidic capillary viscometer. Anal. Chem.
**2006**, 78, 1690–1696. [Google Scholar] [CrossRef] [PubMed] - Han, K.; Zhu, K.; Bahl, G. Opto-mechano-fluidic viscometer. Appl. Phys. Lett.
**2014**, 105, 014103. [Google Scholar] [CrossRef] - Dehestru, G.; Leman, M.; Jundt, J.; Dryden, P.; Sullivan, M.; Harrison, C. A microfluidic vibrating wire viscometer for operation at high pressure and high temperature. Rev. Sci. Instrum.
**2011**, 82, 035113. [Google Scholar] [CrossRef] [PubMed] - Quist, A.; Chand, A.; Ramachandran, S.; Cohen, D.; Lal, R. Piezoresistive cantilever based nanoflow and viscosity sensor for microchannels. Lab Chip
**2006**, 6, 1450–1454. [Google Scholar] [CrossRef] [PubMed] - Zeng, H.; Zhao, Y. Rheological analysis of non-Newtonian blood flow using a microfluidic device. Sens. Actuators A Phys.
**2011**, 166, 207–213. [Google Scholar] [CrossRef] - Chevalier, J.; Ayela, F. Microfluidic on chip viscometers. Rev. Sci. Instrum.
**2008**, 79, 076102. [Google Scholar] [CrossRef] [PubMed] - Pipe, C.J.; Majmudar, T.S.; McKinley, G.H. High shear rate viscometry. Rheol. Acta
**2008**, 47, 621–642. [Google Scholar] [CrossRef] - Guillot, P.; Moulin, T.; Kotitz, R.; Guirardel, M.; Dodge, A.; Joanicot, M.; Colin, A.; Bruneau, C.-H.; Colin, T. Towards a continuous microfluidic rheometer. Microfluid. Nanofluid.
**2008**, 5, 619–630. [Google Scholar] [CrossRef] - Nguyen, N.-T.; Yap, Y.-F.; Sumargo, A. Microfluidic rheometer based on hydrodynamic focusing. Meas. Sci. Technol.
**2008**, 19, 085405. [Google Scholar] [CrossRef] - Choi, S.; Park, J.K. Microfluidic rheometer for characterization of protein unfolding and aggregation in microflows. Small
**2010**, 6, 1306–1310. [Google Scholar] [CrossRef] [PubMed] - Solomon, D.E.; Vanapalli, S.A. Multiplexed microfluidic viscometer for high-throughput complex fluid rheology. Microfluid. Nanofluid.
**2014**, 16, 677–690. [Google Scholar] [CrossRef] - Kang, Y.J.; Ryu, J.; Lee, S.-J. Label-free viscosity measurement of complex fluids using reversal flow switching manipulation in a microfluidic channel. Biomicrofluidics
**2013**, 7, 044106. [Google Scholar] [CrossRef] [PubMed] - Kang, Y.J.; Yoon, S.Y.; Lee, K.H.; Yang, S. A highly accurate and consistent microfluidic viscometer for continuous blood viscosity measurement. Artif. Organs
**2010**, 34, 944–949. [Google Scholar] [CrossRef] [PubMed] - Kang, Y.J.; Yang, S. Integrated microfluidic viscometer equipped with fluid temperature controller for measurement of viscosity in complex fluids. Microfluid. Nanofluid.
**2013**, 14, 657–668. [Google Scholar] [CrossRef] - Fahraeus, R.; Lindqvist, T. The viscosity of the blood in narrow capillary tubes. Am. J. Physiol.
**1931**, 96, 562–568. [Google Scholar]

**Figure 1.**Schematic of the micro-viscometer. The micro-viscometer has a sequence of micro-channel arrays that generate 10 sets of shear rate. The viscosity is derived from the input flow rates for the reference sample fluids and the number of channels filled with both fluids. By varying the channel width at each array, different viscosity values can be obtained. For each array, 100 micro-channels are arranged in parallel. It is designed accordingly to change the width of the channel from the first array to the tenth array to enable each array to form a different shear rate from a single flow condition.

**Figure 2.**Optimal flow rate ratio of the reference and sample fluids. The x-axis represents the viscosity to be measured in the counting section; the y-axis represents the expected relative error that is calculated for the case when a single channel for PBS is mistakenly counted. Flow rate ratio (Q/Q) means the ratio of the reference (Q

_{r}) and sample fluid (Q

_{s}). As the input flow rate of the reference fluid increases relative to the flow rate of the sample fluid, the relative error decreases in the viscosity range above 10 cP. However, as depicted in the inset, the increasing flow rate ratio also induces an elevated relative error in the low viscosity range near 1 cP. Consequently, the optimal condition of the flow rate ratio, which ensures a relative error of less than 5% over the whole range of viscosities, is set to seven (red circle).

**Figure 3.**Parametric study for the design optimization of the micro-viscometer. (

**A**) At a given length of the transient section (1000 μm), it is confirmed that, as the length of the counting section is decreased (from C1 to C3), the relative error significantly decreases. (* p < 0.05, *** p < 0.001) Specifically, the underestimation seems to be alleviated because the hydraulic resistance of the counting section, which is related to the Fahraeus effect, is relatively decreased. (

**B**) At a given counting section length (500 μm), it is confirmed that the relative error significantly decreases as the length of the transient section increases (from T1 to T4). This is also caused by the counting section, which has a relatively low value of hydraulic resistance. (

**C**) When the length of the counting section is reduced and the length of the transient section is simultaneously increased (from C/T1 to C/T3), the underestimation decreases, and eventually the relative error converges. (

**D**) The resistance ratio between the counting and transient sections is expressed as R

_{c}/R

_{t}, represented as the x-axis in the graph. The underestimation due to the Fahraeus effect occurs as the counting section resistance increases. On the contrary, it is confirmed that an accurate viscosity measurement is possible when the resistance ratio converges to be close to zero (y-intersect: 2.5991%).

