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Goodwill and Intangible Assets | 3 Months Ended | 12 Months Ended
--- | --- | ---
Mar. 31, 2023 | Dec. 31, 2022
Goodwill and Intangible Assets |  |
Goodwill and Intangible Assets | 5. Goodwill and Intangible Assets There were no changes in the carrying amount of goodwill for the periods ended March 31, 2023 and December 31, 2022. Intangible assets as of March 31, 2023 and December 31, 2022 consisted of the following: ​
​ | ​ | ​ | ​ | ​ | ​ | ​ | ​ | ​
​ |  | ​ |  | March 31, |  | December 31,
​ | ​ | Useful Lives | ​ | 2023 | ​ | 2022
Patent application costs | ​ | 3 Years | ​ | $ | 405,848 | ​ | $ | 382,285
Trade name and trademarks | ​ | 3 Years | ​ | ​ | 69,223 | ​ | ​ | 68,356
Intangible assets, gross | ​ | ​ | ​ | ​ | 475,071 | ​ | ​ | 450,641
Less: Accumulated amortization | ​ | ​ | ​ | ​ | (236,261) | ​ | ​ | (198,955)
Intangible assets, net | ​ | ​ | ​ | $ | 238,810 | ​ | $ | 251,686
​ Intangible asset amortization expense is recognized on a straight-line basis and intangible asset amortization expense for the three months ended March 31, 2023 and 2022 totaled $37 thousand and $24 thousand, respectively. Estimated future amortization expense of intangible assets is as follows: ​
​ | ​ | ​ | ​
Years Ending December 31, |  | Amount
2023 | ​ | $ | 111,006
2024 | ​ | ​ | 88,818
2025 | ​ | ​ | 38,166
2026 | ​ | ​ | 820
​ | ​ | $ | 238,810
​ | 6. Goodwill and Intangible Assets There were no changes in the carrying amount of goodwill for the years ended December 31, 2022 and 2021. Intangible assets as of December 31, 2022 and 2021 consisted of the following: ​
​ | ​ | ​ | ​ | ​ | ​ | ​ | ​ | ​
​ |  | ​ |  | As of December 31,
​ | ​ | Useful Lives | ​ | 2022 | ​ | 2021
Patent application costs |  | 3 years | ​ | $ | 382,285 | ​ | $ | 207,630
Trade name and trademarks |  | 3 years | ​ |  | 68,356 | ​ |  | 86,999
Intangible assets, gross |  | ​ | ​ |  | 450,641 | ​ |  | 294,629
Less: Accumulated amortization |  | ​ | ​ |  | (198,955) | ​ |  | (92,822)
Intangible assets, net | ​ | ​ | ​ | $ | 251,686 | ​ | $ | 201,807
​ Intangible asset amortization expense is recognized on a straight-line basis and intangible asset amortization expense for the years ended December 31, 2022 and 2021 totaled $107 thousand and $70 thousand, respectively. Estimated future amortization expense of intangible assets is as follows: ​
​ | ​ | ​ | ​
Years Ending December 31, |  | Amount
2023 | ​ | $ | 140,434
2024 | ​ |  | 80,940
2025 | ​ | ​ | 30,312
Total future amortization expense of intangible assets | ​ | $ | 251,686
​

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- Definition The entire disclosure for goodwill and intangible assets. + References Reference 1: http://fasb.org/us-gaap/role/ref/legacyRef -Publisher FASB -Name Accounting Standards Codification -Topic 350 -URI https://asc.fasb.org/topic&trid=2144416 + Details
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