NDsolve Boundary Condition is a Function of the Solution

Question

I am trying to solve something like Fick's Law using NDSolve: $$\frac{\partial \varphi}{\partial t}=\frac{\partial^2 \varphi}{\partial r^2}+F(r,t)$$ Subject to a boundary condition which looks something like: $$\varphi (R,t)=\varphi_0-\int_0^R \varphi(r,t)dr$$ Which is an attempt to simulate the case where whatever is diffusing into the medium of interest is limited in quantity. I assume simply plugging a recursive boundary condition is going to go poorly (Edit: it did go poorly), is there any option here other than writing my own solver?

In specific one particular case could look like: $$\frac{\partial \varphi}{\partial t}=\frac{\partial^2 \varphi}{\partial r^2}+\frac{1}{r^2}\frac{\partial\varphi}{\partial r}(2r+r^3/3-\int\varphi r^2dr)-\varphi$$ In this case $\varphi_0=1$ and the initial condition is $\varphi(0\leq r<R,0)=0$, where $R=1$. The problem should have spherical symmetry so a bound at $r=0$ may result in trouble.