There are three possibilities to evaluate time-dependent field variables, such as velocity, pressure, temperature, humidity, turbulence quantities, etc.: 1) visualization of individual time steps, 2) visualization of changes between time steps, and 3) visualization of developments within finite time intervals. In this work we focus on the first two techniques. The AMIRA software was used for the visualization of single time steps (stationary flow analysis), and the SimVis3 software was used for the exploration of flow dynamics (transient flow analysis).
3.1 Stationary Flow Analysis
To obtain an overview of the simulation results, individual time steps of the inhalation and exhalation phases were selected and the respective flow quantities were visualized. Since all data is encoded on volume cells of an unstructured grid, appropriate visualization methods are to be used. For stationary analysis we employed techniques such as color coding of scalar or vector valued quantities, contour plots, iso-surfaces, vector visualization, line integral convolution (LIC), streamline and particle visualization with an animation of seeded streamlines or particles along respective field lines (cf Fig. 5). These techniques are provided by the AMIRA software  that has already been employed for the geometric modeling in this work.
Fig. 5. Visualization of flow quantities: Relative humidity for an exhalation state (a), flow velocity within the nasal cavity (b) during inhalation (top) and exhalation (bottom), inspiratory flow velocity of up to 15 m/s with color coded stream lines (c), illuminated streamlines with color coded planar LIC visualization (d), and particle visualization along field lines with discrimination of left and right side (e).
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Due to its ease, we made heavy use of the possibility to visualize any quantity on top of arbitrarily oriented planes with an appropriate color coding scheme. In Figure 5 (b), for instance, we see on a section of the nasal cavity that for inhalation (top view) and exhalation (bottom) the main path of airflow differs. Note that the lung induces the flow by creating a reduced pressure, causing air flowing into the nose. Thus, inflow is rather non-directional, whereas outflow is directed due to the shape of the nose. Temperature and humidity are also visualized in this manner (cf Fig. 5 a), enabling us to assess inhalation and exhalation while obtaining a first impression of the scalar distribution and being able to check the correctness of boundary conditions. Iso-surfaces were generated to locate pressure minima, and LIC techniques were best suited for the understanding of flow structures  as shown in Figure 5 (d). A visualization of the flow velocity during inhalation and exhalation using color coded streamlines yields a good overview of spatial flow characteristics for individual time steps. Looking at the velocity field, for example, we can easily grasp the flow structure and see that a maximum of up to 16 m/s is reached within the main nasal cavity (Fig. 5 c). In Figure 5 (b) we used illuminated streamlines in combination with color coded planar cuts, mapping the magnitude of velocity to color values. Even for a single time step the flow field yields enough information to visualize flow characteristics in an animated fashion. Illuminated streamlines can be animated with Amira , as well as particles that - after being seeded - move along the streamlines  to visualize flow direction (Fig. 5 e). Seeding of streamlines or particles can be accomplished, for instance, by locating pressure minima in order to visualize vorticity.
All these techniques allow for a first visual exploration of the data. The challenge is to locate important flow features and to apply ap-propriate visualization techniques to emphasize them. The spatio-temporal relations, however, cannot be easily assessed in this manner. Therefore a transient flow analysis is needed.
3.2 Transient Flow Analysis
Without prior knowledge of relevant features in multi-variate, time-dependent 3-D data, the exploration is a highly demanding task that typically necessitates interactive approaches. On the other hand, visual analysis and data exploration often turns out to be difficult or even impossible without employing semi-automatic visualization methods that require only minimal user interaction. Thus, we employed SimVis, a highly interactive visualization system with basic built-in data analysis tools. Furthermore, we extended SimVis to identify features and to obtain findings from which hypotheses regarding physiological processes of nasal breathing can be derived.
3.2.1 Multiple Linked Views for Feature Selection
The framework being used to perform the transient flow analysis provides multiple linked views for concurrent visualization, exploration, and analysis of multi-variate data. Within the scope of our exploration we utilized multiple 3-D views in combination with several types of attribute views, e.g. scatterplots, (see Fig. 8 c), histograms, parallel coordinates ,  (d), or time series visualizations (e, f) in order to obtain an overview of the flow data. The interactive brushing functionality that is provided by all attribute views enables us to select data regions that exhibit interesting behavior. Our feature selection results in multiple Degree of Interest (DOI) volumes each storing a scalar field that equals 1 in relevant and 0 in negligible regions of the data set. Since flow data usually are smooth, it is difficult to sharply discriminate between relevant and irrelevant data regions. Thus, un-sharp, i.e. non binary DOI values were applied (DOI ∊ [0 …1] ⊂ R) to convey this uncertainty. Moreover, such DOI attributions can be combined using fuzzy logic operations in order to derive even more complex features .
