RBCs or erythrocytes are the most common cellular constituents of blood, and they occupy more than 99% of the particulate matter in blood and 40 to 45% of blood by volume.

Blood is a body uid that is responsible for vital tasks in the human body. The functions of the blood include transport of nutrients, oxygen and waste products such as carbon dioxide out of the organism to expulsion in the kidneys, lungs or spleen; immune function; distribution of heat throughout the organism and blood clotting. The average human has about 4.5-6 liters of blood in his or her body, which gures out approximately 6-10% of his or her body weight.

Blood is composed of blood elements (RBCs, WBCs and platelets) suspended in a liquid. Blood components can be seperated from blood plasma using centrifugal force. The cellular components typically make up about 45% of the whole blood volume, and the liquid as the major part of the blood, called blood plasma makes up 55%. The liquid consists of dissolved proteins, such as serum albumins, brinogen, hormones and nutrients. Below the plasma layer is the bu y coat layer, which consists of WBCs or leukocyte and platelets or thrombocytes, and makes up less than 1% of the total blood volume.

WBCs can be di erentiated into Lymphocytes, Monocytes, Granulocytes including Neutrophil, Eosinophil, and Basophil. The remaining part of the whole blood contains RBCs which make up around 45% of the blood. The volume of RBCs in a donor sample of the blood is named as hematocrit (Ht). Ht values change depending on sex, environment, and health status – in men, the value ranges about 44 5% of the blood volume, and in women about 40 4% of the blood volume. Table 1 presents information on the content of blood.

Composition and content of blood

Red Blood Cells (RBCs)

RBCs or erythrocytes are the most common cellular constituents of blood, and they occupy more than 99% of the particulate matter in blood and 40 to 45% of blood by volume. Mature RBCs are anucleate cells established by a membrane enclosing the uid and cytoplasm. The purpose of RBCs is to deliver oxygen to tissues and CO2 back to the lungs. RBCs travel more than 200 km in the circulatory system during their around 100 days life span. 1 mm3 or 1 l of blood contains 4 to 6 million RBCs.

In their resting state, RBCs have a biconcave shape with a typical diameter of 6-8 m and a thickness of 2-3 m. Evans and Fung observed that an average-sized RBC can more accurately be described by a biconcave-shaped model (see Figure 1.2).

RBCs membrane is a complex structure composed by a phospholid bilayer, an internal cytoskeletal network and membrane proteins (see Figure 2). The membrane and lipid layer skeleton help to maintain the biconcave shape of the cell at rest. Brie y, the lipid layer provides with the membrane its resistance against bending while the cytoskeleton provides with the membrane its resistance against shear deformations.

RBCs are known to exist in other cell structures than the biconcave shape due to genetic or pathological conditions. Figure 3 represents a sequence of distinct subtypes of RBCs.

Cross sectional shapes of the average RBC and other cellular infor mation as reported by Evans and Fung

Schematic representation of the RBCs membrane taken from

2 D cross sections of Discocyte Stomatocyte Echinocyte RBCs taken from

Anemia

Anemia is a manifestation of the insu cient capacity of oxygen carrying RBCs in the body’s organ [44]. It has numerous causes including iron de ciency, nu-tritional de ciencies for folate, vitamin B12, chronic in ammations and parasitic infections which are rather straightforward to treat and cure, to rarer genetic causes, such as sickle cell or beta thalassemia diseases, which result in chronic anemia that requires frequent monitoring.

World Health Organization Global Database on Anemia shows that over two billion people su er from anemia globally [14], more than ten million people die of all kinds of diseases caused by anemia every year. The report provides a comprehensive and comparable assessment of global anaemia prevalence and number of individuals a ected. Anemia have an e ect on 1.62 billion people (95% Con dence Interval (CI): 1.50{1.74 billion), which corresponds to 24.8% of the population (95% CI: 22.9{26.7%). The highest occurence is in preschool-age children (as 47.4%, 95% CI: 45.7{49.1%), and the lowest occurence is in men (as 12.7%, 95% CI: 8.6{16.9%). However, as outlined in Table 1.2, the population group with the greatest number of individuals a ected is non-pregnant women category (as 468.4 million, 95% CI: 446.2{490.6 million).

