## 5 Conclusions  
In this study, we developed and validated an enhanced RC lumped-parameter calorimetric model for AC magnetometry, addressing the challenges of non-adiabatic heat transfer in heating power measurements of magnetic nanoparticles. Our approach improves upon the traditional corrected-slope method by incorporating the undesirable effects of heat flow from the excitation coil to the sample and its surroundings, as well as integrating the thermal parameters of the sample's environment and employing multi-point temperature measurements.  
The calibration protocol of the proposed non-adiabatic measurement system included 19 different heating power excitations, applied using a resistor-based heater with an adjustable resistive heating power and the heating power of the transmission coil. The influence of both these heating sources on the sample and its surroundings was integrated into the model. These measurement configurations were used to determine the model parameters via a global optimization algorithm from MATLAB's global optimization toolbox. The results demonstrate that our refined model significantly reduces measurement errors, particularly at lower heating power values, where external heat transfer effects are most pronounced. Overall, our measurement system achieved an accuracy of less than 2.5% across the entire calibration range (20-200 mW) of heating powers, representing a substantial improvement over the corrected-slope method, which exhibited errors of up to 16% at low power levels.  
The use of a direct current excitation coil during calibration allowed us to isolate and quantify the thermal effects of the coil's heating without interference from eddy currents. This methodological innovation enabled simultaneous calibration measurements with different combinations of both excitation powers, effectively eliminating the additional heat source generated by eddy currents in the heater's power delivery wires during the excitation of the transmission coil with alternating current.  
Our findings provide a more accurate and reliable method for assessing the heating efficiency of magnetic nanoparticles compared to the corrected-slope method. The proposed lumped-parameter thermal model serves as a valuable tool for optimizing hyperthermia applications and advancing research in the field of magnetic nanoparticles.  
Conceptualization, M.M. and P.W.; methodology, M.M. and P.W.; software, M.M.; validation, G.D., W.T.S. and P.W.; formal analysis, M.M., P.W. and G.D.; investigation, X.H. and X.Y.; resources, P.W., M.I. and D.W.; data curation, M.M., P.W. and D.W.; writing--original draft preparation, M.M., W.T.S., P.W. and G.D.; writing--review and editing, P.W., D.W., X.H. and X.Y.; visualization, M.M. and D.W.; supervision, W.T.S.; project administration, W.T.S.; funding acquisition, W.T.S. and P.W. All authors have read and agreed to the published version of the manuscript.  
This work was partially supported by the internal grant for employees of the Warsaw University of Technology supporting scientific activity in the discipline of Biomedical Engineering, Grant No. 3/2022/RNDIB/WETI; the National Natural Science Foundation of China under Grant No. 52207008, and the Open Project of the China Poland Measurement and Control Technology \"the Belt and Road\" Joint Laboratory MCT 202305.  
Not applicable.  
The original contributions presented in this study are included in the article. Further inquiries can be directed to the corresponding author.  
The authors declare no conflict of interest.