3.2.2 Volume Rendering Techniques
For an interactive exploration of complex spatio-temporal data via Scientific Visualization (SciVis) methods such as volume rendering, any feature selection within attribute views necessitates an immediate visual feedback on the respective spatial location. In this regard, the nasal airflow data pose two main challenges: First, the underlying unstructured grid is fairly large (1.3 million tetrahedra and 2.2 million prisms which in total roughly equal 8 million tetrahedra), and second the complex anatomical structure of the nasal airways with all of its cavities may lead to occlusion problems and visually cluttered images (cf Fig. 6, left and right). We tackle these problems by utilizing two different volume rendering techniques. In Fig. 8 (b) a point-based overview rendering of the nasal airway data is shown. In this example, volume cells are represented by Gaussian splats that are blended while being rendered in back to front order. Data regions that have not been selected by the user (i.e DOI = 0) are rendered with low opacity and no color, whereas selected features are displayed less transparently with color that is computed from the data attributes via an appropriate transfer function.
The main advantage of this rendering approach is its performance, as long as the viewing direction remains constant (otherwise an expensive visibility sort becomes necessary). This approach was used to interactively explore the spatio-temporal evolution of selected features and to rapidly switch between different time steps. Main drawbacks of this method are the low rendering quality (circular Gaussian splats only approximate polyhedral cells) as well as the lack of a distinct visualization of object boundaries.
In contrast to the point-based approach, a volume rendering based on ray-casting allows for a visualization of different semi-transparent boundaries. Only regions with DOI > 0 are visualized using volume rendering, whereas unselected portions of the data set are skipped by our ray-casting implementation. Choosing a suitable rendering mode for boundaries is essential as they provide context information that enables us to locate selected features within spatial structures Fig. 6 (left, right) shows two different boundary surface rendering methods: On the left, a transparent Phong shaded surface rendering is used, whereas on the right a shading model with silhouette enhancement is chosen. Obviously, both approaches are not necessarily adequate for the presented case; the surface rendering lacks contrast because the selected feature within the nasal cavity is occluded by multiple irrelevant surfaces. The silhouette rendering, however, introduces visual clutter, thus making it difficult to grasp spatial relations.
Therefore, we chose a combination of both visualization methods by using surface shading for all object boundaries facing away from the observer, and silhouette enhancement for front facing faces only, as shown in Fig. 6 (center). This rendering technique effectively halves the number of surfaces occluding volumetric features while still supplying enough depth queues by rendering back facing surfaces. In addition, silhouette rendering adds detail for front facing surfaces. For a correct composition of transparent surface and volume rendering an extended depth peeling method is applied .
3.2.3 Parallel Coordinates
Parallel coordinates are generally suited for exploring relationships between multiple attributes. Their visualizations, however, tend to become increasingly cluttered the more data items (cells of a grid) are to be considered . In order to reduce visual clutter, inherent to parallel coordinates when visualizing millions of data entries, we propose a technique based on the diffusion of a noise texture via a 2D field of 2 × 2 tensors. This tensor field is constructed by computing , per pixel, with N being the number of lines covering the image pixel and vi the direction of the ith line in image space. The resulting tensor per pixel encodes directional information on all lines passing through that particular pixel as illustrated in Fig. 7: On the left the directions of the lines through a pixel are not uniformly distributed around that pixel; this results in an anisotropic tensor visualized by the red ellipse (each ellipse is constructed from the eigenvectors of the respective tensor). Lines with uniformly distributed directions passing through an image pixel result in an isotropic tensor as illustrated on the right hand side of Figure 7. Simple diffusion is performed on a noise texture via the tensor field in order to create a pattern that is blurred along the predominant line direction at each image pixel (see inlays in Fig. 7). We use 16 iterations with a 7 × 7 Gaussian filter kernel to achieve sufficient image quality retaining interactive frame rates with our current GPU implementation. The main advantage of using this approach is that even when drawing a large number of lines on top of each other (in the example shown in Fig. 8 (d) 21 time steps are visualized resulting in about 75 million lines), the user is still able to recognize trends which otherwise could easily have been missed. Such a scalable implementation of parallel coordinates turned out to be very useful for the visual exploration of the high dimensional attribute space of the nose breathing data set (Fig. 8 d).
Fig. 7. Different line direction distributions through an image pixel result in anisotropic (a) or isotropic (b) tensor representations that can be used for blurring a noise texture in the parallel coordinate view.
