# A Quasi-Mechanistic Mathematical Representation for Blood Viscosity

^{1}

^{2}

^{*}

## Abstract

**:**

^{−1}, root mean square (RMS) error = 0.21 cP). A 5-parameter Modified Krieger Model (MKM5) also demonstrated a very good fit to the data (RMS error = 1.74 cP). These models avoid discontinuities exhibited by previous models with respect to hematocrit and shear rate. In summary, the quasi-mechanistic, Modified-Krieger Model presented here offers a reasonable compromise in complexity to provide flexibility to account for several factors that affect viscosity in practical applications, while assuring accuracy and stability.

## 1. Introduction

#### 1.1. Factors that Contribute to the Viscosity of Blood

_{o}(Hct), in an attempt to make the model continuous, yet a discontinuity still remains for zero shear rate at a hematocrit of 80.4%, which could be problematic for simulation of some pathological conditions such as polycythemia. The elevated viscosity due to the discontinuities at 12.2% and 18% are not as pronounced between shear rates of 1 s

^{−1}to 10 s

^{−1}(not shown in Figure 1), while the discontinuities at 73.1% and 85.6% persists up to 80 s

^{−1}. The discontinuity found in the Das version persists well beyond 1000 s

^{−1}; for example the viscosity at 150 s

^{−1}is 7 times the experimental value. The modified CY model, such as used by Jung et al., [13] and the WS model both exhibit a discontinuity at zero shear rate and the WS model has also a discontinuity at zero hematocrit. Although infinite viscosity could be interpreted as a yield stress, this will prevent the convergence of most numerical schemes except in several simple cases.

#### 1.2. The Krieger Model of Viscosity of Suspensions

## 2. Adaptation of the Krieger Viscosity Model for Blood

_{pl}. The resulting form of the Krieger model becomes:

_{pl}, Hct* and n, can be identified independently and are described in the following three sub-sections.

#### 2.1. Plasma Viscosity

_{t}was determined to be 5.95 for temperatures between 10 °C and 40 °C. The combination of Equations (14) and (16) yields a final expression for plasma viscosity:

#### 2.2. Critical Hematocrit

#### 2.3. Krieger Exponent and Shear Thinning

^{−1}) and the deformation of RBCs for moderate to high shear rates (between 10 s

^{−1}and 150 s

^{−1}). Above this range, blood can be considered Newtonian. The aggregability of RBCs, in turn, is dependent on the concentration of plasma proteins, especially large molecules like fibrinogen and Dextran, the hematocrit, as well as the size, shape, and rigidity of RBCs [6,7,27]. The functional dependence of deformability includes the rigidity of the RBC as well as the hematocrit—since greater packing requires more net energy to deform the cells. In addition, the shear thinning behavior of blood only becomes significant above a specific hematocrit, therefore the Krieger exponent would lend itself to a piecewise definition:

_{∞}contributes to the asymptotic viscosity and n

_{st}refers to the shear thinning component. Rheometric data reported by Brooks et al. [50] exhibits negligible shear thinning for a hematocrit of 12.6% therefore $\overline{\overline{\mathrm{Hct}}}$ was chosen conservatively to be 20%. Regression to these experimental data yielded an exponential dependence of n

_{∞}on Hct:

_{st}was further decomposed into the respective contributions of RBC aggregation and deformability:

_{g}is traditionally 2.0. This form assures that n

_{st}remains bounded at zero shear rate. Although an unbounded zero-shear viscosity has been proposed as a method for modeling yield stress, this discontinuity creates numerically instability and may be better represented through an explicit term in the constitutive law. Due to the limited size of the data set, it was more convenient to combine the components of shear thinning into a single power-law function [8]:

_{1}and B

_{2}are model constants. A Hill model would also be feasible for predicting saturation. The equation can be extrapolated to other molecules through addition of an identical term for each species considered.

