  K. Igusa and G. Todorov introduced two functions $\phi$ and $\psi,$ which are
natural and important homological measures generalising the notion of the
projective dimension. These Igusa-Todorov functions have become into a powerful
tool to understand better the finitistic dimension conjecture.
  In this paper, for an artin $R$-algebra $A$ and the Igusa-Todorov function
$\phi,$ we characterise the $\phi$-dimension of $A$ in terms either of the
bi-functors $\mathrm{Ext}^{i}_{A}(-, -)$ or Tor's bi-functors
$\mathrm{Tor}^{A}_{i}(-,-).$ Furthermore, by using the first characterisation
of the $\phi$-dimension, we show that the finiteness of the $\phi$-dimension of
an artin algebra is invariant under derived equivalences. As an application of
this result, we generalise the classical Bongartz's result as follows: For an
artin algebra $A,$ a tilting $A$-module $T$ and the endomorphism algebra
$B=\mathrm{End}_A(T)^{op},$ we have that
$\mathrm{Fidim}\,(A)-\mathrm{pd}\,T\leq \mathrm{Fidim}\,(B)\leq
\mathrm{Fidim}\,(A)+\mathrm{pd}\,T.$
