micromachines Article Microrheometer for Biofluidic Analysis: Electronic Detection of the Fluid-Front Advancement Lourdes Méndez-Mora 1, * , Maria Cabello-Fusarés 2 , Josep Ferré-Torres 1 , Carla Riera-Llobet 1 , Samantha López 1 , Claudia Trejo-Soto 3 , Tomas Alarcón 2,4,5 and Aurora Hernandez-Machado 1,2,6 1 Department of Condensed Matter Physics, University of Barcelona (UB), 08028 Barcelona, Spain; josep.ferre@fmc.ub.edu (J.F.-T.); crierall7@alumnes.ub.edu (C.R.-L.); samantha.lopezm94@gmail.com (S.L.); a.hernandezmachado@ub.edu (A.H.-M.) 2 Centre de Recerca Matemàtica (CRM), 08193 Bellaterra, Spain; cabello.maria1@gmail.com 3 Instituto de Física, Pontificia Universidad Católica de Valparaíso, Casilla 4059, Chile; claudia.trejo@pucv.cl 4 Catalan Institution for Research and Advanced Studies (ICREA), 08010 Barcelona, Spain; talarcon@crm.cat 5 Departament de Matemàtiques, Universitat Autónoma de Barcelona (UAB), 08193 Bellaterra, Spain 6 Institute of Nanoscience and Nanotechnology (IN2UB), Universitat de Barcelona (UB), 08028 Barcelona, Spain * Correspondence: lmendemo9@alumnes.ub.edu; Tel.: +34-(600)-895-288 Abstract: The motivation for this study was to develop a microdevice for the precise rheological characterization of biofluids, especially blood. The method presented was based on the principles of rheometry and fluid mechanics at the microscale. Traditional rheometers require a considerable amount of space, are expensive, and require a large volume of sample. A mathematical model  was developed that, combined with a proper experimental model, allowed us to characterize the  viscosity of Newtonian and non-Newtonian fluids at different shear rates. The technology presented Citation: Méndez-Mora, L.; here is the basis of a point-of-care device capable of describing the nonlinear rheology of biofluids Cabello-Fusarés, M.; Ferré-Torres, J.; by the fluid/air interface front velocity characterization through a microchannel. The proposed Riera-Llobet, C.; López, S.; Trejo-Soto, microrheometer uses a small amount of sample to deliver fast and accurate results, without needing C.; Alarcón, T.; Hernandez-Machado, a large laboratory space. Blood samples from healthy donors at distinct hematocrit percentages A. Microrheometer for Biofluidic were the non-Newtonian fluid selected for the study. Water and plasma were employed as testing Analysis: Electronic Detection of the Newtonian fluids for validation of the system. The viscosity results obtained for the Newtonian and Fluid-Front Advancement. non-Newtonian fluids were consistent with pertinent studies cited in this paper. In addition, the Micromachines 2021, 12, 726. results achieved using the proposed method allowed distinguishing between blood samples with https://doi.org/10.3390/mi12060726 different characteristics. Academic Editor: Yutaka Kazoe Keywords: rheometer; microrheometer; rheology; hemorheology; viscosity; blood; plasma Received: 9 April 2021 Accepted: 15 June 2021 Published: 20 June 2021 1. Introduction Publisher’s Note: MDPI stays neutral The study of the viscosity of biofluids, such as blood, has been used to understand their with regard to jurisdictional claims in physiological mechanisms and biological functions [1]. Due to their complex properties, published maps and institutional affil- most biological fluids have a non-Newtonian behavior. While a Newtonian fluid shows a iations. viscosity that is independent of the shear rate, non-Newtonian fluids’ viscosity shows a dependence on the applied shear rate. As illustrated in Figure 1, blood exhibits shear-thinning behavior. Shear-thinning fluid viscosity is characterized by a monotonically decreasing dependence on the shear rate. Copyright: © 2021 by the authors. Such behavior is closely related to the dynamics and interactions of red blood cells (RBCs), Licensee MDPI, Basel, Switzerland. the main cellular component of blood. Blood viscosity variation as a function of shear rate This article is an open access article due to rheological properties of RBCs is shown in Figure 1 [2–5]. RBCs have a biconcave distributed under the terms and disklike shape at rest; they aggregate and form rouleaux structures that can, reversibly and conditions of the Creative Commons continuously, disaggregate to single-flowing discocytes for increasing shear rates. This Attribution (CC BY) license (https:// change in microstructure implies a viscosity variation. When RBCs circulate through the creativecommons.org/licenses/by/ vascular system, deformability and dynamics of RBCs account for a further decrease in 4.0/). Micromachines 2021, 12, 726. https://doi.org/10.3390/mi12060726 https://www.mdpi.com/journal/micromachines Micromachines 2021, 12, x 2 of 21 Micromachines 2021, 12, 726 2 of 20 circulate through the vascular system, deformability and dynamics of RBCs account for a further decreaseFurthermore, blood viscosity. in blood viscosity. blood Furthermore, blood viscosity value viscosity strongly value depends onstrongly depends other parameters on other parameters such as temperature, whole blood cellular components (for such as temperature, whole blood cellular components (for example hematocrit value), example and hematocrit hemoglobinvalue), and content [6].hemoglobin content [6]. Figure Figure 1. 1. Viscosity Viscosity profiles profiles of whole blood and plasma. Adapted Adapted from Rosencranz [7]. The gold The gold standard standard for for viscosity viscosity measurements measurements comprises comprises aa widewide range range ofof types types of rheometers (instruments rheometers (instruments that that measure measure the the stress stress andand deformation deformation history history ofof aa material). material). The specific The specific kind kind of rheometer rheometer that can only measure the steady shear viscosity function of aa material of material is is called called aa viscometer viscometer [8]. [8]. Rheometers Rheometers may may bebe differentiated differentiated on on the the type type of of flow they induce; i.e., drag flows and pressure-driven flows. Typical flow they induce; i.e., drag flows and pressure-driven flows. Typical drag-flow rheome- drag-flow rheometers are the ters are coaxial the coaxial cylinder Couette cylinder flowflow Couette and and coaxial cone-plate, coaxial while cone-plate, capillary while rheometers capillary rheom- eters are pressure-driven. The disadvantages of using this type of equipmenttheir are pressure-driven. The disadvantages of using this type of equipment include high include operation costs and the large size of samples they require. The rheometrical their high operation costs and the large size of samples they require. The rheometrical properties of complex fluids have been traditionally studied using such macroscopic properties of complex fluids have been traditionally studied using such macroscopic de- devices. vices.In general, macroscopic rheometers provide less accuracy than their microfluidic counterparts. In general, In macroscopic contrast to classical methods, rheometers the study provide of rheology less accuracy through than microfluidic their microfluidic techniques is a widely used method to analyze the viscosity of complex counterparts. In contrast to classical methods, the study of rheology through microfluidic fluids. Several microfluidic platforms have been developed to reach high confinement techniques is a widely used method to analyze the viscosity of complex fluids. Several levels [9–11] (and thus, high precision), such as broader shear rate crystal microbalance microfluidic platforms have been developed to reach high confinement levels [9–11] (and (QCM) [12], laser- induced capillary wave [13], and the use of multiple microfluidic channels. This relatively thus, high precision), such as broader shear rate crystal microbalance (QCM) [12], la- new field has come up with some viscometers that measure viscosity as a function of ser-induced capillary wave [13], and the use of multiple microfluidic channels. This rela- shear rate and temperature, although most of them are not applied in biofluids. Novel tively new field has come up with some viscometers that measure viscosity as a function microfluidic systems’ accuracy is equivalent to traditional methods, being also preferable of shear rate and temperature, although most of them are not applied in biofluids. Novel for measuring complex viscosity behavior [14] and microflows [15]. A substantial number microfluidic systems’ accuracy is equivalent to traditional methods, being also preferable of viscometers for biofluid characterization exist [16–18]. However, they still present for measuring complex viscosity behavior [14] and microflows [15]. A substantial num- disadvantages in terms of space and sample amount required. ber of viscometers for biofluid characterization exist [16–18]. However, they still present Some new applications use lab-on-a-chip structures from diverse materials aiming disadvantages in terms of space and sample amount required. for miniaturization, such as glass and polydimethylsiloxane (PDMS) [19,20]. Microfluidic Some new applications use lab-on-a-chip structures from diverse materials aiming