## 3. Model Evaluation

#### 3.1. Parameter Estimation

^{−1}to 700 s

^{−1}. (Unless specifically noted, these data were used for all subsequent fits). ANOVA was used to determine the statistical significance and validity of each model constant and to ensure that each contributes to the model and is not simply an additional degree of freedom. These models were also compared with the Quemada model for a more qualitative assessment of fit and to demonstrate the relative order of the root mean square (RMS) error. The effect of hardening cells on the Krieger exponent n was investigated using data from Chien et al. [23] for hematocrits of 13%, 26%, 37%, 44%, 49%, 53%, and 56%. Additional coefficients, namely β

_{agg}, ν

_{agg}, β

_{def}, ν

_{def}, λ

_{agg}, λ

_{def}, B

_{1}, and B

_{2}, were found for the specific case of 45% hematocrit using additional data from Chien et al. using canine blood.

#### 3.2. Sensitivity Study

_{z}(r = R) = 0 at the wall, assuming a no-slip condition and u

_{r}(r) = u

_{θ}(r) = 0. The tube radius, R, was selected to be 0.6 cm. Flow was simulated at three different flow rates, 0.06 L/min, 0.6 L/min, and 6 L/min, equivalent to wall shear rates from approximately 6 s

^{−1}to 600 s

^{−1}. For the purpose of the sensitivity study, the Reynold’s number was limited to the laminar regime (<2500), so that turbulence could be neglected. Equation (25) was numerically solved on a computer workstation using a first-order backwards-difference numerical scheme. The finite difference code was validated using the exact solutions to a Newtonian fluid and the power-law fluid. The solution was evaluated for four bulk hematocrits (20%, 40%, 60% and 75%), assuming a uniform Hct profile. An additional sensitivity study was performed assuming a parabolic hematocrit field:

^{3}) and a nominal condition (dp/dz = −10). The velocity profile resulting from the modified Krieger, Quemada, and the modified Carreau models were compared for each case.

## 4. Results

_{∞}), the constant-n Krieger model (KM) to the best-fit asymptotic Krieger model (AKM) (See Table 3 for parameters). The model with the worst fit to these data was the asymptotic Quemada model (RMS = 2.20 cP) followed by the Quemada model (RMS = 1.36 cP), and the Krieger model assuming n = 2, the value for solid spheres (RMS = 1.22 cP). The AKM showed a significant improvement (p << 0.0001) with the root mean square (RMS) of the residual = 0.212 cP and R

^{2}= 0.99. In addition, the 95% confidence intervals for the coefficients was less than 15%, indicating an overall goodness of fit and independence of coefficients. The AKM was least sensitive to the parameter b, as a 10% change in that parameter resulted in a 35% increase in the RMS value when compared to a and c, which resulted in a 376% and 200% changes respectively.

^{2}value of 0.997 (See Table 3). All three coefficients were altered: a was reduced by 29%, b was reduced by almost an order of magnitude, and c changed in sign causing N to increase exponentially. The optimal critical hematocrit was 58.5% ± 0.8% which agrees well with the value predicted by Carr and Cokelet [24]. It was also found that the AKM model for the hardened cells showed significant (p < 0.0005) improvement vs. the constant-n model, confirmed by the corrected Akaike information criterion δAICc = −50.0.

^{−1}) for the MKM5 to the experimental data of Brooks et al. [49] with constant β and ν. It was found that the parameter a (Equation (19)) was not significantly different from 0 (p = 0.17) and was therefore removed from the model, reducing the number of free parameters to five. It was also shown that ν

_{g}did not vary significantly from the commonly used value of 2 (p = 0.79; δAICc = 3.7). The RMS error was 1.74 cP with an R

^{2}value > 0.98. The maximum absolute error occurred at the shear rates of 1 s

^{−1}and Hct of 67.4%, while the maximum percent error occurred at a shear rate of 17 s

