An Introduction to the Mathematical Modeling ofBlood Clotting

Aaron L. FogelsonDepartments of Mathematics and Bioengineering

University of Utah

IMA 2010 – p. 1/42

Acknowledgements

• Thanks to: National Science Foundation and National Institutes of Health

• Current Math-Coworkers: Robert Guy (UC Davis), Karin Leiderman (Duke),

Lindsay Crowl (SandiaNL), Brittany Bannish, Jim Keener, Mike Kirby (UtahCS), Boyce Griffith (NYU), Richard Hornung (LLNL).

• Former Math-Coworkers: Andrew Kuharsky, Elijah Newren, Nien-Tzu Wang,

Haoyu Yu, Nessy Tania, Eli Bogart

• Experimental Collaborators: John Weisel (U Penn), Vince Turitto (IIT), AlisaWolberg (UNC), Zaverio Ruggeri (Scripps), Gene Eckstein (U Memphis)

IMA 2010 – p. 2/42

Blood Clotting

IMA 2010 – p. 3/42

Blood Clotting

The clotting process involves extremely complex interactions amongmany players and so is prone to break. Too little clotting leads to

bleeding (hemophilia); too much clotting leads to vessel occlusion andheart attack or stroke.

IMA 2010 – p. 3/42

Thrombosis

Thrombosis: intravascular or intradevice blood clot formation

Coronary artery

thrombosis

Retinal artery

thrombosis

Thrombosis on

vascular stent

Thrombosis is an extremely complex dynamic process for whoseunderstanding mathematical modeling and computational simulation are

essential tools.

IMA 2010 – p. 4/42

Hemostasis Overview

Healthy vessel.

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Resting Platelets

Vessel Wall

Fibrinogen

Early events after EC disruption.Resting Platelets

Injury

Activated Platelets

Activator

Platelet aggregation.

Injury

Coagulation and formation of fibrin gel.

Injury

Essential coagulation reactions happen on activated platelets’ surfaces.

Coagulation enzyme thrombin is a potent platelet activator.

IMA 2010 – p. 5/42

Arterial StenosisAtherosclerotic Plaque

Plaque

Recirculation Zone

Max Shear Stress

IMA 2010 – p. 6/42

Arterial Stenosis

Plaque Rupture or Erosion

Plaque

Thrombus

Thrombi formed in experimental stenoses

Strony, Beaudoin, Brands, Adelman, American J. Physiology, 1993.

IMA 2010 – p. 6/42

Venous ThrombosisVenous thrombosis often triggered by endothelial cell activation

due to hypoxia, stasis, inflammatory stimuli.

Coagulation occurs first and fibrin forms below platelets

W.C. Aird, J Thrombosis and Hemostasis, 2007.

IMA 2010 – p. 7/42

Device Thrombosis

Arterial flow conditions, geometric andmaterial perturbation.

Extremely high shear stresses andturbulence.

IMA 2010 – p. 8/42

Flow, Mechanics, and Chemistry

Thrombosis is an extremely complex dynamic process that involves many disparatecomponents and a wide range of both temporal and spatial scales.

Thrombosis involves many types of interactions– Mechanical: Fluid transports cells and chemicals, and exerts forces on cells.

Cells adhering to other cells resist fluid forces and disturb fluid motion.– Biophysical: Cell adhesion is achieved through specific molecular bonds.

Dynamics of bond formation and strength of bonds are important.– Biochemical: Enzyme networks involve reactions in fluid and on surfaces.– Cell Activation: Signal transduction across membrane and intracellularly leads to

cytoskeletal and membrane reorganization and chemical secretion.– Polymerization: Monomers produced on site by an enzyme also produced on site,

assemble into polymer strands which organize into a polymer mesh. The gelstructure influences its mechanics.

Thrombosis involves a wide range of scalesSpatial - nanometer (proteins), micron (platelets), millimeter (vessel)Temporal - milliseconds to minutes.

Thrombosis occurs under a wide range of conditions.

IMA 2010 – p. 9/42

Models of Thrombus Growth

• Models emphasizing interaction between flow and platelet deposition.

• Models emphasizing platelets’ interactions with coagulation chemistry andflow.

