Fibrin Gelation During Blood Clotting

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1 Fibrin Gelation During Blood Clotting Aaron L. Fogelson Department of Mathematics University of Utah July 11, 2016 SIAM LS Conference Boston

2 Acknowledgments Joint work with Jim Keener and Cheryl Zapata-Allegro, drawing on earlier work with Bob Guy Funding from NSF and NHLBI.

3 Blood Clotting Overview Resting Platelets Resting Platelets Activated Platelets Fibrinogen Activator Vessel Wall Injury Injury Injury Platelet aggregation produces platelet plug Coagulation enzyme reactions produce enzyme thrombin Thrombin cleaves fibrinogen to produce fibrin monomers Fibrin polymerizes and gels

4 Fibrin Polymerization Low thrombin concentrations induce coarse fibrin clots. High thrombin concentrations induce fine fibrin clots. Coarse/fine clots differ mechanically and degrade differently. How does thrombin concentration affect clot structure? Goal: develop model of fibrin gelation that gives information about the clot structure to 1 Explain thrombin-concentration-dependent gel structure. 2 Study fibrin gelation during clotting under flow.

5 Kinetic Gelation - The Ziff Model Assume Monomers have f reactive sites Reactive sites combine, forming a polymer chain There are no closed loops C k species consisting of k joined monomers k-mers Chemical reactions: C i + C j k i,j C i+j cf. Ziff and Stell, J. Chem. Phys, 1980.

6 The Chemical Reaction Equations Let c k be the concentration of k-mers dc k dt = 1 2 i+j=k r i r j c i c j r k c k R, dr dt = R2, where r j = (f 2)j + 2 is the number of reactive sites on each j-mer, and R is the total concentration of reactive sites. Remark: Before a gel forms, R(t) = R s (t) k=1 r kc k. Then, the ODEs for c k imply dr s = Rs 2. dt One form of Ziff s model postulates that R satisfies such an ODE for all time even after gelation. We will revisit this later.

7 Approach 1: Moments Define moments M 1 = k kc k, M 2 = k k 2 c k, A = M2 M 1 = k ( ) k kck M 1, Find that dm 1 dt Y = (f 2)M 2 + 2M 1. = 0, dy dt = (f 2)Y 2. Average oligomer size A iff Y. We regard the event that the average cluster size becomes infinite as gelation. For f > 2, Y at finite time t gel. For IVP starting with monomers only, 1 t gel = f (f 2)c 1 (0). Question: What happens after gel time t gel?

8 Approach 2: Moment Generating Function Set g(z, t) = k zr k c k (t) and calculate that g t = 1 2 g z 2 zg z R. Introduce the change of variables W = zr g z. Using PDE for g and find that dr dt = R2, W t + WW z = 0.

9 Quantities of Interest W = zr g z Concentration of reactive sites in solution, R s = k r k c k = g z z=1 Concentration of reactive sites not in solution, i.e., in gel R g = R R s = W z=1. (R g = W z=1 and R s = R R g ). A gel forms at the time t gel for which lim t tgel W z z=1 =. Sol mass density θ s k kc k (t) = 1 ( 2 f W (z)dz W z=1 Only values W (z, t) for 0 z < 1 matter for our purposes. )

10 Determine W Initial data: Monomer only c j (t = 0) = m 0 δ j1 W (z, t = 0) = m 0 f (z z f 1 ). Solve for W (z, t) using the method of characteristics: W Concentration θ s R s θ g R g Z Time W (1, t) = 0 and W z (1, t) < until time t = t gel.

11 Eliminate Gel-Gel Reactions Eliminate gel-gel reactions so gel forms no cycles: dc k = 1 r i r j c i c j r k c k R, dt 2 i+j=k R t = (R 2 R 2 g ), R g = W z=1, W t + WW z = zr 2 g (Non-local) R W Concentration θ s θg R g Z Time

12 Diffusion of Polymer, Time-dependent Source Allow polymer (but not gel) to diffuse with diffusion coefficient D and add time-dependent source of monomer S 1 (t). dc k dt = 1 r i r j c i c j r k c k R + δ k,1 S 1 (t) + D c k, 2 i+j=k W t + WW z = zr 2 g + f (z z f 1 )S 1 (t) + D (W zr g ), R t = (R 2 R 2 g ) + f S 1 (t) + D (R R g ), R g = W z=1 With diffusion in x, MoC in z won t work. Numerical scheme uses centered differencing in x and 2nd-order upwind differencing in z exploiting smoothness of W for z < 1.

13 Fibrin Polymerization Process fibrinogen fibrin monomer a A A a thrombin a A A a polymerize to form protofibril protofibrils form fibers branch points for gel forms What mechanism of branch formation would make the final branch point density sensitive to the thrombin concentration?

