remember that immobilization is good while healing and initial training, but the goal is to return you to your prior level of function. Meaning without the brace. Aside from strengthening, your PT should also have some balance and proprioceptive training incorporated into your exercise program. Proprioceptive receptors are almost always compromised as well when an ankle sprain is incurred. Without the ability to feel where your foot and ankle are in relation to your body your muscles won't be able to react and prevent another ankle sprain (most likely plantarflexion-inversion/toes down and in). Also, when working said therabands, do not go to your end range initially. At least that's what I do with my patients. Work mid range and in a pain free range. Allow your muscled and proprioceptors to know where your midrange is, it's already been taken to end range and most likely beyond that.
If PT doesn't seem to improve your mobility and stability after 4-6 weeks I would suggest the MRI and for sure an orthopedic consult, one that specializes with ankle injuries, I'm not a fan of podiatrists, never recc them and I'm not going to start now. As stated earlier, X-rays only show bone and joint integrity, and MRI will show soft tissue integrity where in this case tendons, ligaments, and muscles. If there is some sort of soft tissue integrity trauma, most likely you'll find yourself in a walking boot for a few weeks then will resume PT again, but at least you'll have the proper papers to excuse you from any high level physical activity required by ROTC. Most tissues tend to heal in about 4-6 weeks with rest and some type of immobilization. Weight bearing joints, like the ankle, can take a bit longer and you can continue to have intermittent swelling. I've seen some ankles swell off and on for up to 1 year.
Good luck with your rehab and hopefully you return to your full prior level of funtion
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