oxygen transport though cappillary with the help of RBC

this is presentation about oxygen transport with red blood cell modelling

1. INSTITUTE OF TECHNOLOGY FACULTY OF ELECTRICAL ENGINEERING DEPARTMENT OF BIOMEDICAL ENGINEERING Course: Physiological modeling and Simulation Title : A dynamic model of oxygen transport from capillaries to tissue with moving red blood cells By TIRUSEW ENGIDAW 1 8/25/2024

2. Contents Introduction Methods Results and discussion Conclusion 8/25/2024 2

3. 1. Introduction Oxygen transport from capillaries to tissues is a fundamental process in the human body for cellular respiration and energy production. This process is driven by the movement of oxygen from RBCs within the capillaries into the surrounding tissues. The primary method for delivering oxygen to tissues is through diffusion from micro vessels. The presence of RBCs leads to a highly heterogeneous blood flow in the microcirculation. This is also essential for meeting the varying energy demands of tissues. 8/25/2024 3

4. 1.1. problem statement Inability to Capture Hemodynamic Effects : most existing models do not reflect the dynamic nature of blood flow and the interactions between RBCs and the static domain. The fixed frame of reference used in many existing models experiences to numerical diffusion, which leads the leakage of hemoglobin out of RBCs. Many traditional models are inflexible regarding geometry, as they assume a constant radial cross-section along the flow direction. This rigidity prevents the simulation of local capillary dilations and other variations observed in vivo . 8/25/2024 4

5. 1.2. Aims and objectives Aim To develop and validate a dynamic model of oxygen transport from capillaries to tissue that incorporates the movement of red blood cells. T o improve the accuracy of simulations by using a conical geometry for the capillary model. 8/25/2024 5

6. Cont. Objectives The primary objective is to develop a comprehensive model that accurately simulates the transport of oxygen in the microcirculation. Q uantify how blood flow heterogeneity affects oxygen partial pressure (PO2) in tissues. Build models that considers the dynamic nature of blood flow and the interactions between RBCs. 8/25/2024 6

7. Cont. To simulate how oxygen is released from RBCs and transported to surrounding tissues during various physiological scenarios. I nvestigates the effects of varying hematocrit levels on oxygen tension and delivery. V alidating the model against experimental data, particularly focusing on the relationship between intravascular PO2 and tissue oxygen levels. 8/25/2024 7

8. 2. Methodology This paper presents a novel computational model designed to simulate oxygen transport from capillaries with moving red blood cells (RBCs). Model Structure: The model is structured around four distinct regions: tissue, capillary wall, plasma, and red blood cells (RBCs). Oxygen transport and consumption are analyzed within this multi-region framework Oxygen Consumption: Oxygen is consumed exclusively in the tissue, while the capillary wall does not consume oxygen. 8/25/2024 8

9. Cont. The model assumes that oxygen is convicted by blood flow in both plasma and RBCs. The low solubility of oxygen in plasma means that most oxygen is bound to hemoglobin within RBCs. Mathematical Relationships: The model employs Henry's law to relate the concentration of dissolved oxygen (C) to its partial pressure (P). The conservation equation for oxygen is formulated in terms of these variables, facilitating the analysis of oxygen dynamics. 8/25/2024 9

10. Cont. Hemoglobin Dynamics : The reaction between oxygen and hemoglobin is described using the Hill equation. Which models the equilibrium curve between oxygen partial pressure and hemoglobin saturation. This approach simplifies the complex interactions into a one-step reaction for the four heme groups of hemoglobin. 8/25/2024 10

11. Cont. To model the reaction rates when oxygen and hemoglobin are in nonequilibrium We followed the approach of Clark function Kinetics of Oxygen Consumption : The model uses first-order Michaelis-Menten kinetics to describe oxygen consumption in the tissue. 8/25/2024 11

12. Cont. Boundary Conditions and Discretization : The model includes specific boundary conditions at the interfaces between different regions: E nsuring continuity of oxygen partial pressure and flux. The discretization of the model is performed in a Lagrangian frame of reference following the moving RBCs to simulate the dynamics of hemoglobin and oxygen transport. 8/25/2024 12

13. Cont. Simulation and Comparison : The model is designed to allow for easy comparison with experimental data This is achieved through a fixed computational domain for the capillaries and tissue While RBCs are modeled in a moving domain. 8/25/2024 13

