4.7.1 Artiﬁcial Scenario
In the artiﬁcial test scenario the size of the computational domain D and the spatial discretization are the same as in Sects. 3.5 and 3.8, i. e. D is of extent 60×60×60 mm3, discretized with a ﬁne grid of 1203 cells. As in Sect. 3.8, for the multi-scale optimization, the ﬁne grid is embedded into an octree grid of grid-level L = 7 (cf. Sect. 3.7), i. e. we consider two coarser grids of 603 and 303 grid cells, respectively. Further, a lesion Dt, which has ellipsoidal shape, is assumed to be located in the center of D with a linear vessel Dv in the vicinity of the lesion. For the perfusion, here, the same values as in Sect. 3.8 are taken (i. e.νcap = 0.01765 s−1, νvessel = 0.05 s−1,ρblood = 1059.0kg/m3 andcblood= 3850.0J/kgK). Also the probe and generator settings are the same as in Sects. 3.5 and 3.8, i. e. a monopolar probe of radius 1.2 mm and 25 mm electrode length is applied, and the electric generator has an inner resistance of 80 Ω, and is set to a maximum power of 80 W. For the
Figure 4.2: The sensitivity of the optimal probe location is investigated for the artiﬁcial scenario with an ellipsoidal lesion and a linear vessel. Left: The PDF of the optimal probe location with distributedσ is shown with a volume rendering. The transfer function uses the color ramp shown on the left such that likely locations appear orange/red whereas unlikely locations appear green. Right: The ellipsoid shown inside the volume rendering cloud reveals a simpler visualization. It is centered at the mean of the distribution and it is oriented and scaled with the eigenvectors and eigenvalues of the distribution’s covariance matrix. Moreover, for better visibility, it is magniﬁed by a factor of 10.
mization, the initial probe position is always located at a distance of 10 mm in each coordinate direction from the center of D and its initial orientation is d= (5,2,3), projected on the sphere (i. e. normalized to length 1).
To guarantee that the size of D does not inﬂuence the result of the optimizer, a comparison between forward simulations using Dirichlet or Neumann boundary conditions at∂D, respectively, has been performed. Both temperature proﬁles diﬀer by less than 0.5 K in the interior of D, i. e. at locations which are more than 10 grid cells (≈ 5 mm) apart from ∂D. Closer to the boundary the temperatures diﬀer slightly more. Hence, it can be concluded that in the vicinity of the lesion the particular choice of boundary condition does not inﬂuence the result signiﬁcantly.
Sensitivity of the Optimal Probe Location
In Sect. 4.4 diﬀerent variants for the analysis of the sensitivity with respect to variations in the material parameters have been described. Note, that since the space of admissible probe placements U is a multi-dimensional space, a visualization and an analysis of the PDFs of the corresponding distributions is not straightforward (cf.
also ), because the PDFs are functionsU →R. In fact, when taking uncertainty inλ and σ into account, we get a six-dimensional stochastic space Γ. On this space a stochastic process has been approximated through collocation that reveals the dependence of the optimal probe placement ¯u w.r.t. variations in the parameters λ and/or σ. The optimal probe placement ¯u lies in again a six-dimensional space U (three dimensional probe location pand three-dimensional orientation vector d).
Now, ﬁrst a visualization that reduces the complexity of the data and allows for an easy perception of the distribution of the optimal probe location ¯p will be
Figure 4.3: For the artiﬁcial scenario the sensitivity of the optimal probe position and orientation w.r.t. variations in the electric conductivity σ is shown. Left: Sensitivity of the optimal probe position visualized through an ellipsoidal representation of the covari-ance matrix (mangiﬁed by a factor of 10). Right: PDF of the optimal probe orientation visualized through a coloring of the sphere. As shown by the color ramp on the right, green colors indicate unlikely orientations, whereas red colors show likely orientations. In both images the RF probe is drawn at the mean of the placement’s distribution, the linear vessel Dv is displayed in red and the ellipsoidal shaped tumor lesionDt is displayed in a transparent gray color.
discussed. The PDF of the optimal probe location is a mappingR3 →Rwhich can be visualized through a volume rendering as shown in Fig. 4.2 (left). There, the PDF of the optimal probe location depending on the three-dimensional vector of random variablesξσ is shown. In terms of a Monte Carlo experiment, this visualization can be explained in the following way: For any realization of the electrical conductivity σ an optimal probe location ¯p is computed (cf. Sect. 4.4.2). In Fig. 4.2 the color codes “how often” a certain probe location is obtained as the optimum.
