The formula you need will depend on exactly what you mean by rate. If you mean that the tumor cells grow at a constant rate of "m" cells per second and die at a constant rate of "n" cells per second, then its easy to see that the number of cells "t" seconds later is P(t)= P(initial) (m-n)t, because after every second there will be (m-n) more cells than before.
However, this is not the sort of growth you would expect from tumor cells. You would expect that if all of the cells are dividing indepently of one another, the rate of growth would depend on the current population. For example, if one cell divides once in one hour, and you only have one cell, the rate of growth for one hour would be 1 cell per hour. Now suppose you have a thousand cells. After one hour they each divide once, so you now have two thousand cells. This means the rate of growth for that hour is 1,000 cells per hour. Notice that the growth rate is propotional to the current population. I have been looking at the growth rate over finite intervalt by taking \frac {\Delta P} {\Delta t} but with more and more cells dividing indepently of one another, growth rates over smaller intervals start making sense and we can make the approximation \frac {\Delta P}{\Delta t} \approx \frac {dP} {dt} Now, if this is the sort of growth we're talking about, then "m" probably doesn't measure new cells per hour, but more likely new cells per hour per existing cell. Likewise "n" would be in cells lost per hour per existing cell. The net growth rate would then be (m-n) cells per hour per existing cell. The growth rate is proportional to the number of existing cells. Since the growth rate is \frac {dP} {dt}, the equation that expresses this idea is \frac {dP} {dt}= (m-n) \times P