IMA 2010 – p. 10/42

Essentials of Platelet Biology

• Platelets do not adhere to intact wall but can adhere to damaged wall.

• Platelets are activated by collagen in damaged wall, and by chemicalsreleased from activated platelets.

• Only activated platelets cohere. Specific ligand-receptor binding.

PLATELET

GPIIb/IIIa

GPIb

vWF

ENDOTHELIALCELL

SUBENDOTHELIUM

FIBRINOGEN

1 µ m

IMA 2010 – p. 11/42

Microscale Platelet Aggregation Modeling

• For small arteries of diameter ≈ 50-100 µm or near-wall region of larger vessels.

• Model involves fluid, unactivated and activated platelets, activating chemical, andelastic links to represent intercellular bonds. Fluid and chemicals treated as continua.Platelets and links are tracked individually.

Fogelson, J Comput. Physics, 1984.Fauci and Fogelson, Comm. Pure Appld. Math, 1992.Fogelson and Guy, Comput Methods Appl Mech Eng, 2008.

IMA 2010 – p. 12/42

Microscale Platelet Aggregation Modeling

Fogelson, Kuharsky, Yu, in Polymers and Cell Dynamics – Multiscale Modeling and Numerical Simulations,2003.

IMA 2010 – p. 13/42

Macroscale Models of Platelet Arterial ThrombosisOur continuum macroscale models arederived from a two-scale network modelthat tracks the distribution of inter-platelet bonds. The continuum modelinstead tracks the additional stresses onthe fluid that stretching and re-orientingthe interplatelet bonds generates. It alsotracks creation and relaxation of stressdue to formation and breaking of inter-platelet (and platelet-wall) bonds.

Platelet Bond Network

Continuum Model Unknowns

u, p fluid velocity and pressureφn, φa non-activated and activated platelet concentrationsc activating chemical concentrationzp, σp concentration of interplatelet bonds, additional mechanical stress (tensor)from interplatelet bonds

Fogelson, SIAM J Appl Math, 1992.Fogelson and Guy, Mathematical Medicine and Biology, 2004.Fogelson and Guy, Comput Methods Appl Mech Eng, 2008.

IMA 2010 – p. 14/42

Continuum Model Equations

Fluid Motion

ρ(ut + u · ∇u) = −∇p + µ∆u+∇ · σp ∇ · u = 0

IMA 2010 – p. 15/42

Continuum Model Equations

Fluid Motion

ρ(ut + u · ∇u) = −∇p + µ∆u+∇ · σp ∇ · u = 0

Platelet and Chemical Transport

(φn)t + u · ∇φn = Dn∆φn − R(c) φn

(φa)t + u · ∇φa = R(c) φn

ct + u · ∇c = Dc∆c + A R(c) φn

IMA 2010 – p. 15/42

Continuum Model Equations

Fluid Motion

ρ(ut + u · ∇u) = −∇p + µ∆u+∇ · σp ∇ · u = 0

Platelet and Chemical Transport

(φn)t + u · ∇φn = Dn∆φn − R(c) φn

(φa)t + u · ∇φa = R(c) φn

ct + u · ∇c = Dc∆c + A R(c) φn

Interplatelet Cohesion

σp

t+ u · ∇σp = σp∇u + (σp∇u)T + αp φ2

a I − β ( E ) σp

zpt + u · ∇zp = αz φ2

a−β ( E ) zp

IMA 2010 – p. 15/42

Continuum Model Equations

Fluid Motion

ρ(ut + u · ∇u) = −∇p + µ∆u+∇ · σp ∇ · u = 0

Platelet and Chemical Transport

(φn)t + u · ∇φn = Dn∆φn − R(c) φn

(φa)t + u · ∇φa = R(c) φn

ct + u · ∇c = Dc∆c + A R(c) φn

Interplatelet Cohesion

σp

t+ u · ∇σp = σp∇u + (σp∇u)T + αp φ2

a I − β ( E ) σp

zpt + u · ∇zp = αz φ2

a−β ( E ) zp

E = Tr(σp)/zp is local average strain-energy per interplatelet bond

IMA 2010 – p. 15/42

Thrombosis after Atherosclerotic Plaque Rupture

IMA 2010 – p. 16/42

Vessel walls in the continuum model

Define density w(x, t) of reactive sites on injured wall.Platelets activated by w and form adhesive links where w > 0.

