Journal of Mathematial Control Siene and Appliations (JMCSA) Vol. Journal 3 No. of Mathematial (July-Deember, Control 017), Siene ISSN and : 0974-0570 Appliations (JMCSA) Vol., No. 1, June 008, pp. 117-1 International Siene Press Effet of Various Parameters of RBC on Oxygen Conentration in Tissues Bimal Kumar Mishra *, Shailendra Singh Shekhawat ** Abstrat: In order to survive, humans have to extrat oxygen from the atmosphere and transport it to their ells where it is utilized for essential metaboli proesses. Tissues differ in their ability to withstand anoxia (lak of oxygen). Initially a lak of oxygen affets organ funtion but with time, irreversible damage is done (within minutes in the ase of the brain) and revival is impossible. In this paper, we studied the effet of RBC ells on oxygen onentration. It is observed that as the length of RBC, hematorit and distane between two RBCs get inreased; the oxygen onentration in tissue also inreases. It has been also investigated that as the veloity of RBC get inreased the onentration of oxygen in tissue dereases. Keywords: Oxygen onentration, RBC ell, Tissue, Miroirulation, Blood flow 1. INTRODUCTION The oupling of fluid dynamis and biology at the level of the ell is an intensive area of investigation beause of its ritial role in normal physiology and disease. Today, the fluid mehanis of ells and the effet of flows on biologial funtion are ative areas of researh, spanning the fields of biophysis, bioengineering, physiology, and biology. Interest in this field has been driven both by the rihness of the fluid-dynami phenomena observed and by their fundamental biologi importane. Oxygen is transported from the air that we breathe to eah ell in the body. Oxygen transport within the body is a multistep proesses. First, during respiration, oxygen is transported by onvetion to the lungs. Next, oxygen passes aross the lung apillaries by diffusion and binds to hemoglobin in red blood ells. Then the oxygen in these ells moves throughout the body by onvetion. One oxygen reahes the tissues, it diffuses through them and within the ells to the aerobi reations that produe energy in the form of adenosine tri phosphate (ATP). In the normal lung, the diffusion of oxygen into the blood is very rapid and is omplete, even if the ardia output is inreased (exerise) and the blood spends less time in ontat with the alveolus. This may not happen when the alveolar apillary network is abnormal (pulmonary disease). However, the ability of the lung to ompensate is great and problems aused by poor gas diffusion are a rare ause for hypoxia, exept with diseases suh as alveolar fibrosis. Blood flow is an integral part of oxygen delivery and the presene of RBCs inreases the resistane to flow [10, 11, 14]. Thus, the rheologial property of RBCs with varying size is important to know the effet on oxygen onentration in tissue. However, larger RBCs are more deformable than smaller ones. Differenes in RBC size are the result of hanges in RBC dimensions whih attended hanges in vessel size, to insure deformation and preserve oxygen onentration effiieny between RBCs and tissue. As a result RBC size has been a entral part of the miroirulatory system. A low ardia output, a low hemoglobin onentration (anemia) or low hemoglobin oxygen saturation will result in an inadequate delivery of oxygen, unless a ompensatory hange ours in one of the other fators. * Mathematis Group, Birla Institute of Tehnology and Siene, Pilani 333 031, India ** Computer Siene and Information Systems Group Birla Institute of Tehnology and Siene, Pilani 333 031, India 107
118 Journal of Mathematial Control Siene and Appliations (JMCSA) Alternatively, if oxygen delivery falls relative to oxygen onsumption the tissues extrat more oxygen from the hemoglobin. In spite of the great importane of oxygen; the stores of oxygen in the body are small and would be unable to sustain life for more than a few minutes. If breathing eases, oxygen stores are limited to the oxygen in the lung and in the blood. Intraellular energy prodution, a prerequisite for all physiologial ellular reations, is dependent on ontinual absorption of oxygen through respiration and transport from the alveoli and eventually into the mitohondria. The enormous quantity of the metabolized oxygen is utilized by the body for energy prodution as a vital ondition for sound ell funtions and regenerative proesses. Several models of the oxygen exhange through the blood gas barrier have been published [6, 7, 1]. In all these works, the oxygen partial pressure profile (PO ) along the pulmonary apillary was determined by means of iterative proedures. In fat, sine the mathematial desription of the oxygen transport proess from the alveoli to the apillaries is ompliated by the reversible binding between hemoglobin and oxygen, it is very diffiult to determine a losed form for the oxygen partial pressure profile along the pulmonary apillaries. As a onsequene, in omplex models of the human respiratory system [1, 3, 8] the oxygen partial pressure at the end of the pulmonary apillaries is generally assumed equal to the alveolar oxygen partial pressure profile. We try to derive expressions for the effet of length of RBC, hematorit, veloity of RBC and distane between two RBCs on oxygen onentration, whih will lead toward more physiologial explanation of the flow of blood in tissues.. MATHEMATICAL MODEL FORMULATION When oxygen reahes the tissue, equilibrium favors the dissoiation of oxygen from the hemoglobin due to the lower oxygen levels and higher onentration of dissolved oxygen levels in tissue. Oxygen transport from the oxygenated red ells to the tissues is not simply