techniques that use shear as a mean to understand viscosity include pressure sensing, for miniaturization, such as glass and polydimethylsiloxane (PDMS) [19,20]. Microfluidic flow-rate sensing, surface tension, coflowing streams, and diffusion-based and velocimetry- techniques based sensing that[1].useTheshear as a mean to proposed microrheometer understand by viscosity Solomon include [21] usedpressure a mobile sensing, phone flow-rate sensing, surface tension, coflowing streams, and camera to detect the fluid-front advancement through a microchannel. Srivastava diffusion-based and veloci- [22], metry-based on the other sensing [1]. The microrheometer hand, proposed a microrheometer proposed in which bythe Solomon [21] used measurement of aviscosity mobile phone was based camera to detect thewith on monitoring fluid-front a camera advancement the capillarythrough a microchannel. pressure-driven movement Srivastava of fluid [22], on the other hand, proposed a microrheometer in which sample, and shear rate varied with time. The device fabrication was not simple, the measurement of and vis- cosity requiredwasmuchbased on monitoring more with a camera time and specialized the capillary equipment pressure-driven and materials to produce, movement than our of fluid sample, and shear rate varied with time. The device PDMS on glass microchannels. By contrast, the method of detection in our device is fabrication was not simple, based and required much more time and specialized equipment and materials on an array of electrodes that automatically detect the advancement of the fluid. By adding to produce, than our PDMS on glass microchannels. By contrast, the method of detection electronic sensing, we do not need further video processing and analysis. Additionally, our in our device is Micromachines 2021, 12, 726 3 of 20 method has been demonstrated to measure viscosity of blood and differentiate between samples at distinct hematocrit levels. The scale reduction allows for an increase in sensitivity and accuracy. Other factors such as hematocrit level, plasma, and coagulation could affect blood viscosity. On that matter, point-of-care devices such as thromboelastograph (TEG) (Haemoscope Corpora- tion, Niles, IL, USA) and the ROTEM thromboelastometer (Pentapharm GmbH, Munich, Germany) [23,24] have led the research toward successful results for blood and plasma [25]. The technology presented here consists of a front microrheometer mainly composed of three parts: a microfluidic consumable that comprises a microchannel with an inlet and an outlet; the electronic setup, including pump, electronic detection, and data acquisition; and a mathematical model to properly convert obtained data into viscosity measurements. The electronic detection is performed with a sensor array (of metallic electrodes) located along the microchannel. This technology allows us to characterize the rheological properties of the fluid as a function of the fluid-front advancement, using a small amount of sample, reducing space required, and finally minimizing time and material consumption. The tested samples come into contact only with the walls of the microfluidic channel; therefore, there is neither a need for cleaning protocols, nor for the use of cleaning substances. Unlike other rheometers that require extensive cleaning of their components, the microfluidic chip is intended to be used as a consumable. The consumable is disposed of after each experiment. Thanks to the number of pairs of electrodes (24), situated at each side of the microchannel, it is possible to obtain a high number of data points, thus enhancing the accuracy of measurements. Finally, the integration of electronic detection into microfluidic devices improves the reliability of measurements. The general goal of this study was to develop the method and model for a mi- crorheometer capable of characterizing the viscosity of biofluids by analyzing the fluid-front advancement, and reducing the required sample’s amount, space, and time consumption. First, we characterized the viscosity of healthy blood samples at distinct hematocrit lev- . els. The viscosity of blood obtained at γ = 1 was 12.2 mPa·s for the sample at 50% Ht; 10.67 mPa·s for 42% Ht; and 8.81 mPa·s for 35% Ht. These results were consistent with results obtained by other authors [26–28]. This constitutes a necessary benchmark to, in the near future, compare and detect rheological anomalies associated with hematological diseases, since the shape and biomechanical properties of RBCs can be altered in several diseases related to hematological disorders [29]. Second, we validated the new technology by comparing its results to data obtained using traditional equipment. We compared our viscosity results to results obtained using a rotational rheometer (Malvern Kinexus Pro+, Malvern Instruments Limited, Worcestershire, UK). The viscosity value obtained for blood at 42% Ht using this macrorheometer for comparison was 10.69 mPa·s. 2. Materials and Methods 2.1. Mathematical Model The development of mathematical models describing the Newtonian and non-Newtonian characteristics of blood as a complex fluid [30,31] are very important for understanding viscosity behavior at the microscale, and determine how different blood conditions affect the viscosity values. Deformation of RBCs at high shear rates and the formation of aggre- gates in the form of rouleaux at low shear rates are essential aspects to keep in focus when developing models for blood [32,33]. The linear displacement of a fluid layer with respect to another in the time interval dt allows obtaining the rate of displacement (or velocity). If we consider a fluid flow moving into the x direction between two plates, the only nonzero velocity component is vx , which → varies in the z direction due to the interaction with the planes, being v = {vx (z), 0, 0}. The shear rate is the rate of change of velocity at which one layer of fluid passes over an adjacent layer. The shear produced between the layers depends on the z dimension, and is: . γ(z) = (dvx (z))/dz (1) An easy method to study fluid flow consists of the study of the fluid-front (fluid–air interface) advancement. In a microfluidic channel, the front velocity, h (t), is the change in position of the fluid, h(t), through time along the microfluidic channel. The term h(t) is defined as the average position of the front h( t ) = ∑ h (t) , where h (t) is the flu- Micromachines 2021, 12, 726 id-front position with respect to z, as shown in Figure 2. One of the advantages 4 of 20 of working with the mean front velocity is that its value is independent of the z position. The theoretical model used assumes that velocity inside the microchannel behaves ac- cording to Darcy’s An easy methodlaw. In past to study experiments fluid [34],ofTrejo flow consists C. observed the study that this theoretical of the fluid-front (fluid–air description of the velocity of the fluid inside the microchannel held for. microchannels of interface) advancement. In a microfluidic channel, the front velocity, h(t), is the change 1 mm width and heights over 150 um. To test distinct aspect ratios, the authors consid- in position of the fluid, h(t), through time along the microfluidic channel. The term h(t) ered microchannels of different heights: b = 300 μm, b = 200 μm, b = 150 μm, and b = 50 is defined as the average position of the front h(t) = n1 ∑jn=1 hj (t) , where hj (t) is the μm; width w = 1 mm; and length 𝑙 = 4cm. For heights lower than 150 um the model is fluid-front position with respect to z, as shown in Figure 2. One of the advantages of not longer valid. working with the mean front velocity is that its value is independent of the z position. The By studying the fluid/air interface, the shear rate (γ) can be defined as the normal- theoretical model used assumes that velocity inside the microchannel behaves according to Darcy’s of ization theInmean law. front velocity, past experiments h(t), [34], according Trejo C. observedto the thatmicrochannel height, this theoretical b, as de- description of fined by Equation (1) (see Figure 2). the velocity of the fluid inside the microchannel held for microchannels of 1 mm width and heights over 150 um. To test distinct aspect ratios,hthe authors considered microchannels of different heights: b = 300 µm, b = 200 µm, bγ ==150b µm, and b = 50 µm; width w = 1 mm; (2) and length lc = 4 cm. For heights lower than 150 um the model is not longer valid. Figure Figure 2. The velocityvv(z) Thevelocity x (z)isisthe thex xcomponent component ofof velocity velocity inside inside thethe microfluidic microfluidic channel channel as aas a function of function height ((z); of height z); h(t) indicates the mean position of the fluid front. The h(t) indicates The channel channel dimensions dimensions satisfy the satisfy the relation b/ω ≪ relation b/ω  1.1. . By Thestudying viscositythe offluid/air interface,fluids non-Newtonian the shear can rate γ can beusing be described defined as the models different normaliza- ac- . cording to the characteristics of each fluid. The most commonly used are the tion of the mean front velocity, h(t), according to the microchannel height, b, as defined by Car- reau–Yasuda Equation andFigure (1) (see power-law 2). models [35]. The power-law model, also known as the Os- . waldt–De Waele fluid model, describes the . viscosity h of a fluid on dependence of its shear rate: γ= (2) b The viscosity of non-Newtonian fluidsη(γ)can =m be γdescribed using different models accord- (3) ing to the characteristics of each fluid. The most commonly used are the Carreau–Yasuda where the prefactor m corresponds to the value of viscosity at γ = 1. The exponent n in- and power-law models [35]. The power-law model, also known as the Oswaldt–De Waele dicates the nature of the viscosity of the studied fluid. Depending on the value of n, the fluid model, describes the viscosity of a fluid on dependence of its shear rate: behavior of a fluid is characterized as Newtonian or non-Newtonian. Values of n 1 correspond to shear-thinning behavior, .  