Modelling Blood Flow in the Veins - Research Paper Example

MODELLING BLOOD FLOW IN THE VEINS (Using Non-Newtonian Model Blood Flow in the Veins) By Student’s Name Course Tutor Institution Department Date Contents 1.0ABSTRACT 3 2.0INTRODUCTION 3 3.0 PREVIOUS INVESTIGATIONS 5 4.0PROPOSED RESEARCH 7 4.0.1 Using Non-Newtonian Model Blood Flow in the Veins 7 4.1 Literature Review 7 4.1.1 Hemorheology 8 4.1.2 Blood Function 9 4.1.3 Blood Constituents 9 4.1.4 Non-Newtonian Flow 10 4.1.5 Rheological Parameter of Blood 11 4.2 Non-Newtonian Model 12 4.2.1 Power Law Model 13 4.2.2 Casson Model 14 4.2.3 Effect of Non-Newtonian Flow 14 5.0 RECOMMENDATION 15 6.0 CONCLUSION 15 1.0 ABSTRACT The paper is about modelling the movement of blood in the veins with the aid of the non-Newtonian models to explain the concept through research. Most of the previous research carried out on the flow of blood in the veins was driven by different medical conditions which require the knowledge of fluid flow. It will also detail the dynamics of the fluid in the veins (blood) which play a significant role in the modelling. Under this topic of discussion blood, rheology and its property (non-Newtonian) will need explanation and the different modelling methods presented which takes a look at it as a mixed liquid. The structure of the veins and characteristics of the blood help in the understanding of the dynamic flow of blood in the veins under low pressure towards the heart. 2.0 INTRODUCTION Caro et al. (1978, pg 12) argue that the modelling methods gave rise to the view of blood circulation within the different vessels. One of the first scholars to discover this was Harvey who announced the presence of a cardiovascular circulatory system strongly. The author continues to argue that the discovery took place almost half a century before the light microscope invention took place. Weeks and Alcamo (2008, pg. 40) detail the history of the first light microscope which was held in the year 1674 by Anton Van Leeuwenhoek who provided the first working light microscope which until now plays a significant role in defining blood flow and circulation. The authors continue to outline others who made a contribution in that era, Marcello Malpighi who coincidentally made the same discovery in the year 1675 and often credited with the discovery of the capillary vessels by the use of a microscope. Costanzo (2010, pg. 123) explains how the pressure flow of blood in veins is lower when compared to arteries and its structure plays a significant role in the completion of the flow back to the heart for re-oxygenation. Veins are transportation vessels of blood with relatively thin walls and a valve defined as flap-like, with a projection (of the valves) oriented inwards to its lining. Mollica et al. (2007, Pg. 194) detail the functions of the valves which involve the prevention of backflow of blood in the vein. Together with the skeletal muscles around the vein, the valves are also important in the flow of blood because of its property of opening and closing during blood flow. Costanzo (2010 pg. 100) details the mean diameter of the vein approximated at 5.0 millimetre and its mean wall thickness at 0.5 millimetres and are important in the determination of the resistance of blood flow. The author continues to detail these structural characteristics to help in the understanding of the non-Newtonian properties of blood which is a major factor in the modelling process. Using non-Newtonian model has an advantage over other methods because the structure of the vessel and the liquid characteristics are the main factors to consider in the absence of external factors. 3.0 Aims and Objectives The primary research goal is to understand better why the Non-Newtonian model is best suited to explain blood flow in the veins due to lack of pressure from the heart, unlike its counterparts, the arteries, which its flows require the need of the pressure generated by the heartbeat. One of the primary objectives of the research is to find out the non-Newtonian property of blood and whether these characteristics can be utilised to understand further how blood flows in the veins. Another aim includes the finding out of other relevant models which are used but at the same time find their weaknesses. 