How to predict data that follows a log logistic distribution?

I want to predict surgery durations, using independent variables such as: procedure, doctor, age, prior estimations etc.

I already tried linear regression and loglinear regression (with the function stepwiselm), this works! But as the graph shows, the best probability distribution is the loglogistic distribution. Which functions are there to predict the surgery duration if it follows a loglogistic distribution? So my goal is not to find the average and standard deviation, but to predict future data with factors.

I already checked how to do it with Maximum Log Likelihood, or with Survival analysis, but it is difficult to understand and does not always include the log logistic distribution.

Thanks in advance!