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I was searching for an answer that "why do we need intrinsic pathway when there is fast extrinsic pathway" and i found the answer in the following link:

It says that the factor X produced in the extrinsic pathway turns off the extrinsic pathway and activates the intrinsic pathway. As the extrinsic pathway is turned off so, we need intrinsic pathway to continue the clotting process. But if the extrinsic pathway is stopped then how do we get prothormbin time because prothrombin time is for extrinsic pathway?

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In order to understand the importance of both pathways, you first need to be clear about what is meant by them respectively. Clotting activation from tissue factor (TF) is known as the extrinsic pathway, so you've got that one right.

The intrinsic pathway doesn't involve FX

The intrinsic pathway does not (as often misunderstood) refer to the clotting cascade downstream of factor X (FX) which involves FV, FVIII and thrombin. Rather, the intrinsic pathway is a system which is able to initiate the clotting cascade without any tissue factor being present. It is triggered by the assembly of high molecular weight kininogen, prekallikrein and FXII on collagen. This is also known as contact activation, and sets off a small cascade producing active FIX (FIXa), which can then assemble with circulating FVIIa in order to produce FXa and thereby set off the rest of the clotting system.

The extrinsic and intrinsic pathway are separate ways towards a "common pathway"

So think of it this way: The extrinsic pathway produces FXa in one way (TF), the intrinsic pathway produces FXa in a different way (HMWK+prekallikrein+FXII). There isn't really any interaction between these two. Once FXa is produced from either of these two, it will start to set off what you could call the "common pathway", i.e. it will produce thrombin, which will activate co-factors that will then lead to more production of FXa and propel the entire thing forward faster and faster until it reaches what we know as the thrombin peak.

The intrinsic pathway is not important for haemostasis in humans

Aside from this, you should know that the intrinsic pathway exists, but it doesn't bear much significance in a living human. This is demonstrated by genetic FXII deficiency - you would expect it to produce haemophilia if FXII and therefore the intrinsic pathway was important (just like FVIII and FIX), but it doesn't (read here:

The principle behind the initiation pathways and the common pathway

So to summarise: You fell victim to a misunderstanding and a bad explanation on that site you linked. The extrinsic pathway doesn't "turn itself off", and the intrinsic pathway doesn't "take over". What really happens in a human is that the intrisic pathway doesn't really do anything. Rather, the extrinsic pathway will produce a little bit of active FXa. But that is enough to produce a little bit of thrombin. And that bit of thrombin will then produce small amounts of FIXa and co-factors (FV and FVIII). These will start to produce more FXa, which will produce more thrombin, which will produce more FIXa and co-factors, and so on.

What the article you refer to is trying to mention are the inhibitory systems that will be set off alongside all of this. Think about it - if thrombin increases its own production perpetually, how does it know when to stop? These systems which ensure that clotting does stop are known as anticoagulation pathways. They are just as amazing as the coagulation pathways themselves, and they are extremely well-geared towards shutting off exactly the right clotting factors at exactly the right time, to ensure that a clot forms in the wound but only there and nowhere else.

Edit: It turns out people do use more complicated definitions of "intrinsic" and "extrinsic" - the problem being once again that everybody can define words as they like. This is rather based on the distinguishing of the two ten-ases, TF-FVII (extrinsic tenase) and FIX-FVIII (intrinsic tenase). In this understanding, the physiological process does indeed start off by the extrinsic pathway (i.e. mechanisms which generate extrinsic tenase), which will kick off the amplification cascade of the intrinsic pathway (i.e. which produces intrinsic tenase). It is true that this way, the contribution of the "extrinsic pathway" is minor once the "intrinsic pathway" is active, but even so, it does not "shut itself off". What turns the extrinsic pathway off is the layer of material which gets deposited on top of the exposed subendothelial surface, preventing access to TF!

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