Global anaemia prevalence and number of individuals a ected in the worldwide

Sickle Cell Anemia

Sickle Cell Anemia, an inherited RBC disorder, was de ned as the rst molecu-lar disease [54]. The simple point mutation in beta globin generates a Hb molecule that polymerizes under low oxygen tension changing the discoid morphology of the RBC to the shapes that gave the disorder its name, an altered the cell rigid-ity, altered cell surface, adhesion, a short life span, and intravascular hemolysis and in ammation [6, 7, 33, 55{58] (see Figure 1.7). The polymerization of de-oxygenated hemoglobin S (HbS) promotes RBC sickling. Hence, the process of morphology change (sickling) of RBC from sickle cell patients (SS-RBCs) is the fundamental pathophysilogical hallmark of the vasculopathy, and organ damage that de nes sickle cell disease (SCD).

Kaul et al. revelaed four subtypes of sickle RBCs in terms of their rheological and hemodynamic characteristics [59]. Figure 1.8 presents four subtypes of sickle RBCs as I = reticulocytes, II = discocytes, III = dense discocytes and IV = ir-reversibly sickled cells under oxygenated and deoxygenated conditions. However, throughout this thesis, we have characterized the RBC morphological pro le as discocyte, sickle, elongated, as well as irregularly or abnormal RBC shaped cells, including echinocytes, holly-leaf, and granular structures of SS-RBCs upon de-oxygenation, obtained from IFC’s bright- and dark- eld data [12].

Sickled RBCs have reduced deformability and increased adhesiveness to the vascular wall, hence triggering frequent and recurrent vaso-occlusions. Although sickling is a reversible process upon re-oxygenation, RBC deformability remains reduced even when sickle cells are normally oxygenated. This reduction in RBC deformability under normal oxygen levels can be due to the persistence of HbS polymers, the low solubility of HbS and RBC membrane damages. The decreased deformability and increased fragility of sickled RBCs are at the origin of the en-hanced hemolysis in SCD patients [60{63]. Perhaps more importantly, the rate by which sickling takes place is signi cant. A decrease in the rate of polymerization-induced sickling is thought to have a profound e ect on the clinical outcome of patients.

Optical Modeling Techniques of RBCs

The analysis of a light scattering pattern by a single biological cell has been re-ported by using various electromagnetic numerical methods such as Mie-series [88], physical-geometric-optics [89], discrete-dipole [25] approximations, anoma-lous di raction [90], Lorentz-Mie [91], and discrete sources method [92] theories, nite-di erence time-domain (FDTD) [93], T-matrix [10], multilevel fast multi-pole algorithm [94] and integral equation methods. Surface integral equations are one of the promising methods in the solution of electromagnetic scattering problems. Scattering cross section values of a single biological cell pro-vide essential information on the morphological properties of the cell. RBCs are assumed to be spherical, elliptical or biconcave, and their electromagnetic prop-erties are considered as homogenous, depending on the method. Previous studies have mostly relied on the simulation of healthy RBCs with typical shapes, and have paid less attention to pathological RBCs with deformed shapes.

Boundary integral equation (BIE) techniques are one of the most popular computational tools in the scattering of waves by 2-D homogeneous dielectric bodies, being more economic than volume IE. The spurious eigenvalues are absent for the Muller Boundary Integral Equation (MBIE), which is a pair of coupled second-kind IEs for the eld components tangential to the scatterer contour [99, 100]. In this thesis, a boundary integral equation method (MBIE), which enables fast and e cient procedures for spherical and non-spherical biological particles with a Nystrom-type meshless discretization, was used to solve biological cells in health and disease conditions [12]. In our case, when the boundary is smooth and has a regular 2 -periodic parametrization, the Nystrom-type discretization [99, 101{103] using trigonometric approximation of integrand functions is more convenient, as it leads to exponential convergence of numerical solution.

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