Pre-surgery planning tool for estimation of resection volume to improve nasal breathing based on lattice Boltzmann fluid flow simulations

The output of segmentation defines the fluid domain of the simulation. The resolution for the LB simulations, 0.234 mm, was found by a mesh convergence study [12]. CT data were isotropically resampled (python ndimage.interpolation.zoom [17]) to 0.234 mm, thresholding at −300 Hounsfield [18, 19] units for airway segmentation, and binarized. Inlet and outlet boundary conditions for nostrils and nasopharynx were set as a sphere with a diameter of 70 mm overlapping both nostrils and a cuboid (60 × 40 × 30 mm³), respectively. The sphere’s center was defined in coronal slices just anterior to the tip of the nose. Region growing with a seed point inside the nasal airway passage was used to segment air. In the most inferior axial slice, the centroid of the air voxels defined the midpoint of the cuboid. The surface voxels of sphere and cuboid were saved as label maps as boundary conditions for LB simulations.

At the cuboid a Dirichlet velocity boundary condition (Sailfish CFD: NTRegularizedVelocity [6]) of bilateral inhalation flow rate of 600 ml/s was set, the pressure boundary condition on the sphere was set to ambient pressure (Sailfish CFD: NTDoNothing [6]), on solid surfaces the NTWallTMS [6] boundary condition was used. All voxels were initialized with \(\vec{v} = \vec{0}\) m/s, simulation stopped when the airflow was fully developed, i.e. the pressure drop between inlets and outlet, after 0.0125 s [12]. The flow is characterized by a Reynolds number (Re) [9] of about 2800 based on the hydraulic diameters at the nostrils and the volume flux. The large eddy simulation (LES) turbulence model Smagorinsky [6] (with constant cs = 0.14, found in [12]) was used to simulate the transitional and unsteady features of nasal airflow [20] with the D3Q19 lattice element [6]. Computational time on a NVIDIA^® RTX 2080 TI was < 5 min. All LB simulations were performed with 13,605 time steps and a mesh with about 9 million cells.

Velocity streamlines were calculated with Paraview’s stream tracer tool [21]. Continuous streamlines from inlet to outlet were generated, integration direction was set to both, integrating with Runge–Kutta 4–5. The streamline analysis is based on the last simulation result of the unsteady simulation with a maximum streamline length of 0.3 m, exceeding the size of the nasal cavity [22]. With Paraview option “Seed Type: Point Source” [21] 532 streamline start points, found in a preliminary study (Fig. 4), were randomly positioned within the sphere at the nostrils.

A high pressure gradient region indicates a constriction [9], a region with increased nasal resistance. A differential pressure criterion \(\frac{{{\text{d}}p}}{{{\text{d}}l}}_{{{\text{streamline}}\;{\text{healthy}}}}\) (p … pressure, l … streamline) along the streamlines was determined in a preliminary study of five healthy persons without nasal septum deviation to define a baseline, Fig. 3. When the differential pressure at a streamline locally exceeds \(\frac{{{\text{d}}p}}{{{\text{d}}l}}_{{{\text{streamline}}\;{\text{healthy}}}}\) a HPGR is identified at this position.

The pressure gradient was iteratively changed over the extracted streamlines [−5, 0] Pa/mm in steps of −0.1 Pa/mm to find the critical locations. Figure 3 shows local pressure variations. HPGRs based on initial fluid flow simulation are used for optimization. Desired surgery points were selected by the user in the set of HPGRs to define the regions of interests (ROI) (see Fig. 2) with a graphical user interface, Fig. 10.

Every HPGR is placed in a ROI and an optimization cube (OC) of 10 mm side length, Fig. 2b. During optimization, the wall (see Fig. 2a,b—blue line) is moved inside the ROI in surface normal direction towards the boundary. Laplace filtering (scipy.ndimage.filters.laplace [17]) detects edges of the dataset; adding this edge the “wall” is moved by one voxel, yielding resection volume 1 (see Fig. 2b).

Initially ROI coincides with OC. In every iteration ROI size increases by 0.468 mm. HPGRs are recalculated at every optimization step and considered for optimization inside the ROIs (see Fig. 2c). Every iteration is initialized with the initial simulation result, new volumes with \(\vec{v} = \vec{0}\) m/s; LB simulation stopped when pressure drop between inlets and outlet became fully developed after 0.00625 s. Optimization stops when < 5 HPGRs are found in all ROIs. Figure 2c shows the resulting resection surface after two iterations.

The solutions of the simulations are required to be mesh independent [23]. In all simulations the surface averaged pressure \(\frac{{\smallint p {\text{d}}A}}{A}\) at the investigation planes (see Figs. 5, 6) for every patient was used. The grid convergence index (GCI) [23] was used to determine the adequate lattice resolution and was used in this study.

Table 1 shows the results of the mesh independence study (for details please see [12]) for the five patients with pre- and post-surgery LB simulations with 0.0125 s integration time. All simulations showed asymptotic grid convergence with p > 1 and for all simulations \({\text{MI}}:\, = \,\frac{{gci_{23} }}{{r^{p} gci_{12} }} \approx 1,\) mesh independence is valid [23].

