## 4 Discussion
### Limitations  
Several limitations of the proposed PAC methods were observed in this study. The methods fail for bone inclination angles larger than 7\({}^{\circ}\) with respect to the beam axis, as well as for the bones with high porosity values (20% or more). For bone models with tilt angles larger than 7\({}^{\circ}\), most backside echoes were not captured by the receiver array resulting in a much smaller DAS beamformed signal and the transition from convex to concave shape of the backside signal wavefront disappeared. Subsequently, the zero-crossing point could not be retrieved (Figure A4c in Appendix D). As the bone surface inclination in the imaging plane can be reliably reconstructed, the application of the PAC methods can be easily restricted to locations, in which the surface inclination is within \(\pm\)7\({}^{\circ}\). Second, the simulation study was restricted to one scan position for one multi-focus measurement, while the ex vivo measurement performed the multi-focus measurement at 128 scan positions along the lateral distance; therefore, future in silico studies should simulate multi-focus measurements with more scan positions along the lateral distance and include simulation models with real bone curvature, tilt, and porosity. Moreover, compound imaging with beam steering [15; 46] should be used to ensure that the bone area of interest is probed with sufficiently small beam inclinations. Third, for high porosity values, large amounts of scattering of ultrasound waves resulted in a strong attenuation and distortion of the backside signal, yielding an imprecise estimation of the confocal backside position (Figure A5c in Appendix D).  
Another limitation is the use of simplified bone models. For in vivo transition, the effect of heterogeneous cortical pores and heterogeneous backside surface on the phase aberration should be investigated. Cortical pores lead to increased scattering, and therefore, increased phase aberration, which could be corrected with cross correlation analysis. Furthermore, the effect of changes regarding the speed of sound in soft tissue should be considered in the future, based on realistic simulation models. Conventional image reconstruction assumes an invariant speed of sound of 1540 m/s. Although the higher and variable velocity in bone was considered, soft tissue velocities can also vary by up to 10% between subjects depending on the relative distribution of skin, fat, and connective tissue along the bone length [16]. This leads to additional wave distortion, defocusing of bone regions, and misalignments of beamformed signals. [PERSON] et al. [47] showed on a tissue-mimicking phantom that a speed of sound error up to \(\pm\)8% degrades the lateral resolution of the image by up to a factor of three. The mismatch between the assumed and actual speed of sound could be compensated for by evaluating the focus quality using the coherence factor proposed by [PERSON] et al. [48] or by using the minimum average sum of absolute differences between all pre-beamformed radio frequency channel data proposed by [49]. [PERSON] et al. proposed an autofocused method to estimate the optimal speed of sound of the overlaying soft tissue [50]. Additional phase aberration corrections by tissue structure may improve lateral resolution, signal quality and the accuracy and precision of the measured time-of-flight through the bone for in vivo transition measurements.  
Another limitation of this study is the non-repeated measurement design. Only one simulation was performed for each simulation model and the ex vivo measurement was performed once; therefore, for the reproducibility and precision of the multi-focus method for realistic bone simulations, ex vivo and in vivo measurements should be investigated in the future, by repeating the measurements by repositioning of the transducer between each measurement.