Mimic Non-Newtonian Shear-Thinning Behaviors of Blood Using Synthetic Analogs Under Various Thermal Conditions

1. Introduction

Blood is a complex biofluid with non-Newtonian properties, exhibiting both viscoelastic characteristics and shear-thinning behaviors, meaning its viscosity decreases as the shear rate increases [1,2]. These rheological properties significantly influence hemodynamics in both neurovascular and cardiovascular systems, affecting physiological and pathological behaviors [3,4]. Additionally, blood viscosity is temperature-dependent, with variations in temperature having a substantial impact on blood viscosity and hemodynamics within vessels [4].

To replicate realistic blood rheological behaviors, blood analogs are often used in in-vitro experiments, particularly in 3D-printed vascular models. Traditional Newtonian fluids (e.g., water or glycerol solutions), which assume a constant blood viscosity of approximately 3.0~4.0 mPa⋅s, fail to capture some critical hemodynamic characteristics such as instantaneous wall shear stress, oscillatory shear index, and relative residence time, thus limiting their effectiveness in realistic simulations of blood flow [4,5,6,7,8]. Therefore, the development of non-Newtonian blood analogs that accurately replicate the rheological behavior of human blood under variable temperature conditions is crucial for advancing biomedical research, medical device development, educational demonstrations, and hemodynamic modeling.

Realistic blood analogs must not only exhibit non-Newtonian behavior but also account for the effects of temperature variations. The analog must behave similarly to real blood across a broad range of thermal conditions. For instance, hypothermia (body temperature < 310.15 K) increases blood viscosity and alters its shear-thinning behavior, potentially exacerbating circulatory complications [9], while hyperthermia (body temperature > 310.15 K) decreases viscosity and changes its viscoelastic features [10,11]. These temperature-induced changes have important implications for applications such as cardiopulmonary bypass systems [12], cryopreservation protocols [13], and thermal ablation therapies [14]. Despite this, most existing blood analogs do not consider temperature as a variable, leading to oversimplified experimental or computational models [5,15,16,17,18,19].

To address knowledge gaps in non-Newtonian blood analogs suitable for in-vitro experiments under various thermal conditions, this work presented the design and synthesis of two types (Type-I and Type-II) of temperature-tunable non-Newtonian blood analogs that mimic human blood’s rheological properties across a clinically relevant temperature range of 295-312 K [20]. Type I was intended for general use where velocity field visualizations/measurements are not required, while Type-II was specifically designed for particle image velocimetry (PIV) measurements to investigate hemodynamics in vascular models. Both analogs were based on a solution of 800 ppm xanthan gum (XG), composed of XG power deionized (DI) water, which holds the capability to produce non-Newtonian shear-thinning properties similar to those of real blood [21,22]. To create Type-I blood analogs, the 800 ppm XG solution was diluted with DI water to obtain concentrations ranging from 120 ppm to 600 ppm, in 20 ppm increments. Type-II blood analogs were created similarly, but with the addition of 450 ppm fluorescent red polyethylene microspheres (FRPM). The fluorescent light from the particles due to the laser excitation enables the velocity profile capture in PIV tests. Progress in this research could assist scientists and engineers in the blood research community to (1) test medical devices (e.g., stents, catheters, pumps) under physiologically realistic temperature scenarios, (2) study disease mechanisms (e.g., thrombosis, atherosclerosis, cerebral aneurysms) influenced by temperature gradients, and (3) validate computational models that incorporate thermal effects on blood flows.

