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According to Sigma, the solubility of deoxycholic acid is:

0.24 g/L in water at 15C

According to the FDA, Kybella (trademark) is a 1% water solution of deoxycholic acid. How did they do this?

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According to the sigma document you referenced (emphasis added by me):

"The free acid is soluble at 0.24 g/L in water at 15ºC, while the sodium salt is soluble at >333 g/L in water at 15ºC. Therefore, pH control is very important to the use of this product. Aqueous solutions precipitate as the pH is lowered to 5."

According to the FDA document (emphasis added by me):

"Each 2 mL vial of KYBELLA® (deoxycholic acid) injection contains 20 mg synthetic deoxycholic acid as the active ingredient and the following inactive ingredients: benzyl alcohol(18 mg), dibasic sodium phosphate (2.84 mg), sodium chloride (8.76 mg), sodium hydroxide(2.86 mg) in water for injection, USP. Hydrochloric acid and additional sodium hydroxide are added as necessary to adjust the formulation to pH 8.3."

The solubility of 0.24 g/L in water at 15ºC is determined by dissolution of the free acid in pure water. As you dissolve an acid in pure water, the pH will get progressively lower, and since this compound is less soluble at lower pH, it limits it's own solubility in pure water.

The sodium hydroxide in the KYBELLA is a strong base, which raises the pH and, when mixed with the deoxycholic acid, forms an aqueous sodium salt and water via an acid-base reaction. The conjugate base of deoxycholic acid in aqueous solution is deoxycholate. This portion of the compound remains unchanged, regardless if it is part of a free acid or a sodium salt. The sodium phosphate is a buffering compound which helps stabilize the pH and allows for finer adjustment to control the final pH to ensure it remains soluble.

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