Computational Fluid Dynamics has become relevant in the study of hemodynamics, where clinical results are challenging to obtain. This paper discusses a 2-Dimensional transient blood flow analysis through an arterial bifurcation for patients infected with the Coronavirus. The geometry considered is an arterial bifurcation with main stem diameter 3 mm and two outlets. The left outlet (smaller) has a diameter of 1.5 mm and the right outlet (larger), 2 mm. The length of the main stem, left branch and right branch are fixed at 35 mm, 20 mm and 25 mm respectively. Viscosity change that occurs in the blood leads to different parametrical changes in blood flow. The blood flow towards the smaller branch is significantly affected by the changed blood viscosity. Extended regions of high pressure and increased velocity towards the larger outlet are obtained. The Time Averaged Wall Shear Stress (TAWSS) for the corona affected artery is found to be 10.4114 Pa at a 90° angle of bifurcation as compared to 2.45002 Pa of the normal artery. For varying angles of bifurcation, an angle of 75° was found to have a maximum Time Averaged Wall Shear Stress of 2.46076 Pa and 10.42542 Pa for normal and corona affected artery, respectively.

The world is witnessing and combating a difficult situation created by the Severe Acute Respiratory Syndrome Coronavirus 2 (SARS-CoV-2). Around 220 million of the global population is infected, of which 3.9 million succumbed to death so far [

The novel form of Coronavirus has severe effects on haematological parameters. Endothelial dysfunction, platelet activation, hyper-viscosity, and blood flow aberration are the factors leading to thrombosis. Various studies show that blood viscosity in the case of COVID-19 patients has increased due to multiple reasons, such as elevated concentrations of acute phase reactants and hypergammaglobulinemia. Hyper-viscosity of blood may lead to severe clinical conditions such as myocardial infarction (MI), venous thrombosis, and venous thromboembolism. The changes in the physical parameters of blood due to Coronavirus infection are still open for further investigation. One possibility is that the Coronavirus directly affects the endothelial cells that line the blood vessels providing a smooth passage for blood to flow. Otherwise, the immune system that gets triggered as a result of the infection sparks clot formation [

Vascular endothelial cells experience two distinctive forces; cyclic strain due to the transmural pressure and shear stress due to the frictional force generated by fluid flow. Mean fluid flow rate, the physical dimensions of blood vessels, and blood viscosity contribute to the shear stress associated. Both strains cause appreciable cell deformation in the endothelial cells [

Computational Fluid Dynamics (CFD) comes handy for in-depth analysis of flow parameters of different fluids that are difficult to achieve in real world applications. Both laminar and turbulent models are effectively utilized in different commercially available CFD codes. Mesh refinement and validation are extremely important in any numerical simulation study [

The curvature of the coronary arteries significantly impacts the haemodynamic parameters associated with it. Kashyap, Arora and Bhattacharjee studied the effect of individual branch curvatures on occurrence of atherosclerosis using Computational Fluid Dynamics. The arteries were modelled using idealized geometries. The curvature is found to have increased the low Wall Shear Stress the artery walls are undergoing. The 3-dimensional Cartesian Navier–Stokes equations was solved using Fluent to analyse the blood flow through various branch curvatures from 0° to 60°. Non-Newtonian rheological models such as the Carreau and Carreau-Yasuda give reasonably accurate predictions of the flow field [

Rabbi, Laboni and Arafat in their study considered blood as a non-Newtonian fluid governed by continuity equation and Navier Stokes equation. Time Averaged Wall Shear Stress can provide meaningful insights to any cardio-vascular complexities developed in a patient. Idealized and patient specific Computer Models were analysed for Time Averaged Wall Shear Stress and other parameters. Higher exposure to Wall Shear Stresses were identified for increased angles of bifurcations [

