10. Warrant Liabilities

Warrants issued in connection with the Company's July 2016 equity public offering and modified in the Company's December 2016 equity public offering are classified as liabilities as opposed to equity due to their settlement terms. These warrants are non-cash liabilities and the Company is not required to expend any cash to settle these liabilities. The fair value of these warrants were recorded on the balance sheet at issuance and the warrants were marked to fair value at each financial reporting period, with changes in the fair value recorded as a gain or loss in the statement of operations. The fair value of the warrants is determined using the Black-Scholes option pricing model, which requires the use of significant judgment and estimates for the inputs used in the model. The warrants issued in connection with the Company's August 2011 equity public offering expired in August 2016. The following reflects the weighted-average assumptions for each of the periods indicated:

	Year Ended December 31,
	2017			2016			2015

Risk-free interest rate		1.53	%		0.90	%		0.57	%
Expected dividend yield		0	%		0	%		0	%
Expected lives		0.55			1.23			0.59
Expected volatility		96.7	%		119.1	%		61.7	%
Number of warrants classified as liabilities		2,834,246			4,752,512			1,061,976
Gain on warrant liabilities	$	1,367,777		$	3,827,617		$	4,437,628

The dividend yield assumption of zero is based upon the fact the Company has never paid cash dividends and presently has no intention of paying cash dividends. The risk-free interest rate used for each grant is equal to the U.S. Treasury rates in effect at the time of the grant for instruments with a similar expected life. The expected lives are based on the remaining contractual lives of the related warrants at the valuation date. The Company's computation of expected volatility is based on the historical daily volatility of its publicly traded stock.

In 2017, 1.2 million warrants expired, and 0.9 million warrants were exercised resulting in the issuance of 0.9 million shares of the Company's common stock. In 2016, 4.8 million warrants in connection with the July 2016 equity offering were issued.