• Ei tuloksia

The effect and compensation of modelling errors in the speed of sound in the so-lution of the inverse problem were studied with 2D simulations in various sensor geometries. The modelling error was generated by modelling a heterogeneous speed of sound as constant or piece-wise constant in the inverse problem. In addition, this error was compensated by solving the inverse problem utilising the Bayesian ap-proximation error approach. As a reference, the solution using the accurate speed of sound distribution was computed. In each case the whole posterior density was inspected and relative errors of the estimates were calculated. In addition, marginal densities were computed.

In the data simulation, a square domain with a side length of 10 mm and dis-cretisation of 300×300 square pixels (pixel side length of 33µm) was used. The initial pressure and speed of sound distribution that were employed to generate measurement data are shown in Figure 4.13. Sensors were arranged around the do-main (full view), on two adjacent sides of the dodo-main (two side) or on one side of the domain (one side). The measurement data was simulated using thek-space time domain method using 4001 time steps at a temporal sampling rate of 333 MHz. The measured signal was downsampled to 66 MHz and a 1% Gaussian noise was added to it.

In the inverse problem, the domain was discretised into 200×200 square pixels (pixel side length of 50µm). The posterior density was solved using the Ornstein-Uhlenbeck prior with the meanηp0 =5, standard deviationσp0 =5 and characteris-tic length scalel =2 mm and the correctly modelled noise statistics. If the accurate speed of sound was used (AFM), the mean and covariance of the posterior density was computed using (3.5) and (3.6), respectively. In the case of the inaccurate speed of sound, the posterior density was solved using (3.20) and (3.21). If the modelling errors were ignored (IFM) i.e. e = 0, then ηn = ηe and Γn = Γe were used in the computations. In the case of error modelling (IFM&EM), an approximation of mod-elling error (mean and covariance) were computed using (3.22), (3.23) and (3.24), respectively, based on 10000 samples that were drawn from the teaching distribu-tions of the initial pressure and speed of sound. Furthermore, marginal densities were calculated in two locations (a square and circle in Figure 4.13) using (3.9).

The results show that the inverse problem of PAT is sensitive to errors arising from uncertainties in the speed of sound (Figures 4.14 and 4.15). The sensitivity increases as the inverse problem becomes more ill-posed i.e. sensor geometry turns more limited view. The inaccurate modelling of the speed of sound yields the inac-curate posterior densities. That is, the numerical values and shapes of inclusions in the estimated posterior mean were reconstructed inadequately and the correspond-ing uncertainty estimates were not feasible meancorrespond-ing that the posterior uncertainty was underestimated. In addition, the relative errors increase significantly (Table 4.3). However, modelling of the errors improved the solution of the inverse prob-lem. Artefacts in the mean of the posterior density are reduced and the posterior density is widen such that the corresponding uncertainty estimates are meaningful.

In addition, the relative errors decrease significantly. However, the accuracy of the solution is not in the same level as in the case of the true speed of sound is used.

In addition, the results of the soft tissue mimicking simulations in publication IV show that the Bayesian approximation error modelling is able to compensate only for small speed of sound variations that can be expected in practice since the speed of sound is usually relatively well known.

Figure 4.13: The simulated (true) initial pressure distribution (left image) and the speed of sound distribution (right image). The initial pressure distribution is given in arbitrary units and the speed of sound distribution is given in units of m/s.

The square and circle in the left image indicate the locations where the marginal densities are plotted.

Table 4.3: The relative errors Ep0(%) of the estimated mean of the posterior ob-tained using AFM, IFM and IFM&EM in the full view, two side and one side sensor geometries.

AFM IFM IFM&EM

Full view 4 31 12

Two side 7 60 22

One side 15 129 33

Figure 4.14:The posterior mean (top block) and standard deviation (bottom block) obtained using AFM (first row), IFM (second row) and IFM&EM (third row). The columns from left to right represent the full view (first column), two side (second column) and one side (third column) sensor geometries. The red dots in the first row images indicate the locations of the sensors.

