Margination of a leukocyte in a model microvessel

Size: px

Start display at page:

Download "Margination of a leukocyte in a model microvessel"

Priscilla McKinney
5 years ago
Views:

1 Margination of a leukocyte in a model microvessel Jonathan B. Freund Mechanical Science & Engineering University of Illinois at Urbana-Champaign J. B. Freund p.1/40

2 Inflammation Response Leukocyte (white cell) recruitment to endothelium typically in post-capillary venules ( 10µm to 100µm) Transport & Margination Firm Adhesion Rolling Capture 30µm 7µm followed by emigration between endothelial cells J. B. Freund p.2/40

3 Inflammation Response Usually it s a good thing: fighting infection, etc. J. B. Freund p.3/40

4 Inflammation Response Usually it s a good thing: fighting infection, etc. but it can be a bad thing... Rheumatoid arthritis Multiple sclerosis Atherosclerosis Ischemia reperfusion J. B. Freund p.3/40

5 Inflammation Response Biochemistry/molecular biophysics well studied Binding in two stages rolling capture stage by selectin binding chemoattractants from endothelium activate integrin immobilization by integrin binding Selectin Integrin (Springer, 1995) J. B. Freund p.4/40

6 Hydrodynamics Multiple stages involve hydrodynamics Margination the transport of the leukocyte to the wall not a simple Stokes flow effect leukocytes are rigid (symmetric) must involve multi-body flow dynamics Adhesion must resist flow interactions with red cells probably important J. B. Freund p.5/40

7 Hydrodynamic Effects On-wall probability sensitive to shear rate (flow rate) Abbitt & Nash (2001) in vitro: channel flow Firrell & Lipowsky (1989) in vivo: rat mesentery Mechanism? J. B. Freund p.6/40

8 RBC Aggregation At low strain rates, RBCs aggregate (in athletic species) RBC aggregation augmentation promotes margination/adhesion Pearson & Lipowsky (2000), in vivo experiments, rat mesentery Abbitt & Nash (2003), in vitro experiments Is aggregation necessary? Direct observations do not suggest significant agg. needed But 50kDa dextran (Dx50), which inhibits aggregation, still decreases adhesion (Pearson & Lipowsky 2000)??? J. B. Freund p.7/40

9 Background: the F-L effect Key microcirculation feature: Fåræus-Lindqvist effect RBCs cluster toward vessel center Reduced net resistance relative to Poiseuille flow J. B. Freund p.8/40

10 Background: the F-L effect Key microcirculation feature: Fåræus-Lindqvist effect RBCs cluster toward vessel center Reduced net resistance relative to Poiseuille flow µ/µplasma r (µm) Experiment: Long et al., PNAS (2004) J. B. Freund p.8/40

11 Background: the F-L effect Key microcirculation feature: Fåræus-Lindqvist effect RBCs cluster toward vessel center Reduced net resistance relative to Poiseuille flow Leukocyte margination seems counter to F-L effect J. B. Freund p.8/40

12 Questions Summary Do hydrodynamics alone marginate? Aggregation necessary? Dependence on RBC flexibility? Source of shear-rate/flow-rate dependence? Role of cell-free F-L layer? Pries Mechanisms for Dx50 adhesion inhibition? J. B. Freund p.9/40

13 Simulation Model J. B. Freund p.10/40

14 Simulation Model Blood: cellular suspension in Stokes flow Matched interior/exterior cell viscosity Two dimensional J. B. Freund p.11/40

15 Simulation Model Blood: cellular suspension in Stokes flow Matched interior/exterior cell viscosity Two dimensional Not a quantitative model of the microcirculation but... includes several key components will reproduce key phenomena J. B. Freund p.11/40

16 Simulation Model Finite-deformation massless shell model for cell walls reasonable model for RBCs very simple cells linear constitutive model ( ) ds m = M(κ κ o ) τ = T 1 ds o membrane traction on fluid σ = tτ s + ( ) m s s n m bending moment τ tension M bending modulus T tension modulus κ curvature s arc length κ o reference curvature s o referential arc length J. B. Freund p.12/40

17 Boundary Integral Formulation Boundary integral formulation (e.g. Pozrikidis et al.) u j (x) = U + 1 σ j (y(s))s ij (x(s) y(s)) ds (4πµ) Ω Ω σ j S ij all surfaces j-th component of membrane traction fundamental solution for Stokes operator J. B. Freund p.13/40

18 Surface Discretization Evenly spaced points in referential s o coordinate: x m = x(s m o ) Assume x m are interpolated by harmonics x m = N/2 1 l= N/2 ˆx l e ik ls m o Efficient for given accuracy for smooth shapes Dealiasing Consistent accurate quadratures: N a > N integrand harder to resolve than cell shape only N restricts time step J. B. Freund p.14/40

19 Fast Particle-Mesh Methods Particle-Mesh-Ewald (PME) or Particle-Part./Particle-Mesh (P 3 M) Darden et al. (1993), Essmann et al. (1995), Metsi (2000), Saintillan et al. (2005), Hockney & Eastwood (1988) FFTs on mesh: O(N log N) relatively small coefficient Significantly faster than multipole methods for typical N Standard for electrostatics in molecular biophysics codes J. B. Freund p.15/40