while. nn −1 1 corresponds to shear-thickening η γ = mγ (3) behavior. On the other hand, Newtonian fluids have a value of n = 1. . where the prefactor m corresponds to the value of viscosity at γ = 1. The exponent n indicates the nature of the viscosity of the studied fluid. Depending on the value of n, the behavior of a fluid is characterized as Newtonian or non-Newtonian. Values of n < 1 correspond to shear-thinning behavior, while n > 1 corresponds to shear-thickening behavior. On the other hand, Newtonian fluids have a value of n = 1. Using the Stokes equation, it is possible to calculate the flow through a known geome- try as a function of the average velocity. In most cases, microfluidic systems are composed of pressure sources and microfluidic channels connected by cylindrical tubing, creating coupled fluidic systems. The flows passing through two coupled geometries are equivalent due to the mass conservation principle. The mathematical deductions concerning these calculations are detailed in Appendix A.1. In a closed system composed of a pressure source, a fluid reservoir, and the coupling of tubing with a microchannel of rectangular cross-section, the pressure inside the rectangular microchannel, ∆P, is the summation of all the pressures in the system, as shown in Figure 3. equivalent due to the mass conservation principle. The mathematical deductions con- cerning these calculations are detailed in Appendix A.1. In a closed system composed of a pressure source, a fluid reservoir, and the coupling of tubing with a microchannel of rectangular cross-section, the pressure inside the rec- Micromachines 2021, 12, 726 tangular microchannel, ΔP, is the summation of all the pressures in the system, as shown 5 of 20 in Figure 3. pressures involved Figure 3. Diagram of the pressures involved in in the the system. system. As properly deduced deduced in inAppendix AppendixA.2, A.2,the theeffective effective pressure pressure (P (P)eff is) defined is defined as as the the pressure pressure resulting resulting fromfrom the combination the combination of pumping of the the pumping system system pressure, pressure, the cap- the capillary illary pressure, pressure, and theandhydrostatic the hydrostatic pressure pressure intowhole into the the whole system system shown shown in Figure in Figure 3 3 (see (see Equation (A25) Equation (A.25) deduced . deducedin Appendix A.2). Effective pressure P has in Appendix A.2). Effective pressure P has an algebraic eff an algebraic de- dependence pendence with withγ: γ: .n Peff = K(m, n)γ (4) P = K(m, n)γ (4) where all the independent variables are grouped in K, which depends on the fluid proper- where ties all the (m and independent n) and variables the geometrical are grouped parameters in K, which of the system shown depends in Equation on (4): the fluid properties (m and n) and the geometrical parameters of the system shown in Equation (4): 2 n !  n ωbωb 1 1 11 K(m, n ) = m2l K(m, n) = m2l + 3 (5) nn + 3 t (5) πr2πr r1+rn Accordingtotothethe mass conservation principle, in our Q system, Q = Q. → ωbh 2 = According mass conservation principle, in our system, c = Qt → ωbh = π r vt π r v (Appendix A.2). This is why we can write velocity inside the tubing as a function (Appendix A.2). This is why we can write velocity inside the tubing as a function of the . of the velocity measured inside the microchannel, h: velocity measured inside the microchannel, h: 8l bω v = 8l η t bω . h (6) vt = η πr h (6) πr4 2.2. Experimental 2.2. Experimental Model model 2.2.1. General Case: Non-Newtonian Fluids We used We usedaacoupled coupledsystem systemininwhich which a tube a tube waswas connected connected to atomicrochannel a microchannel struc- structure. The ture.effective pressure, The effective pressure, P , in Peff , acting the system acting (see Figure in the system (see3) caused Figure 3)fluid to flow caused fluidthrough to flow the coupled through the system. coupledThrough system. electronic detection,detection, Through electronic the mean front velocity the mean was front measured velocity was along the microfluidic channel. By calculating velocity values and the channel measured along the microfluidic channel. By calculating velocity values and the channel dimensions, shear-rate dimensions, values were obtained. shear-rate values . By obtained. were plotting PBy eff vs. plotting P vs. shear rate, and fitting shear a curve rate, for thea and fitting relation curve forbetween Peff and the relation γ, wePobtained: between and γ, we obtained: .n P ==AAγ Peff γ (7) (7) Terms A and n can be obtained from Equation (6). By equating experimental data in Equation (5) with the mathematical model in Equations (3) and (6), we could calculate the value of m as follows: A m=  2 n   n (8) ωb 1 1 2lt πr2 r1+n n + 3 Once n and m were obtained, the viscosity values of a fluid were calculated using the power-law model in Equation (2). 2.2.2. Newtonian Fluids A Newtonian fluid has a constant viscosity. In opposition to non-Newtonian fluids, it does not depend on the applied pressure. Analogously, as it has been done in the general case, the mathematical model for Newtonian fluids (n = 1) is shown in Appendix A.3. Micromachines 2021, 12, 726 6 of 20 Pressure for the entire system is described in Equation (8), which has a linear depen- dence on the mean front velocity. . Peff (m, n = 1) = K(m, n = 1)γ (9) For the Newtonian case, K can also be expressed in terms of the geometry of the system and the experimentally obtained front velocity: 8ωb2 lt K(m, n = 1) = m (10) πr4 Along the same lines, the experimental data of the shear rate values obtained from applying a known pressure to a Newtonian fluid have the linear relation shown in Equation (10): . Peff = Aγ (11) Using the experimental results alongside the mathematical model, m can be calculated as: A m ( n = 1) = (12) 8ωb2 lt πr4 In Newtonian fluids for which n = 1, the power-law equation for viscosity in Equation (2) can be re-written as: η=m (13) 2.3. Experimental Method The experiments were performed using microfluidic chips made of polydimethyl- siloxane (PDMS) attached to glass substrates 2.6 × 7.6 cm2 by using plasma bonding;  schematics of the final structures are shown in Figure 4a. The glass substrates had a printed pattern of 24 pairs of gold electrodes, which were in contact with the flowing fluid. The electrodes had a 350 µm separation in between, and were in the form of 4 groups of 6 pairs located along the length of the channel. The distance between each set was 8.5 mm (Figure 4b). The channel width was ω = 1000 µm, the channel height was b = 300 µm, Micromachines 2021, 12, x and the length from inlet to outlet was lc = 4 cm. Commonly, PDMS,7glass, of 21 and Cole Parmer Tygon tubing are employed due to their good biocompatibility, low cost, and high adaptability. (a) (b) Figure Figure 4. Schematics 4. Schematics of the microfluidic of the microfluidic consumable. consumable. (a) The microfluidic (a) The microfluidic channel madechannel of PDMSmade of PDMS sealed on a on sealed glass substrate. a glass The channel substrate. dimensions The channel are ω = 1000 dimensions are ω =b1000 μm, = 300 μm,band µm, L =µm, = 300 4 cm. and Lc = 4 cm. (b) Gold electrodes (b) Gold printed electrodes on the on printed glass thesurface glass and beneath surface andthe PDMSthe beneath layer. PDMS layer. The The experimental setupsetup experimental used to carry used to out theout carry experiments was composed the experiments of a pres-of a pressure was composed sure pump (Dolomite Fluika) connected to a closed fluid reservoir. From the reservoir, pump (Dolomite Fluika) connected to a closed fluid reservoir. From the reservoir, Tygon Tygon tubing with an internal radius r = 127 μm and length l = 20 cm was connected tubing with an internal radius r = 127 µm and length lt = 20 cm was connected to the to the microchannel. The pressure exerted on the fluid was set using a simple graphic microchannel. The pressure exerted on the fluid was set using a simple graphic interface, interface, which controlled the pressure pump’s performance. A schematic representa- tion of the experimental setup described is presented in Figure 5. Different pressures for the Fluika Pump were set to run the