Increased blood flow velocity indicates the decreased blood flow due to reduced lumen diameter. The results obtained from such CFD simulations can provide a vast database for prognosis in the case of COVID-19 related cardiovascular issues. The endothelial cells mediate the exchange of substances between the bloodstream and neighbouring tissues. A high value of wall shear stress leads the endothelial cells to an inactive state. The blood lumen struggles to maintain its regular shape and high stresses lead to their rupture. The peak systole and diastole phases show the occurrence of flow reversal near bifurcations and blocked areas. Non-Newtonian Carreau model is used by Pandey et al. [

The main disadvantage of the clinical methods is that they can only find out the flow velocity for analyzing the significant cardio problems in COVID-19 patients. The pressure, velocity, and wall shear distributions can provide vital information for early diagnosis, and possible vide CFD simulations. Identifying the areas of critical interest would provide an opportunity for medical practitioners to streamline the treatment methods, thereby saving valuable time on the due course.

Through a detailed literature review, the following are observed:

Idealized geometry may be utilized for simplification of blood flow problem through human arteries and avoid patient-specific variations.

Blood is considered as a non-Newtonian fluid for most analysis and is governed by the continuity equation and Navier Stokes equation.

Artery wall is supposed to be rigid and still produces meaningful results.

Thus, the current work discusses these flow parameters associated with the changes in blood viscosity during Coronavirus infection using a 2-Dimensional transient simulation.

The schematic of the geometry used for 2D transient simulation of blood flow through a bifurcating artery in ANSYS fluent is shown in

The angle A° depicts the angle of bifurcation. Idealized geometry rather than patient-specific models (obtained from CT scans) is used so as to simplify the problem and reduce the uncertainties associated. The wall of the artery is assumed to be fixed, rigid and non-porous as in many previous studies [

The results are validated using an already published numerical model. Carreau viscosity model with pulsating inlet velocity using a user-defined function was used to analyze flow dynamics of a normal bifurcated artery. As part of coronavirus infection, the hemodynamic changes, as predicted by Joob et al. [

Blood is generally agreed as a non-Newtonian fluid. The Carreau model is one of the most commonly used models for the prediction of blood flow dynamics. The properties of blood used for analysis are listed in

Property | Normal artery | Corona affected artery |
---|---|---|

Density (kg/m^{3}) |
1060 | 1060 |

Zero Shear Viscosity (Pa-s) (μ_{0}) |
0.056 | 0.0724 |

Infinite Shear Viscosity (μ_{∞}) (Pa-s) |
0.0035 | 0.0182 |

Power Law Index (n_{c}) |
0.3568 | 0.3568 |

Time Constant, λ (s) | 3.313 | 3.313 |

The viscosity values for COVID-19 patients are as suggested by Ray et al. [

The viscosity model suggested by Carreau-Yasuda model is shown below [

where,

μ_{c} = effective dynamic viscosity (Pa-s)

n_{c} and a are indexes determined from experiments.

In the simulations presented in this work, the power index a is fixed to be 2. This reduces the Carreau- Yasuda model in

Continuity and Incompressible Navier Stokes equation are the governing equations to be solved for the particular problem concerned [

Continuity equation:

Navier-Stokes equation:

^{3}), p = blood pressure (Pa), u = blood velocity vector (m/s), μ_{c} = effective dynamic viscosity (Pa-s). The applied boundary conditions are listed in

Boundary | Condition |
---|---|

Inlet | Pulsatile velocity with a peak velocity of 0.5 m/s at diastole and 0.1 m/s constant velocity at systole ( |

Outlet 1 (Larger Branch) | Gauge Pressure 13332 Pa |

Outlet 2 (Smaller Branch) | Gauge Pressure 13332 Pa |

Arterial Wall | Stationary, no slip, rigid and non-porous |

The geometry used for both the normal and corona affected artery is the same. The geometry is discretized into 23015 elements and checked for mesh refinement. Transient simulation of the discretized model is carried out for 50 number of time steps, each 0.01 s size. Simple pressure-velocity coupling method is used and second order discretization is applied for both pressure and momentum. Three different meshes were analysed for a bifurcating angle of 90°. The data for which is tabulated in

Mesh | Number of elements | Maximum wall shear stress (Pa) | Difference percentage (%) |
---|---|---|---|

Mesh 1 | 18734 | 5.9800 | — |

Mesh 2 | 23015 | 5.9960 | 0.2671 |

Mesh 3 | 26031 | 6.0052 | 0.1532 |

Mesh 2 with 23015 elements is selected for further simulation and analysis. The number of elements is a trade-off between accuracy and time required for analysis.

