# Microfluidic Simulation and Optimization of Blood Coagulation Factors and Anticoagulants in Polymethyl Methacrylate Microchannels

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## Abstract

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## 1. Introduction

^{3}/s and the entropy generation rate is minimized. Therefore, genetic algorithm can be optimized for unknown parameters to obtain better results [13]. The inertial force of fluid is small due to the diameter of the microfluid channel, which is on the order of 10

^{−6}m, resulting in a Reynolds number between 0.01 and 100 [8]. Under this Reynolds number condition, laminar flow of the fluid occurs in the microchannel. Fluids can be mixed into this flow only through diffusion. In biomedical testing, increasing the efficiency of the mixing of fluid with the sample and the reagent pretreatment is necessary. In this work, we use species transport and chemical reaction, which involves an unknown reaction rate constant and reaction order, and use a genetic algorithm to calculate these parameters.

## 2. Methodology

## 3. Governing Equations and Boundary Conditions

#### 3.1. Equation of Continuity

#### 3.2. Momentum Equation

#### 3.3. Species Transport Equations

_{j}and the chemical reaction rate R

_{j}are expressed as follows:

_{l}is an objective function coefficient corresponding to the i

^{th}variable; dA is the average mass fraction of the prothrombin activator and fibrin; and wl is the weighting mass fraction of the prothrombin activator and fibrin. The following parameters are selected for the genetic algorithm: population size of 100; chromosome length of 10; crossover probability of 0.75; and mutation probability of 0.05. In addition, to simulate the flow of red blood cells (RBC) and white blood cells (WBC) inside the microchannel, particles are added to the flow field. Moreover, the transport behavior of the particles in the microchannel is observed using the dynamic mesh technique and Newton’s second and third laws of motion. In this study, to reduce the convergence time and complexity of the simulation, the particles are divided into three groups. Each group has approximately 3–10 RBCs, and the groups do not affect each other.

#### 3.4. Boundary Equations

## 4. Results and Discussion

_{1}-k

_{3}and n

_{1}-n

_{5}values after 20 s, because different k

_{1}–k

_{3}and n

_{1}–n

_{5}values affect the prothrombin activator, thrombin, and fibrin contents in the microchannel. The small area in the panel on the left-hand side of Figure 3 indicates that the blue zone contains the blood inlet side and mixing area. The brown zone is the reagent inlet side, which indicates that the mass fraction of the prothrombin activator decreases because the k

_{1}and n

_{1}values are small. The rate of decrease of n

_{1}is higher than that of k

_{1}. Figure 3 shows that initially the prothrombin activator in blood side and mass fraction is 100% before the reaction; then, prothrombin gradually descends in the microfluidic channel. At x = 0, it immediately reacts with the reagent, so the reaction is complete before x < 5. Since the prothrombin reaction is related to R1 (Equation (18)), the k

_{1}and n

_{1}will affect the time of prothrombin activator in the microchannel. From Figure 3, it can be found that when n

_{1}is equal to 2.0, the reagent will flow to the blood side and react with prothrombin rapidly, hence it will complete reaction when x < 5. As can be seen in Figure 4, the tendency of the thrombin content to decrease after an increase in response does not occur in the mixed zone. Therefore, it is speculated that the prothrombin activator has completely reacted with thrombin before it enters the mixed zone. The larger the k

_{1}value and the smaller the n

_{3}value, the slower the rate of thrombin formation in the flow channel. As indicated on the left of Figure 4, the increase in magnitude when k

_{1}= 0.1 and 0.001 is almost the same as for 1.0 and 0.1, indicating that there should be a limiting value for k

_{1}. As depicted on the right of Figure 4, the maximum value is obtained at x = −10 mm when n

_{3}= 2, which indicates that the values of k

_{1}and n

_{3}have a considerable effect on the distribution of thrombin in the microchannel. According to Equation (17), the thrombin content is proportional to the fibrin content. Thrombin is the intermediate product of prothrombin and fibrinogen. Prothrombin activator starts to produce thrombin in the blood side of the microchannel at the beginning. When the reagent and the prothrombin activator react completely, the thrombin will react only with fibrinogen, hence thrombin will decrease in the microchannel. When n

_{3}= 2, which indicates that thrombin will react with fibrinogen to produce fibrin, hence, the thrombin will decrease at x > 0. As depicted on the left in Figure 5, when k

_{2}= 0.001, the rate of thrombin generation is low, which results in the fibrin mass fraction being almost equal to 0. Conversely, part of the thrombin content continues to react to form fibrin in the mixed zone when k

_{2}= 0.1. When x < 0, the fibrin mass fraction increases along with the microchannel, and when it is not mixed with reagents and chemical reaction (x > 0), it starts to decrease along the microchannel.

_{1}–k

_{3}obtained using the genetic algorithm were 0.987, 0.43, and 0.877, respectively, and those obtained for n

_{1}–n

_{5}were 0.548, 1.631, 1.23, 0.459, and 1.341, respectively. These results are indicative of the behavior of the prothrombin activator and fibrin in the microchannel. As can be seen in Figure 6, the chemical reaction with the anticoagulant does not occur at 1 s, because the prothrombin activator has yet to enter the mixed zone at this time. The prothrombin activator reacts with the anticoagulant within 3–5 s, and the reaction rate is high. Although a high prothrombin activator content is present in the blood at the side channel at 3 s, the activity of the prothrombin activator decreases after 5 s. Finally, the mass fraction of the prothrombin activator is nearly stable.