^{−1}and a Hct of 8.25%. The 95% confidence intervals for the model parameters were 11.3% for the parameter c. The 5-parameter modified Krieger model MKM5 was least sensitive to λ, as a 10% change in λ resulted in only a 19.5% increase in the RMS value, while similar deviations resulted in changes greater than 80%. The modified Krieger model showed similar accuracy as the Quemada model, despite fewer parameters (RMS error = 1.74 vs. 4.07). Both models showed similar accuracy for low hematocrit (<25%). The modified Krieger model showed better overall fit for moderate hematocrit values when compared to the Quemada model, with the opposite trend for high hematocrit (>55%). The R

^{2}value for the Quemada model was 0.914. The initial form of the MKM5 allowed the shear-thinning behavior due to aggregability to be treated independently from the effect of cell deformation. This is illustrated in Figure 4 which shows the viscosity function consisting of Equations (18)–(22). The values for β were within 3% when comparing aggregation to deformation; but the values for relaxation time (λ) and the exponent (ν) differed by factors of 5 and 4 respectively. The relationship of fibrinogen and the low shear viscosity is illustrated in Figure 5. The optimal values for parameters B

_{1}and B

_{2}are provided in Table 3.

#### Sensitivity Study of Shear Thinning Behavior in Tube Flow

^{6}elements. All three results exhibited the expected velocity blunting phenomenon, which was considerably more pronounced at low flow rate (Figure 6a) compared to high flow rate (Figure 6b.) Irrespective of the flow rate, the MKM5 and Quemada models were very similar, with a maximum difference of centerline velocity less than 1%. The modified Cross model (used by Jung) showed more blunting than both the modified Krieger (MKM5) and Quemada models for all three conditions. Similar results were found when varying the bulk hematocrit, but differences were found to be greater (maximum differences 4%). The introduction of a parabolic hematocrit profile to each of the three models demonstrated a more pronounced blunting effect on the velocity profile (see Figure 7). For the high-flow case (not shown) the centerline velocity was reduced by 36% compared to the Newtonian profile, and 25% compared to the MKM5 shear-thinning model with uniform Hct.

## 5. Discussion

^{−1}) velocity profile blunting due to shear thinning is still observable in tube flow. The consequence of these errors propagates into the calculation of derived quantities of interest, such as shear stress, hemolysis, and transport of leukocytes, platelets, and various chemical species. Moreover, the asymptotic blood viscosity is not a universal constant. It depends on several blood parameters (hematocrit, plasma viscosity, and RBC deformability) that vary from person to person. Micro-scale flow, such as in micro-fluidic devices or small arterioles may introduce inhomogeneity of hematocrit (e.g., due to plasma skimming), and therefore in these situations it is advantageous to employ a hematocrit-dependent formula for viscosity. Recent interest in pediatric medical devices, cryogenic surgery, the use of drag reducing polymers, plasma dilution, and other areas of study can also benefit from a viscosity model of blood that can accommodate localized variations in hematocrit, temperature, and/or protein concentration.

_{1}, b

_{2}, n

_{1}, n

_{2}, n

_{3}are empirical constants. This would result in a 9-parameter modified Krieger model (MKM9). Yet additional coefficients would be needed to explicitly include the influence of red blood cell deformability on high-shear viscosity and aggregability on low-shear viscosity. Although these effects are non-negligible, they would require a rather extensive set of experiments, in the manner described by Chien et al. [23], to identify the functional relationships. These properties are also inter-dependent, since RBC aggregation is affected by RBC deformability and shape, as well as the concentration of fibrinogen and other large molecules. RBC shape, in turn, could include functional dependence on the osmolarity of plasma (or suspension medium in vitro) and temperature. In summary, the number of parameters needed to account for all known dependencies and inter-dependencies can quickly become unwieldy with respect to the quantity of experimental data needed to identify them uniquely, and avoid over-fitting. Therefore, it is logical to limit the complexity of the model to those factors that are relevant for a given application.