14. 2.1. Methods and strategies Existing models for oxygen transport from capillaries to tissue generally employ two distinct approaches. The first class of models focuses on the tissue and does not represent individual RBCs. Instead, they employ a boundary condition at the capillary wall for accounting oxygen transport from the capillary. Most models use MTC that relates the PO2 drop from the RBC to the oxygen flux across the capillary wall. 8/25/2024 14

15. Cont. The second approach models intravascular oxygen transport in more detail and can be used to compute MTCs. Accurate MTC estimates require discrete RBCs to be modeled as opposed to a continuous hemoglobin solution. Most models with individual RBCs carry out computations in the frame of reference of the erythrocyte. 8/25/2024 15

16. Cont. New proposed model is based on a single equation of oxygen for all regions, that is: Where D is the diffusion coefficient and v the advection velocity Hemoglobin saturation is governed by the equation. Where DHb is the diffusivity of hemoglobin in RBCs 8/25/2024 16

17. Cont. Lagrangian frame of reference F or the fixed frame of reference motivated by previous work experiences flux boundary condition for hemoglobin out of RBC. The computational grid or mesh moves along with the RBCs, rather than staying fixed in place. This means that the mesh "follows" while RBCs move through the bloodstream. This approach ensure the no-flux boundary condition for hemoglobin out of RBC. 8/25/2024 17

18. Cont. Overlapping Meshes in axisymmetry The two meshes, Ω and Ωrbc, overlap in the region being simulated. This overlap allows for interaction between the moving RBCs (Ωrbc) and the stationary tissue and plasma (Ω). The model can efficiently handle the complex interactions between the RBCs and the stationary elements. The moving mesh adapts to the RBCs' position, while the fixed mesh remains unchanged, simplifying the computation. 8/25/2024 18

19. Cont. Hemoglobin Saturation (S): This value is initially calculated on the moving mesh Ωrbc, which represents the RBCs. Interpolation of S from Ωrbc to Ω: After computing S in the RBCs, this information is interpolated from the Ωrbc to the Ω. This step ensures that the fixed mesh has updated information on oxygen levels within the RBCs as they move through the domain. PO2: This represents the concentration of oxygen in the blood plasma, outside the RBCs. 8/25/2024 19

20. Cont. Interpolation of PO2 from Ω to Ωrbc: Similarly, the PO2 of oxygen, calculated on the Ω, is interpolated to the Ωrbc. This ensures that the RBCs (Ωrbc) have updated information on the surrounding oxygen levels . The interpolations is to accurately simulate the exchange of oxygen between the RBCs and the surrounding blood plasma. 8/25/2024 20

21. Cont. A xially symmetric geometry Blood vessels are typically cylindrical in shape (like arteries and veins) A nd the flow of blood through them can often be approximated as having rotational symmetry around the centerline of the vessel. This allows for simplification in modeling and computation . C one-shaped domain Instead of a cylinder, they employed a cone-shaped domain with different radii at the proximal (arteriolar and distal (venular) ends. 8/25/2024 21

22. Cont. Modeling blood flow in three dimensions (3D) can be computationally expensive. By assuming axisymmetry, the problem is reduced from 3D to 2D problem in the radial and axial directions. A cone-shaped domain is used to model in narrowing vessels, and asses how blood flow changes in response to varying vessel diameters. 8/25/2024 22

23. Result and discussion Oxygen Tensions Measurement The study simulated oxygen tensions inside capillaries and surrounding tissue using two-photon phosphorescence lifetime microscopy in rodent models. The model successfully captured the dynamics of oxygen transport in the microcirculation. 8/25/2024 23

24. Cont. Quantification of PO2 This review characterized intracapillary oxygen tensions using three key metrics: RBC PO2 (maximal oxygen tension at the erythrocyte membrane) And mean PO2 (average between two erythrocytes), and inter-RBC PO2 (minimal PO2 between two RBCs). The average EAT (Erythrocyte-Associated Tension) amplitude was found to be 29.7 mmHg, with RBC PO2 at 50.8 mmHg, inter-RBC PO2 at 21.1 mmHg, and mean PO2 at 27.4 mmHg. 8/25/2024 24

