Trust-regions provide a globally convergent framework for managing inexactness in computed quantities throughout the optimization process. Of particular interest is inexactness due to adaptive discretizations in PDE-constrained optimization. In this talk, we discuss existing and new conditions for objective function and gradient inexactness which guarantee first-order convergence. Many existing conditions for objective function inexactness depend on constants that are explicitly tied to algorithmic parameters and have hard upper bounds. For application problems, such hard upper bounds are often impossible to satisfy. We present simple modifications to these existing conditions which allow for satisfaction of the original inexactness conditions after finitely many optimization iterations. Moreover, we present new conditions that depend only on positive constants. These constants are independent of the iteration count. We discuss how these new and modified conditions can be satisfied in practice and present a PDE-constrained optimization example in which the PDE coefficients are uncertain.