_{∞}to the associated asymptotic data (Figure 2) yielded a different set of parameters (a = 1.7, c = 6.06) than when the full model was fit to the full set of data (Figure 3), (a = 0.0, c = 2.87.) This is also illustrated for Equation (7) in Table 1 where a single equation represents six different models with drastically different parameters (Table 3).

^{−1}to 5 s

^{−1}, compared to the data sets shown in Figure 2 and Figure 3 where the maximum shear rate was two orders greater. This demonstrates both the limitations of the model, and a call for a richer set of experimental data. Future work should also include validation beyond the simple viscometric flows that were used to calibrate the model. By considering a diversity of practical applications, it would elucidate anomalies introduced by an overly-simplified viscosity model and how they can be corrected by quasi-mechanistic models, such as those presented here.

## 6. Conclusions

## Acknowledgments

## Author Contributions

## Conflicts of Interest

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**Figure 1.**The zero-shear viscosity prediction of the original Quemada model, the modified Quemada model, and the Krieger model with n = 2. Discontinuities and regions where the viscosity is inversely proportional to hematocrit were found for both forms of the Quemada model.

**Figure 2.**Comparison of regression of asymptotic viscosity vs. hematocrit with the Asymptotic Krieger model (AKM), Krieger model (n = 2), the Quemada model, and the asymptotic Quemada model (k = k

_{∞}) to the experimental data of Brooks et al. (asterisks) [49]. (Shear rate > 100 s

^{−1}).

**Figure 3.**A comparison of shear thinning according to the MKM5 (

**a**) and the Quemada model (

**b**) vs. the experimental data of Brooks et al. [49] (symbols) for seven values of hematocrit and spanning four decades of shear rate.

**Figure 4.**Contribution of aggregation and cell deformation on viscosity for canine red blood cells (RBCs) at a hematocrit of 45% and a temperature of 37 °C. The lines for each case are generated from the modified Krieger model with the appropriate optimal parameters. Experimental data from Chien [50].

**Figure 5.**The effect of fibrinogen concentration on the viscosity of 45% canine blood at a temperature of 37 °C. Comparison of experimental data from Chien [23] with the modified Krieger model.

**Figure 6.**The fully-developed, steady-state velocity profile for Poiseuille flow of 40% hematocrit blood in a tube. Results compare the Newtonian solution (Parabolic), with the Modified Krieger model, the modified Quemada Model, and the model used by Jung et al. at a flow rate of 0.06 lpm (

**a**) and 6 lpm (

**b**).

**Figure 7.**Comparison of the fully-developed, steady-state velocity profile for Poiseuille flow of blood assuming a parabolic hematocrit profile vs. a uniform profile. The results of the MKM5 model and Quemada model overlap.

**Table 1.**Models of blood viscosity including a list of model features. X indicates the model has the feature and—indicates the model lacks the feature. (LS—low shear, HS—high shear).

Model | Equation | Shear Rate | Hematocrit | Proteins | Temperature | Asymptote LS/HS | Continuous | Legend |
---|---|---|---|---|---|---|---|---|

Newtonian [8] | $\mathsf{\eta}=0.0345\mathrm{cP}$ | – | – | – | – | –/X | X | (1) |

Asymptotic [9] | $\mathsf{\eta}=[{\mathsf{\eta}}_{0}+{\mathsf{\eta}}_{1}\mathsf{\phi}-{\mathsf{\eta}}_{2}{\mathsf{\phi}}^{2}+{\mathsf{\eta}}_{3}{\mathsf{\phi}}^{3}]P$ | – | X | – | – | –/X | X | (2) |

Power-Law [8] | $\mathsf{\eta}=k{\dot{\gamma}}^{n-1}$ | X | – | – | – | –/– | – | (3) |

Generalized Power-Law [10] | $\mathsf{\eta}=k(\dot{\gamma}){\dot{\gamma}}^{n(\dot{\gamma})-1}$ | X | – | – | – | –/X | – | (4) |