2. Materials and Methods

2.1. Developing a Shear-Rate and Temperatur-Dependent non-Newtonian Blood Viscosity Model

The generated Type-I and Type-II blood analogs were compared with our previously developed shear-rate and temperature-dependent non-Newtonian blood viscosity model [4,23], which has been employed by many other research groups. To ensure the paper is self-contained, we presented the deduction process to assist the blood analog research community in better understanding the intrinsic coupling mechanisms between shear-thinning behavior and thermal effects on blood viscosity [4], that is,

$$\mu =a{\dot{\gamma }}^{b-1}{e}^{\left(\alpha \left({\displaystyle \frac{1}{T}}-{\displaystyle \frac{1}{\delta T+\epsilon }}\right)+\beta T\right)},T[K]\in [~295,~310].$$

(1)

In Equation (1), $\mu$ is blood viscosity, $a$ is the consistency index with $2.05\times {10}^{12}$ $mPa·s^b$, b is the power-law index registered as 0.685, and $T$ is the blood temperature in $K$. In addition, $\alpha$, $\beta$, $\delta$, and $\epsilon$ are temperature index constants assigned by 4945.4 $\mathrm{K}$, -0.083 $K^{-1}$, 1.53 and -162.22 $K$, respectively. The rest of this section shows the details of the developing process. The process of original data collection can be accessed from our previous publication [4].

In general, the non-Newtonian blood viscosity model can be expressed by a power-law [24]:

$$\mu =a{\dot{\gamma }}^{b-1}$$

(2)

The real blood viscosity was measured using an IKA ROTAVISC lo-vi viscometer (Staufen, Germany) under four designated temperatures [4], in which the measured viscosities under corresponding temperatures are shown in Figure 1. Integrating the generalized power-law viscosity model (i.e., Equation (2)) with the collected experimental data, four non-Newtonian power-law viscosity models can be regressed under the corresponding four temperatures, respectively. Specifically, the equations are, i.e.,

$${\mu }_{\begin{array}{c}T=295.5K\\ \end{array}}={a_1\dot{\gamma }}^{b_1-1},a_1=41.927\times {10}^{-3}Pa·s^{b_1},b_1=0.635$$

(3)

$${\mu }_{\begin{array}{c}T=300.2K\\ \end{array}}={a_2\dot{\gamma }}^{b_2-1},a_2=32.948\times {10}^{-3}Pa·s^{b_2},b_2=0.643$$

(4)

$${\mu }_{\begin{array}{c}T=305.8K\\ \end{array}}={a_3\dot{\gamma }}^{b_3-1},a_3=18.817\times {10}^{-3}Pa·s^{b_3},b_3=0.710$$

(5)

$${\mu }_{\begin{array}{c}T=310.3K\\ \end{array}}={a_4\dot{\gamma }}^{b_4-1},a_4=12.763\times {10}^{-3}Pa·s^{b_4},b_4=0.753$$

(6)

The correlation coefficients $R$ for the regressed Equations (3)-(6) are 0.975, 0.995, 0.990, 0.995 under the temperatures of 295.5$,$ 300.2, 305.8, and 310.3 $K$, respectively (see Figure 1).

.

Considering thermal influences on blood viscosity, it was assumed that the blood viscosity model consisted of two variables, i.e., the shear rate term $\eta \left(\dot{\gamma }\right)$ and the temperature index $H\left(T\right)$, in which $H\left(T\right)$ is based on the Arrhenius law [25,26,27,28]. The equation was expressed as

$${\mu }_{\left(\dot{\gamma },T\right)}=\eta \left(\dot{\gamma }\right)H\left(T\right)$$

(7a)

$$\eta \left(\dot{\gamma }\right)=a{\dot{\gamma }}^{b-1}$$

(7b)

In Equation (7b), $b$ was calculated by averaging the consistency indexes in Equations (3)-(6), i.e.,

$b={\displaystyle \frac{b_1+b_2+b_3+b_4}{4}}=0.685$To find the consistency index $a$ in Equation (7b), an interim consistency index $a^{\text{'}}$ was introduced in this study to present its relationship with temperature (see Figure 2). Using the exponential regressed strategy, $a^{\text{'}}$ was regressed as a power-law form, that is,

$$a^{\mathrm{\text{'}}}=a{e}^{-0.083T},a=2.05\times {10}^{12}mPa·s^b.$$

(8)

In Figure 2, the correlation coefficient $R$ of Equation (8) for the interim consistency index $a^{\text{'}}$ is 0.987 under measured temperatures from 295

to 310$K$.