3.0 PREVIOUS INVESTIGATIONS Over the years a model is known as micropolar fluid, a proposal by Eringen (1966, pg. 14) in the sixties, has been used to explain blood flow. It borrowed most of its characteristics from the fluid dynamic models and termed as an extension of the fluid dynamic model. The assumptions made on the model include a medium which is continuous but also considers micro-rotation (denoted as w) of the molecules which is not the same as the vorticity of the flow in the vessel. When micro-rotation vector occurs, there is the formation of coupled stresses and anti-symmetric stresses. It, therefore, leads to the model needing two constitutive equations which include the equation for shear stress for a continuous medium and the second the couple stress. Many papers are available in texts, and electronic which try to explain the modelling of blood using this technique, and some of the considerations include; the flow of blood described as pulsatile and steady, coefficients of the micropolar materials by Bugliarello and Sevilla and the pulsatile blood flow phenomena concerning the blood vessels hydraulic impedance. The results from different experiments prove that this model is suitable for small vessel flow and mean shear rate. It not only proves not essential in the vein flow circulation but also lacks the aspect of its right structure to show the blood flow phenomena clearly. Another common model is the mathematical model argued by Ottesen and Danielson (2000, pg. 29) which establishes blood flow from mathematical formulas and numerical methods which are too complex. The approach is complex in its computational domain and looks at the vessels nature regarding it deformability and a multi-scale approach utilised to handle the different scales. A zero-dimension of the domain (spatial) is the result of the combination of a three-dimension model which involves the modelling of the vessels walls and the fluid, one dimension model which averages the cross-sectional area of the vessel and an average of the lumped up perimeter model. Although all the above factors put into consideration, most models under the mathematical model utilise the one-dimensional model termed as the model that plays an important role. One of the best examples of the one-dimensional model derived from the proposal by Schaaf and Albrecht. The method mainly focused on the arteries, and the results obtained from the method of characteristics. A more sophisticated model of the already established models by the two previous scholars was the one presented by Avolio. He introduced a complete one-dimensional model which also looked at arteries which comprised of arterial segments adding up to 128. One of the most successful one-dimensional models was proposed by Stergiopulos et al. which paved the way for many types of research on the theoretical model of blood flow. Matthys et al. is another scholar who successfully used the one dimension model to better help in the understanding of the models. All of the above scholars have gone to great lengths to use a mathematical model to show blood flow mainly in the arteries because of its original walls which allow some of the assumptions under different sub-models to show blood flow. Some of the specific one-dimensional models include the Venous system modelling and the closed-loop model of the cardiovascular system. 4.0 PROPOSED RESEARCH 4.0.1 Using Non-Newtonian Model Blood Flow in the Veins It is the best method to utilise in the blood flow modelling of the veins because its study encompasses a wider range of vessels regarding structure and size, as compared to previous studies. One of the shortcomings of the other models includes the limited research information on the characteristic of blood as a non-Newtonian fluid which is in agreement with Chhabra and Richardson (2008, pg. 32). The authors argue that most of the previous models make an assumption on the characteristic of blood (as a Newtonian Fluid) which produces lower accuracy in modelling as compared to the Non-Newtonian Model which considers the actual nature of blood in a confide tube. Even though many scholars tries to incorporate the non-Newtonian characteristic of blood in their research, they fail to fully employ the scientific aspect of the Non-Newtonian Model using Cason Model and Power Law Model. Other models such as the numerical model depend on forces such as pressure from the heart and the vascular walls of the vessels, especially arteries, to explain the modelling of blood flow. Most of the previous research are based on the modelling of arterial systems with respect to the forces within it and the elasticity of the thick walls. 4.1 Literature Review The struggle to understand blood flow inspired the modelling of the vascular systems to understand best vessel and blood-related diseases and how to stop them. One of the biggest contributors to the research was by Popel and Johnson (2005, pg. 58) who reviewed the important experiments and theories that co-relate to the haematology and the microcirculation of the blood mechanically. The mechanics of the research involved the flow of red blood cells, found in blood, through suspension of narrow tubes, which have a similarity to the movement in vein system, capillaries, and arteries. Takashi (2013, pg. 20) in his analysis theorised the investigation of the non-Newtonian flow model with the help of mathematical models through the power law model. Power law model is relevant for the determination of velocity flu, flow and wall shear stress. The factors previously mentioned are critical to the establishment of flow and the general recirculation process. Collins and König (2012, pg. 69) also contributed to the non-Newtonian model flow of blood through the generalization