experiment, ranging from 500 Pa to 5000 Pa. Each electrode was connected via a pin array box to a National Instruments control card. As soon as the pressure began to be applied, electronic reading was activated by the con- Figure 4. Schematics of the microfluidic consumable. (a) The microfluidic channel made of PDMS sealed on a glass substrate. The channel dimensions are ω = 1000 μm, b = 300 μm, and L = 4 cm. (b) Gold electrodes printed on the glass surface and beneath the PDMS layer. The experimental setup used to carry out the experiments was composed of a pres- Micromachines 2021, 12, 726 7 of 20 sure pump (Dolomite Fluika) connected to a closed fluid reservoir. From the reservoir, Tygon tubing with an internal radius r = 127 μm and length l = 20 cm was connected to the microchannel. The pressure exerted on the fluid was set using a simple graphic interface, which controlled which controlled the pressurethe pump’s pressureperformance. pump’s performance. A schematic A schematic representa- representation of the tion of the experimental experimental setup described setup described is presented is presented in Figure 5. inDifferent Figure 5.pressures Different for pressures for the Fluika the PumpFluika Pump were set towere run settheto run the experiment, experiment, ranging fromranging from 500 Pa to 500 5000Pa Pa.to Each 5000 electrode Pa. Each electrode was connected was connected via a pinvia a pin array boxarray to a box to a National National InstrumentsInstruments control control card. Ascard. soonAs as soon as the pressure began to be applied, electronic reading was activated by the con-a the pressure began to be applied, electronic reading was activated by the controller of myRIO troller ofNational a myRIOInstruments card. This card. National Instruments tool communicated with the fluidic This tool communicated pump with the and fluidic the electronic pump and the reading electronic pins through reading pinsa through computer. The sample a computer. Thefluid came sample outcame fluid from outthe reservoir from at the setatpressure the reservoir the set and wentand pressure directly wentinto the channel. directly into the As the fluid channel. As came into the fluid contact came with into the electrodes, contact the time required with the electrodes, the time forrequired the fluid for front thetofluid reachfront each toelectrode pair reach each inside the microfluidic channel was obtained. Using the time data and the electrode pair inside the microfluidic channel was obtained. Using the time data and the geometry of the microfluidic system, it was possible to calculate the fluid-front velocity between geometry of the microfluidic system, it was possible to calculate the fluid-front velocity electrode pairs through between the microchannel. electrode pairs through the microchannel. Figure Figure5.5.Experimental Experimentalsetup setupfor forlaboratory laboratoryuse. use. Ten samples of Ten of500 μLfrom 500µL fromeach eachdifferent testtest different fluid werewere fluid usedused during this study: during (fluid this study: standard (fluid MGVS60, standard deionized MGVS60, water (DI deionized waterwater), plasma,plasma, (DI water), and blood andfrom healthy blood from donors). healthy We selected donors). We MGVS60, a viscosityastandard selected MGVS60, viscositytested in strict standard accordance tested in strictwith ASTM D2162, accordance with the primary method for viscosity standards calibration that is typically used for the cal- ibration and verification of viscosity-measuring equipment. Its viscosity is 6.00 mPa·s at 24 ◦ C. Deionized water was selected because its viscosity is constant and known: η = 1.01 mPa·s [36]. Blood samples and their respective plasma were provided in 10 mL tubes by Banc de Sang i Teixits (Barcelona, Spain), from 10 anonymous healthy donors. The use of these samples was authorized by the Bioethics Committee of the University of Barcelona (IRB 00003099). To obtain the plasma samples, we centrifuged whole blood samples at 2500 rpm for 5 min. The plasma that rested at the top was collected with a pipette. Samples were prepared at three different hematocrit concentrations, Ht (%) of 50%, 42%, and 35%. These were prepared by taking plasma previously separated from whole blood and then adding to it the desired volume of RBCs corresponding to each hematocrit percentage. The hematocrit selected for further comparison to a benchtop device was 42% Ht, consistent with the levels found in healthy adults. All the tests were performed at room temperature (25 ◦ C) and, for the case of blood, within two days of extraction, to avoid cell damage from aging [28]. At the beginning of each experiment, the DI water was evaluated to guarantee that the setup was properly calibrated. 3. Experimental Results and Discussion For a correct rheological characterization of the fluid, it was necessary to properly determine the pressures inside the system (Figure 3). The pressure set for the pumping source Pp was automatically recorded. Considering that a sample of 500 µL inside a Micromachines 2021, 12, 726 8 of 20 reservoir supposes a height, H = 5 mm, the contribution generated by the hydrostatic pressure (PH = ρgH) can be obtained. Parameters for the different tested fluids at room temperature can be found in Table 1. The material used for the microchannel was hy- drophobic, meaning that it opposed the fluid-front advancement. Fluid-front images at different pressures were obtained by using a camera and an inverted microscope. The contact angle, θ, was extracted optically via Image J free software. The surface tension (τ) of the standard fluid was experimentally measured with a capillary tube with an internal diameter of 1.15 mm [37,38]. By using Equation (A20) deduced in Appendix A.2, it was possible to calculate Pcap for a rectangular cross-section channel at each measured pressure by analyzing the fluid-front contact angle. The contact angle also depended on the fluid being analyzed and the total pressure applied. The curvature of the fluid–air interface was expected to be almost constant for water, standard MGVS60, and plasma; however, it changed as the pressure was increased when measuring blood (Table 2). Table 1. Characteristic values for water, MGVS60 plasma, and blood. kg N ρ( m3 ) PH (Pa) τ( m ) Sample θ(◦ ) at 25 ◦ C at H = 5 mm at 25 ◦ C Water 1000 49.00 0.072 102.42 MGVS60 1134 55.56 0.107 106.90 Plasma 1025 50.22 0.057 102.27 Blood 1050 51.45 0.058 * (Table 2) * Contact angle for blood was nonconstant at different pressures. See Table 2. Table 2. Contact angle and capillary pressure at each pump pressure applied for healthy blood at 42% hematocrit and RT (25 ◦ C). Pcap was obtained using Equation (A20). Pp (Pa) θ(◦ ) Pcap (Pa) 500 103.24 115.11 1000 106.94 146.46 1500 109.60 168.62 2000 111.76 186.35 2500 113.62 201.37 3000 115.27 214.54 3500 116.76 226.35 5000 120.64 256.22 When studying blood-front behavior, dependence between the applied pressure and the dynamic contact angle measured was observed. Hence, Pcap depended on the applied pressure, Pp . To obtain the contact angle of blood inside our microfluidic channel, we conducted a separate experiment for blood at 42% Ht. By using a microscope (Optika XDS-3, Optika SRL, Bergamo, Italy) and a high-speed camera (Photron Fastcam SA3, Photron Limited, Tokyo, Japan), we captured images of the fluid front at different set pressures, Pp . We measured the contact angle using the angle-measurement tool in Image J, as depicted in Figure A2. Calculation of Pcap was done as per Appendix A.2. The obtained contact angle when changing the applied pressure and thus, the capillary pressure obtained can be seen in Table 2. The contact angles of blood used in this paper have been considered independent of the Ht (%) value. To validate our setup and the use of electronic detection as a proper method to determine viscosity for Newtonian fluids, two different strategies were followed: first, a Newtonian standard fluid and DI water with reported viscosity value were evaluated. Second, the obtained results with our technology were compared to the results obtained with a commercial macrorheometer, the Malvern Kinexus Pro+ (Malvern Instruments Limited, Worcestershire, United Kingdom). 