Then, taking the natural logarithm in both sides of Equation (7a) yields:

$$\ln \left({\mu }_{\left(\dot{\gamma },T\right)}\right)=\ln \left(\eta \left(\dot{\gamma }\right)\right)+\ln \left(H\left(T\right)\right)$$

(9)

Since $\eta \left(\dot{\gamma }\right)$ is a shear rate dependent variable solely, a plot of $\ln \left({\mu }_{\left(\dot{\gamma },T\right)}\right)$ vs. $\ln \left(H\left(T\right)\right)$can be obtained using two different temperatures ($T_m$ and $T_{\mathrm{n}}$, with m and n represent two different thermal data points, m=1, 2, 3, and n=2, 3, 4) under the same shear rate $\dot{\gamma }$ listed as follows:

$$\ln \left({\mu }_{\left(\dot{\gamma },T_{\mathrm{m}}\right)}\right)-\ln \left({\mu }_{\left(\dot{\gamma },T_{\mathrm{n}}\right)}\right)=\ln \left(H\left(T_{\mathrm{m}}\right)\right)-\ln \left(H\left(T_{\mathrm{n}}\right)\right)$$

(10)

Based on Arrhenius law and integrating$e^{-0.083T}$ (i.e., $e^{\beta T}$ ) in Equation (8), we can define $H\left(T\right)$ as:

$$H\left(T\right)=e^{\alpha \left({\displaystyle \frac{1}{T}}-{\displaystyle \frac{1}{T_{Ref.}}}\right)+\beta T}$$

(11)

In Equation (11), $\alpha$ is a temperature index constant related to the activation energy, and $\beta$ is another temperature index constant registered as -0.083 $K^{-1}$ (see Equation (8)). Additionally, $T_{Ref.}$ is an introduced reference temperature. Integrating Equation (11), Equations (10) can be rewritten as follows:

$$\ln \left({\mu }_{\left(\dot{\gamma },T_{\mathrm{m}}\right)}\right)-\ln \left({\mu }_{\left(\dot{\gamma },T_{\mathrm{n}}\right)}\right)-\beta (T_{\mathrm{m}}-T_{\mathrm{n}})=\alpha \left({\left({\displaystyle \frac{1}{T}}-{\displaystyle \frac{1}{T_{Ref.}}}\right)}_{T=T_{\mathrm{m}}}-{\left({\displaystyle \frac{1}{T}}-{\displaystyle \frac{1}{T_{Ref.}}}\right)}_{T=T_{\mathrm{n}}}\right)$$

(12)

Temporarily, $T_{Ref.}$ is assumed as a constant value which will be further corrected after the temperature index $\alpha$ is decided. Thus, Equation (12) can be simplified as:

$$\ln \left({\mu }_{\left(\dot{\gamma },T_{\mathrm{m}}\right)}\right)-\ln \left({\mu }_{\left(\dot{\gamma },T_{\mathrm{n}}\right)}\right)-\beta (T_{\mathrm{m}}-T_{\mathrm{n}})=\alpha \left({\displaystyle \frac{1}{T_{\mathrm{m}}}}-{\displaystyle \frac{1}{T_{\mathrm{n}}}}\right)$$

(13)

Further, $\alpha$ can be decided by Equation (14), that is,

$$\ln \left({\mu }_{\left(\dot{\gamma },T_{\mathrm{m}}\right)}\right)-\ln \left({\mu }_{\left(\dot{\gamma },T_{\mathrm{n}}\right)}\right)-\beta (T_{\mathrm{m}}-T_{\mathrm{n}})=\alpha \left({\displaystyle \frac{1}{T_{\mathrm{m}}}}-{\displaystyle \frac{1}{T_{\mathrm{n}}}}\right)$$

(14)

To facilitate the calculations and expressions, we define:

$$∆T_{\mathrm{m}-\mathrm{n}}={\displaystyle \frac{1}{T_{\mathrm{m}}}}-{\displaystyle \frac{1}{T_{\mathrm{n}}}}$$