25. Cont. The graph shows instantaneous longitudinal profiles on the capillary centerline And at various radial distances from the capillary wall. 8/25/2024 25

26. Cont. Longitudinal Variations The study observed that the maximal PO2 in plasma occurs at the rear side of the RBC membrane. with significant variations in intracellular PO2 along the capillary length. In RBCs close to the arteriolar end of the domain, the intracellular PO2 variation exceeds 30 mmHg and decreases to 15 mmHg at the venular end. These strong intravascular oxygen variations extend to the nearby tissue 8/25/2024 26

27. Cont. Instantaneous linear density fluctuations: affects inter-RBC PO2 is clearly showed. Since short RBC spacings cause higher inter-RBC PO2 values, The EAT amplitude drops when the instantaneous linear density increases. I ndicating a complex relationship between RBC spacing and oxygen transport Geometric Considerations result The results indicated that a cone-shaped geometry for capillaries provided a better match with experimental data compared to a cylindrical geometry. 8/25/2024 27

28. Cont. Figure shows time-averaged oxygen partial pressures for the cone-shaped geometry and for a cylinder with equal tissue volume. Since RBC PO2 declines faster than inter-RBC PO2, the EAT amplitudes also decrease along the capillary. 8/25/2024 28

29. Cont. A is simulated PO2 values collected during 3 s at 30 m from the capillary entrance. Circles: RBC PO2; triangles: inter-RBC PO2; crosses: difference between inter-RBC PO2 and tissue PO2 at 10 m from the capillary wall. B is measurement: Top bar: RBC PO2; shaded bar: inter-RBC PO2. 8/25/2024 29

30. Cont. This figure also shows the difference between inter-RBC PO2 and tissue PO2 at 10 m from the capillary wall as a function of linear density. For linear densities lower than 0.25, this difference stays below 2.0 mmHg. For high hematocrit values, this gap exceeds 10 mmHg. Thus their results indicate that inter-RBC PO2 may significantly exceed tissue PO2 for high linear densities. 8/25/2024 30

31. Discussion Oxygen transport from a capillary with moving RBCs to the surrounding tissue has been simulated in an axisymmetric cone-shaped geometry. Their average measured EAT amplitude was 33.5 mmHg, and similar amplitudes were obtained in the first section of our sample capillary. At 30 m from the capillary entrance, the simulated EAT amplitude was 33.6 mmHg. Close to the venular end, RBC PO2 was lower due to oxygen consumption in the tissue, which gave rise to smaller EATs (25 mmHg). Therefore, their average EAT amplitude of 29.7 mmHg over the nine sampled positions is slightly lower than other works. 8/25/2024 31

32. Cont. The relationship between intracapillary oxygen tensions and tissue PO2 was also examined. Moreover, the influence of hematocrit fluctuations was not examined in this part of the experiment. Therefore, their simulations indicate that inter-RBC PO2 is similar to tissue PO2 only close to the capillary or at low linear densities. Since concentration gradients drive molecular diffusion They suggest that inter-RBC PO2 is on average higher than tissue PO2 far away from capillaries 8/25/2024 32

33. Cont. Their simulation setup with RBCs moving through a fixed capillary allows the computation of longitudinal oxygen gradients. Motivated by the fact that capillary segments with high oxygen tensions can supply a correspondingly large tissue volume, They used a cone-shaped geometry similar to other work. They compared results obtained with this geometry and with a simple cylindrical domain to the data, longitudinal PO2 variations were measured in individual capillaries. 8/25/2024 33

34. Cont. Even though model for oxygen transport was applied to a simple axisymmetric geometry. The numerical algorithm is independent of the domain topology and can be extended to realistic capillary networks provided velocities of single RBCs are known. This can be achieved by coupling their method with a detailed model of RBC transport. This combined approach will remove the need for separately computed mass transfer coefficients and is suitable for investigating unsteady scenarios. 8/25/2024 34

35. Conclusion In conclusion, we have developed a new model of oxygen transport from capillaries with moving RBCs based on overlapping grids. We successfully validated it against experimental data acquired in the rodent brain. EATs and longitudinal gradients of PO2 could be reproduced using a cone-shaped geometry. Instantaneous variations of hematocrit were shown to cause considerable fluctuations of oxygen tension in the tissue. 8/25/2024 35

36. Thank you! 8/25/2024 36