## 4 Discussion  
We have developed and validated a computationally efficient 3-D DBIM algorithm for microwave head imaging. Our DBIM implementation uses the FISTA solver for the linear inverse problem, which has shown advantages over traditional CGLS solvers in our previous work [58]. The 3-D DBIM-FISTA algorithm relies on an in-house 3-D FDTD forward solver implemented on GPU, which is equally accurate but runs considerably faster than previous implementations with commercial software Acceleware. Our implementation combined this 3-D GPU-based FDTD solver with the inversion code on MATLAB via MEX functions.  
Inspired by microwave head imaging applications, we used CST-calculated as well as experimental data to validate the 3-D DBIM-FISTA algorithm. We implemented a simplified 3-D inverse algorithm that considers a scalar Green's function for linearly polarized antennas, such as the printed monopoles used by our system. Importantly, we showed that this approximation does not lead to worse performance than a vectorial Green's function implementation for the considered head imaging scenarios. Our numerical studies employed different numerical head models in CST based on the Zubal head phantom. Our experimental study examined a simpler, homogeneous head model, with the purpose to validate the 3-D implementation rather than examining more complex head models.  
Our results showed that reconstructions are of similar quality to those produced by a 2-D version of the algorithm for cases where the data processed by the 2-D imaging algorithm is of sufficient quality. To show that this may not always be possible, we considered cases where the target was not centered at the same transverse plane as the antenna ring and reflections from the interface of a headband with air were significant. Comparison of 3-D and 2-D reconstructions for these imaging scenarios showcased the advantages of imaging in 3-D. Moreover, these cases also demonstrated that terminating the imaging headband with absorbing material can improve drastically the 3-D array's imaging performance.  
Future work will focus further on reconstructions with experimental data inspired by the outcomes of this study. In particular, using prior information to improve reconstruction accuracy in realistic clinical settings requires further investigation. This prior information is not only required for dealing with the complexity of the brain's heterogeneous tissues, but also to estimate accurately the head boundary, which constitutes the reconstruction domain of our MWT approach. Moreover, the issue of patient movement should also be investigated in clinical settings. Motion artefacts are an important issue in imaging techniques which require long data acquisition times such as MRI, and a MWT system must be designed to minimise their impact by ensuring very short data acquisition times.  
Conceptualization, P.L. and P.K.; methodology, P.L.; software, P.L.; validation, P.L.; formal analysis, P.L.; investigation, P.L. and P.K.; resources, P.K.; writing--original draft preparation, P.L.; writing--review and editing, P.L. and P.K.; supervision, P.K.; project administration, P.K.; funding acquisition, P.K. All authors have read and agreed to the published version of the manuscript.  
This research was funded in part by UK's Engineering and Physical Sciences Research Council (EPSRC) under project grant EP/R013918/1.  
Not applicable.  
The authors declare no conflict of interest. The funders had no role in the design of the study; in the collection, analyses, or interpretation of data; in the writing of the manuscript, or in the decision to publish the results.