(15)

Integrating the experimental data shown in Figure 1 and Equation (8), $\alpha$ can be decided with the corresponding shear rate $\dot{\gamma }$, shown in Table 1

Averaging the values in Table 1, the temperature index of $\alpha$ can be obtained, i.e.,

$$\alpha =4945.4K$$

The next step was to regress $T_{Ref.}$ in Equation (11). Integrating Equations (7a), (7b), (8), and (11), we can obtain:

$$T_{Ref.}={\displaystyle \frac{1}{\frac{1}{T}-\frac{1}{\alpha }\left(\ln \left({\mu }_{\left(\dot{\gamma },T\right)}\right)-ln\left(a{\dot{\gamma }}^{b-1}\right)+0.083T\right)}}$$

(16)

Using the viscosities and shear rates under corresponding temperatures, the calculated results of $T_{Ref.}$ based on Equation (16) were exhibited in Table 2. Averaging $T_{Ref.}$ values under the corresponding four thermal conditions, we can obtain

$${T_{Ref.}}_{T=295.5K}=291.0679K{T_{Ref.}}_{T=300.2K}=298.5440K$$

$${T_{Ref.}}_{T=305.8K}=306.9594K{T_{Ref.}}_{T=310.3K}=313.9530K$$

Averaging the calculated $T_{Ref.}$ in Table 2 and employing linear regression method, the relationship between the reference temperature $T_{Ref.}$ and $T$ (see Figure 3) can be expressed by

$$T_{Ref.}=\delta T+\epsilon$$

(17)

where $\delta$ and $\epsilon$ are temperature constants registered as 1.53 and -162.22 $K$, respectively.

Finally, we can obtain the non-Newtonian blood model under various thermal conditions, shown in Equation (1). Figure 4 shows the correlation coefficients $R$ between Equation (1) [4] and experimental measurements, which are 0.99, 0.99, 1.00, and 0.98 at T = 310. 3, 305.8, 300.2, and 295.5 $K$, respectively. The results indicate that the developed non-Newtonian model successfully presents blood non-Newtonian shear-thinning properties under various thermal conditions. In this study, the developed Equation (1) was employed as the baseline equation to estimate the accuracy of generated Type-I and Type-II blood analogs by minimizing the root mean square deviation (RMSD).

2.2. Blood Analog Generation

2.2.1. Materials

As shown in Figure 5, two synthetic blood analogues, i.e., Type-I and Type-II were generated to replicate the shear-thinning properties of blood under the designated thermal conditions from 295 to 312 $K$. Type-I was intended for general use and consisted of DI water (supplied by the Chemistry Department at the Wright State University, Dayton, OH, USA) and xanthan gum (XG) powder (Kate Naturals, Harrisonburg, VA, USA). Type-II was designed for the use of PIV experiments to study hemodynamics in vessels or heart models, which require transparent blood analog with seeded particles. Type-II analog was a mixture of DI water, XG power, and FRPM particles (Cospheric LLC, Somis, CA, USA), with particle size from 10 to 45 $\mu m$.

To prepare blood analogs, the process began by mixing boiling DI water with XG powder to create the original base XG solution, with a concentration of 800 ppm. This solution was then diluted by DI water to generate desired concentrations for Type-I blood analogs at specified thermal conditions. For Type-II blood analogs, DI water and FRPM solution were added to the XG solution to achieve designated concentrations and ensure 450 ppm seeded FRPM particles for PIV experiments.

It is worth mentioning that the mixing of DI water and XG powder was a delicate and technically challenging process, as it requires complete dissolution of XG powders without the formation of clumps, particles, or bubbles. To ensure the consistency and accuracy of the blood analogs, three batches of original blood analogs were created. The viscosities of these batches were measured under shear rates ranging from 1 to 250 s^-1, and the discrepancy was controlled under 3%. In the preparation of Type-II blood analogs, a raw FRPM particle solution at 4500 ppm was prepared. Similar to the XG solution, three batches of the FRPM particle solution were made, with differences among them controlled within 3%.