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3.2.4 Visual Exploration
To gain insight into multi-variate, spatio-temporal data a generic work-flow can be followed which closely matches the information seeking mantra: "Overview first, zoom and filter, then details-on-demand" as proposed by Shneiderman . At first, the user typically needs to get a general idea of all potentially interesting data attributes and their spatial distribution within the simulated domain. This can be achieved by setting up an exploration session as shown in Fig. 8. As a starting point simple selections (e.g. high or low attribute values) are defined by the user within a particular attribute view, whereas all other linked views are simultaneously updated.
This initial step is illustrated in Fig. 8 where relatively cold air (up to 23 .5° C) is selected by the lower brushing rectangle (Fig. 8 c) to obtain an overview of the aerodynamic heating during the inhalation phase of the first breathing cycle. An additional selection of warmer air (between 29° and 31° C) was defined in order to highlight regions which have already been heated by the mucous walls, but still being colder than the air within the paranasal cavities. Both selections do represent a feature that can be explored in all other views shown in Fig. 8. The 3-D visualization in Fig. 8 (a) displays these features 450 ms after the first breathing cycle has started. The two parts of the selected data region (cold and heated air) can be differentiated by their coloring, where blue-green corresponds to colder flow and red represents the warm air. It can be observed that the main portion of cold air travels along one distinct path as indicated by the dotted line and is heated approximately 4 .5° C during its traversal. Note that a reservoir of warm air is located below the dotted line as indicated by the label. Further exploration of this feature will be conducted in subsequent sections.
A second notable structure can be seen on the left hand side of Fig: 8 (a) behind the nasal valve: The geometry of the nasal isthmi (the narrowest passage between nostrils and nasal atrium) appears to result in vortical flow which again leads to early mixing of warm and cold air, as indicated by the black arrow. The overview visualization presented in Fig. 8 (b) illustrates the same features as in (a), but with flow velocity mapped to color. Focusing on high velocity regions of the selected data, the main trajectory of the inhaled air can be discerned. Note that in the region of the olfactory epithelium the airflow velocity during the inhalation phase remains relatively low, resulting in a preservation of humidity and warmth which again are optimal conditions for the reaction between scent molecules and sensory receptors. Furthermore, the relatively high air velocity in the middle passage of the nasal cavity suggests that due to the resulting pressure drop within the inhalation phase a ventilation of the frontal and maxillary sinuses occurs which are connected to the nasal cavity via small ducts.
The temporal development of the selected features has not yet been investigated in detail. Even though the scatterplot that was used to select the cold and warm air is capable of displaying a selected time-interval, the overall evolution of features over time is difficult to comprehend. Thus, a time series visualization similar to approaches proposed by Akibay and May  and Muigg et al.  has been used to plot the selected feature over time within the context of an additional data attribute (cf Fig. 8). Here, for each volume cell the actual relative humidity (e) and the pressure (f) are plotted over time (x-axis). Color indicates to which degree a curve is selected (red equals fully selected and white represents unselected data) whereas brightness represents the number of curves overlapping each other. The vertical yellow line indicates the current time step and is linked to both time series views in correspondence to the 3-D visualizations where the respective state of the selected data is displayed.
Fig. 8. Overview of a visual exploration session: Different attribute views (c), (d), (e), and (f) are used to interactively select data regions of interest which can be examined and modified in all other visualizations. The 3-D views (a) and (b) display the selections within their spatial context.
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Since the SimVis framework does not yet support the selection of semantically different portions of a flow domain (e.g., environment or nasal airways) the time series view for the pressure attribute (Fig. 8 f) was used to exclude the anterior inflow region from all selections by employing an inverted time step brush that is represented by the pink rectangle. This brush defines a DOI function which is 0 if the associated scalar attribute at a particular time step lies within a user defined interval (in this case a relative pressure close to 0 Pa) - and 1 otherwise. As a reference, the time step of maximal negative pressure at the interface to the lung was selected since it exhibits the strongest pressure gradient between the nasal airways and the environment.
The humidification of inhaled air is another important physiological function of the nasal airways Fig. 8 (e) gives an overview of the temporal development of the relative humidity within the upper respiratory tract. By investigating its progression, which is indicated by a visualization approach based on line integral convolution, a steep drop in relative humidity can be observed which stabilizes after approximately 200 ms. Note that the warm/cold air feature previously defined in the scatterplot is shown as two distinct red regions in the respective visualization. The region of higher humidity corresponds to the warm air selection whereas the drier air corresponds to the selected region with lower temperature. After 700 ms the relative humidity increases again since low pressure from the lung diminishes (see Fig. 8 f).