2500 113.62 201.37 3000 115.27 214.54 3500 116.76 226.35 5000 120.64 256.22 Micromachines 2021, 12, 726 To validate our setup and the use of electronic detection as a proper method to de- 9 of 20 termine viscosity for Newtonian fluids, two different strategies were followed: first, a Newtonian standard fluid and DI water with reported viscosity value were evaluated. Second, the obtained results with our technology were compared to the results obtained withFrom the data obtained a commercial (applied macrorheometer, thepressure Malvernand time),Pro+ Kinexus knowing theInstruments (Malvern device geometry, resistances, and the pressures Limited, Worcestershire, United added to the system, it was possible to plot the effective Kingdom). From pressure as athe data obtained function (applied of shear pressure rate for and time), knowing the device geometry, each sample. resistances, A linearand the fit of pressures the added to data obtained the system, when plottingit the wasapplied possiblepressure to plot the effective with an external . . pressure as a function of shear rate for each sample. pressure pump Pp as a function of h or γ is shown in Figure 6. It was used to determine A linear fit of the data obtained when plotting the applied pressure with an external Peff for Newtonian fluids, as detailed in Equation (A33), as: pressure pump (P ) as a function of h or γ is shown in Figure 6. It was used to deter- mine P for Newtonian fluids, as detailed in Equation . (A.33), as: 2 8ωblt 8ωb lt . Peff (n = 1) = m8ωbl4 h = m 8ωb γ (14) P (n = 1) = m πr h = m πrl4 γ (14) πr πr 6000 100000 10⁵ · Peff= 824,44γ1,0174 5000 10000 Effective Pressure , Peff (Pa) Effective Pressure , Peff (Pa) 10⁴ 4000 · Peff = 196.05γ1.0651 Peff = 824.44γ· 1.0174 1000 10³ Peff = 196.05γ·1.0651 3000 10² 100 2000 10¹ 10 1000 DI Water MGVS60 DI Water MGVS60 0 10⁰ 1 0 5 10 15 20 25 1 10 100 Shear rate (s-1) Shear rate (s -1) (a) (b) Figure Figure 6. Effective 6. Effective pressure pressure vs.vs. shearrate shear rateobtained obtained for for the the Newtonian Newtonian calibration fluids calibration studied. fluids Newtonian studied. fluids: fluids: Newtonian DI DI water and MGVS60: (a) linear scale and (b) log-log scale. water and MGVS60: (a) linear scale and (b) log-log scale. As shown in Figure 6 and detailed in Table 3, in both cases, 𝑛 was very close to 1, As shown in Figure 6 and detailed in Table 3, in both cases, n was very close to 1, which indicated that the fluid was Newtonian, compatible with the behavior expected for which indicated that the fluid was Newtonian, compatible with the behavior expected both. for both. Table 3. Viscosity obtained using the developed microrheometer (with the corresponding n values obtained) and a commercial macrorheometer and electronic detection methods. The samples analyzed were DI water and MGVS60, a commercial standard fluid. Viscosity (mPa·s) Viscosity (mPa·s) Sample n Microrheometer Macrorheometer DI water 1.08 ± 0.03 1.017 1.02 ± 0.01 MGVS60 5.87 ± 0.05 1.065 5.88 ± 0.02 From these data, as previously explained, the viscosity could be obtained by us- ing Equation (13). These results were compared to those obtained with a commercial macrorheometer (see Figure 7). Water has been found to exhibit a constant viscosity of η ≈ 1.002 mPa·s [39]. The same results were obtained with both rheometers (differing from each other by 3%). The viscosity obtained with the standard MGVS60 was around η ≈ 5.55 mPa·s using the microrheometer, while the macrorheometer delivered a viscosity of 5.8 mPa·s. The expected value for this fluid was 6.00 mPa·s. These results validated that the accuracy of the new electronic sensing method was adequate (equivalent to standard macrorheometer measures) regarding the measurement of viscosity in an automatic, fast, and effective way. Micromachines 2021, 12, x 10 of 21 Micromachines 2021, 12, 726 From these data, as previously explained, the viscosity could be obtained by 10 using of 20 Equation (13). These results were compared to those obtained with a commercial macrorheometer (see Figure 7). 7 10 6 5 Plasma Viscosity (mPa·s) Viscosity (mPa·s) DI Water 4 DI Water (Macrorheometer) 1 3 MGVS60 Plasma MGVS60 (Macrorheometer) DI Water 2 DI Water (Macrorheometer) 1 MGVS60 0 MGVS60 (Macrorheometer) 0.1 0 5 10 15 20 25 1 10 100 Shear Rate (s⁻¹) Shear Rate (s⁻¹) (a) (b) Figure 7. Viscosity vs. shear shear rate rate using using the the microrheometer microrheometer for for DI DI water water (green) (green) and standard MGVS60, MGVS60, and using the macrorheometer: (a) linear scale and (b) log–log scale. macrorheometer: (a) linear scale and (b) log–log scale. Water Once thehas measurement been found to exhibit system awas constant viscosity validated for of η≈1.002 mPa·s Newtonian fluids[39]. withThe same known s 2021, 12, x 11 of 21 results were obtained characteristics, with bothwas the experiment rheometers repeated(differing fromand with plasma eachblood otheratbydifferent 3%). The%vis- Ht values.obtained cosity The results arethe with shown in Figure standard 8. MGVS60 was around η ≈ 5.55 mPa · s using the mi- crorheometer, while the macrorheometer delivered a viscosity of 5.8 mPa·s. The expected 6000 value for this fluid was 6.00 mPa · s. These 5000 results validated that the accuracy of the new electronic sensing method was adequate (equivalent to standard macrorheometer measures) regarding the measure- Effective Pressure , Peff (Pa) ment 4000of viscosity in an automatic, fast, and effective way. Table 30003. Viscosity obtained using the developed microrheometer (with the corresponding n values obtained) and a commercial macrorheometer and electronic detection methods. The samples ana- Blood at 50% Ht 2000were DI water and MGVS60, a commercial standard fluid. lyzed Blood at 42% Ht 1000 Viscosity (mPa Blood at 35%s) Ht Viscosity (mPa s) Sample 𝒏 Microrheometer Plasma Macrorheometer DI0 water 0 5 1.08 ± 0.0310 15 1.017 1.02 ± 0.01 MGVS60 5.87 Shear rate ± 0.05 (s¯¹) 1.065 5.88 ± 0.02 8. Effective Figurevs. Figure 8. Effective pressure shear ratepressure of blood,vs. shear rate of blood, a non-Newtonian a non-Newtonian fluid, fluid, at 50% Ht, 42% Ht, 35%at 50% Ht, 42% Ht, 35% Once the measurement system was validated for Newtonian fluids with known Ht; and Ht; and plasma, plasma, a Newtonian a Newtonian fluid. fluid. characteristics, the experiment was repeated with plasma and blood at different % ht. The behavior ofvalues. theThe The shear results behavior rate Pare as of the eff shown shear in applied to Figure rate Peff8.applied as fluid the varianceto is the fluid variance shown in Figureis shown in Figure 8. Non-Newtonian fluids (in this case, blood as a shear-thinning 8. Non-Newtonian fluids (in this case, blood as a shear-thinning fluid) showed a nonlin- fluid) showed a nonlinear relation between the pressure and their shear-rate ear relation between the pressure and their shear-rate response, while plasma showed response, while a plasma showed a Newtonian behavior. Using the experimentally obtained Newtonian behavior. Using the experimentally obtained values alongside Equations (6) values alongside Equations (6) and (7) for the and (7) for the non-Newtonian non-Newtonian case, and Equationscase, and (11) (10) and Equations for the(10) and (11)cases, Newtonian for the Newtonian cases, the value of K could be calculated. Then, m and n the value of K could be calculated. Then, m and n were obtained. The values for each areThe values for each were obtained. arereferences shown in Table 4 (see shown in in Table 4 (see C); Appendix references water and in MGVS60 AppendixvaluesB); water andin- are also MGVS60 values are cluded. also included. Table 4. Values for prefactor m and n for Newtonian and non-Newtonian fluids. Table 4. Values for prefactor m and n for Newtonian and non-Newtonian fluids. Sample m (Pa·s) n Sample m(Pa·s) n *Blood at 50% ht. *Blood at 50% Ht 0.0122 0.0122 0.8500 0.8500 *Blood at 42% Ht 0.0107 0.8519 *Blood at 42% ht. *Blood at 35% Ht 0.0107 0.0088 0.8519 0.8720 *Blood at 35% ht. MGVS60 0.0088 0.0058 0.8720 1.0174 ± 0.002 MGVS60 Plasma 0.0058 0.0018 ± 0.00021.0174 ± 0.002 1.0131 ± 0.028 Plasma Water 0.0018 ± 0.0002 0.0010 ± 00101 1.0131 ± 0.028 1.0651 ± 0651 . Water *Blood viscosity values presented γ = 1. 0.0010 ±at00101 1.0651 ± 0651 *Blood viscosity values presented at γ = 1. With the effective pressure, and after obtaining m and n (Table 4), viscosity plots were obtained. Viscosity was calculated using Equation (2) (for non-Newtonian fluids: blood at different Ht (%). levels) and Equations (11) and (12) (for Newtonian fluids: Micromachines 2021, 12, 726 11 of 20 With the effective pressure, and after obtaining m and n (Table 4), viscosity plots were obtained. Viscosity was calculated using Equation (2) (for non-Newtonian fluids: blood at different Ht (%). levels) and Equations (11) and (12) (for Newtonian fluids: plasma). As depicted in Figure 9, plasma exhibited Newtonian behavior with a constant viscos- ity, between 1.50–1.81 mPa·s [40]. Blood exhibited shear-thinning behavior, as its viscosity decreased as the shear increased. Note that the viscosity of blood at 42% Ht (see Figure 9) . varied from 12 to 7 mPa·s within the γ range evaluated, with a value of viscosity 10.67 mPa·s . at γ = 1 s−1 ; blood at 50% Ht had a viscosity of 12.2 mPa·s; and blood at 35% Ht had the lowest viscosity of the three tested hematocrit levels, 8.8 mPa·s. At a constant shear rate, the viscosity of blood increased as the hematocrit grew. This also demonstrated that our new device could be used to distinguish between different hematocrit levels. The hematocrit selected for further comparison to a benchtop device was 42%, consistent with the levels found in healthy adults. The viscosity obtained by this method was 10.69 mPa·s. Micromachines 2021, 12, x for blood at 42% Ht. This can be compared to results also obtained under room-temperature12 of 21 . − 1 conditions by consulted authors, who situated the viscosity of blood at γ = 1 s between 9–10 mPa·s [40]. Dispersion in the measurements at the low shear-rate values observed in Figure tervals9for would be reduced healthy by increasing blood could the number be obtained, which of samples would studied. indicate whatWith theabehavior sufficientlyof large N, a general curve and validity intervals for healthy blood could be obtained, which the viscosity must be as a function of the shear rate in a healthy patient (at a constant would indicate hematocrit what the behavior of the viscosity must be as a function of the shear rate in value). a healthy patient (at a constant hematocrit value). 