Modelling blood coagulation

Part I: the biological background

Prague, August 2011

Antonio Fasano Dipartimento di Matematica U. Dini, Univ. Firenze

Istituto di Analisi dei Sistemi ed Informatica “A. Ruberti” - CNR Viale Manzoni 30, Roma, 00185, Italy

Survey paper:

A. Fasano, A. Sequeira, R. Santos.

Blood coagulation: a puzzle for biologists, a maze for mathematicians.

MODELLING PHYSIOLOGICAL FLOWS D. Ambrosi, A. Quarteroni, G. Rozza (Editors), Springer Italia. Chapt. 3, (2011) 44-77

When a lesion is produced on a blood vessel an impressively complex machine is set in motion

leading to coagulation. The result is the

formation of a clot (or thrombus) sealing the wound.

Blood coagulation goes in parallel with the

antagonist process (fibrinolysis) whose goal is the dissolution of the clot.

A two-step process

• primary hemostasis: platelets bind to von Willebrand Factor and collagen at the wound site, forming the so-called “white thrombus”

• secondary hemostasis: goes through a chemical cascade in which many “Factors” intervene

clot remains confined

The hemostatic system is in a state of permanent background activity

but also normally inhibited

an engine permanently on a brake constantly applied

Maria Spelterini crossing Niagara Falls (July 7, 1876)

Difference: blood coagulation has to

be fast .

Similarity: no mistakes are allowed

Correct path for blood coagulation is narrow

thrombosis

What a pulmonary embolism can produce

(courtesy of Dr. Jeremi Mizerski)

Bleeding disorders

Thrombocytopenic purpura (due to low platelets count)

FIBRIN

PLATELETS (linked among themselves and to fibrin)

RBC

WBC

Clots are many components systems

Some history

The Yellow Emperor (Huang Di)

Before 2600 B.C.

Symptoms attributable to arterial thrombosis

Hippocrates ( 460 370 B.C.)

With the term leucophlegmatia describes limbs swelling

Hippocrates’ humoral theory: blood, phlegm, black bile, yellow bile

Aristotle (384-322 B.C.)

Blood coagulation needs some “fibrous material” and is due to heat loss

A fibrous component of clots was isolated by Marcello Malpighi (1628-1694)

Galen of Pergamon

Aelius (Claudius) Galenus (129-200?)

Coined the word thrombosis

(from the Greek thrombos = clot)

He sketched an erroneous scheme of blood circulation.

William Harvey (1578-1657)

Exercitatio anatomica de motu cordis et sanguinis in animalibus (1628)

Firts systematic description of blood circulation

Robert Hooke (1635 –1703)

Discovered what he called cells in thin slices of cork (1665)

Anthony Leeuwenhoek (1632-1723)

Greatly improved the microscope.

He investigated on RBC’s (1674)

(previously identified in frogs by Jan Swammerdam (1658 ))

Jean-Louis Petit (1674–1750)

French surgeon

Connected the formation of clots to the process of

hemostasis

The three elements

Hypercoagulability Hemodynamic changes (stasis, turbulence) Endothelial injury/dysfunction

are today known as the “Virchow triad”

Rudolf Virchow (1821-1902)

described pulmonary embolism in 1846

Giulio Bizzozero (1846 – 1901)

Max Johann Sigismund Schultze (1825-1874)

The discovery of platelets (1865)

Platelets have a number of receptors on their membrane which intervene in many processes:

• aggregation,

• binding to specific molecules,

• reacting to stress,

• etc.

Platelets perform a number of operations

They possess two families of granules: -granules and dense ()-granules, able to release various substances when platelets are “activated”, able to activate more platelets (*)

They can synthesize factors important in the coagulation process

They can modify their cytoskeleton to assume different shapes

(*) ASPIRIN exterts its anticoagulant action at this stage

activated platelets

Red Blood Cells (RBC): diam. 8 m, concentration 56/mm3, lifespan 120 days (approximate data), no nucleus

Platelets: diameter of 24 m, life span 5-9 days, discoid shape (at rest), concentration 1.54105 /mm3, no nucleus

star shaped platelets (rolling)

adventitia Tunica media

intima

Erik Adolf von Willebrand (1870-1949), Finland

Identified (1924) the bleeding disorder later called von Willebrand disease (vWD)

Today we know that vWD is due to deficiency or dysfunction of the so-called von Willebrand Factor (vWF): it is released by Weibel-Palade bodies

vWF is one of the many “Factors” entering the coagulation process

Some of the Factors have been labelled by Roman numbers

in the order of their discovery: FI – FXIII

Usually they come in pairs: inactivated (usually a zymogen = enzyme precursor) and activated (usually an enzyme)

FI = fibrinogen FIa = fibrin : the polymer making the clot skeleton

FII = prothrombin FIIa = thrombin (has a key role)

There are many more !!

