Department of Chemical EngineeringUniversity of California, Los Angeles

2003 AIChE Annual Meeting

San Francisco, CANovember 17, 2003

Nael H. El-FarraPanagiotis D. Christofides

James C. Liao

Computational Modeling & Simulation Computational Modeling & Simulation of Nitric Oxide Transport-Reaction in of Nitric Oxide Transport-Reaction in

the Bloodthe Blood

• Nitric oxide (NO) : active free radical

Immune response

Neuronal signal transduction

Inhibition of platelet adhesion & aggregation

Regulation of vascular tone and permeability

• Versatility as a biological signaling molecule

Molecule of the year (Science, 1993)

Nobel Prize (Dr. Ignarro, UCLA, 1998)

• Need for fundamental understanding of NO regulation

Distributed modeling

IntroductionIntroduction

• Complex mechanism:

Release in blood vessel wall

Diffusion into surrounding tissue

Blood pressure regulation

Diffusion into vessel interior

Scavenging by hemoglobin

Trace amounts can abolish NO

• Paradox: how can NO maintain its biological how can NO maintain its biological

function ?function ?

Barriers for NO uptake

NO Transport-Reactions in BloodNO Transport-Reactions in Blood

Vessel wall

Barriers for NO Uptake in the Blood

(1)(2)

(3)(4)

Previous Work on Modeling NO Transport

• Homogenous models:

Blood treated as a continuum

e.g., Lancaster, 1994; Vaughn et al., 1998

• Single-cell models:

Neglects inter-cellular diffusion

e.g., Vaughn et al., 2000; Liu et al., 2002

• Survey of previous modeling works (Buerk, 2001)

• Limitations:

Population of red blood cells (RBC) unaccounted for

Cannot quantify relative significance of barriers

Present Work

• Objectives:

Develop a detailed multi-particle model to describe NO transport-reactions in the blood

Use the developed model to investigate sources for NO transport resistance

Boundary layer diffusion (RBC population)

RBC membrane permeability

Cell-free zone

Quantify barriers for NO uptake

(El-Farra, Christofides, & Liao, Annals Biomed. Eng., 2003)

Abluminal region (smooth muscle)

Endothelium (NO production)

R

R+

Physical Dimensions:

R=50 m, =2.5 m

Geometry of Blood Vessel

Blood vessel lumen

• Steady-state behavior:

Small characteristic time for diffusion/reaction

(~10 ms)

• NO diffusivity independent of concentration or position

NO is dilute

• Isotropic diffusion

• Convective transport of NO negligible

Axial gradient small vs. length of region emitting NO

• Hb is main source of NO consumption

Negligible reaction rates with O2

Modeling Assumptions

Surrounding tissue (Abluminal region):

Vessel wall (Endothelium):

Vessel interior (lumen):

• Governing Equations:

Mathematical Modeling of NO Transport

Mathematical Modeling of NO Transport

• Continuum model (Basic scenario):

Spatially uniform NO-Hb reaction rate in vessel

• Particulate model:

Barriers for NO uptake:

Red blood cells (infinitely permeable)

RBC membrane permeability

Cell-free zone

• Transport resistance analysis

• Numerical solutions thru finite-element algorithms

Adaptive mesh (finer mesh near boundaries)

Overview of Simulation ResultsM

odel Com

plexity grows

• NO distribution in blood vessel and surrounding tissue

Simulations of Continuum Model

Radial variations of mean NO concentration

Simulations of Continuum Model

• Hemoglobin “packaged” inside permeable RBCs

Inter-cell diffusion (boundary layer)Abluminal region

Extra-cellular space

Intracellular space

Endothelium

Effect of Red Blood Cells

Simulations of Basic Particulate Model

• NO distribution in blood vessel and surrounding tissue

• Blood hematocrit determines number of cells

~ 45-50% under normal physiological conditions

Simulations of Basic Particulate Model

Radial variations of mean NO concentration for homogeneous & particulate models

Effect of RBC Membrane Permeability

Abluminal region

EndotheliumIntracellular space

Extra-cellular space

Simulations of Particulate Model+Membrane

Radial variations of NO concentration for homogeneous, particulate & particulate+RBC membrane models

Simulations of Full Particulate Model

NO concentration profiles for homogeneous, particulate, particulate+membrane, & full particulate models

• Computation of mass transfer resistance

Quantifying NO Transport Barriers

Relative Significance of Transport Barriers

• Fractional resistance is a strong function of blood hematocrit:

Membrane resistance dominant at high Hct.

Extra-cellular diffusion dominant at low Hct.

Extracellular diffusion

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

45% 25% 15% 5%

Blood hematocrit

RBC membrane

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

45% 25% 15% 5%

Blood hematocrit

Acknowledgements

• Mathematical modeling of NO diffusion-reaction in blood

• Diffusional limitations of NO transport:

Population of red blood cells

RBC membrane permeability

Cell free zone

• Relative significance of resistances depends on Hct.

• Practical implications:

Encapsulation of Hb in design of blood substitutes

• NSF and NIH

Conclusions

ECEC ECEC

Stationary Flow

RBC RBC

Effect of Blood Flow

• Creates a cell-depleted zone near vessel wall (~2.5 m)