Fig 1.
Harp seal maxilloturbinate tomograms ordered from youngest to oldest.
The tomograms are reoriented through the densest part of the maxilloturbinate masses, and the specimens’ relative ages are based on measured condylobasal lengths, CBL. Maxilloturbinate bones shown in black, airways in white. Enclosing nasal cavity walls not shown. values given below are relative to adult size (
). Scale bar 50 mm. (a) Juvenile (specimen 7357),
. (b) Juvenile (specimen 7498),
. (c) Juvenile (specimen 7360),
. (d) Adult (specimen 7495),
.
Fig 2.
CT reconstruction of the maxilloturbinate region of the adult harp seal (specimen 7495), sectioned parasagittally.
Blue line indicates approximate location of reoriented cross section reproduced in Fig 1d.
Fig 3.
We model the two-dimensional backbone of a maxilloturbinate mass as a series of connected nodes.
The nodes are constrained within a closed circular boundary representing the nasal cavity wall (thick line) of radius B. The first node (a) is fixed to the boundary. Neighbour nodes (a and b, b and c, etc.) are kept within a certain distance of each other by means of an overdamped spring force along the connection. Non-neighbour nodes (b and d, a and c, etc.) repel each other with a characteristic minimum distance
. New nodes grow from branch tips (e.g. at d). Except for a, all nodes are repelled from the boundary.
Fig 4.
Time series of modelled growth with our model.
Boundary circle (nasal cavity walls, with radius B) shown as a thick line. The node tree is rooted to the left side of the boundary (red arrows). The final result shows longer branches towards the right side and edges of the system due to forking being turned off and branches only elongating postnatally (i.e. beyond ).
values correspond to rescaled simulation time (i.e. relative age, 0-1). (a)
. (b)
. (c)
. (d)
. (e)
. (f) t∗ = 12.8 × 10−2. (g)
.
Fig 5.
Sample simulation results from alternative branch growth models.
(a) Hannezo’s branching and annihilating random walk model [21]. (b) DLA [22].
Fig 6.
Evolution of porosity (ϕ) as a function of rescaled CBL (relative age; ) measured in seal skulls.
Dotted trend lines are fitted via linear regression of actual measurement values in harp seals (blue) and grey seals (red). The black line corresponds to the mean porosity computed through simulation runs, where (t in growth timesteps).
Table 1.
Measurements in seal tomograms and model results (mean ± standard deviation).
Fig 7.
Evolution of the rescaled hydraulic diameter () and complexity (
) in both simulations and seal tomograms as a function of simulation timestep and relative age (
).
Grey seal values (red stars) measured by [2] ( values rescaled by a measured nasal cavity width of respectively 29.0 m and 45.1 m). (a) Rescaled hydraulic diameter. (b) Complexity.
Fig 8.
Backbone dimension () as a function of relative age (
).
Shaded grey area represents the limits of computed at each timestep over 100 individual simulation runs.
Fig 9.
Seal tomogram and simulation Strahler diagrams.
The root node (base of the root branch w = H, in dark red) is selected in tomograms to correspond with the maxilloturbinate root. Branches with low Strahler orders are shown in colder colours (), with warmer colours corresponding to higher orders. (a) Juvenile harp (specimen 7357). (b) Adult harp (specimen 7495). (c) Juvenile grey (specimen SE1). (d) Model
Fig 10.
Strahler number (H), branching ratio () and length ratio (
) as functions of relative age (
).
Shaded grey area represents the limits computed at each timestep over 100 individual simulation runs. (a) Strahler number. (b) Branching ratio. (c) Branch length ratio.
Fig 11.
Modelled development of a Mediterranean monk seal maxilloturbinate.
Monk seal (specimen NHMO M115) tomogram reproduced from [2] under the terms of the Creative Commons CC BY license (scale bar 50 m), juxtaposed to final model result obtained for a target porosity . (a) Mediterranean monk seal. (b) Model.
Fig 12.
Simulation outcomes obtained for two variants of our model: the removal of density-based growth and the removal of branch avoidance. Red dots indicate points of collision between branches. Simulations were stopped beyond 50 branch intersections for clarity. (a) No density-based growth. (b) No branch avoidance.
Fig 13.
Simplified geometry of a channel of dimensions ℓ × h × b with pressure gradient ΔP and shear stress on the walls σ.