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. 2022 Oct 7;12:16829. doi: 10.1038/s41598-022-21269-2

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© The Author(s) 2022

Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article's Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article's Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/.

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A weighted average of MNAE (a) for estimated bi-exponential parameters, MNAE for the fraction, short T_1ρ, and long T_1ρ versus K. The optimized sampling schedules resulted in lower errors for the entire range, and the MCRLB optimized schedule was the top performer. Interestingly, the CRLB schedule performed better for the estimation of long T_1ρ. This is probably due to the longer TSLs employed. In (b) Weighted average of MNAE for estimated stretched-exponential parameters, MNAE for the T_1ρ*, and β versus K. Errors are much lower than for the bi-exponential model, and so is the gain over the non-optimized schedule. The differences between CRLB and MCRLB schedules also become smaller. The black dashed line indicates the minimum error obtained using the non-optimized schedule. The optimized schedules can reach this line before the non-optimized schedules, leading to lower acquisition times.