To create Type-I blood analogs, the 800 ppm XG solution was diluted by DI water to form a 16 ml solution with the XG concentration ranging from 120 to 600 ppm, in increments of 20 ppm. For Type-II blood analogs, the 800 ppm XG solution was diluted in the same manner to form a 18 ml solution, to which 2 ml of 4500 ppm FRPM solution was added, resulting in a total volume of 20 ml. Such a step was to ensure all Type-II blood analogs were uniformly seeded with the same number (450 ppm) of FRPM particles, satisfying the requirements for PIV tests to secure accurate flow characteristics in the tested vessel or heart models.

2.2.2. Experimental Setups and Measurements

The prepared solutions were thoroughly mixed by stirring in the beaker (see Figure 5) and then allowed to stand for sufficient time to ensure the complete dissipation of any visible bubbles. Once the solution was bubble-free, it was transferred to the fluid container and placed in a custom-design water bath. The water bath maintained the blood analog at a stable environmental temperature of 293.15 K. Viscosity measurements were conducted using an IKA Rotavisc lo-vi viscometer (IKA-Werke GmbH & Co. KG, Staufen im Breisgau, Germany). Prior to testing, the stability of IKA Rotavisc lo-vi device was verified by rotating the bottom standers to ensure proper functionality.

During the viscosity measurement process, two independent operators performed the tests. Each operator recorded viscosity readings twice, and the average of these measurements was taken. To minimize random errors, the results from both operators were compared systematically. Discrepancies, if any, were addressed by additional experiments to ensure the accuracy and reliability of the data. For each blood analog of a given concentration, the measured viscosity data were compared with viscosity values calculated using Equation (1). This comparison was used to match the analog’s viscosity with that of human blood at a specific body temperature. For the matching process, temperatures from 292 to 320 K were searched in increments of 0.01 K for the best match by minimizing the root mean square deviation (RMSD) which was calculated by

$$RMSD=\sqrt{{\displaystyle \frac{{\sum }_{n=1}^N{\left({\mu }_{exp.}-{\mu }_{equation\left(1\right)}\right)}^2}{N}}}$$

(18)

where ${\mu }_{exp.}$is the experimental measured values, ${\mu }_{\mathit{E}\mathit{q}\mathit{u}\mathit{a}\mathit{t}\mathit{i}\mathit{o}\mathit{n}\left(1\right)}$ is the calculated viscosity using Equation (1) under certain temperatures, N is the number of data points. The averaged relative discrepancy $\overline{D}$ in the viscosity comparison between the developed blood analog and Equation (1) was calculated by

$$\overline{D}={\displaystyle \frac{{\sum }_{n=1}^N\left|\frac{{\mu }_{exp.}-{\mu }_{Equation\left(1\right)}}{{\mu }_{Equation\left(1\right)}}\right|}{N}}\times 100\%$$

(19)

3. Results

3.1. Type-I Blood Analog

The results presented in Figure 6 (a) to 6(y) illustrate the relationship between shear rates and viscosity of Type-I blood analogs, with XG concentrations varying in increments of 20 ppm. These measurements were taken at a constant room temperature of 293.15 K, with the original base solution of 800 ppm XG serving as a reference. The data demonstrate that all generated Type-I blood analogs exhibit shear-thinning rheological properties similar to those of real blood, where viscosity decreases as the shear rate increases. By minimizing $RSMD$between experimental measurements and the developed Equation (1), we derived the relationship between temperature and XG concentration for Type-I blood analogs, as shown in Figure 7. The results revealed a strong linear correlation between temperature and XG concentration $c_{XG}$, with a registered coefficient of determination $R^2=0.9828$ and $\overline{D}$ < 5%. Specifically, this relationship can be expressed by the following equation

$$c_{XG}=-16.317T+5170.10,T[K]\in [295,312]$$

(20)

Equation (20) provides a formula to generate blood analogs for in-vitro experiments that require a realistic shear-thinning rheological environment, simulating blood flow behaviors at various “human body” temperatures from 295 to 312 K.