Figure 4.15: The marginal probability densities of the posterior densities at loca-tions denoted by a square (first row) and circle (second row) in Figure 4.13. The columns presents results obtained using the full view (first column), two side (sec-ond column), and one side (third column) sensor geometry. Shown in the graphs are the true initial pressure (vertical black line) together with the marginal densities of the posterior density obtained using AFM (blue solid line), IFM (red dotted line), and IFM&EM ( black dashed line). Note: The true value does not lie within the principal support of the IFM distribution (second and third column images in the second row), which illustrates the infeasibility of the posterior uncertainty in the case of modelling errors.

5 Discussion and conclusion

In this thesis, the Bayesian approach to the inverse problem of PAT was employed.

First, a solution method to the inverse problem of PAT based on the Bayesian frame-work was developed. The approach was evaluated with numerical simulations, and further validated using experimental data. Secondly, a method to compensate for uncertainties in the speed of sound in PAT utilising the Bayesian approximation error approach was described and studied with numerical simulations.

One of the attractions of the Bayesian approach is the quantitative information that it provides. That is, the method can be used to provide a probability distribu-tion with the mean and standard deviadistribu-tion of the initial pressure in each element of the domain. Thus, the uncertainty of the estimates, i.e. the reliability of the re-constructed images, can be assessed. Since the posterior density is based on the measurements, model, and prior information, uncertainties in these will also in-fluence the uncertainty estimates. As it was shown in the thesis, uncertainties in modelling can result in misleading uncertainty estimates. Thus, more research is required for interpretation of when uncertainty estimates can be regarded safe, see e.g. [200]. In the thesis, the uncertainty estimates are based on the standard devi-ations of the posterior density. In the future, an utilisation of the whole posterior covariance in the computation of the uncertainty estimates could be studied.

Other attractions of the Bayesian approach is an opportunity to incorporate prior information to estimation of the initial pressure. The relevance of proper prior in-formation increases as the inverse problem becomes more ill-posed. In the future, utilising other prior models could be studied to better incorporate, for example, anatomical information. Examples of other priors include, for example, total varia-tion, structural, anatomical and sample based prior. In fact, including total variation and structural prior information in regularization have been found to provide good photoacoustic images [45, 101, 104].

The results of this thesis show that by utilising the Bayesian approach, the solu-tion of the inverse problem can also be obtained when the problem is ill-posed i.e.

in limited view and sparse angle measurement geometries. Thus, the approach is suitable for practical applications. In addition, the suitability of the approach was further improved by compensating modelling errors using the approximation error modelling. However, in this thesis, only modelling errors caused by the uncertainties and inaccuracies in the speed of sound were taken into account and uncertainties in other acoustical parameters were not studied. Similar procedure as described in publication IV could be used to compensate errors caused by uncertainties in other acoustic parameters as well. Despite the promising simulation results, the experimental verification of the approximation error approach is still required.

The Bayesian approach can be computationally expensive, since forming of large matrices or solving of a large system of equations is required. However, it could be possible to utilise a model reduction (for example a coarse discretisation or trun-cated expansions), for example, using the Bayesian approximation error modelling to decrease the memory requirements and speed up the computations. In addition, the utilisation of efficient optimisation algorithms could speed up solving the linear

systems. In the approximation error modelling, the computation of the samples of the approximation error is time consuming, but it needs to be carried out only once and it can be done offline (before the computation of the posterior density). On the other hand, sampling could be speeded up by utilising parallel computing. Fur-thermore, the possibility of utilising machine learning in the approximation error approach could be studied.

Other tomographic imaging modalities could take an advantage of the developed methods. One of these modalities could be thermoacoustic tomography (TAT) that is closely related to PAT [29, 34]. In TAT, an initial pressure distribution is induced by an excitation of microwave radiation. Since the acoustic process is the same in PAT and TAT, they are nearly identical from a mathematical perspective. Thus, the methods of this thesis could be applied in TAT with relatively small modifications.

Another modality that could benefit from the developed methods is QPAT, where concentrations of light absorbing molecules are estimated from PAT images [201].

In QPAT, it is essential that PAT images are quantitative and accurate, and the meth-ods developed in this thesis can address to these requirements. It has been shown that, an incorporation of the uncertainty information of the initial pressure into the inverse problem of QPAT improves the accuracy of the solution [188].

In conclusion, the research presented in this thesis develops computational meth-ods to the inverse problem of PAT. The developed methmeth-ods not only improve the quality of the PAT images but also give the reliability of these images. In addition, the Bayesian approach enables incorporating statistical models for model errors and approximations.

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