Let $\Sigma$ be a complete fan in $N_{\mathbb{R}}$ and $\Psi$ a support function on $\Sigma$. There is a bijection between equivalence classes of semipositive toric models of $(X_{\Sigma},D_{\Psi})$ and rational piecewise affine concave functions on $M_{\mathbb{R}}$, with effective support $\Delta_{\Psi}$.