Modify transport equations for {φn φa c}:

(φn)t + u · ∇φn = Dn∆φn − R(c)φn − Rw(w)φn

New equations for additional stress σw due to platelet-wall elastic links:

σwt

+ u · ∇σw = σw∇u + (σw∇u)T + αw φa w I − βw σw

Add ∇ · σw as a new forcing term in the fluid equations:

ρ(ut + u · ∇u) = −∇p + µ∆u + ∇ · σp + ∇ · σw

IMA 2010 – p. 17/42

Simulated Thrombosis after Plaque Rupture

• Flow accelerates through stenosis (constriction).

• Higher shear stress in stenosis.• Recirculation zone downstream of stenosis.

Red – viscous oozy thrombus.Yellow – solid thrombus.

Grey – activating chemical concentration above threshold.

IMA 2010 – p. 18/42

Thrombus Grows to Occlude Vessel

Movie: s50a54r.mov

IMA 2010 – p. 19/42

Effect of Location and Flow

Movie: s50updown.mov

IMA 2010 – p. 20/42

Model’s Strengths and Limitations

Strengths:

The model can track thrombus growth from inception to occlusion of vessel.

The model captures stress-mediated remodeling and breakup of a thrombus.

Limitations:Model cannot capture events on physiological timescales. It must speed

process substantially. This is because platelets move at the same velocity asthe fluid.

Solution:Develop multiphase model in which fluid and aggregated platelets move in

different velocity fields.

IMA 2010 – p. 21/42

Coagulation Reactions

XaXIXa IX

TF

Subendothelium

VII VIIa

TF TF:VIIa

VII

• Vessel SurfaceTF:VIIa

IMA 2010 – p. 22/42

Coagulation Reactions

XaXIXa IX

TF

Subendothelium

Activated Platelet

X

Xa

Va

V Va

Prothrombin

THROMBINVa:Xa

VIIIa

VIIIa

VIII V

Va

VII VIIa

TF TF:VIIa

VIIIa:IXa

VII

VIII

Fibrinogen Fibrin • Vessel SurfaceTF:VIIa

• Platelet SurfaceVIIIa:IXaVa:Xa

IMA 2010 – p. 22/42

Coagulation Reactions

XaXIXa IX

TF

Subendothelium

Activated Platelet

X

Xa

Va

V Va

Prothrombin

THROMBINVa:Xa

VIIIa

VIIIa

VIII V

Va

VII VIIa

TF TF:VIIa

VIIIa:IXa

VII

VIII

Unactivated Platelet Fibrinogen Fibrin

• Vessel SurfaceTF:VIIa

• Platelet SurfaceVIIIa:IXaVa:Xa

• FeedbackThrombin

IMA 2010 – p. 22/42

Coagulation Reactions

ATIII

ATIII ATIII

TFPI

APC

APC APC

XaXIXa IX

TF

Subendothelium

Activated Platelet

X

Xa

Va

V Va

Prothrombin

THROMBINVa:Xa

VIIIa

VIIIa

VIII V

Va

VII VIIa

TF TF:VIIa

VIIIa:IXa

VII

VIII

Unactivated Platelet Fibrinogen Fibrin

• Vessel SurfaceTF:VIIa

• Platelet SurfaceVIIIa:IXaVa:Xa

• FeedbackThrombin

• Chemical InhibitionTFPIATIIIAPC

IMA 2010 – p. 22/42

Coagulation Reactions

ATIII

ATIII ATIII

TFPI

APC

APC APC

XaXIXa IX

TF

Subendothelium

Activated Platelet

X

Xa

Va

V Va

Prothrombin

THROMBINVa:Xa

VIIIa

VIIIa

VIII V

Va

VII VIIa

TF TF:VIIa

VIIIa:IXa

VII

VIII

Unactivated Platelet Fibrinogen Fibrin

• Vessel SurfaceTF:VIIa

• Platelet SurfaceVIIIa:IXaVa:Xa

• FeedbackThrombin

• Chemical InhibitionTFPIATIIIAPC

• Physical Inhibitionby plateletdeposition onsubendothelium

IMA 2010 – p. 22/42

Some Questions

• How can the clotting system be rapidly and powerfully switched on when needed, yetbe off when not needed?