Walburn-Schneck (WS) [11] | $\mathsf{\eta}={C}_{1}\mathrm{exp}({C}_{2}\mathsf{\phi})\cdot \mathrm{exp}({C}_{4}(TPMA/{\mathsf{\phi}}^{2})){\dot{\gamma}}^{-{C}_{3}\mathsf{\phi}}$ | X | X | X | X | –/– | – | (5) |

Asymptotic Power-Law | $\mathsf{\eta}=k{\dot{\gamma}}^{n-1}+{\mathsf{\eta}}_{\infty}$ | X | X | X | X | –/X | X | (6) |

Cross, Modified Cross, Simplified Cross, Carreau, Carreau-Yasuda, Jung et al. [8,10,12,13] | $\mathsf{\eta}={\mathsf{\eta}}_{\infty}+\mathsf{\Delta}\mathsf{\eta}\frac{1}{{\left(1+{(\lambda \dot{\gamma})}^{m}\right)}^{a}}$ | X | – | – | – | X/X | X | (7) |

Powell-Eyring [8] | $\mathsf{\eta}={\mathsf{\eta}}_{\infty}+\mathsf{\Delta}\mathsf{\eta}\frac{{\mathrm{sinh}}^{-1}(\lambda \dot{\gamma})}{\lambda \dot{\gamma}}$ | X | – | – | – | X/X | – | (8) |

Modified Powell-Eyring [10] | $\mathsf{\eta}={\mathsf{\eta}}_{\infty}+\mathsf{\Delta}\mathsf{\eta}\frac{\mathrm{ln}(1+\lambda \dot{\gamma})}{{\left(\lambda \dot{\gamma}\right)}^{m}}$ | X | – | – | – | X/X | – | (9) |

Yeleswarapu [14] | $\mathsf{\eta}={\mathsf{\eta}}_{\infty}+\mathsf{\Delta}\mathsf{\eta}\frac{1+\mathrm{ln}(1+\lambda \dot{\gamma})}{1+\lambda \dot{\gamma}}$ | X | – | – | – | X/X | X | (10) |

Quemada [15] | $\mathsf{\eta}={\mathsf{\eta}}_{pl}{(1-0.5K(\mathsf{\phi},\dot{\gamma})\mathsf{\phi})}^{-2}$ | X | X | – | – | –/X | – | (11) |

Krieger [16] | $\mathsf{\eta}={\mathsf{\eta}}_{pl}{(1-\frac{\mathsf{\phi}}{{\mathsf{\phi}}^{*}})}^{-N}$ | – | X | – | – | X/X | X | (12) |

**Table 2.**Coefficients for blood viscosity models presented in Table 1.

Model | Coefficient Values | Ref. | Model | Coefficient Values | Ref. |

Power-law | k = 0.42, | [8] | Walburn-Schneck (WS) | C_{1} = 0.00797, | [11] |

C_{2} = 0.0608, | |||||

n = 0.61 | C_{3} = 0.00499, | ||||

C_{4} = 14.59 L/g | |||||

Asymptotic | η_{0} = 0.014175, | [9] | Modified Cross | η_{∞} = 3.5 cP, | [8] |

η_{1} = 0.05878, | Δη = 52.5 cP, | ||||

η_{2} = 0.1598, | λ = 3.736 s, | ||||

η_{3} = 0.31964 | m = 2.406, | ||||

a = 0.254 | |||||

Cross | η_{∞} = 3.5 cP, | [8] | Carreau-Yasuda | η_{∞} = 3.5 cP, | [8] |

Δη = 52.5 cP, | Δη = 52.5 cP, | ||||

λ = 1.007 s, | λ = 1.902 s, | ||||

m = 1, | m = 1.25, | ||||

a = 1.028 | a = 0.7588 | ||||

Carreau | η_{∞} = 3.5 cP, | [8] | Modified Power-Erying | η_{∞} = 3.5 cP, | [10] |