16 100 Blood at 50% Ht Blood at 50% Ht 14 Blood at 42% Ht Blood at 42% Ht 12 Blood at 42% Ht (Macrorheometer) Blood at 42% Ht (Macrorheometer) Blood at 35% Ht Blood at 35% Ht Viscosity (mPa·s) Viscosity (mPa·s) 10 Plasma Plasma 8 10 6 4 2 0 1 0 5 10 15 20 25 1 10 100 Shear Rate (s⁻¹) Shear Rate (s⁻¹) (a) (b) Figure 9. Figure 9. (a) (a) Viscosity Viscosity vs. vs. shear rate of blood, a non-Newtonian fluid, at 50% Ht, 42% Ht, and 35% Ht. Blood Blood atat 42% 42% Ht. Ht using the macrorheometer (white circles). Viscosity vs. shear rate of plasma, a Newtonian fluid. (b) Viscosity vs. shearshear using the macrorheometer (white circles). Viscosity vs. shear rate of plasma, a Newtonian fluid. (b) Viscosity vs. rate rate of of non-Newtonian non-Newtonian and Newtonian and Newtonian fluidsfluids in log–log in log–log scale.scale. Regarding the rheological properties of RBCs, RBCs, blood showed showed higher higher resistance resistance to to . . flow at low γγ values. values. In contrast, at higher values of γ, γ, blood blood showed showed aa moremore Newtonian Newtonian behavior, which was reflected in the nearly flat region on the viscosity vs. shear rate plot. This distinct shear-thinning curve behavior could only be properly observed observed when when shear shear rates as low as as 11ss−1were werereached. reached.For Forthis thisreason, reason,it itwas wasimportant important totochoose a range choose of a range . pressure of that pressure thatallowed for for allowed γ regime thisthis γ regime to be to available. be available.Residual fit for Residual fit both methods for both was methods a useful was way way a useful to determine the accuracy to determine differences the accuracy between differences our technology between and aand our technology com-a commercial rheometer mercial rheometer (see (see Figure Figure 10). 10). Residuals, R, for the micro- and macrorheometer were calculated as the difference be- 14 tween experimentally obtained viscosity, 0.6 η, and predicted viscosity, ηPredicted (Equation (14)). 0.5 12 Viscosity values from experiment R 0.4 = η − ηPredicted (15) 10 0.3 The theoretically predicted viscosity 0.2 of blood, ηPredicted , for both methods was ob- 8 Residual tained by calculating viscosity at each 0.1 shear rate, using the fit of viscosity vs. shear rate 6 0.0 -0.1 0 5 10 15 20 25 4 Macrorheometer -0.2 2 -0.3 Macrorheometer Microrheometer -0.4 Microrheometer 0 -0.5 0 2 4 6 8 10 12 14 Shear Rate (s⁻¹) Shear Rate (s⁻¹) (a) (b) Figure 9. (a) Viscosity vs. shear rate of blood, a non-Newtonian fluid, at 50% Ht, 42% Ht, and 35% Ht. Blood at 42% Ht. using the macrorheometer (white circles). Viscosity vs. shear rate of plasma, a Newtonian fluid. (b) Viscosity vs. shear rate of2021, Micromachines non-Newtonian 12, 726 and Newtonian fluids in log–log scale. 12 of 20 Regarding the rheological properties of RBCs, blood showed higher resistance to flow at low γ values. In contrast, at higher values of γ, blood showed a more Newtonian from plots inwhich behavior, Figurewas 9. For the micro- reflected in theand macrorheometer, nearly flat region onthethecalculation wasshear viscosity vs. maderate using plot. Equations (15) and (16) respectively: This distinct shear-thinning curve behavior could only be properly observed when shear rates as low as 1s were reached. For this reason, .it−was 0.148 important to choose a range of pressure that allowed for this ηPredicted γ regime to= micro be10.674 γ Residual fit for both methods available. (16) was a useful way to determine the accuracy differences. − between 0.146 our technology and a com- η = 10.687γ (17) mercial rheometer (see FigurePredicted 10). macro 14 0.6 0.5 12 Viscosity values from experiment 0.4 10 0.3 0.2 8 Residual 0.1 6 0.0 -0.1 0 5 10 15 20 25 4 Macrorheometer -0.2 2 -0.3 Macrorheometer Microrheometer -0.4 Microrheometer 0 -0.5 0 2 4 6 8 10 12 14 Predicted values for viscosity Shear rate (s¯¹) (a) (b) Figure Figure 10.10. (a).(a). Experimentally Experimentally obtained obtained viscosity viscosity values values vs. predicted vs. predicted values values for both for both devices devices usingusing blood.blood. (b) Residuals (b) Residuals for for macro and microrheometers. macro and microrheometers. AResiduals, R, for was good correlation the observed micro- and macrorheometer when were calculated plotting experimentally obtainedasvalues the difference for vis- between cosity experimentally and theoretically obtained predicted viscosity, values usingη,theand predictedmodel power-law viscosity, shown η in Figure(Equa- 10a, tionconfirming thus 14). that the experimentally obtained data was close enough to theoretically expected values. In addition, residuals’ plotted vs. predicted values in Figure 10b show R=η−η (15) that the macrorheometer residuals were larger than those obtained with our microrheome- ter. The accuracy of the prediction for the macrorheometer in this experiment was lower than the accuracy of our microrheometer by three orders of magnitude. Residuals for our microrheometer were closer to zero. At these low shear rates, from 1 to 25 s−1 , our technology performed with higher accuracy. This could be explained by the fact that blood . is a complex fluid of the shear-thinning type. We decided to work on this γ window to obtain the nonlinear viscosity of blood and to avoid hemolysis. Our microrheometer had the advantage of being able to operate under low shear conditions with high stability. This makes it suitable for applications such as the analysis of blood, where low-shear-rate performance is necessary. At the shear-rate values required to observe the non-Newtonian behavior of blood, the microrheometer results presented herein had a smaller deviation. This can be attributed to the high stability at low shear rates. These low shear rates could be reached thanks to the size reduction achieved by using a microchannel. Moreover, a benchtop rheometer for laboratory use also requires a rather large sample of blood. For the macrorheometer experiments, the sample amount was 1.19 mL for each run. The microrheometer required samples of approximately 250 µL; this constituted 21% of the sample required for the benchtop device. 4. Conclusions The technology presented here was based on the detection of the advancement of the fluid front using electrodes printed underneath a microfluidic channel. The accuracy of the method was tested by measuring the viscosities of plasma, DI water, and a commercial viscosity standard. Experiments to measure human blood viscosity also were performed. Unlike other microfluidic capillary viscometers, the microrheometer presented here was based on the detection of the fluid/air interface as it advanced through a microchannel. Through the experimental method described, it was possible to produce a series of shear Micromachines 2021, 12, 726 13 of 20 rates that constituted a proper range to observe how the viscosity of non-Newtonian fluid changed as the applied pressure increased. Effective pressure, the pressure acting on the fluid is composed of all the interactions between the pressures acting in the system. The characterization of the non-Newtonian behavior was obtained by applying the power-law model in Equation (2). The experimentally obtained results included a shear- rate window range of less than 1 s−1 up to 100 s−1 for both Newtonian and non-Newtonian fluids. The values obtained for viscosity—1.05 mPa·s for water and 1.8 mPa·s for plasma— fell in line with the values found in the literature [36,39]. Alternatively, we used the calibration standard MGVS60 to test the accuracy of the setup and obtained a viscosity value of 5.88 mPa·s, showing a difference of less than 2% from the target viscosity indicated by the manufacturer at 24 ◦ C. On the other hand, blood results showed shear thinning behavior, due to the presence of mainly red blood cells, under normal morphological conditions [2]. This confirmed that the experimental conditions were adequate to observe the shear-thinning behavior and the action of red blood cells on the viscosity of the whole . fluid. The viscosity of blood at γ = 1 was 12.2 mPa·s for the sample at 50% Ht; 10.67 mPa·s for 42% Ht; and 8.81 mPa·s for 35% Ht. In addition, the viscosity value obtained for blood at 42% Ht using a macrorheometer for comparison was 10.69 mPa·s. As expected, as the hematocrit level increased, so did the value of viscosity. We therefore concluded that the new method proposed in this article is a reliable and accurate device that exhibits a degree of sensitivity capable of differentiating between these distinct hematocrit percentages. In addition, we measured the viscosity of DI water, MGVS60, and blood using a macroscopic rheometer for comparison purposes. Focusing on non-Newtonian fluids, we compared the behavior of the residuals of blood viscosity obtained with both rheometers. Throughout this study, it has been demonstrated that the microrheometer using electronic detection of the fluid front can deliver viscosity results comparable to those obtained using traditional equipment, while saving time and reducing the sample size and space required. At the same time, this technique permits a better control at low shear rates for shear-thinning fluids, such as blood. The development of this experimental and theoretical method for blood will be useful in the near future to determine the presence of RBC abnormalities through the rheological characterization of blood. It could be useful for medical diagnosis in the study of diseases associated with changes in blood viscosity, operating in a wide range of shear rates [41]. Author Contributions: Conceptualization, A.H.