You have two possible options here:

1. Patient satisfaction

This is a compound noun, composed of two nouns positioned one after the other. The second noun is the more general one, while the first - which is always singular and has no apostrophe - is the more specific. 'Customer satisfaction' is a similar compound noun, as is 'patient adherence'.

English is full of compound nouns of this type. Think of wine glass ( a glass specifically for wine) or toothbrush (a brush specifically for teeth).

2. Patients' satisfaction

This is a different construction grammatically. It is not a compound noun. Note that the first noun is plural (because it refers to patients in general) and it is in the genitive case - hence the apostrophe after the 's'. This noun phrase is a condensed form of 'the satisfaction of patients''.

"Improving patient's satisfaction" is not a correct option. If you were to write this, people would presume you were writing in note form and were omitting the word 'the' ( or 'a' or 'your', for example) before the singular noun.

I hope that all makes sense.

Thank you Su.Ki for you comment and explanation, that would help me have a basic idea so I can investigate more and find out when to use one or the other. Thank you!