We completed a private placement in September 2011 for the sale of units consisting of shares of common stock and warrants to purchase our common stock. Both the common shares and the warrants contain anti-dilutive, or down round, price protection. We recorded derivative liabilities related to the down round price protection on the common shares and the warrants.

The down round price protection on the common shares expired in August 2012, and the down round price protection for the warrants terminates when the warrants expire or are exercised.

Our derivative liabilities at December 31, 2014 and March 31, 2015 both relate to these warrants.

The fair values of our derivative liabilities are estimated at the issuance date and are revalued at each subsequent reporting date using a Monte Carlo simulation discussed below.

At March 31, 2015 and December 31, 2014, we recorded current derivative liabilities of $24,334 and $42,659, respectively, which are detailed by instrument type in the table below.

The net change in fair value of the derivative liabilities for the three months ended March 31, 2015 and 2014 were gains of $18,325 and $30,079, respectively.

The following table presents the derivative liabilities by instrument type as of March 31, 2015 and December 31, 2014:

The following table presents details of our derivative liabilities from December 31, 2014 to March 31, 2015:

An independent valuation expert calculated the fair value of the compound embedded derivatives using a complex, customized Monte Carlo simulation model suitable to value path dependent American options. The model uses the risk neutral methodology adapted to value corporate securities. This model utilized subjective and theoretical assumptions that can materially affect fair values from period to period.

Key inputs and assumptions used in valuing our derivative liabilities are as follows:

For issuances of warrants: