Note 4 - Warrants

6 Months Ended

Jun. 30, 2024

Notes to Financial Statements

Warrants Disclosure [Text Block]

4. Warrants

Liability Classified Warrants

The Company uses the Black-Scholes option pricing model (BSM) to determine the fair value of its warrants at the date of issue and outstanding at each reporting date. The risk-free interest rate assumption is based upon observed interest rates on zero coupon U.S. Treasury bonds linearly interpolated to obtain a maturity period commensurate with the term of the warrants. Estimated volatility is a measure of the amount by which the Company's stock price is expected to fluctuate each year during the expected life of the warrants. Only the volatility of the Company's own stock is used in the Black-Scholes option pricing model. 

The assumptions used in determining the fair value of the Company's outstanding liability classified warrants are as follows:

	June 30, 2024	December 31, 2023
Risk-free interest rate	4.3% to 5.2%	3.8% to 5.4%
Volatility	87.9% to 94.9%	79.5% to 108.7%
Expected life (years)	0.6 to 4.5	0.3 to 5.0
Dividend yield	—%	—%

A summary of the Company's liability classified warrant activity during the six months ended June 30, 2024 and related information follows: 

	Number of Shares			Range of Warrant Exercise				Weighted Average		Weighted Average Remaining Contractual
	Under Warrant			Price per Share				Exercise Price		Life (Years)
Balance at January 1, 2024		1,082,895		$	9.60	$	157.50	$	24.32		5.1
Expired warrants		(75,909	)	$	99.00	$	99.00	$	99.00		—
Balance at June 30, 2024		1,006,986		$	9.60	$	94.50	$	—		4.4
Exercisable at June 30, 2024		1,006,986		$	9.60	$	94.50	$	15.31		4.4

For a summary of the changes in fair value associated with the Company's warrant liability for the six months ended June 30, 2024, see Note 2 - Basis of presentation, principles of consolidation and significant accounting policies - Fair Value of Financial Instruments.

Equity Classified Warrants

In March 2024, the Company granted equity-classified warrants to purchase up to 3,334 shares of Company common stock with a ten-year term and an exercise price of $9.15. The warrants vest annually over four years while services are being performed.

At June 30, 2024, the Company had 63,105 equity classified warrants outstanding and 40,658 warrants were exercisable. At  December 31, 2023, the Company had 289,276 equity classified warrants outstanding and 266,350 warrants were exercisable.

The Company recorded stock compensation expense for the non-employee consulting agreements of $35,000 and $46,000 for the three months ended  June 30, 2024 and 2023, respectively, and $68,000 and $92,000 for the six months ended June 30, 2024 and 2023 , respectively. At  June 30, 2024, there was $309,000 of unrecognized stock compensation expense related to the Company's equity classified warrants.