The pressure-time curves shown in Fig. 8 (f) enable us to relate time steps of interest to the entire breathing cycle. Looking at the visualization it becomes apparent that after a short period of time (roughly 150 ms) the basic distribution of pressure values remains constant with respect to scaling (i.e., there are no curves crossing each other). This means that after an initial turbulent phase between exhalation and inhalation, flow features, such as vortices, remain fairly stationary.
In Fig. 8 (d) our parallel coordinates implementation is used to gain an overview of the selected data regions in multiple additional data dimensions. Like in all other attribute views currently selected features are highlighted in red whereas the remaining data is shown in white/grey. The axes correspond to the scalar attributes pressure, velocity, mass fraction of humidity, temperature, relative humidity, and time. At a first glance, a strong correlation between mass fraction of humidity and temperature can be recognized (this assumption is further confirmed by the horizontally blurred noise indicating that most of the lines are parallel to the x-axis). In addition, at least the selected data regions exhibit a correlation between temperature and relative humidity. Furthermore, a negative correlation between flow velocity and mass fraction humidity can be observed. As with the scatterplot we also explored the temporal distribution of the data values by interactively selecting time intervals for visualization. The region of the data set outlined in yellow and marked with "environment" in Fig. 8 (d) corresponds to the anterior inflow region that, as mentioned earlier, was excluded from the selections via a time step brush (Fig. 8 f).
3.2.5 In-Depth Visual Analysis
After an initial exploration of the nasal airflow data, the next step of our investigation corresponds to the "zoom and filter" portion of Shneiderman's mantra. In order to gain more insight into nasal breathing we use our analysis framework to derive additional information such as spatial/temporal gradients or even more complex flow parameters such as λ2 . This information can then be used to further restrict/modify previously defined features, thus enabling us to hypothesize about phenomena being observed in the course of our exploration.
Below we present an in-depth analysis of two distinct features: Fig. 9 shows an overview of three different time steps: 100 ms, 600 ms, and 1 s after the beginning of the inhalation phase which lasts roughly 1.05 s. On the very right of Fig. 9 an additional inferior view is shown 600 ms after the beginning of the inhalation phase. In the middle row a selection is depicted which is based on the warm air feature that was explained in the previous section. This selection has been further restricted to low velocity regions and temperature is mapped to color. The accompanying inlets show the parallel coordinates views at corresponding time steps that were used to define the selection via two brushing rectangles (indicated in yellow). In the top row of images the warm/slow air feature was further restricted to vortical regions within the flow, as defined by λ2. Color is mapped to the strength of vorticity indicating weak (green) and strong (red) vortical air motion. The inlets shown in the top row provide an overview of all vortical regions without any restriction to warm/slow flow regions.
Fig. 9. Different time steps of a feature defined by low flow velocity and relatively high temperature in the middle row. In the top row the feature is further restricted to vortical regions as defined by the λ2 criterion. On the very right an additional inferior view illustrates how vortex regions can be related to heat distribution. The parallel coordinate plots show the data at corresponding time steps.
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The first finding discussed in this section is the early mixing of cold and warm air behind the nasal valve. It is notable that this process is much stronger on the left side of the nose than on the right. This can be seen clearly at 600 ms when comparing the selected regions indicated by (L) and (R). The difference is caused by a slight asymmetry in the nasal anatomy as shown in the close-up image. Here, two different (weak) vortices, which are indicated as grey dotted lines, originate behind the nasal isthmi. On the left side of the nose the progression of the vortex points slightly up which results in a stronger expression of this feature. This demonstrates how minor deviations in the nasal geometry may already have a major effect on the airflow, which has to be taken into account for therapy planning in rhinosurgery.
Another finding that has already been hinted at in the previous overview is related to the existence of a warm air reservoir in the nasopharynx, visible in all three time steps in Fig. 9 (see red ellipses). This is remarkable since all inhaled air must pass through this region and it was assumed that its relatively high speed does not promote any warming. However, all selected regions shown in the middle row of Fig. 9 do represent warm but slow moving air (as selected in the parallel coordinates view). This indicates that most of the fast moving cold air is traveling along the trajectory already identified in Fig. 8 (a). A main cause for this seems to be a complex system of vortices located in the upper regions of the nasopharynx marked by the blue ellipse in the close-up picture on top at 600 ms. It shows vortical regions without the restriction to warm/slow air with color indicating vortex strength. Below the vortex system, that is transporting fast/cold air, the aforementioned slowly moving air is warmed up. This process is further supported by a secondary system of two parallel vortices indicated by the ellipses colored in orange (Fig. 9). The vortices originate at the transition between nasopharynx and the inferior meatus on both sides of the nasal airways (axes indicated as stippled lines). Note how they constitute the shape of the regions of slow moving warm air.