# New paper on liver regeneration

I have a new paper that has just appeared in Biophysical Journal entitled “A model of liver regeneration ” by Furchgott, Chow and Periwal.  The liver has this remarkable property that if a portion of it is removed up to a critical fraction, it will grow back to approximately 10% of its original size.  The restoration does not recreate the original morphology of the liver but it does restore function.  Although this has been known since ancient times, it is still a puzzle as to exactly how the liver does this, especially how it knows to stop growing when it is back to its original size and why it does not oscillate.  Our paper proposes a simple model to explain it.

The bulk of the liver is comprised of a cell called the hepatocyte.  It performs most of the necessary functions of the liver like synthesizing proteins and detoxifying the blood.  In a healthy state, almost all of the hepatacytes are in a “quiescent” state (i.e. they are not dividing).  However, if  a fraction of the liver is removed then the cells in the quiescent state enter a primed (P) state where they can either continue transforming into a fully active replicating (R) state or return to the quiescent (Q) state.  Our model consists of three ordinary differential equations for these three populations of cells.  Transitions between the cellular states are governed by secreted cytokines and growth factors.

The full model with the biological details is in the paper but the dynamics of liver regeneration can be understood geometrically with a reduced two dimensional model for Q and R having the form

$\dot Q = k Q(R-f(Q))$
$\dot R = cf(Q)Q - R$

where f(Q) is a decreasing sigmoidal function, which is equal to zero for some value of Q.  The phase-plane for the model looks like this:

The healthy liver lies at rest on an invariant manifold (a line attractor) on the Q axis (i.e. for Q  values where f(Q) is zero) so it can exist at a range of sizes.  Removing a fraction of the liver corresponds to moving Q to a lower value.  If Q is reduced (i.e. more of the liver is removed) such that it crosses the R-nullcline then Q will become activated and transform into R cells, which start to divide and increase on their own. (In the full model this is caused by an increase in cell signaling molecules due to the injury).  When R becomes large enough, the active cells start to return to quiescence and Q starts to increase, which in turn causes R to decrease even more.  Thus the liver trajectory in the Q-R plane increases quickly and then returns to rest at some point on the invariant manifold and just as is observed in experiments, the more the liver is initially reduced the larger it ends up after recovery.  There is also a separatrix such that if too much of the liver is removed then the liver cannot recover, which is also observed in experiments.  In summary, the liver is an excitable system.  Reducing the size below a threshold induces it to grow back to its resting state.