  We consider ancient solutions to the mean curvature flow in
$\mathbb{R}^{n+1}$ ($n \geq 3$) that are weakly convex, uniformly two-convex,
and satisfy derivative estimates $|\nabla A| \leq \gamma_1 |H|^2, |\nabla^2 A|
\leq \gamma_2 |H|^3$. We show that such solutions are noncollapsed. As an
application, in arbitrary codimension, we consider compact $n$-dimensional ($n
\geq 5$) solutions to the mean curvature flow in $\mathbb{R}^N$ that satisfy
the pinching condition $|H| > 0$ and $|A|^2 < c(n) |H|^2$, $c(n) =
\min\{\frac{1}{n-2}, \frac{3(n+1)}{2n(n+2)}\}$. We conclude that any blow-up
model at the first singular time must be a codimension one shrinking sphere,
shrinking cylinder, or translating bowl soliton.