-M.; investigation, L.M.-M.; supervision, C.T.-S., A.H.-M. and T.A.; validation, C.R.-L.; visualization, J.F.-T. and S.L.; writing—original draft, L.M.-M.; writing—review and editing, M.C.-F. All authors have read and agreed to the published version of the manuscript. Funding: This research was funded by Generalitat de Catalunya, under grants number 2018 DI 068 and 2018 DI 064; Ministry of Economy and Competitivity (MINECO) under grants number MTM2015- 71509-C2-1-R, MDM-2014-0445 and FIS2016-78883-C2-1P; Ministerio de Ciencia e Innovación (Spain) under grant number PID2019-106063GB-100; AGAUR (Generalitat de Catalunya) under project 2017 SGR-1061; ANID/PCI CONICYT (Chile) under project MEC80180021. Institutional Review Board Statement: The study was conducted according to the guidelines of the Declaration of Helsinki, and approved by the University of Barcelona’s Bioethics Commission (CBUB) (Project 160016 and date of approval 1 March 2016). Informed Consent Statement: All the samples were obtained by Banc de Sang I Teixits with the informed consent of the subjects. All the donors are anonymous and healthy. The samples were analytical surplus blood without information or personal data. The blood samples were going to be extracted for routine analysis purposes regardless of this study. The results herein obtained would not affect clinical decisions about the individuals’ care. Data Availability Statement: Data is contained within the article. Acknowledgments: Lourdes Méndez-Mora and Josep Ferré-Torres acknowledge support from Generalitat de Catalunya under the program Doctorat Industrial (2018 DI 068) and (2018 DI 064). T.A. acknowledges the Spanish MINECO for funding under grant MTM2015-71509-C2-1-R. Tomas Micromachines 2021, 12, 726 14 of 20 Alarcón acknowledges further support from the Ministry of Economy and Competitivity (MINECO) for funding awarded to the Barcelona Graduate School of Mathematics under the “Maria de Maeztu” program; grant number MDM-2014-0445. Tomas Alarcón has been partially funded by the CERCA Programme of the Generalitat de Catalunya. Aurora Hernandez-Machado acknowledges support from MINECO (Spain) under project FIS2016-78883-C2-1P, Ministerio de Ciencia e Innovación (Spain) under project PID2019-106063GB-100, and AGAUR (Generalitat de Catalunya) under project 2017 SGR-1061. Claudia Trejo-Soto and Aurora Hernandez-Machado acknowledge partial support from ANID/PCI CONICYT (Chile) under project MEC80180021. Along the same lines, the authors extend their gratitude toward Rheo Diagnostics S. L. for making this research possible. Conflicts of Interest: The authors declare no conflict of interest. Appendix A Appendix A.1. Flow Inside a Known Geometry The flow passing through a known geometry can be obtained as the product of the mean velocity of the fluid multiplied by the cross-section area. The model described herein corresponds to a structure formed by a tubing of length, lt , and internal radius, rt , connected into a microfluidic channel of dimensions ω and b. The resistance inside a known geometry is defined by the relation Rt = ∆P Q . The pressure loss inside the tubing is: 8lt ∆Pt = η vt (A1) r2 . According to the mass conservation principle, in our system, Qc = Qt → ωbh = π r2 vt ; therefore, we write velocity inside the tubing as a function of the velocity measured inside the microchannel: 8lt bω . vt = η h (A2) πr4 The flow can also be obtained from the integration process of mean velocity along with the geometry; a cylinder (the flow into a tube or pipe) or between parallel walls (a rectangular cross-section channel that satisfies the relation b/ω  1). Using cylindrical coordinates of the Stokes [42] equation, an expression for ∆Pt can be obtained, utilizing the average flow velocity inside the tube, vt :   1 ∂ ∂vt ηr = ∇Pt (A3) r ∂r ∂r ∆Pt where ∇Pt = lt and ∆Pt is the pressure inside the cylindrical geometry. Defining shear . ∂v . . n −1 rate as γ = ∂r and using the power-law model in Equation (2) (η γ = mγ ), the previous equation can be written as follows: 1 1 ∆Pt ∆Pt   . n ∂vt n 1 γ= r ⇒∆ = rn (A4) 2lt m ∂r 2lt m By integrating this expression while imposing a non-slip boundary condition, vz (r = R) = 0, we obtain the velocity component vz (r) : 1 ! ∆Pt  n 1  1 1  vz (r) = 1 r n +1 − R n +1 (A5) 2lt m n +1 Now, an equation for the flow rate, Qt , can be defined: Z2π Zr 1 ! ∆Pt  0 0 1 n 1 n +1 2  Qt = dθ vz r dr = r πr 1 (A6) 2lt m n +3 0 0 Micromachines 2021, 12, 726 15 of 20 To determine the flow in a rectangular microchannel, we first need to know the velocity of a plane Poiseuille flow, vx (z), using the non-slip boundary condition at the position of the solid plates vx (z = ±b/2) = 0: 1/n !   1 +1 ! ∆p  1 1 b n vx ( z ) = z n +1 − (A7) mh(t) 1 2 n +1 where ∆P is the pressure inside the microchannel and h(t) is the position of the fluid interface inside the microchannel. Micromachines 2021, 12, x 16 of 21 The resulting equation for the flow rate inside the microchannel is: b  1   1 +2 ! ∆P  n b n 1 Z 2 Q=ω vx (z)dz = 2ω (A8) −b ΔP mh (t) b 2 1 1 +2 Q=ω 2 v (z)dz = 2ω n (A.8) mh(t) 2 1 +2 n where Q is the flow inside the channel and Qt is the flow inside the tubing. The term lt where Q is the flow inside the channel and Q is the tubing length and r is its internal radius. ∆P is the pressurethe is the flow inside tubing. inside theThe term l rectangular is the tubing length and r is its internal radius. ΔP is the pressure inside the rectangular geometry and ∆Pt isand geometry theΔPpressure inside the cylindrical geometry. These parameters can be is the pressure inside the cylindrical geometry. These parameters can seen in Figure A1a,b. be seen in Figure A1a,b. (a) (b) Figure A1. representation Figure A1. (a) Schematic (a) Schematic representation of a Poiseuille of a Poiseuille flow between flow between two parallel two parallel wallswalls separated separated at aatdistance a distanceb.b.The The mean mean . front velocity, h, and the flow inside the microfluidic channel is  . Q = hωb . (b) Schematic representation of Poiseuille flow in front velocity, h, and the flow inside the microfluidic channel is Q = hωb . (b) Schematic representation of Poiseuille flow a cylindrical tube of radius r and length l . The velocity, v , is the mean velocity of the fluid inside the cylindrical tube and in a cylindrical tube of radius Q is the r and flow inside thelength tube (Qlt .=The πr vvelocity, ). is the mean velocity of the fluid inside the cylindrical tube and Qt is the flow inside the tube (Qt = πr2 vt ). Given current micromanufacturing techniques, the microchannels are mainly rec- Giventangular. currentMicrofluidic systems oftentechniques, micromanufacturing need pneumatic the systems to apply pressure. microchannels To make are mainly rectan- the connections gular. Microfluidic between systems oftenthe pneumatic need systems pneumatic and thetomicrochannels, systems apply pressure.tubesTo are used,the make with different radii depending on their purpose. For this reason, this study defined a connections between the pneumatic systems and the microchannels, tubes are used, with mathematical model describing the flow inside a coupled structure (cylindrical different radii depending on their purpose. For this reason, this study defined a mathe- tube–rectangular channel). Due to the mass conservation principle, the flows passing matical model describing through the geometries two coupled flow insideareaequivalent. coupled structure (cylindrical This equivalency can betube–rectangular expressed as: channel). Due to the mass conservation Q = Q principle, → ωbh = π the r vflows passing through two(A.9) coupled geometrieswhere are equivalent. ω and b areThis equivalency the width canofbe and height theexpressed as: channel, respectively; and v is the ve- locity inside the tubing. . From Equation (A.9),Qthe Qt → ωb = velocity inside h= r2 vt can be written as: theπtubing (A9) hωb (A.10) where ω and b are the width and height vof=the πrchannel, respectively; and vt is the velocity inside the tubing. From A.2 Equation Pressure(A9), Dropthe velocity Inside inside the a Microfluidic tubing can be written