The Time-averaged Wall Shear Stress (TAWSS) of the model analysed is found out using the equation:

where, T is the total time elapsed during one pulse (0.5 s) and WSS is the Area Weighted Wall Shear Stress (Pa) during each time step under consideration. The Wall Shear Stress during the entire period of one pulse for a healthy artery is plotted in

The area under the Wall Shear Stress plot is found out using Origin graphing software and is substituted in

Using Ansys Fluent, the numerical model was analysed for one cycle of the pulsatile flow (0.5 s) and pressure gradient, velocity vectors, wall shear stress and TAWSS on the healthy and corona affected arteries were studied. The pressure distribution, velocity streamlines and wall shear stress at t = 0.05 s for the normal and corona affected arteries with a bifurcation of angle 90° are compared in

It is seen from the above plots that there is an increase in the velocity of blood flowing through the large branch in the case of the artery affected by Coronavirus during the systolic phase. The flow velocity in the smaller branch is less than that of a normal healthy artery during the systolic but greater during the diastolic phase. This is verified from the velocity streamlines too. The flow through the smaller branch decreases significantly during the pumping phase. It may lead to an acute shortage of oxygen and nutrients in the corresponding organs. The instantaneous velocity plots for Outlet 1 (left branch) and Outlet 2 are further plotted in

The uniformity of pressure distribution is affected in the diseased artery. Extended regions of high pressure, when compared to the normal healthy artery, are seen. The total pressure distribution across the arteries during one pulse is plotted in

The TAWSS of both arteries are found out as described in

The increase in wall shear stress can cause damage to the endothelial cells lining the blood vessels by increasing the chances of plaques and there by narrowing the lumen. The angles of bifurcation were changed to 30°, 60°, 75°, 105° and 120° for understanding the change in TAWSS the arteries get subjected to and is plotted in

It is seen that the TAWSS is maximum for an angle of bifurcation of 75°. The same trend is observed in the case of corona infected artery too. Along with the angular variations, the curvature of bifurcation may also affect the distribution of shear stresses along the artery wall. The complexity involved due to changes in curvature is not covered under the present study.

This study presented a 2D transient numerical simulation of arterial bifurcation in patients affected by Coronavirus. The viscosity changes in blood due to infection imparts considerable changes in the blood flow parameters such as velocity, pressure, and wall shear stress. Various blood parameters for varying angles of bifurcation from 30° to 120° for both normal and diseased conditions were analysed and validated using existing literature. The following conclusions are made after studying the transient blood flow in detail:

Bifurcations and junctions are prone to severe hemodynamic changes as a result of the increase in viscosity of blood. The outlet velocity towards the large branch is increased and towards the smaller branch outlet is considerably decreased during the systole.

The total pressure towards the larger outlet of the bifurcation is increased and towards the smaller outlet is decreased during the systole.

The Time Averaged Wall Shear Stress (TAWSS) for the corona affected artery is 10.4114 Pa at a 90^{0} angle of bifurcation as compared to 2.45002 Pa of the normal artery. Damages are caused to the endothelial lining of the blood vessel due to sustained higher shear stresses. It may lead to plaque formation and narrowing of the lumen leading to severe cardiac risks or destabilization of existing plaques, both leading to adverse health conditions.

It is seen that the TAWSS is maximum at an angle of bifurcation of 75° and has a value of 2.46076 Pa and 10.42542 Pa for normal and corona affected artery respectively.

Reproducible results are obtained using commercially available viscometers for measuring whole blood viscosity, which when combined with Computational Fluid Dynamics simulation results can provide more information to crucial health conditions of a coronavirus infected person. Further studies are to be carried out by modelling blood vessels from real-life cases, and with other specific viscosity models.