• How does blood flow affect coagulation?

• How is coagulation localized to the site of injury?

• How does platelet deposition influence coagulation?

• How does coagulation affect platelet deposition.

• Can the players and the roles ascribed to them in the literature explain observedbehaviors?

• What factors limit the growth of a clot?

IMA 2010 – p. 23/42

Model: Coagulation and Platelet Deposition

ATIII

ATIII ATIII

TFPI

APC

APC APC

XaXIXa IX

TF

Subendothelium

Activated Platelet

X

Xa

Va

V Va

Prothrombin

THROMBINVa:Xa

VIIIa

VIIIa

VIII V

Va

VII VIIa

TF TF:VIIa

VIIIa:IXa

VII

VIII

Unactivated Platelet Fibrinogen Fibrin

h_

L

SE

PLT

kflow flowk

FLOW Advection and Diffusion

• Fluid and surface phase reactions

• Unactivated and activated platelets

• Simple transport by mass transfer.

• Platelet deposition.

• Vessel surface reactivities tocoagulation and platelets aremodel parameters.

• Well-stirred reaction zone (ODEs).

Kuharsky and Fogelson, Biophysical Journal, 2001.Fogelson and Tania, Pathophysiology of Thrombosis and Hemostasis, 2005.

IMA 2010 – p. 24/42

Spatial-Temporal Model Players and Behavior

• Platelets - unactivated mobile and activated mobile platelets move with the fluid and‘diffuse’; activated platelets bound to subendothelium or to other activated plateletsare stationary; any platelet can bind to SE, only activated platelets can bind to otheractivated platelets.

• Chemicals - some chemicals are bound to tissue factor molecules on thesubendothelium and are involved in reactions there;- some chemicals are in the fluid and move by advection and diffusion and areinvolved in fluid-phase reactions, fluid-phase chemicals may bind to receptors on theSE or to receptors on activated platelets;- chemicals bound to the surfaces of activated platelets are involved in reactionsthere, they may unbind moving into the fluid phase;- the controlled availability of receptors on the SE or activated platelet surfaces is animportant element of the system.

• Fluid - a prescribed flux of fluid is driven through the domain; bound platelets offerresistance to fluid motion

Leiderman and Fogelson, Mathematical Medicine and Biology, 2010.

IMA 2010 – p. 25/42

Subendothelial-bound ChemicalsAt points of the subendothelial surface,

∂ese7

∂t= kone7(TF − e

se,tot7

− zse,tot7

) − koffese7

| {z }

Binding with TF

+ kcat[Zse7 : E10] + kcat[Zse

7 : E2]| {z }

Activation by Xa or thrombin

+ (k− + kcat) [Z10 : Ese7 ] − k+z10ese

7| {z }

Activation of X to Xa

+ (k− + kcat) [Z9 : Ese7 ] − k+z9ese

7| {z }

Activation of IX to IXa

+ k−[TFPI : E10 : Ese7 ] − k+[TFPI : E10]ese

7| {z }

Binding with TFPI:Xa

− kadh(x) (P se,a + P se,u + P b,a) ese7

| {z }

Coverage by Platelet Deposition

(Note: ese7 is TF:VIIa density on subendothelium.)

IMA 2010 – p. 26/42

Fluid-phase Chemicals

In the fluid,

∂e10

∂t= −u · ∇e10 + ∇ · (D∇e10)

| {z }

Transport by advection and diffusion

+ (k−

+ kcat)[Z7 : E10] − k+z7 e10| {z }

Activation of VII

− kin e10 + k−

[TFPI : E10] − k+ [TFPI] e10| {z }

Inhibition by ATIII or by binding to TFPI

− kone10

`Nb

10P b,a + Nse10P se,a − z

ptot10

− eptot10

´+ koff e

b,a10

| {z }

Binding to platelet receptor for X and Xa

(Note: e10 is [Xa] in fluid.)

