The answer depends a lot on how you want to implement your model: either in all of $\mathbb{R}^3$ or in a finite ball of radius $R>0$.
Case 1 - All of $\mathbb{R}^3$
In this case you want to solve
$$
\begin{cases}
\partial_t u(x,t) = \Delta u(x,t) & \text{ for } x \in \mathbb{R}^3, t >0 \\
u(x,0) = u_0(x),
\end{cases}
$$
where
$$
u_0(x) =
\begin{cases}
c_0 & \text{if }|x| \le r_0 \\
0 &\text{otherwise.}
\end{cases}
$$
This can be solved with the usual heat kernel via convolution:
$$
u(x,t) = \int_{\mathbb{R}^3} \frac{1}{(4\pi t)^{3/2}} e^{-|x-y|^2/(4t)} u_0(y)dy = \frac{c_0}{(4\pi t)^{3/2}} \int_{B(0,r_0)} e^{-|x-y|^2/(4t)} dy.
$$
The resulting solution will be radial, bounded, smooth, and decay at infinity.
Case 2 - The ball $B(0,R)$
Obviously we need $R>r_0$ here. In this case the key thing is that you need to choose boundary conditions to enforce on $\partial B(0,R)$, i.e. for $x \in \mathbb{R}^3 $ such that $|x|=R$. The two standard choices would be the Dirichlet condition, where you specify the value of $u$, and the Neumann condition, where you specify the value of $\nabla u \cdot \nu$ (the normal derivative). Based on what you're trying to model, I would guess that the Neumann condition is more appropriate. If you specify that the solution satisfies $\nabla u \cdot \nu =0$ then this means that there is no "flux of the drug" out of the domain, which is quite reasonable. Moreover, since your initial data is radial this can be written as $\partial_r u =0$ for $|x|= R$.
You thus want to solve the $3D$ radial heat equation:
$$
\begin{cases}
\partial_t u(r,t) = \partial_r^2 u(r,t) + \frac{2}{r} \partial_r u(r,t) & \text{for }0 < r < R, t>0 \\
\partial_r u(R,t) = 0 & \text{for } t >0 \\
u(r,0) = u_0.
\end{cases}
$$
This problem can be solved by using separation of variables. You assume an ansatz $u(r,t) = S(r) T(t)$ and plug in and use ODE theory to find an infinite family of solutions. Then you represent the solution to your problem as an appropriate linear combination of these solutions. For details I will refer you to this post. In it they consider different initial conditions, but the analysis of the separation of variables is the same. The post also restricts to $R=1$, so you will need to modify things a bit to handle general $R$.