as: System In a closed system composed of a pressure . source, a fluid reservoir, and the coupling of tubing with a microchannel of rectangular hωb cross-section (geometries illustrated in Figure A1a,b), the pressure inside v t = the rectangular microchannel, ΔP, is the summation(A10) of πr2 all the pressures in the system, as shown in Figure 3, is expressed in Equation (A.11): Appendix A.2. Pressure Drop InsideΔP = P + P − ΔP a Microfluidic −P System (A.11) where P indicates the values set for the pumping source, ΔP is the pressure drop due In a closed system composed of a pressure source, a fluid reservoir, and the cou- to the coupled tubing, P is the hydrostatic pressure, and P is the capillary pressure. pling of tubing with All these a microchannel concepts of rectangular are introduced below. cross-section (geometries illustrated in The hydrostatic pressure, P is the one that originates from the fluid height (the difference between the fluid reservoir height and the measurement site height). It can be calculated by multiplying the fluid height inside the reservoir relative to the microfluidic channel, density of the fluid, and gravity: P = ρgH. Micromachines 2021, 12, 726 16 of 20 Figure A1a,b), the pressure inside the rectangular microchannel, ∆P, is the summation of all the pressures in the system, as shown in Figure 3, is expressed in Equation (A11): ∆P = Pp + PH − ∆Pt − Pcap (A11) where Pp indicates the values set for the pumping source, ∆Pt is the pressure drop due to the coupled tubing, PH is the hydrostatic pressure, and Pcap is the capillary pressure. All these concepts are introduced below. The hydrostatic pressure, PH is the one that originates from the fluid height (the Micromachines 2021, 12, x 17 of 21 difference between the fluid reservoir height and the measurement site height). It can be calculated by multiplying the fluid height inside the reservoir relative to the microfluidic channel, density of the fluid, and gravity: PH = ρgH. Capillary Capillary pressure, Pcap,, is pressure, P is defined definedasasthe the resistance resistance of hydrophobic of the the hydrophobic channel channel walls walls to the fluid advancement. P is dependent on the contact angle and to the fluid advancement. Pcap is dependent on the contact angle and the surface tension of the surface tension of τthe the fluid, fluid,isτdependent (which (which is dependent on temperature). on temperature). Considering Considering the geometrical the geometrical parameters parameters of a section of a rectangular rectangular section channel, withchannel, height bwith heightω, and width b as and widthinω,Figure shown as shown A2, Pcapin Figure can beA2, P can be deduced. deduced. Figure FigureA2. A2.Top Topview viewofofthe contact the angle contact of of angle a fluid–air interface a fluid–air inside interface a microfluidic inside channel a microfluidic of of channel width ω and height b. width ω and height b. With the With the fluid fluid front in in Figure FigureA2 A2expressed expressedininterms termsofoftwo tworadii: r1 ,rshown radii: , shown in the top in the view, and r top view, and2 r , the lateral view counterpart, we get: , the lateral view counterpart, we get: 1 1  P = τ + 1 1 (A.12) Pcap =r τ rr + r (A12) 1 2 This is needed to determine the value of the curvature radii, r and r . Following the This is representation geometrical needed to determine of the the value of the rectangular curvature channel cross-section radii, r1 in and r2 . Following Figure A2, the the geometrical tangent curve to representation the radius on aof the rectangular circumference hascross-section an angle of 90°,channel in Figure and the contactA2, the angle tangent curve to the radius on a circumference has an angle of 90 ◦ , and the contact angle of of the front can be written as: the front can be written as: θ = 90 + α◦ (A.13) θ = 90 + α (A13) Then: Then: 𝜔 ω = 𝑟 sin=𝛼r1 sin α (A.14) (A14) 2 2 ω  ◦  = r1 sin θ − 90 (A15) 2 ω𝜔 = 𝑟 sin(𝜃 − 90 = r (sin θ cos 90 −) sin 90 cos θ) ◦ ◦ (A.15) (A16) 22 1 ω = −r1 cos θ (A17) 2 𝜔 = 𝑟 (sin 𝜃 cos 90 − sin 90 cos 𝜃) (A.16) 2 ω = −r cos θ (A.17) 2 Considering the geometrical parameters of a rectangular section channel, with height b and width ω, r can be written as: Micromachines 2021, 12, 726 17 of 20 Considering the geometrical parameters of a rectangular section channel, with height b and width ω, r1 can be written as: ω r1 = − (A18) 2 cos θ Since the contact angle on the side of the surface is also θ, and using the same trigono- metric as before, r2 is: b r2 = − (A19) 2 cos θ Finally, replacing r1 and r2 in Equation (A12), capillary pressure can be calculated using:   1 1 Pcap = −2τ cos θ + (A20) b ω The value of capillarity is always positive, since the surface is hydrophobic, and the contact angle is always θ = π/2+α. The term ∆P can be also understood as the product of the hydraulic resistance inside the microchannel, S (dependent on m, n and the channel geometric parameters ω and b) and the channel flow, Q. Using Equation (A8), ∆P is:  1+2n  n n 1 2 1 ∆P = SQ = mh(t) + 2 Qn (A21) (2ω)n b n  .  Considering the flow dependence on front velocity Q = hωb :   1+ n  n . 2 1 n ∆P = mh(t) +2 h (A22) b n ∆Pt , similarly to ∆P, can be expressed as the product of St and Qt n . St is the hydraulic resistance represented by the tubing; it depends on m, n, and the tubing parameters r, lt . Qt is the flow inside the tubing:  n   n 1 1 1 ∆Pt = St Qt n = m2lt +3 Qt n (A23) πr2 r1 + n n Taking into account the flow dependence on flow velocity inside the tube Qt = πr2 vt  and the relation seen in Equation (A10), for the coupled system in this study, ∆Pt can be written as follows in Equation (A24). Due to mass conservation inside the coupled system as shown in Figure 5, ∆Pt depends on m, n, the tubing, and the microchannel parameters:  n   n . ωb 1 1 n ∆Pt = m2lt +3 h (A24) πr2 r1+n n According to mass conservation, and taking Equation (A21) and Equation (A23) into account, an effective pressure (Peff ) defined as the pressure drop caused by the pumping system, the capillary pressure, and the hydrostatic pressure into the whole system can be determined by: Peff= Pp + PH −Pcap n i . n n n (A25) 2 1+ n h   = ∆Pt + ∆P = m 2lt ωb 1 1 1  πr2 r1 + n n +3 + h(t) b n +2 h Moreover, Equation (A25) allows for the evaluation of the case in which one of the main terms (relative to the coupled structures of the tubing and microchannel) is much Micromachines 2021, 12, 726 18 of 20 larger than the other (relative to the microchannel only). Peff is constant and not time- dependent. Then, when the relation in Equation (A26) is satisfied,  n   n   1+ n  n ωb 1 1 2 1 2lt +3  h(t) +2 (A26) πr2 r1+n n b n The term on the right side can be neglected and the expression for Peff is no longer dependent on time (t). Now, Equation (A25) can be simplified to: n  n . !n  n ωb2    ωb 1 1 n 1 1 .n Peff = m2lt +3 h = m2lt + 3 γ (A27) πr2 r1+n n πr2 r1 + n n . . γ was described in Equation (1) as the ratio between the mean front velocity, h, and channel height, b. All the independent variables can now be expressed in the form of a constant K, defined as: !n  n ωb2  1 1 K(m, n) = m2lt + 3 (A28) πr2 r1 + n n This depends on the fluid properties (m and n) and geometrical parameters of the sys- tem. Now, Equation (A27) can be written as follows, where K depends on the geometrical properties of the setup and the fluid viscosity value: .n Peff = K(m, n)γ (A29) Appendix A.3. Newtonian Fluids For the case of Newtonian fluids (n = 1), Equation (A22) and Equation (A24) from Appendix A.2 are simplified to: 8ωblt . ∆Pt (n = 1) = m h (A30) πr4 12h(t) . ∆P(n = 1) = m 2 h (A31) b Due to mass conservation, the pressure for the entire system can be described (as it was in Equation (A25)) as follows: 12h(t) .   8ωblt Peff (n = 1) = m + h (A32) πr4 b2 As evaluated in the non-Newtonian case, Equation (A32) allows evaluating when the term relative to the coupled geometry is much larger than the term that only depends on the microchannel, 8ωblπr4 t  12h2(t) ; then, the time dependence of the equation also can b be neglected in the case of the Newtonian fluids. This allows for Equation (A32) to be expressed as: 8ωblt . 8ωb2 lt . Peff (n = 1) = m h = m γ (A33) πr4 πr4 Just as before, and now for the Newtonian case, K can be expressed in terms of the geometry of the system and the experimentally obtained front velocity: 8ωb2 lt K(m, n = 1) = m (A34) πr4 Micromachines 2021, 12, 726 19 of 20 The relation between the pressure and the mean front velocity follows a linear behavior. The equivalent expression for Equation (A29) for Newtonian fluids is: . Peff (m, n = 1) = K(m, n = 1)γ (A35) Appendix B. Viscosity Data Viscosity η Fluid Temperature(◦ C) Reference (m × Pas) MGVS60 24 6.00 [Paragon Scientific, Ltd.] Blood plasma 20 1.50–1.81 [Baskurt, 2007] [36] Water 20 1.002 [Bruus, 2008] [35] Blood at 44% Ht 22 9–10 at (1 s−1 ) [Thurston and Henderson, 2006] [40] References 1. Gupta, S.; Wang, W.S.; Vanapalli, S.A. Microfluidic viscometers for shear rheology of complex fluids and biofluids. Biomicrofluidics 2016, 10, 043402. [CrossRef] [PubMed] 2. 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