## 5. Conclusions

- Through simulation, it can be found that the chemical reaction of prothrombin in blood is a factor. When the reaction rate is k
_{1}, and reaction order n_{2}, n_{3}increase, prothrombin will be quickly depleted, causing the drug to flow to the blood end, but too slow the time of PT is extended. The reaction rate and reaction order can be obtained more accurately by genetic algorithm. The chemical reaction rate constants indicate that the reaction rate of prothrombin activator is faster than thrombin, fibrin activator is faster than thrombin, and prothrombin activator is faster than fibrin. - Predicted by simulation and experimental results, high blood concentration (>65%) is more accurate in predicting PT.
- In order to observe whether RBCs and WBCs are obstructed in the microchannel, the PT time cannot be accurately and effectively evaluated. Therefore, the dynamic mesh method can clearly determine the time for RBCs and WBCs to pass through the microchannel and can predict the dynamic behavior of blood and coagulants. The white blood cell flow time is 11.7 s and there are no obstructions.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## Nomenclature

Nomenclature | |

V | velocity |

$\dot{{\mathrm{m}}_{\mathrm{ij}}}$ | mass from phase i to phase j |

X, Y | mass fraction |

M_{w} | molecular weight |

P | static pressure |

J | diffusion flux |

R | net rate of production by chemical reaction |

D_{j,m} | diffusion coefficient for species j in the mixture |

N | number of chemical species in the system |

N_{r} | number of chemical species in reaction r |

${\mathrm{v}}_{\mathrm{j},\mathrm{r}}^{\prime}$ | stoichiometric coefficient for reactant j in reaction r |

${\mathrm{v}}_{\mathrm{j},\mathrm{r}}^{\u2033}$ | stoichiometric coefficient for product j in reaction r |

M_{j} | symbol denoting species j |

${\mathrm{k}}_{\mathrm{f},\mathrm{r}}$ | forward rate constant for reaction r |

${\mathrm{k}}_{\mathrm{b},\mathrm{r}}$ | backward rate constant for reaction r |

${\mathrm{C}}_{\mathrm{k},\mathrm{r}}$ | molar concentration of each reactant and product species k in reaction r |

${\mathsf{\eta}}_{\mathrm{k},\mathrm{r}}^{\prime}$ | forward rate exponent for each reactant and product species k in reaction r |

${\mathsf{\eta}}_{\mathrm{k},\mathrm{r}}^{\u2033}$ | backward rate exponent for each reactant and product species k in reaction r |

t | time |

g | gravity acceleration |

k_{1}–k_{3} | chemical reaction rate |

n_{1}–n_{5} | chemical reaction order |

${\mathrm{w}}_{\mathrm{l}}$ | weighting mass fraction |

${\mathrm{f}}_{\mathrm{l}}$ | objective function coefficient |

dA | average mass fraction |

Greek letters | |

α | volume fraction |

β | interphase momentum exchange coefficient |

μ | viscosity |

τ | stress tensor |

ρ | density |

Subscripts | |

i | phase, species |

j, k, l | species |

r | reaction |

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**Figure 3.**Distribution of the mass fraction of the prothrombin activator for (

**a**) various k

_{1}and (

**b**) n

_{1}.

**Figure 6.**Mass fraction of the prothrombin activator in the microchannel at various times (

**a**) 1 s, (

**b**) 3 s, (

**c**) 5 s, and (

**d**) 20 s.

**Figure 7.**Mass fraction of fibrin in the microchannel at various times (

**a**) 1 s, (

**b**) 3 s, (

**c**) 5 s, and (

**d**) 20 s.

**Figure 9.**Dynamic behavior of WBCs and RBCs in the microchannel of (

**a**) whole flow, (

**b**) 0 s, (

**c**) 4.3 s, (

**d**) 7.2 s, (

**e**) 10 s and, (

**f**) 11.7 s.

**Table 1.**Comparison between point of care prothrombin time POC PT [30] and simulated PT.

Blood Volume (%) | Mean POC PT (Min, Max) (s) | Mean Lab PT (Min, Max) (s) | Simulated PT (s) |
---|---|---|---|

100 | 12.0 (11.5, 13.0) | 11.7 (10.7, 13.4) | 12.3 |

75 | 14.6 (13.6, 15.9) | 13.3 (12.7, 14.6) | 14.5 |

65 | 16.2 (15.1, 17.4) | 14.7 (13.8, 15.4) | 15.5 |

55 | 20.1 (17.8, 24.3) | 16.1 (14.3, 17.3) | 21.0 |

40 | 41.8 (35.6, 60.6) | 21.8 (17.7, 23.4) | 21.0 |

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**MDPI and ACS Style**

Immanuel, P.N.; Chiu, Y.-H.; Huang, S.-J.
Microfluidic Simulation and Optimization of Blood Coagulation Factors and Anticoagulants in Polymethyl Methacrylate Microchannels. *Coatings* **2021**, *11*, 1394.
https://doi.org/10.3390/coatings11111394

**AMA Style**

Immanuel PN, Chiu Y-H, Huang S-J.
Microfluidic Simulation and Optimization of Blood Coagulation Factors and Anticoagulants in Polymethyl Methacrylate Microchannels. *Coatings*. 2021; 11(11):1394.
https://doi.org/10.3390/coatings11111394

**Chicago/Turabian Style**

Immanuel, Philip Nathaniel, Yi-Hsiung Chiu, and Song-Jeng Huang.
2021. "Microfluidic Simulation and Optimization of Blood Coagulation Factors and Anticoagulants in Polymethyl Methacrylate Microchannels" *Coatings* 11, no. 11: 1394.
https://doi.org/10.3390/coatings11111394