Hey Karen,
Please believe me, I know what you're going through. I went through a similar scenario July - September of last year with airline travel to Zurich, Hamburg, Reno, Seattle and Ketchikan, Alaska. You have the advantage of already starting the anti-inflammatory regimen so have a leg up on what I was going through. I got nailed on nearly every flight...
What I'd suggest at this point is hang in there and don't let CH change your plans... You'll get through this one way or another. The first thing I'd do is have my blood tested for 25(OH)D levels. This will let you know where you stand with respect to how much longer it's going to take for the anti-inflammatory regimen's vitamin D3 to achieve a therapeutic level of 25(OH)D.
The second thing I'd do is start on an anti-inflammatory and alkaline forming diet. The following links will point you in the right directions.
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I'd also avoid food stuffs containing gluten and peanuts until you get your CH totally under control. While you may not be having a noticeable allergic reaction to these food types with clear or apparent symptoms, it's still possible you're having a reaction and although low, it's high enough to affect the frequency and severity of your CH.
Finally, I've found that ordering oxygen cylinders to have them available at travel destinations is easy and comforting. Between LifeGas - OxyTravel™ and Apria I've had oxygen cylinders waiting for me at all my travel destinations across Europe, Alaska and here in the lower 48. The important thing and a MUST... is to have a copy of your Rx for oxygen therapy and that it states it is for your CH, so you can fax or scan to pdf and email to the provider.
I also ask for the aluminum M60 or aluminum M-size cylinders. At 23 lbs, the aluminum M60 is light enough to carry from room to room and they hold 1700 liters of oxygen so are good for 8 to 9 aborts. The aluminum M weighs in at 41 lbs so is still luggable, but with a capacity of 3450 liters, it's good for up to 20 aborts...
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Taking my oxygen regulator and mask through TSA security is usually event free. I tell the TSA attendant at the security check point there's an oxygen regulator in my carry-on bag as I push it towards the the X-Ray machine. They may want to look at it so I keep the regulator and mask in ziploc bags on top if they want to open the carry-on bag.
That leaves the actual flights with the inevitable CH that hit. Expect them to occur and plan accordingly. You've got the fast acting oral triptans, I carried a nasal spray applicator for each leg of air travel. I also alerted the flight attendants while boarding that I'm a cluster headache sufferer and will need a couple glasses of ice and an ice bag when one hits.
I've also found that hyperventilating with air at forced vital capacity tidal volumes can help. It won't stop the CH, but it can help hold the pain in check and keep it from becoming a head-banger.
With luck, your CH will start responding to the anti-inflammatory regimen making most of the actual travel preparations a secondary safety net.
Take care and please keep us posted,
V/R, Batch