Δη = 52.5 cP, | Δη = 52.5 cP, | ||||

λ = 3.313 s, | λ = 2.415 s, | ||||

m = 2, | m = 1.089 | ||||

a = 0.3216 | |||||

Powell-Erying | η_{∞} = 3.5 cP, | [8] | Yeleswarapu | η_{∞} = 5.0 cP, | [14] |

Δη = 52.5 cP, | Δη = 68.6 cP, | ||||

λ = 5.383 s | λ = 14.81 s | ||||

Simplified Cross | η_{∞} = 5 cP, | [12] | -- | ||

Δη = 125 cP, | |||||

λ = 8.0 s, | |||||

m = 1, | |||||

a = 1 | |||||

Model | Coefficient Forms | Coefficients | Ref. | ||

Generalized Power-Law | $\begin{array}{l}k={\mathsf{\eta}}_{\infty}+\mathsf{\Delta}\mathsf{\eta}\mathrm{exp}(-{\overline{\dot{\gamma}}}_{1}\mathrm{exp}(b/\dot{\gamma}))\\ n=1+\mathsf{\Delta}n\mathrm{exp}(-{\overline{\dot{\gamma}}}_{2}\mathrm{exp}(d/\dot{\gamma}))\end{array}$ | η_{∞} = 3.5, Δη = 25, Δn = 0.45, a = 50, b = 3, c = 50, d = 4. | [10] | ||

Jung et al. Modified Carreau | $\begin{array}{l}a=\left({a}_{1}\mathsf{\phi}+{a}_{2}{\mathsf{\phi}}^{2}+{a}_{3}{\mathsf{\phi}}^{3}\right){k}_{o}\\ \mathsf{\Delta}\mathsf{\eta}={\mathsf{\eta}}_{pl}(1+{d}_{1}\mathsf{\phi}+{d}_{2}{\mathsf{\phi}}^{2}+{d}_{3}{\mathsf{\phi}}^{3})\\ {k}_{o}=1+Ko\left[\mathrm{ln}\left(\mathrm{ln}\left(1+{\left(\lambda \dot{\gamma}\right)}^{2}\right)\right)/\mathrm{ln}\left(1+{\left(\lambda \dot{\gamma}\right)}^{2}\right)-1\right]\end{array}$ | η_{∞} = 0, λ = 0.1101 s, $\dot{\gamma}\ge $ 6 s ^{−1}Ko = 0, a _{1} = 0.1752, a _{2} = 0.4123, a _{3} = −0.4046, d _{1} = 16.305, d _{2} = −51.213, d _{3} = 122.28, $\dot{\gamma}<$6 s^{−1}ko = 1, a _{1} = 0.8907, a _{2} = −1.0339, a _{3} = −0.4456, d _{1} = 9.7193, d _{2} = −22.454, d _{3} = 70.782 | [13] | ||

Quemada Model (Cokelet) | $\begin{array}{l}K=\frac{{k}_{o}+{k}_{\infty}\sqrt{\dot{\gamma}/{\dot{\gamma}}_{c}}}{1+\sqrt{\dot{\gamma}/{\dot{\gamma}}_{c}}}\\ {k}_{o}=\mathrm{exp}({a}_{o}+{a}_{1}\mathsf{\phi}+{a}_{2}{\mathsf{\phi}}^{2}+{a}_{3}{\mathsf{\phi}}^{3})\\ {k}_{\infty}=\mathrm{exp}({b}_{o}+{b}_{1}\mathsf{\phi}+{b}_{2}{\mathsf{\phi}}^{2}+{b}_{3}{\mathsf{\phi}}^{3})\\ {\dot{\gamma}}_{c}=\mathrm{exp}({c}_{o}+{c}_{1}\mathsf{\phi}+{c}_{2}{\mathsf{\phi}}^{2}+{c}_{3}{\mathsf{\phi}}^{3})\end{array}$ | a_{o} = 3.874, a _{1} = −10.41, a _{2} = 13.80, a _{3} = −6.738, b _{o} = 1.3435, b _{1} = −2.803, b _{2} = 2.711, b _{3} = −0.6479, c _{o} = −6.1508, c _{1} = 27.923, c _{2} = -25.60, c _{3} = 3.697 | [17] | ||