HPGRs were identified by a preliminary study on anonymous CT datasets of five patients with and five without septum deformation (Fig. 3). The onset is the first appearance of HPGRs, at ≂2.5 Pa/mm (patients 1, 2 and 4) and < ≂5 Pa/mm for the rest. Healthy persons show an onset of HPGRs at smaller values, see Fig. 3. The HPGRs for healthy individual were averaged, thick black curve. 25 HPGRs at −1.1 Pa/mm were chosen as a surgery criterion (Fig. 3) as this created a regime (arrow in Fig. 3) that visually separated patients from healthy individuals well.

In a second preliminary study, the number of streamlines was determined to cover a sufficient amount of the resection volume per optimization iteration. Streamlines number was varied from 0 to 10,000 in steps of 50 and was randomly placed at the inlet sphere. 9000 streamline starting points gave a stationary solution (Fig. 4) within 5 min per patient at 532 streamline start points, at least 50% of the total amount of resection volume is covered within 20 s. As a trade-off between speed and accuracy 532 streamline starting points were chosen for the simulations. This is further justified as the first steep increase of detected resection cells is covered. A larger number of starting points were not deemed feasible for this initial investigation.

LB simulation results on preoperative CT data of the five patients with nasal septum deviation are shown in Fig. 5. The color gradient depicts the static pressure drop originating from anatomical constrictions. Investigation planes were chosen at coronal positions with high concentration of HPGRs (Fig. 5). These positions were varied to test the proposed optimization.

Postoperative CT data were LB simulated for cross section comparison, Fig. 6. The overall pressure drop \(\Delta p\) between nostrils and nasopharynx on pre-, post-, and virtual surgery data sets are summarized in Table 2. The change predicted by the surgical planning method (p_pre – p_{virtual surgery}) is related to results based on the pre-, and postoperative LB simulations. The ratio relates planned to achieved results.

Virtual surgery LB solutions with a local critical pressure gradient of −1.1 Pa/mm yielded an average pressure drop between nostrils and nasopharynx of −27.6 Pa. Pre-surgery LB simulations of patients 2 and 4 showed overall pressure drops of ~ 40 Pa between nostril and nasopharynx, compared −36 Pa, the average of five healthy individuals. Most of the HPGRs were found at the vestibulum nasi; at patient 2 only a small amount of HPGRs was determined at the turbinate. According to the pre-surgery pressure drop in Table 2 and Fig. 5, patients 1 and 3 seem to have a “medium-severe breathing problem”; patient 5 the “most severe” problem with a too small airflow cross section throughout the nasal channel.

Figure 6 shows the coronal slices defined in Fig. 5. Background images are either based on pre- (a) or postoperative (c) CT data. (b) shows the calculated resection volume, “virtual surgery” in yellow. Between 10–29 optimization iterations were necessary to reach the optimum. The actually resected volume is the Boolean difference between initial pre-surgery CT data and the final optimization result. Every optimization step takes about 3 min of computational time. Red surfaces show predicted resection areas that were actually resected. Patient 1 had a nasal airflow problem at the middle meatus, the ostium and the duct of the maxillary sinus. Optimization suggested resection there. The postoperative CT shows an increased airflow cross section. In patient 2 the nasal septum was straightened. Similar airflow cross section between virtual surgery and post-surgery was determined (see Table 3). In patient 3 optimization suggests an air-flow cross-section increase on the left nasal cavity, which was also confirmed by postoperative CT dataset. The right nasal cavity was not changed by optimization; however, in the postoperative CT dataset airflow cross section is increased there, too. Furthermore, postoperative CT data of patient 3 show a swollen middle nasal concha. The cross section was similar to the pre-surgery CT dataset. For patient 4 only minor corrections are performed close to the nostrils. Patient 5 had a nasal airflow problem at the inferior nasal meatus, which is also confirmed by the postoperative CT. Pre-surgery CT data revealed a reduced airflow cross section due to a reactively swollen nasal mucosa, a short-term clinical side effect. Figure 7 shows the airspace cross-sectional area versus coronal distance from the nostrils. For the cross-sectional area evaluation, the sphere at the nostrils, the outlet cuboid, the frontal sinus and the maxillary sinuses were not considered. The coronal investigation planes (Figs. 5, 6) of patients 1 to 5 are at coronal distances from the nostril of 57, 22, 26, 0 and 59 mm, respectively. In the post-surgery model at patient 3, at coronal distance 25–65 mm from nostril, the swollen state of the nasal cavity is identifiable due to a reduced airspace cross-sectional area. Results at coronal distance > 75 mm show differences between pre–post airspace cross-sectional areas partly due to the simple segmentation approach implemented. Here the sphenoid sinus was not removed from the investigation as no coronal investigation plane was positioned there; it was removed surgically.

Table 3 shows the airspace cross-sectional area (A) evaluations on the investigation planes of Fig. 6. Simulated resection volume agrees within 15% with postoperative data.

Coronal slice positions: Patient 1 and 5: the posterior part of the nasal cavity, including the tails of the turbinates; Patient 2 and 3: anterior part of the nasal cavity including the heads of the turbinates and the infundibulum; Patient 4: area underneath the bony and cartilaginous vault, the attic; [24]