of the power law and the elastic capability of the vascular walls and its characteristic of deformability. The author also argues on the mathematical methods that aid in the design of non-Newtonian models through research in non-Newtonian fluids which follow the feature of the power law model and at the same time addressing the issue of shear in the blood vessels. The above scholars set the pace for the research together with others who add value to the already explained data. 4.1.1 Hemorheology Henry et al. (2011, pg. 747) define hemorheology as the branch of science which looks into the mechanics of blood flow and the different processes within its flow. One of the processes involves the deformation of the blood as it flows due to the constituent of the blood The study of hemorheology consists of the microscopic and macroscopic inspection of blood and its properties. 4.1.2 Blood Function Blood movement around the body is for the sole purpose of oxygen exchange, nutrient transportation, tissue repair and waste removal (carbon dioxide). The heart and the mechanical properties of the blood are responsible for its movement 4.1.3 Blood Constituents Blood, as described by (Popel and Johnson, 2005 pg. 47), states that blood constitutes of concentrated, suspended matter which comprises of the red blood cells (RBC) or commonly referred to as erythrocytes, white blood cells (WBC) or commonly known as leukocytes and the blood platelets. Plasma, one of the constituents of blood, comprises of water mostly and plays a major role in the transportation of dissolved substances. The red blood cells derive their name from the red colour of haemoglobin which is highly concentrated in the erythrocytes together with the mineral elements and water. Kawthalkar (2013, pg. 489) describes these cells function to mostly involve the exchange and transportation of oxygen and carbon dioxide in the different blood vessels. The concentration of haemoglobin significantly affect the blood rheological properties and commonly known as Hematocrit (Ht). The Leukocytes on the other side are vital tools in fighting infection in the body through the destruction of bacteria and virus and the formation process of antibodies. The effect of the white blood cells on the blood rheology is negligible, except in the case of minuscule vessels. Moore et al. (2016, pg. 50) discuss platelets, commonly known as thrombocytes, have a composition of a third of its total comprising of solid blood element. They are round, tiny and lack a nucleus because they are not intact cells but a cellular fragment. Platelets are also responsible for stopping bleeding in the case of an injury or open wound. 4.1.4 Non-Newtonian Flow Chhabra and Richardson (2008, pg. 15) describe the non-Newtonian flow and define it as the fluid flow that does not comply with the characteristics of a Newtonian fluid, that is, it has the typical shear thinning as it flows. The above diagram represents the flow of fluid in a confined tube with parallel plates during a non-Newtonian flow. F is the force and if applied gives the following equation: () Under the condition of a steady flow as is in the veins, and as it continues to decrease then the equation becomes The basic equation for the representation of a non-Newtonian Fluid is given by: Η in the above equation represents apparent viscosity, a function of shear rate represented by: The above basic equations create a background for the research proposal of using non-Newtonian models together with the properties bound to it. To understand blood flow on its non-Newtonian characteristics of a fluid, it is necessary at rheological parameter of blood as a background to the model. 4.1.5 Rheological Parameter of Blood The heart is responsible for the energy found in the blood from the each beat. A portion of the energy is either stored in the blood or dissipated as rearrangement, orientation and the deformation of cells take place. Rheological parameters as argued by Chhabra and Richardson (2008, pg. 7) detail the physically interpreted features of blood into an assessment of its viscosity. It is an evaluation of the dissipated energy rates from the processes of cell deformation and sliding. Another physical assessment is the elasticity which defines the primarily the elastic storage of energy from the red blood cells kinetic distortion. These two factors, viscosity, and elasticity are important in the determination of the pressure require during blood flow. 4.2 Non-Newtonian Model The behaviour of blood under this model is defined to have the characteristics of shear thinning non-Newtonian behaviour. Many models try to explain this behaviour and in the process have to take into account that the blood viscosity is much higher at low strain rates as compared to high strain rates. Signer, Kee, and Chhabra (1999, pg. 254) argue that the models do not only show viscosity but also the different ranges of strain rates which showcase the transition of blood from high viscous state to low, ∂μ/∂γ Read More