3.2. Type-II Blood Analog

Similarly, Figure 8 (a) - (y) present the experimental results of measuring viscosities of generated Type-II blood analogs across various shear rates, with XG concentrations varying in increments of 20 ppm. These measurements were conducted at a controlled room temperature of 293.15 K, and the raw solution used for Type-II blood analogs had the same XG concentration of 800 ppm as Type-I blood analogs. The FRPM particle concentration in Type-II blood analogs was controlled at 450 ppm, ensuring consistency in PIV tests by maintaining a uniform particle concentration to avoid errors due to varied particle density in captured images. As shown in Figure 8 (a) - (y), the results demonstrate that all generated Type-II blood analogs exhibit shear-thinning rheological properties of real blood, where the viscosity decreases as shear rate increases. By minimizing $RSMD$between experimental measurements and the developed Equation (1), it was found that XG concentration $c_{XG}$ is linearly related to human-body temperature. This relationship was highlighted by an orange line in Figure 7, with $R^2=0.9642$ and $\overline{D}$ < 5%. Specifically, the relationship can be expressed by the following equation

$$c_{XG}=-16.317T+5170.10,T[K]\in [295,312]$$

(21)

Equation (21) provides a formula to generate Type-II blood analogs which replicate the shear-thinning properties of real blood and ensure accurate hemodynamic patterns in in-vitro neurovascular and cardiovascular models specifically for PIV tests.

4. Discussion

The results presented above demonstrate that both Type-I and Type-II blood analogs exhibit a linear relationship between XG concentration and body temperature. As temperature decreases, the internal resistance of the blood increases due to the rise in viscosity [4]. Additionally, as shown in Figure 7, at the same temperature, the XG concentration of the Type-II blood analog is higher than that of Type-I blood analog. This difference is attributed to the addition of 450 ppm of FRPM particle solution, which further dilutes the XG solution, requiring a higher XG concentration to ensure that the viscosity of Type-II blood analogs matches the viscosity of the corresponding Type-I blood analogs.

Despite our team’s best efforts to minimize the errors and uncertainties during the measurements, discrepancies were still observed in the formulas in the two types of blood analog under varying thermal conditions, as depicted in Figure 7. These discrepancies were primarily attributed to systematic uncertainties in the viscometer, especially for low shear rates, as illustrated in Figure 9. It was found that although the uncertainty decreases with the increasing shear rate, it remains significant with the shear rates from 1.22 to 12.22 s^-1. In this range, the possible uncertainties in viscosity measurements range from 1.19 to 11.99 mPa⋅s. For instance, for one blood analog with 120 ppm XG and 450 ppm particle, we measured an average viscosity of 25.5 mPa⋅s at a shear rate of 1.22 s^-1. At this shear rate, the systematic uncertainty is 11.99 mPa⋅s, which can substantially impact the accuracy of the formulas used to generate blood analogs at the designated temperatures.

To reduce uncertainties for the regression models, we made efforts to improve the agreement by omitting the first few data points, ranging from 1 to 10 specifically. Figure 10 (a) to 10(j) display the relationship between temperature and XG concentration for both Type-I and Type-II blood analogs, with designated initial data points skipped. The linearly regressed models for each situation were given in Equations (22-31) and Equations (32-41) for Type-I and Type-II blood analogs, respectively.

$$c_{XG}=-\left[\begin{array}{c}17.615\\ 18.292\\ 19.015\\ 19.635\\ 20.116\\ 20.427\\ 20.782\\ 21.173\\ 21.568\\ 21.918\end{array}\right]T+\left[\begin{array}{c}5647.0\\ 5867.4\\ 6099.7\\ 6298.4\\ 6455.0\\ 6557.7\\ 6670.8\\ 6794.5\\ 6919.5\\ 7030.7\end{array}\right],T[K]\in [290,315], \text{Type-I}$$

