  In the self-similar scenario for galaxy cluster formation and evolution, the
thermodynamic properties of the X-ray emitting plasma can be predicted in their
dependencies on the halo mass and redshift only. However, several departures
from this simple self-similar scenario have been observed. We show how our
semi-analytic model $i(cm)z$, which modifies the self-similar predictions
through two temperature-dependent quantities, the gas mass fraction $f_g=f_0
T^{f_1} E_z^{f_z}$ and the temperature variation $f_T=t_0 T^{t_1} E_z^{t_z}$,
can be calibrated to incorporate the mass and redshift dependencies. We used a
published set of 17 scaling relations to constrain the parameters of the model.
We were subsequently able to make predictions as to the slope of any observed
scaling relation within a few percent of the central value and about one
$\sigma$ of the nominal error. Contextually, the evolution of these scaling
laws was also determined, with predictions within $1.5 \sigma$ and within 10
percent of the observational constraints. Relying on this calibration, we have
also evaluated the consistency of the predictions on the radial profiles with
some observational datasets. For a sample of high-quality data (X-COP), we were
able to constrain a further parameter of the model, the hydrostatic bias $b$.
By calibrating the model, we have determined that (i) the slopes of the
temperature dependence are $f_1=0.403 (\pm0.009)$ and $t_1=0.144 (\pm0.017)$;
and that (ii) the dependence upon $E_z$ are constrained to be $f_z=-0.004 (\pm
0.023)$ and $t_z=0.349 (\pm 0.059)$. These values permit one to estimate
directly how the normalizations of a given quantity $Q_{\Delta}$ changes as a
function of the mass (or temperature) and redshift halo in the form $Q_{\Delta}
\sim M^{a_M} E_z^{a_z} \sim T^{a_T} E_z^{a_{Tz}}$, in very good agreement with
the current observational constraints.