14 Monomer Binding Reactions Fibrin monomers bind center to end and end to center in separate steps.

15 Monomer Binding Reactions Fibrin monomers bind center to end and end to center in separate steps. Model first path as bimolecular reaction Model second path as trimolecular reaction

16 Fibrin Reactions Elongation C m,b denotes oligomer with b branches and m + 2b monomers. Two oligomers bind to create larger oligomer: C m1,b 1 + C m2,b 2 k l Cm1 +m 2,b 1 +b 2 Lose two reaction sites per elongation reaction.

17 Fibrin Reactions - Branching Three oligomers bind to create larger oligomer with new branch: C m1,b 1 + C m2,b 2 + C m3,b 3 k b Cm1 +m 2 +m 3 2,b 1 +b 2 +b 3 +1 Lose three reaction sites per branching reaction.

18 Fibrin Reaction Kinetics Two types of reactions lead to elongation or branch formation C m1,b 1 + C m2,b 2 C m1,b 1 + C m2,b 2 + C m3,b 3 k l Cm1 +m 2,b 1 +b 2 k b Cm1 +m 2 +m 3 2,b 1 +b 2 +b 3 +1 dc mb dt = S 10 δ m1 δ b0 + k l 2 + k b 6 bj =b mj =m bj =b mj =m r b1 r b2 c m1 b 1 c m2 b 2 k l r b c mb R r b1 r b2 r b3 c m1 b 1 c m2 b 2 c m3 b 3 k b 2 r bc mb R 2, R t = 2S 10 k l R 2 k b 2 R3, Number of reactive sites on C mb oligomer is r b = b + 2.

19 Fibrin Reaction Kinetics Two types of reactions lead to elongation or branch formation C m1,b 1 + C m2,b 2 C m1,b 1 + C m2,b 2 + C m3,b 3 k l Cm1 +m 2,b 1 +b 2 k b Cm1 +m 2 +m 3 2,b 1 +b 2 +b 3 +1 dc mb dt = S 10 δ m1 δ b0 + k l 2 + k b 6 bj =b mj =m bj =b mj =m r b1 r b2 c m1 b 1 c m2 b 2 k l r b c mb R r b1 r b2 r b3 c m1 b 1 c m2 b 2 c m3 b 3 k b 2 r bc mb R 2, R t = 2S 10 k l R 2 k b 2 R3, Number of reactive sites on C mb polymer is r b = b + 2.

20 Fibrin Reaction Kinetics Two types of reactions lead to elongation or branch formation C m1,b 1 + C m2,b 2 C m1,b 1 + C m2,b 2 + C m3,b 3 k l Cm1 +m 2,b 1 +b 2 k b Cm1 +m 2 +m 3 2,b 1 +b 2 +b 3 +1 dc mb dt = S 10 δ m1 δ b0 + k l 2 + k b 6 bj =b mj =m bj =b mj =m r b1 r b2 c m1 b 1 c m2 b 2 k l r b c mb R r b1 r b2 r b3 c m1 b 1 c m2 b 2 c m3 b 3 k b 2 r bc mb R 2, R t = 2S 10 k l R 2 k b 2 R3, Number of reactive sites on C mb polymer is r b = b + 2.

21 Gel Time Branch Concentration Structure Parameter Branch Formation Rates Branching Structure vs Supply Rate M = mb, A = m,b(m+2b)c 1 M (m+2b) 2 c mb, A as t t gel. m,b Derive and solve ODEs up to t gel for quantities of interest. R = (b + 2)c mb, B = bc mb, O = M c 10. m,b m,b Experiments with different constant monomer supply rates S monomer 2 monomers 3 monomers all monomers all branches Supply Rate Supply Rate Time Gel time decreases with S 10. B increases and structure parameter O/B decreases with S 10. Almost all branch formation involves at least one monomer.

22 PDE Form of Fibrin Equations Generating function g(x, t, y, z) = m,b y m z b+2 c mb (x, t). W = zr g z y=1, R g = W z=1, R s = R R g, V = g zy y=1. With diffusion, source of monomer, and no gel-gel reactions, { kl W t = 2 W 2 + k b 6 ( ) 3 k b ( zr W R 2 R 2 )( g z 2 R zw )} 2 z ) ) + D ( ) W zr g + k l zrg 2 k ( b 2 z R 3 ( 3R s Rg 2 + Rg 3 xx R t = k l (R 2 Rg 2 ) k { } b R 3 (3R s Rg 2 + Rg 3 ) + 2S 10 + D(R s ) xx 2 { (kl V t = W k b 2 (zr W )2 + k b 2 z(r2 Rg 2 ) ) } V z ( + k b (zr W ) 2 (R W z ) ) 2S 10 z + DV xx.