(22-31)

$$c_{XG}=-\left[\begin{array}{c}18.372\\ 19.426\\ 20.576\\ 21.513\\ 22.207\\ 22.654\\ 23.080\\ 23.477\\ 23.829\\ 24.148\end{array}\right]T+\left[\begin{array}{c}5806.1\\ 6141.3\\ 6503.3\\ 6798.6\\ 7021.4\\ 7166.4\\ 7301.9\\ 7428.5\\ 7541.3\\ 7643.9\end{array}\right],T\left[K\right]\in \left[290,315\right], \text{Type-II}$$

(32-41)

For Type-I blood analogs, the $R^2$ of the regressed Equations (22)-(31), were registered as 0.9888, 0.9941, 0.9955, 0.9959, 0.9963, 0.9963, 0.9967, 0.9970, 0.9972, and 0.9974, respectively, for datasets where the number of 1 to 10 initial data points were skipped correspondingly. For Type-II blood analog, the $R^2$ for the corresponding Equations (32)-(41) were registered as 0.9926, 0.9940, 0.9939, 0.9946, 0.9957, 0.9974, 0.9976, 0.9976, 0.9978, and 0.9980, respectively, when the number of 1 to 10 initial data points are excluded correspondingly. The best fit was achieved by excluding the first 10 data points at lower low shear rates. These initial data often exhibited steep changes and large uncertainties, which could otherwise impact the accuracy of the regression models.

However, it is important to note that while skipping more data points results in better regressed equations, as indicated by the $R^2$ values, over-fittings can introduce new challenges. Figure 11 (a) to 11(d) illustrate the comparisons between regression equations and the baseline Equation (1) for both blood analog types at two representative temperatures (i.e., T = 295 K and 312 K), with the first 10 data points excluded. The results show that for shear rate greater than 30 s^-1, there is a good match between the regression equations and the baseline Equation (1). However, noticeable discrepancies appear in the viscosity values for shear rates ranging from 0 to 30 s^-1. As more initial points skipped, the discrepancies in viscosity values at low shear rates become more pronounced. These differences can be attributed to the intrinsic physical properties of the XG solution. By skipping fewer initial data points, a better match can be achieved in the viscosity comparison at shear rates of 0 -30 s^-1. However, the difference becomes more significant at higher shear rate such as greater than 30 s^-1. Therefore, we recommend that researchers need to first identify the shear rate range relevant to their experiments and then select the appropriate regression formulas (i.e., Equations (20)- (41)) to generate desired blood analogs for their specific applications.

Additionally, this study did not simulate blood cells explicitly. Although FRMP particles in Type-II blood analog partially represent blood cells, their concentration of 450 ppm is much lower than the red blood cell (RCB) concentrations in humans ranging 4.7 to 6.1 million ppm for males, 4.2 to 5.4 million ppm for females, and 4.0 to 5.5 million ppm for children [30]. Future studies could incorporate alternative materials, such as polysaccharide particles, to more accurately mimic blood cells. This would allow for more realistic blood analog generation and enable further investigations of hemodynamic behaviors associated with the pathophysiology of neurovascular and cardiovascular diseases.

5. Conclusions

In this study, we presented a synthetic method for developing a non-Newtonian human blood model (Equation (1)) that explicitly captures shear-thinning properties as a function of temperature, based on real blood viscosities. Using Equation (1) as the baseline, we successfully generated two types of blood analogs, i.e., Type-I and Type-II, for different purposes across a temperature range of 295-312 K, both exhibiting the same shear-thinning rheological properties. Type-I blood analog, composed solely of XG solution, was designed for general use; while Type-II blood analog, a mixture of XG solution and FRMP particles, was developed for PIV measurements to capture hemodynamic characteristics in in-vitro cardiovascular and neurovascular models. Both blood analogs demonstrated a linear relationship between XG concentration and body temperature. Although omitting the initial few data points improved the matching performance of the regression models at high shear rate (> 30 s^-1), this approach introduced potential errors in predicting viscosities at lower shear rate (< 30 s^-1). Therefore, it is crucial for researchers to select appropriate formulas based on their experimental shear rate range to ensure accurate blood analog generation.