the opposite of the proess ourring in the lungs. Beause the kinetis of dissoiation differs from that of binding, and the tissue thikness is muh greater than the thikness of the alveolar epithelium and apillary endothelium. Oxygen delivery to tissues involves the dissoiation of oxygen from hemoglobin, the diffusion and onvetion of oxygen through the plasma, diffusion through the ells and tissue, and finally, the reation of oxygen in the mitohondria as part of aerobi metabolism. As we know that RBCs are indeed a enterpiee in the evolution of the miroirulation. It is onsidered apillary to be homogeneous for oxygen transport; but, in reality, the apillaries ontain disrete red blood ells (RBCs), and this disreteness (Length of RBC, veloity of RBC, Distane between two RBCs) will affet oxygen onentration in the tissue. Physiology of oxygen onentration in tissue is depited in Fig.1. Figure 1: Oxygen Conentration in Tissue In order to develop a model of oxygen transport to tissues, it is neessary to model a simplified organization of apillaries. The geometry of the problem is depited in Figure. 108
Effet of Various Parameters of RBC on Oxygen Conentration in Tissues 119 Figure : Geometry of the Problem In the present paper, to onsider the oxygen transport in the tissue, Krogh model is onsidered under the following assumptions [5, 9, 13]: i. The apillaries are arranged in an ordered array that an be represented by a entral apillary surrounded by a ylinder of tissue. ii. iii. The oxygen hemoglobin dissoiation reation is at equilibrium and ours uniformly throughout the blood. Oxygen release in plasma ours uniformly. iv. The mass transfer resistane offered by the ell free fluid layer and endothelium is negleted. v. Axial diffusion in tissue and blood is negligible. vi. Oxygen uptake in tissue an be represented by zeroth- order kinetis, and the update is uniform throughout the tissue. vii. Other reations of oxygen in tissue are negligible. Let us onsider, a steady state, no onvetion in the tissue and a mass balane on steady state. The one-dimensional oxygen diffusion and reation in the tissue yields the following relation [4]: DO d dco r RO r dr dr (1) Although oxygen onsumption generally follows Mihaelis-Menton kinetis, the reation rate is assumed to be zero order. This assumption is based on the small value of the Mihaelis onstant (K m = 0.67 µm to.7 µm) relative to average oxygen onentrations in tissue. That is, only when the oxygen onentrations are lose to zero, the rate deviate from zeroth order kinetis. At the apillary surfae, the tissue onentration is in equilibrium with the oxygen onentration in plasma: R = R, CO = CR () As is noted subsequently, the oxygen onentration in the apillary is, in general, a funtion of axial distane along the apillary. At R 0, however, there is a plane of symmetry between the two ylinders. Thus, the flux is zero: dco At r = R 0, DO dr = 0 (3) 108
10 Journal of Mathematial Control Siene and Appliations (JMCSA) The solution of equation (1) is where, A and B are onstants. ROr CO Aln r B (4) 4DO When the boundary onditions (from equations () & (3)) are applied, the onentration profile in the tissue is given as: CO 1 RO R o r R 4 ln r CR CRDO R0 R o R (5) The group o RO R 4CR DO represents a dimensionless reation rate and is denoted by. So, o RO R (6) 4CR DO Take * = r/r 0 and R * = R /R 0, equation (5) beomes CO CR * * r 1 ( r R ) ln R * * (7) Now, kl DO (8) From equations (6) and (8), we find = ROR0 4kCRL Ø (9) The relation between hematorit, length of RBC and distane between two RBCs [] is given by L Ht L z (10) Pelet number is defined as vz Pe (11) DO Now, from equations (5), (9), (10) and (11), we observe the effet of length of RBC, distane between two RBCs, hematorit and veloity of RBC on onentration of oxygen. 109
Effet of Various Parameters of RBC on Oxygen Conentration in Tissues 11 3. DISCUSSION It is found that RBCs are a primitive harater in blood flow. Thus, we are interested to find the link between RBC size and oxygen onentration in tissue. We hypothesize that RBC size effets the oxygen onentration in tissue. An analytial solution is used to onsider the effet of RBC on oxygen onentration from the apillary to the tissue. In order to get a physiologial insight into the effet of length of RBC, distane between two RBCs, hematorit and veloity of RBC on onentration of oxygen, the following physiologial values are taken: r* = r/r 0 = 0.10, R * = R /R 0 = 0.05, Ø = 1, k = 18 se 1, D0 = 6 10 6 m se 1 = 0.01 (a) (b) () (d) Figure 3 (a), (b), (), (d): Represents the Variation of Normalized Conentration of Oxygen against Length of RBC, Distane Between Two RBCs and Hematorit and Veloity of RBC Respetively We found a positive orrelation between length of RBC, distane between two RBCs, hematorit and the oxygen onentration in tissue. But we found a negative orrelation between veloity of RBC and the onentration of oxygen in tissue. From our study, we observe that as the length of RBC, distane between two RBCs and hematorit is inreased, the oxygen onentration in tissue also inreases (Fig. 3(a), Fig. 3 (b), Fig. 3()). Also as the veloity of RBC inreases, the onentration of oxygen in tissue dereases (Fig. 3(d)). REFERENCES [1] Artioli E., Avanzolini G., Barbini P. and Gnudi G., A four-parameter linear model for analyzing ardio-respiratory data in post-operative ardia patients, Med. Eng. Phys., 16: 484 91, 1994. [] Cees Bost and Louis Hoofd, The effet of separate red blood ells on apillary tissue oxygenation alulated with a numerial model, IMA Journal of Mathematis Applied in Mediine & Biology, 13: 59 74, 1996. 110
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