A deep understanding and analysis of the three-dimensional PDF can be achieved only by an interactive three-dimensional display of the data. Therefore, the approach described in Sect. 4.4.3 can be used to provide a simpler and better perceptible visualization. From the probability distribution of the optimal probe location the ﬁrst and the second moment, corresponding to the mean and the covariance, are computed. Then, an eigenanalysis allows to draw an ellipsoid centered at the mean, oriented with the eigenvectors, and scaled with the square root of the eigenvalues of the covariance matrix. Indeed, as shown in Fig. 4.2 (right) the ellipsoid (magniﬁed by a factor of 10 for reasons of better visibility) is aligned with the PDF. It can be interpreted as a principal component analysis of the PDF: large eigenvalues imply that the distribution is wide (has a high variance) in the corresponding direction.
Moreover, in Fig. 4.3 (left) the ellipsoid is embedded in the surrounding artiﬁcial anatomy. There, the sensitivity of the optimal probe placement with respect to variations in σ (left) is visualized. We see that the cigar-shaped ellipsoid is aligned with the linear vessel.
However, as mentioned earlier, an interpretation of this results must be done with care since a high sensitivity of the optimal probe position (as seen here) can also be
Figure 4.4:The 60◦C iso-surface of the temperature is shown for the optimal probe place-ment obtained in collocation point 5 (left), collocation point 16 (middle) and collocation point 23 (right) of the sensitivity analysis of the probe placement w.r.t. uncertainties in σ. In all three cases the lesion is ablated completely.
−1 −0.5 0 0.5 1
Figure 4.5: Approximation of the graph of the objective functionf(x) where the position x varies along the eigenvector corresponding to the largest eigenvalue of the covariance matrix (longest principal axis of the ellipsoid).
the result of a ﬂat energy graph of the objective functionf (cf. Sect. 4.4, p. 67). In such a case, not only variations in the tissue parameters, but also small changes in e.g. the numerical settings can inﬂuence the optimal probe location strongly. In fact, the situation of the artiﬁcial scenario induces such a false sensitivity. The reason is that the same fraction of the lesion can be ablated for many diﬀerent placements of the probe. Thus, for many diﬀerent probe locations, the value of the objective function is very similar. The energy graph of the objective function is ﬂat and the minima found by the optimizer for diﬀerent realizations of the material parameters can be far apart from each other. Indeed, as shown in Fig. 4.4 there are diﬀerent placements which allow for a complete ablation of the ellipsoidal shaped lesion in the artiﬁcial example.
In order to diﬀerentiate between these two cases, and to detect false sensitivities, one can e. g. analyze the values of the objective function f. In Fig. 4.5 (left) a one-dimensional interpolation of the energy in direction of the largest eigenvalue of the covariance-matrix (i. e. in direction of the largest principal axis of the ellipsoid) is depicted. This interpolation is obtained by the same means as the approximation of the stochastic processes: Here, a one-dimensional collocation table with approxi-mation order k = 3 containing 9 collocation points has been used. At these points
the energy f was evaluated for a probe location and material parameters which lie central in the PDF. Indeed, Fig. 4.5 (left) shows a relatively ﬂat graph and just relatively small variations of the objective function’s value. Thus the strong sensitivity shown by the PDF and the ellipsoidal representation of the covariance matrix in Figs. 4.2 and 4.3 (left) may be due to an insensitivity of the objective function w.r.t. the probe location. Hence, as seen here for the artiﬁcial scenario, the local graph of the objective function is a useful tool for evaluating sensitivity distributions.
Sensitivity of the Optimal Probe Orientation
The visualization of the PDFs of the optimal probe orientation is much easier, since the orientation lies on the two dimensional sphere S2. In Fig. 4.3 (right) a PDF of the optimal probe orientation is shown by a color coding of the sphere. Also for the probe orientation, a false sensitivity can be caused by an insensitivity of the objective function w.r.t. rotations of the probe (e. g. spherical lesion with no vascular structures). Again, an analysis of the local behavior of the energy graph can help to rule those cases out.