Quemada Model (Das) | As above, except: ${k}_{o}={a}_{0}+\frac{2}{{a}_{1}+\mathsf{\phi}}$ | as above, except: a _{o} = 0.275363 and a _{1} = 0.100158 | [18] |

**Table 3.**The best fit parameters for the three modified Krieger models with 95% confidence intervals for the parameters as determined from Monte Carlo simulation.

Model Parameter | AKM | AKM (Fixed Cells) | MKM5 | MKM9 | Agg/Def. Model | Fg Model |
---|---|---|---|---|---|---|

a | 1.70 (1.66–1.75) | 1.06 (0.944–1.18) | 0 | 0.686 (0.34–1.03) | 0.0974 | 1.30 |

b | 9.86 (8.63–11.1) | −0.226−(0.201–0.251) | 8.71 (7.85–9.57) | 11.8 (4.11–19.5) | n/a | n/a |

c | 6.07 (5.59–6.55) | −1.69−(1.54–1.85) | 2.87 (2.55–3.20) | 8.60 (3.42–13.8) | n/a | n/a |

β | n/a | n/a | 8.23 (7.85–8.60) | n/a | n/a | n/a |

λ | n/a | n/a | 108 (106–110) | 136 (120–152) | n/a | n/a |

ν | n/a | n/a | 0.134 (0.122–0.146) | n/a | n/a | n/a |

b_{1} | n/a | n/a | n/a | −9.11 −(10.7–7.48) | n/a | n/a |

b_{2} | n/a | n/a | n/a | 13.0 (12.2–13.9) | n/a | n/a |

n_{1} | n/a | n/a | n/a | 0.180 (0.090–0.269) | n/a | n/a |

n_{2} | n/a | n/a | n/a | −0.170 −(0.304–0.035) | n/a | n/a |

n_{3} | n/a | n/a | n/a | 0.124 (0.073–0.174) | n/a | n/a |

β_{agg} | n/a | n/a | n/a | n/a | 4.27 | n/a |

λ_{agg} | n/a | n/a | n/a | n/a | 24.1 | 16.0 |

ν_{agg} | n/a | n/a | n/a | n/a | 0.380 | 0.0895 |

β_{def} | n/a | n/a | n/a | n/a | 4.36 | n/a |

λ_{def} | n/a | n/a | n/a | n/a | 5.44 | n/a |

ν_{def} | n/a | n/a | n/a | n/a | 0.120 | n/a |

B_{1} | n/a | n/a | n/a | n/a | n/a | 6.26 |

B_{2} | n/a | n/a | n/a | n/a | n/a | 5.54 |

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**MDPI and ACS Style**

Hund, S.J.; Kameneva, M.V.; Antaki, J.F.
A Quasi-Mechanistic Mathematical Representation for Blood Viscosity. *Fluids* **2017**, *2*, 10.
https://doi.org/10.3390/fluids2010010

**AMA Style**

Hund SJ, Kameneva MV, Antaki JF.
A Quasi-Mechanistic Mathematical Representation for Blood Viscosity. *Fluids*. 2017; 2(1):10.
https://doi.org/10.3390/fluids2010010

**Chicago/Turabian Style**

Hund, Samuel J., Marina V. Kameneva, and James F. Antaki.
2017. "A Quasi-Mechanistic Mathematical Representation for Blood Viscosity" *Fluids* 2, no. 1: 10.
https://doi.org/10.3390/fluids2010010