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Note 7 - Stock-Based Compensation Expense | 6 Months Ended | 12 Months Ended
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Jun. 30, 2025 | Dec. 31, 2024
Notes to Financial Statements |  |
Share-Based Payment Arrangement [Text Block] |
6. | Stock-Based Compensation Expense
Stock-based compensation expense related to stock options is recognized on a straight-line basis over the requisite service period for the award and is allocated to research and development expense or general and administrative expense based upon the classification of the individual to whom the award is granted. Stock-based compensation expense related to stock option grants was $292,740 and $585,484 during the three-month and six-month periods ended June 30, 2025, respectively, as compared to $101,640 and $205,209, respectively, during the same periods of 2024. As of June 30, 2025, there is $1,894,027 of unrecognized compensation expense that we expect to recognize over a weighted-average period of 2.3 years. We occasionally issue shares of our restricted common stock for consulting and other services and recognize the related expense over the terms of the related agreements. During the three-month and six-month periods ended June 30, 2024 we recorded stock-based compensation expense of $-0- and $54,167, respectively, associated with common stock issued for consulting services. No such expense was recorded during the 2025 periods. |
7. | Stock-Based Compensation Expense
Stock-based compensation expense related to stock options is recognized on a straight-line basis over the requisite service period for the award and is allocated to research and development expense or general and administrative expense based upon the classification of the individual to whom the award is granted. We also have issued shares of restricted common stock to consultants and recognize the related expense over the terms of the related agreements. We use the Black-Scholes model for determining the grant date fair value of our stock option grants. This model utilizes certain information, such as the interest rate on a risk-free security with a term generally equivalent to the expected life of the option being valued and requires certain other assumptions, such as the expected amount of time an option will be outstanding until it is exercised or expired, to calculate the fair value of stock options granted. We granted no stock options during 2023. The significant assumptions we used in our fair value calculations for stock options granted during 2024 were as follows:
Weighted average risk-free interest rates |  |  | 3.8 | %
Expected dividend yield |  |  | 0.0 | %
Expected life of option (in years) |  |  | 7.0 |
Expected volatility |  |  | 204.2 | %
The weighted-average grant date fair values of stock options granted during 2024 and 2023 were $2.22 and $-0-, respectively. As of December 31, 2024, there is $515,969 of unrecognized compensation expense that will be recognized over a weighted-average period of 1.8 years. The following table summarizes our total stock-based compensation expense for employees, directors and consultants for the years ended December 31, 2024 and 2023:
|  | 2024 |  |  | 2023 |
Stock options: |  |  |  |  |  |  |  |
Research and development |  | $ | 222,202 |  |  | $ | 291,094 |
General and administrative |  |  | 252,275 |  |  |  | 564,030 |
Total stock option expense |  |  | 474,477 |  |  |  | 855,124 |
Stock awards: |  |  |  |  |  |  |  |
General and administrative |  |  | 54,167 |  |  |  | 219,833 |
Total stock-based compensation expense |  | $ | 528,644 |  |  | $ | 1,074,957 |

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