**Figure 4.**Numerical and experimental demonstrations of the micro-viscometer. (

**A**) For the given flow rate conditions (0.1 and 1.0 mL/h), the captured images of 10 counting sections showing the fluidic boundaries are depicted. The number of channels filled with the sample fluid (red) range from a maximum of 50 to a minimum of 39 in the low shear rate regime (14.2–145.4/s), and to a minimum of 34 in the high shear rate regime (168.8–1668.3/s). (

**B**) Captured images showing fluidic boundaries in the 10 counting sections. The number of channels filled with the blood sample decreased from a maximum of 50 to a minimum of 41 (for the flow rate of 0.1 mL/h), and decreased from a maximum of 43 to a minimum of 35 (for the flow rate of 1.0 mL/h). (

**C**) The viscosity values from the numerical study (blue circle) are obtained from two flow rate conditions, which can be fitted with 9.69 (for k) and −0.14 (for n) using the power-law model. Also, 20 viscosity values (red circle) obtained from the experiment are fitted using the power-law model with 9.84 (for k) and −0.13 (for n). (

**D**) It is confirmed that the numerical result shows relative errors of 0.2% (for k) and 6.8% (for n) and the experimental result shows relative errors of 1.79% (for k) and 5.30% (for n). This implies that the micro-viscometer design concept enables accurate measurement of the shear-thinning viscosity.

**Figure 5.**Experimental reliability test of the micro-viscometer. (

**A**) Repeated viscosity measurements were performed (N = 7) with the proposed micro-viscometer. Each measurement provides very similar results (averaged standard deviation of 4.25% for seven sets of viscosity measurements). The result also shows a very similar shear-thinning viscosity of the blood compared to the result from the rotational viscometer. (

**B**) Relative errors for the two indices (k and n) are 1.88% and 6.30%, respectively. (

**C**,

**D**) According to the t-test, there are no statistically significant differences (p > 0.05) for 14 data points, and the averaged relative error is 1.56%. Although the remaining six data points show a significant difference (p < 0.05), the averaged relative error is 5.35%. This implies that the micro-viscometer shows reliability in repeated viscosity measurements.

**Figure 6.**Viscosity measurement with variation in hematocrit levels. (

**A**) Viscosity measurements of the three samples with different hematocrit levels (35%, 45%, and 55%) were performed. The results show 5.36, 8.98, and 13.52 (for k) and −0.08, −0.11, and −0.18 (for n), respectively. Thus, an increase in both indices proves that blood samples with higher hematocrit levels have higher viscosities as well as higher magnitudes of shear-thinning behavior. The relative errors are 5.5%, 6.4%, and 5.6% (for k), and 10.4%, 6.4%, and 6.6% (for n), respectively. (

**B**) A comparison of the 20 data points obtained from each sample show average relative errors of −1.72 ± 4.21%, −2.90 ± 4.03%, and −0.09 ± 3.61%, respectively. Consequently, it is demonstrated that the micro-viscometer recognizes viscosity changes due to the variation in hematocrit levels.

Property | Density [kg/m^{3}] | Viscosity [cP] | Mass Flow Rate [kg/s] | |
---|---|---|---|---|

Flow Rate 1 | Flow Rate 2 | |||

Reference fluid | 1000 | 0.001 | 1.95 × 10^{−7} | 1.95 × 10^{−6} |

Sample fluid | 1025 | k = 9.6685 n = −0.132 | 2.78 × 10^{−}^{8} | 2.78 × 10^{−7} |

Type | Counting Section | Transient Section |
---|---|---|

C1 | 2500 | 1000 |

C2 | 1500 | |

C3 | 500 | |

T1 | 500 | 1000 |

T2 | 2000 | |

T3 | 3000 | |

T4 | 4000 | |

C/T1 | 2000 | 2000 |

C/T2 | 1000 | 3000 |

C/T3 | 500 | 4000 |

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**MDPI and ACS Style**

Kim, B.J.; Lee, S.Y.; Jee, S.; Atajanov, A.; Yang, S.
Micro-Viscometer for Measuring Shear-Varying Blood Viscosity over a Wide-Ranging Shear Rate. *Sensors* **2017**, *17*, 1442.
https://doi.org/10.3390/s17061442

**AMA Style**

Kim BJ, Lee SY, Jee S, Atajanov A, Yang S.
Micro-Viscometer for Measuring Shear-Varying Blood Viscosity over a Wide-Ranging Shear Rate. *Sensors*. 2017; 17(6):1442.
https://doi.org/10.3390/s17061442

**Chicago/Turabian Style**

Kim, Byung Jun, Seung Yeob Lee, Solkeun Jee, Arslan Atajanov, and Sung Yang.
2017. "Micro-Viscometer for Measuring Shear-Varying Blood Viscosity over a Wide-Ranging Shear Rate" *Sensors* 17, no. 6: 1442.
https://doi.org/10.3390/s17061442