At the vascular wall,

−D∂e10

∂y=

8>><

>>:

kcat[Z10 : Ese7 ]

| {z }

Activation by TF:VIIa

+`k− + kcat

´[Zse

7 : E10] − k+ zse7 e10

| {z }

Binding to and activation of TF:VII

on the subendothelium

0 elsewhere

IMA 2010 – p. 27/42

Platelet-bound Chemicals

∂em2

∂t= kone2

`Nb

2 P b,a + Nse2 P se,a − zmtot

2 − emtot2

´− k

off2

em2

| {z }

Binding with receptors on platelet surface

+ kcat [Zm2 : PRO]

| {z }

Activation of thrombin by Prothrombinase

+ (k− + kcat) [Zm5 : Em

2 ] − k+zm5 em

2 + (k− + kcat) [Zm8 : Em

2 ] − k+zm8 em

2| {z }

Activation of Va and VIIIa by thrombin

(Note: em2

is concentration of platelet-bound thrombin.)

IMA 2010 – p. 28/42

Platelet Motion and BehaviorWe track the number densities of four classes of platelets: mobile unactivated, mobileactivated, platelet-bound activated and subendothelium-bound activated:

∂P m,u

∂t= −∇ ·

˘W (φT )

`u P m,u − D ∇P m,u

´ ¯

| {z }

Transport by advection and ‘diffusion’

−kadh(x) {P semax − P se,a}P m,u

| {z }

Adhesion to subendothelium

−{A(e2) + A(adp)}P m,u

| {z }

Activation by thrombin or ADP

∂P m,a

∂t= −∇ ·

˘W (φT )

`u P m,a − D ∇P m,a

´ ¯−kcoh g(η) Pmax P m,a

| {z }

Cohesion to bound platelets

−kadh (x) {P semax − P se,a}P m,a + {A(e2) + A(adp)}P m,u

∂P b,a

∂t= −kadh(x)

`P se

max − P se,a´

P b,a + kcoh g(η) Pmax P m,a

∂P se,a

∂t= kadh(x)

`P se

max − P se,a´ `

P m,a + P m,u + P b,a´

IMA 2010 – p. 29/42

Platelet SizeAlthough platelets are described by number density functions, the size of platelets istaken into account in four ways in the model:

∂P m,u

∂t= −∇ ·

˘W (φT )

`u P m,u − D ∇P m,u

´ ¯

| {z }

Transport by advection and ‘diffusion’

−kadh(x) {P semax − P se,a}P m,u

| {z }

Adhesion to subendothelium

−{A(e2) + A(adp)}P m,u

| {z }

Activation by thrombin or ADP

∂P m,a

∂t= −∇ ·

˘W (φT )

`u P m,a − D ∇P m,a

´ ¯−kcoh g(η) Pmax P m,a

| {z }

Cohesion to bound platelets

−kadh (x) {P semax − P se,a}P m,a + {A(e2) + A(adp)}P m,u

∂P b,a

∂t= −kadh(x)

`P se

max − P se,a´

P b,a + kcoh g(η) Pmax P m,a

∂P se,a

∂t= kadh(x)

`P se

max − P se,a´ `

P m,a + P m,u + P b,a´

IMA 2010 – p. 30/42

Fluid Motion

ρ ( ut + u · ∇u ) = −∇p + µ∆u − µ α(φB) u, (1)

∇ · u = 0. (2)

Brinkman term used to model resistance to flow provided by bound platelets:

α(φB) =αmax(φB)2

(φB0 )2 + (φB)2

, where φB(x, t) =

P b,a + P se,a

Pmax(3)

is the sum of the number densities of platelet- and subendothelium-bound

platelets at that location divided by the maximum number density of platelets

that is possible. We use φB0 = 0.5.

IMA 2010 – p. 31/42

Platelet Binding Region

∂η

∂t= Dη ∆ η − γ η + γ

` P b,a + P se,a

Pmax

´

Choose Dη and γ so that (Dη γ)1/2 = O(platelet diameter)

Thrombus

Binding Region

0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1

η

g(η)

Platelet binding rate is kcoh g(η) Pmax P m,a.

IMA 2010 – p. 32/42

Platelet-flux Limitation

To prevent platelets from entering a smallvolume already filled with platelets, wedefine the flux of platelets by:

W (φT )`uP m,u − D ∇P m,u

´

where`uP m,u − D ∇P m,u

´is evaluated

on an edge of the volume and W (φT )

is evaluated on the side of the edge intowhich this vector points.