20 Configuration and Cases J. B. Freund p.16/40

21 Flow Configuration L Periodic W 2r l r o I.C. l o J. B. Freund p.17/40

22 Parameters Fixed, anticipated important r l leukocyte radius r o RBC initial radius l o RBC reference length ρ hematocrit (0.45, 0.33, 0.20) µ viscosity W channel width Fixed, anticipated unimportant for selected values M l leukocyte moment modulus T l leukocyte tension modulus L channel length Varied M T U RBC moment modulus RBC tension modulus driving flow J. B. Freund p.18/40

23 Constant Groups r 2 o T M = 50 [100] RBC property l o = 1.6(2π) r o r l = 1.75 r o L = 27 r o W = 7.8 r o RBC property leukocyte radius channel length channel width J. B. Freund p.19/40

24 Time Scales (Variable) Advection: τ adv = r o U m Shear: τ sh = 1 where σ m = du σ m dy w Relaxation: τ rlx = µr o T J. B. Freund p.20/40

25 Cases µu T τ rlx τ sh τ rlx τ adv Case = µr oσ m = µu m = r oσ m T T u m N N a a a a a a b c d d d e f f f Copyright J B. Freund, J. B. Freund p.21/40 τ adv τ sh

26 Basic Results J. B. Freund p.22/40

27 Visualizations/Animations µu T = 0.05 Floppier, Constant U µu T = 0.2 µu T = 0.8 J. B. Freund p.23/40

28 Mean Velocity Profile 2 Blood cell flow Wall forces only Parabolic fit u(y)/um y/w J. B. Freund p.24/40

29 Velocity Profile Bluntness 1 1 u(y)/um u(y)/um y/w Slow flow Fast flow (same cell properties) y/w Slow Flow Fast Flow Long (2005); mouse blood 29.7µm ( 15r o ) glass tube J. B. Freund p.25/40

30 Leukocyte Path: M, T = consts Faster, Constant T and M y(t)/ro y(t)/ro y(t)/ro y(t)/ro x(t)/r o J. B. Freund p.26/40

31 Leukocyte Path: M, T = consts Faster, Constant T and M y(t)/ro y(t)/ro y(t)/ro y(t)/ro x(t)/r o P(y) J. B. Freund p.26/40

32 Near-Wall Probability Probability d < d c = 0.4r o : 1 P w = dc 0 P(d)dd 0.8 Pw d µu m 0.1 T Qualitatively similar to experiments J. B. Freund p.27/40

33 Near-Wall Probability τadv/τsh = roσw/um slower/less Newtonian floppier τ rlx /τ adv = µu m /T P w Insensitive to RBC stiffness Sensitive to profile bluntness...? J. B. Freund p.28/40

34 Most Probable Luk. Location Pd y/r o τ rlx /τ adv = µu m /T = ; 0.017; 0.039; 0.077; 0.11 Constant cell properties ρ Leukocyte sits at edge of cell free layer J. B. Freund p.29/40

35 Cell Free Layer Thickness d/ro T and M constant data d u 0.43 m µu c /T Lubrication scaling d u for constant force? J. B. Freund p.30/40

36 Single-Cell Lift Test F (constant) u cell d J. B. Freund p.31/40

37 Cell Free Layer Thickness d/ro T and M constant data d u 0.43 m single-cell data d u 0.5 m Why is leukocyte at same height? µu c /T J. B. Freund p.32/40

38 3D H. Zhao, A. Isfahani, J. B. Freund Mechanical Science & Engineering University of Illinois at Urbana-Champaign J. B. Freund p.33/40

39 3D Spherical harmonics with dealising GMRES for mismatched viscosity implicit system J. B. Freund p.34/40

40 Deformation Relaxation (π θ) 100 E B = θ t (ms) Relaxation times with expected bending moduli match experiments of Bronkhorst et al. J. B. Freund p.35/40

41 Cellular flow in cylindrical tubes D = 11.4µm H T = 30% D = 20.0µm H T = 30% J. B. Freund p.36/40

42 Effective Viscosity λ = 5 λ = 1 µeff u = 1 u = D(µm) J. B. Freund p.37/40

43 Complex Geometries J. B. Freund p.38/40

44 Conclusions More probable on-wall corresponds to longer periods on-wall Margination does not require RBC agglomeration Relatively insensitive to RBC stiffness Leukocyte most probably at edge of cell-free layer Cell-free layer thickness scales nearly as u. Lubrication with constant wall-ward force? Emerging Picture: lubrication forces lift RBC putting leukocyte into less stable configuration consistent with Dx50, which increases µ plasma 3D: preliminary validations underway J. B. Freund p.39/40

45 Role of Leukocyte Flexibility Pd(y) d/r o Leukocyte three time stiffer than leukocyte Insensitive to (small) leukocyte flexibility Lubrication forces lift RBC destabilizing leukocyte J. B. Freund p.40/40

HOW GOOD ARE THE FITS TO THE EXPERIMENTAL VELOCITY PROFILES IN VIVO?

HOW GOOD ARE THE FITS TO THE EXPERIMENTAL VELOCITY PROFILES IN VIVO? Aristotle G. KOUTSIARIS 1,3,*, Sophia V. TACHMITZI 2, Athanassios D. GIANNOUKAS 3 * Corresponding author: Tel.: ++30 (2410) 555278;