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Forecasting can be done using any length of time series. For example, if I have a set of data {1, 10, 19, 28}, then I can be pretty sure that the next value in the set is going to be 37 (because there is a strong pattern here: 10=1+9, 19=10+9, etc.).
So if you have a strong signal, then even if you don't have a very long sequence, you can get a pretty accurate forecast.
The question becomes: what type of time series forecasting model should you use?
I would avoid any type of neural network here (the Data Science forum often talks about neural networks of some type). Your data are not rich enough to support estimation of the hundreds (to millions!) of parameters that such models require.
Instead, I would try something fairly simple to start, like a moving average or perhaps an ARIMA-type model.
Remember, each patient is independent (hopefully!) but each measurement per patient is dependent (i.e. if you have patient 1 at time 1, patient 2 at time 2, and patient 3 at time 3, each patient time measurement is dependent on the patient). So if you want to forecast "the value at 7:30 AM regardless of the patient", that's different from forecasting "patient 1's value at 10 pm".
Hope that at least gives you a starting point!