0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1

φT

W(φ

T)

IMA 2010 – p. 33/42

Clotting Story - Dependence on TF

5 10 15 20 25 30

10−2

100

102

Thr

ombi

n C

once

ntra

tion

(nM

)Tissue Factor Density (fmol/cm 2)

MaximumAverage

Bound Platelets Thrombin (nM)

Time elapsed = 600

0

2

4

6

x 10−4

0

20

40

0

200

400

0

200

400

IMA 2010 – p. 34/42

Clotting Story - Why a threshold at low TF?

Pathway 1

Flow

Activated Platelet

Subendothelium

Prothrombin

TF:VIIa

IX IXa X Xa

Va:XaVIIIa:IXa

VIIIa VIII

XaX

THROMBIN Possible fates of factor IXa made by TF:VIIa:• Washed away by flow,• Inactivated by AT-III,• Bind loosely to activated platelet,• Bind firmly to activated platelet in complex withcofactor VIIIa.

The models show that to get a substantial production of thrombin, there had to be timewindow before the platelets covered the SE during which there is BOTH a significantamount of IXa in the fluid and VIIIa bound to platelets.

5 10 15 20 25 30

10−2

100

102

Thr

ombi

n C

once

ntra

tion

(nM

)

Tissue Factor Density (fmol/cm 2)

MaximumAverage

0 10 20 30 40 50 6010

−3

10−2

10−1

100

101

102

TF , (fmol/cm2)

[Thr

ombi

n], (

nM)

100500 1500

IMA 2010 – p. 35/42

Clotting Story - Why a plateau at high TF?

Va:Xa

P5 P10

Prothrombinase

P5

Va

P8 P9

VIIIa:IXaTenase

P10

Xa +

P10

X

Activated Platelet

5 10 15 20 25 3010

−8

10−6

10−4

10−2

100

102

Ave

rage

Ten

ase

Con

cent

ratio

n (n

M)

Tissue Factor Concentration (fmol/cm 2)5 10 15 20 25 30

10−6

10−4

10−2

100

102

104

Ave

rage

Pro

thro

mbi

nase

Con

cent

ratio

n (n

M)

Tissue Factor Concentration (fmol/cm 2)

Tenase

0

0.5

1

x 10−6

0.511.522.5x 10

−3

0.5

1

1.5

Time elapsed = 600

0

5

10

Empty X/Xa binding sitesTotal available P10 binding sites (nM)

Time elapsed = 600

1

2

3

4

0

1

2

3

4

0

1

2

3

4

0

1

2

3

4

ProthrombinasePRO

Time elapsed = 600

0

5

10

15

x 10−5

2468

0

100

200

0

100

200

IMA 2010 – p. 36/42

Transport Around and In Clot

Movie: transport2.mov

IMA 2010 – p. 37/42

Tenase and Prothrombinase

Movie: e9-e8m-ten-pro-15.mov

IMA 2010 – p. 38/42

Platelet Activation

Movie: PltActivation.mov

IMA 2010 – p. 39/42

Two Clots Separated by ECs

Movie: TwoClotsEC-e2-plts.mov

IMA 2010 – p. 40/42

Some Major Model Results

• The key to turning on thrombin production is establishment of sufficient enzyme activityon platelet surface before platelets cover subendothelium.

• The primary obstacles to turning system on are physical (flow and platelet deposition)not chemical inhibitors.

• Flow makes location of reactions very important and reduces role of chemical inhibitionprocesses that might seem important without flow.

• Transport within the growing thrombus may be very important in determining theeventual size and structure of the thrombus.

IMA 2010 – p. 41/42

Concluding Words

Capturing the interplay between physical and chemical processes is critical tounderstanding thrombosis.

Mathematical modeling and computational simulation are essential tools for studying thisinterplay and for gaining insight into how the clotting system functions as an integrateddynamical system.

IMA 2010 – p. 42/42

Concluding Words

Capturing the interplay between physical and chemical processes is critical tounderstanding thrombosis.

Mathematical modeling and computational simulation are essential tools for studying thisinterplay and for gaining insight into how the clotting system functions as an integrateddynamical system.

I’m done. Thank you and I’m happy to try to answer questions.

IMA 2010 – p. 42/42