23 PDE Form of Fibrin Equations Generating function g(x, t, y, z) = m,b y m z b+2 c mb (x, t). W = zr g z y=1, R g = W z=1, R s = R R g, V = g zy y=1. With diffusion, source of monomer, and no gel-gel reactions, { kl W t = 2 W 2 + k b 6 ( ) 3 k b ( zr W R 2 R 2 )( g z 2 R zw )} 2 z ) ) + D ( ) W zr g + k l zrg 2 k ( b 2 z R 3 ( 3R s Rg 2 + Rg 3 xx R t = k l (R 2 Rg 2 ) k { } b R 3 (3R s Rg 2 + Rg 3 ) + 2S 10 + D(R s ) xx 2 { (kl V t = W k b 2 (zr W )2 + k b 2 z(r2 Rg 2 ) ) } V z ( + k b (zr W ) 2 (R W z ) ) 2S 10 z + DV xx.

24 PDE Form of Fibrin Equations Generating function g(x, t, y, z) = m,b y m z b+2 c mb (x, t). W = zr g z y=1, R g = W z=1, R s = R R g, V = g zy y=1. With diffusion, source of monomer, and no gel-gel reactions, { kl W t = 2 W 2 + k b 6 ( ) 3 k b ( zr W R 2 R 2 )( g z 2 R zw )} 2 z ) ) + D ( ) W zr g + k l zrg 2 k ( b 2 z R 3 ( 3R s Rg 2 + Rg 3 xx R t = k l (R 2 Rg 2 ) k { } b R 3 (3R s Rg 2 + Rg 3 ) + 2S 10 + D(R s ) xx 2 { (kl V t = W k b 2 (zr W )2 + k b 2 z(r2 Rg 2 ) ) } V z ( + k b (zr W ) 2 (R W z ) ) 2S 10 z + DV xx.

25 Auxiliary Quantities Polymer branch concentration B s = mb bc mb Polymer mass density θ s = mb (b + 2m)c mb Total branch concentration B, total mass density θ Gel branch concentration B g = B B s θ s (x, t) = 4 B s (x, t) = 2 θ t = D(θ s ) xx + S 10 B t = D(B s ) xx + k b 6 (R3 (3R s R 2 g + R 3 g )) W (x, t, z ), dz 1 W (x, t, z )dz W (x, t, 1) 0 V (x, t, z )dz 2W (x, t, 1)

26 Fibrin Model Results Monomer supplied near x = 0 at exponentially decreasing rate 4 x = 0 t = θ g Concentration R s θ s B s Time θ g R g B g Concentration B g x R g

27 Coagulation Reaction System Fibrin monomers are produced by coagulation system.

28 Vascular Injury Model Flow Injury Model Assumptions Prothrombin P and fibrinogen F enter upstream and move by advection and diffusion. P is converted to thrombin E on injury at rate α P K m+p. E moves by advection and diffusion, is degraded, and converts F into fibrin monomers C 1,0 at rate S 10 = kcat EF K m+f. Fibrin polymerizes as before and fibrin oligomers move by advection and diffusion. Gel does not move.

29 Diffusion But No Flow (Top) Thrombin (Middle) Fibrinogen (Bottom) Monomer Source Rate S 10 = kcat EF K m+f

30 Diffusion But No Flow Gel Time

31 Diffusion But No Flow (Top) Pore Size (Bottom) Fiber Diameter

32 Fibrin Gelation with Flow With flow, gel forms far downstream of injury. Need flow obstacle for fibrin formation over injury. Flow Injury Insert porous bead representing platelet aggregate.

33 Fibrin Gelation with Flow Monomer Source Rate Gel Mass Density

34 Final Words Have developed model of fibrin polymerization in which the resulting gel structure is a function of the thrombin concentration. Have coupled this model to reduced coagulation model to begin to study fibrin gelation under flow during blood clotting. Model gives reasonable times for gel formation, pore size, fiber diameters, and predicts need for shelter to allow fibrin gelation at injury. Model can be combined with our more comprehensive models of coagulation and platelet deposition to study the clotting response to vascular injury. References Ziff and Stell, J. Chem. Phys, Guy, Fogelson and Keener, MMB, Fogelson and Keener, PRE, Fogelson and Keener, SIAP, Zapata-Allegro, U of U PhD thesis, 2016.

35 How valid are mean-field gel models? k i,j The chemical reactions C i + C j C i+j Use Gillespie algorithm to simulate dynamics with large number of monomers. Second Moment of Oligomer Distribution x Time Largest Oligomer Size x Time Obvious transitions at gel time predicted by continuum model.

36 Which post-gel model is most reasonable? Model with no gel-gel reactions matches stochastic model results It like stochastic model does not allow cycles even in large oligomers (gel) Monomer Concentration Reaction Site Concentration Time Time

37 Fibrin at Base of Thrombus

38 Introduction Kinetic Gelation Model Fibrin in Neeve s Experiments Fibrin Polymerization Vascular Injury Model

39 Introduction Kinetic Gelation Model Fibrin Polymerization Fibrin in Wolberg s Experiments Vascular Injury Model