They act through a chemical cascade

Rome (1958). Committee to number coagulation Factors

A first scheme for blood coagulation, proposed in 1905 by Paul Morawitz, was based on just four factors:

• prothrombin FII

• thrombin FIIa

• fibrinogen FI

• fibrin FIa

They do intervene in the last steps of the cascade

The 3-pathway Cascade Model was proposed in 1964 independently by

and by

It has remained unquestioned till very recently

The revolution of the last decade

etc…

Blood coagulation has two stages:

• primary hemostasis

• secondary hemostasis (cell-based model)

Primary Hemostasis

Platelets rolling on the blood vessel wall adhere at the lesion site to von Willebrand Factor and to Collagen

This is a very complicated 2-step process Two kinds of receptors intervene

(fast-reversible/slow-irreversible)

we omit the details

vWF can be released by activated EC’s even when damage is limited (Z. Xu et al. Soft Matter 5 (2009) 769-779)

Secondary hemostasis

(the cell-based model)

The triggering event is the exposure to

blood of the Tissue Factor (TF) contained

in the endothelium, which combines in a

complex with FVIIa circulating in very small quantities.

Initiation

End.Cell +TF

FVIIa FVII

End.Cell +TF

FVIIa FIX

FIXa

FX

FXa

FV

FVa

diffuses to platelets

End.Cell +TF

FXa+FVa: prothrombinase

FII FIIa Thrombin (small amount)

excess FXa inactivated

Lesion site

FVIIa available in small amounts in circulating blood

Activates complex TF-FVII

TF-FVII

Amplification

Small amount of thrombin and of FIXa available

FIIa breaks vWF

FVIII

vWF

FVIII

Platelet FV

FVa

+ stress

Cross links among platelets

FXI

FXIa

FIIa

FVIIIa

produces more FIXa

Platelets are activated and release the contents

of granules

Three actions of thrombin

1

2

3 and

activated platelet

FVIIIa + FIXa tenase

Propagation

FX

FVa + FXa prothrombinase

FII

FIIa more FVa is produced ETC.

To fibrin production …

FIIa

FI

FIa

FXIII

FXIIIa

Fibrin network

cross links

Fibrin production

Consolidation

thrombin

Endothelium TM

termination

TAFI (protects fibrin)

PC APC PS Va

VIIIa

AT III

TFPI

FIXa, FXa, FXIa, FXIIa, (FIIA)

FXa, TF+FVIIa

Thrombin Activatable Fibrinolysis Inhibitor

Tissue Factor Pathway Inhibitor

Heparin enhanced

fibrinolysis

Plasminogen (accumulated during the thrombus growth)

TAFI (Thrombin Activatable Fibrinolysis Inhibitor)

tPA urokinase

Plasmin

Grown fibrin network

fibrinolysis

(some fragments may recombine)

(retarded by plasmin inhibitors)

slowly released by endothelium

(positive feedback: produces tPA)

(tissue Plasminogen Activator)

WARNING

The preceding description has been simplified

• there are many more platelets activators (ADP, TxA2, etc.)

• vitamin K has an important role in catalyzing most of

the “activations”

• platelets may produce FXI and even TF !!!

WARNING

The preceding description has been simplified

• there are many more platelets activators (ADP, TxA2, etc.)

• vitamin K has an important role in catalyzing most of

the “activations”

• platelets may produce FXI and even TF !!!

A further revolution??

Papers reporting evidence of platelet-derived TF

G. Davì, C Patrono. Platelet activation and atherothrombosis. N.Engl. J. Med. 357 (2007) 2482-2494. O. Panes, V. Matus, C.G. Sàez, T. Quiroga, J. Pereira, D. Mezzano. Human platelets synthesize and express functional tissue factor. Blood 109 (2007) 5242-5250.

fibrin fibres are formed through a multi-step process

For details about fibrin and fibrin fibres formation see A.L. Fogelson, J.P.Keener. Toward an understanding of fibrin branching structure. Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 81 (2010) 1-24

A way of understanding how each element works is to study the

bleeding disorder associated with its

deficiency or dysfunction

The list of bleeding disorders is impressively long

For instance

Hemophilia A, B, C are caused by deficiency of

Factors VIII, IX, XI respectively

The inadequacy of the 3-pathway cascade model

High Molecular Weight Kininogen

(slow activator of FXII, produced by

platelets)

Kallikrein: fast activator of FXII

From now on the process is similar to the cell-based

model

FVIII bypassed by the extrinsic pathway

In the cascade model

• the intrinsic and extrinsic pathway can independently produce coagulation

• the extrinsic pathway bypasses FVIII

Experimental facts:

• FXII deficiency mild (or no) bleeding disorder

• FVIII deficiency is the cause of Hemophilia A

We conclude that

• the extrinsic pathway cannot be correct

• the intrinsic pathway either is not present or at most it gives a small contribution

FXII can autoactivate in the presence of

artificial surfaces

(coagulation on implanted artificial bodies is a matter of great concern)

Modelling bleeding disorders is important in order to model therapies

What is the effect of Aspirin

Coumadin etc. ?

How to prevent coagulation after the implantation of stents?

Clotting is essentially related with the blood flow conditions

• thrombi in arteries are different from those in veins • altered flow conditions can cause thrombosis: remarkable examples are

atrial fibrillation Deep Vein Thrombosis (DVT)

Hence, whatever model is taken for the biochemical process has to be coupled with blood flow

Blood rheology is another very complicated field …