1-parameter NFW dark matter profile This should be a straightforward exercise, but I'm struggling. Help is much appreciated

I would like to parameterize a Navarro-Frenk-White profile with one parameter, using the correlation between total mass and the concentration. 

As a starting point, I have the density in terms of two parameters: the total dark matter mass inside the virial radius $r_{vir}$, and the concentration parameter $c=r_{vir}/r_{s}$:

$\rho_{DM}(r) = \frac{M_{DM}}{4\pi A(c)}\frac{1}{r(r_{s}+r)^{2}}$

where the function $A(c)$ is given by

$A(c) = \ln{(1+c)} - \frac{c}{1+c} $

Using the correlation between $M_{DM}$ and $c$ (from e.g. Napolitano et al. 2005)

$c(M_{DM}) \approx 9.195 \left(\frac{M_{DM}}{h^{-1}10^{12} M_{\odot}} \right)^{-0.094}$

it should then in principle be possible to parameterize $\rho_{DM}$ as a function of $M_{DM}$ only. This is how they do it in e.g. Williams et al. 2009.

However, I'm not managing to reproduce that. Sure, given the correlation above, $M_{DM}$ determines $A(c)$, but won't $\rho_{DM}$ still depend on $r_{s}$?
You are missing one last ingredient: the correlation between the virial radius $r_{\mathrm{vir}}$ and the virial mass $M_{\mathrm{vir}}$. The idea is that the orbital time scale of a particle orbiting in the halo becomes roughly equal to the age of the universe at $r_{\mathrm{vir}}$. This implies a relation between the (mean) density of the halo and of the universe:
$$
    \bar{\rho}_{\mathrm{vir}} \equiv \frac{3M_{\mathrm{vir}}}{4\pi r_{\mathrm{vir}}^3} = \Delta \rho_{\mathrm{crit}}(z)
$$
where $\bar{\rho}_{\mathrm{vir}}$ is the halo's mean density interior of $r_{\mathrm{vir}}$, $\rho_{\mathrm{crit}}$ is the critical density of the universe (as function of redshift $z$), and $\Delta$ a parameter that depends on the cosmology. Often $\Delta=200$ is used and the corresponding quantities labeled $r_{200}$ and $M_{200}$ instead. See also this Wikipedia page.

So the way to construct a halo is as follows:


Pick $M_{\mathrm{vir}}$ (or $M_{200}$) and redshift $z$ (and your favorite cosmology)
Use above relation to find $r_{\mathrm{vir}}$ (or $r_{200}$)
Use the relation from Napolitano et al. (2005), or equivalent, to find $c$ and hence $r_s$
Obtain $\rho_{\mathrm{DM}}$ (if you still need it)