Consider a four parameter hill function as follows to describe a growth curve.
$$ f(t) = y_0 + \frac{a \times t^b}{c^b + t^b} $$
The area under the curve is estimated as follows.
$$ AUC = \int_{t=0}^{t_{max}}f(t) $$
Which is described as
$$ AUC = \frac{\sum_{t=0}^{t_{max \times 10}} \left [ y_0 + \frac{a \times (t/10)^b}{c^b + (t/10)^b} \right ]}{10} $$
How the above expression can be explained?
Also mean growth time (not rate) is described as
$$ MGT = \frac{\frac{\sum_{t=0}^{t_{max \times 1000}} \left [ \left ( y_0 + \frac{a \times (\frac{t}{1000})^b}{c^b + (\frac{t-1}{1000})^b} \right )-\left ( y_0 + \frac{a \times (\frac{t-1}{1000})^b}{c^b + (\frac{t-2}{1000})^b} \right ) \right ]}{1000}}{a} $$
How to explain MGT in terms of calculus?