If the "tumor grade" is somwhat ordered (for example 1 = worst, ..., 4 = best) you should use ordered logistic regression; multinomial logistic regression is similar except that it assumes that there is no order in the categories of the outcome variable (i.e., the categories are nominal, e.g., type of sandwitch: chicken, beef, fish). If you use the multinomial model when there in order in the categories, the information contained in the ordering is lost.

I second Noel. And in this case it seems highly likely that the tumor is graded w r t size of something similar that suggest the coding to be ordered -> ordinal regression. As for model checking, you would want to apply at least the same model checking Tools as for logistic regression using for instance generalized residuals.