Annals of Botany 108: 1001– 1011, 2011
doi:10.1093/aob/mcr067, available online at www.aob.oxfordjournals.org
PART OF A SPECIAL ISSUE ON FUNCTIONAL– STRUCTURAL PLANT MODELLING
A mathematical framework for modelling cambial surface evolution using a level
set method
Damien Sellier 1,*, Michael J. Plank 2 and Jonathan J. Harrington 1
1
New Zealand Forest Research Institute Ltd, Private Bag 3020, Rotorua 3046, New Zealand and 2Department of Mathematics
and Statistics, University of Canterbury, Private Bag 4800, Christchuch, New Zealand
* For correspondence. E-mail [email protected]
Received: 29 November 2010 Returned for revision: 13 January 2011 Accepted: 11 February 2011 Published electronically: 5 April 2011
† Background and Aims During their lifetime, tree stems take a series of successive nested shapes. Individual tree
growth models traditionally focus on apical growth and architecture. However, cambial growth, which is distributed over a surface layer wrapping the whole organism, equally contributes to plant form and function. This study
aims at providing a framework to simulate how organism shape evolves as a result of a secondary growth process
that occurs at the cellular scale.
† Methods The development of the vascular cambium is modelled as an expanding surface using the level set
method. The surface consists of multiple compartments following distinct expansion rules. Growth behaviour
can be formulated as a mathematical function of surface state variables and independent variables to describe
biological processes.
† Key Results The model was coupled to an architectural model and to a forest stand model to simulate cambium
dynamics and wood formation at the scale of the organism. The model is able to simulate competition between
cambia, surface irregularities and local features. Predicting the shapes associated with arbitrarily complex growth
functions does not add complexity to the numerical method itself.
† Conclusions Despite their slenderness, it is sometimes useful to conceive of trees as expanding surfaces. The
proposed mathematical framework provides a way to integrate through time and space the biological and physical
mechanisms underlying cambium activity. It can be used either to test growth hypotheses or to generate detailed
maps of wood internal structure.
Key words: Dynamic model, level sets, surface growth, vascular cambium, wood formation.
IN T RO DU C T IO N
Woody tissue is formed on the inner surface of the secondary
meristem. This meristem, often referred to as the vascular
cambium, envelops the non-elongating parts of trees. Over the
life of a tree, the development of the vascular cambium is
subject to many influences, including water availability (Abe
et al., 2003) mechanical stress (Linthillac and Vesecky, 1981),
photoperiod (Larson, 1994), auxin and carbohydrates (Uggla
et al., 2001), and temperature (Gričar et al., 2007). By impacting
on cambial activity, those variables affect the amount of wood
produced as well as the structure and macroscopic properties
of that wood. The strong dependency on the physical environment places trees among the most accurate climate archives
(Hughes, 2002). Environmental fluctuations result in xylem
being a highly heterogeneous material (Downes et al., 2009).
Genetic factors also contribute to patterns of variation in wood
properties (Zobel and Van Buijtenen 1989). To a large extent,
heterogeneity is beneficial to trees. For example, the formation
of reaction wood allows an active control of tree stem orientation
to minimize the action of gravitational forces. In contrast, many
technological issues in wood processing originate from the
aforementioned heterogeneity.
Heterogeneity occurs at different physical scales from treewide to within-ring patterns of variation. Such patterns also
differ with every wood property and are direction-dependent,
whether longitudinal, radial or tangential. To understand how
those multiscale and anisotropic patterns arise, secondary
growth processes must be very finely described. On the other
hand, modelling wood formation is better performed at the scale
of the whole organism if it is to assist in attaining better appreciation of both tree function and wood technological performance.
Simulating wood structure at tree scale, however, is challenging
because of its multiscale and anisotropic features.
Tree growth models can be split into three main categories.
Those models serve different purposes depending on whether
they investigate architecture, cambial growth or visual
display of trees.
Architectural models typically focus on primary growth processes (Prusinkiewicz and Lindenmayer, 1990; de Reffye
et al., 1997). At the organism level, transverse dimensions
are frequently overlooked (Le Roux et al., 2001;
Prusinkiewicz, 2003). While the skeletal view seems appropriate for investigating light interception and space colonization,
it is more questionable for other aspects such as water transport
and biomechanics. Models intended for forestry applications
(Perttunen et al., 1998; Fernández et al., 2010) are an exception in that they describe annual ring structure.
At the other hand of the spectrum are microscale models of
the cambium (Downes et al., 2009). They capture at various
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1002
Sellier et al. — A surface growth model of the cambium
degrees of detail the intricate processes that control the development of the secondary meristem in conifers (Deleuze and
Houllier, 1998; Fritts et al., 1999; Vaganov et al., 2006;
Hölttä et al., 2010) and angiosperms (Drew et al., 2010).
What these models lack, however, is the ability to operate at
a large scale. Ring structure or a radial cell file are simulated
at one position of the stem whereas cambial activity is distributed (and variable) over the entire tree surface. There is a
missing link between cambial growth and tree shape.
A third type of common tree model, aimed at delivering
realistic-looking trees, including the external surface, can
be found in computer graphics. Parametric surfaces
(Bloomenthal, 1985) and mesh-based approaches (Tobler
et al., 2002) have been employed to create smooth stem –
branch junctions. Galbraith et al. (2004) used implicit surfaces
to include surface irregularities and non-smooth features such
scars and bark ridges. Surfaces are static and generated procedurally during growth. Computer graphic models share two traits.
First, they generate a tree outer surface corresponding to a predetermined skeletal representation of the structure. Secondly, the
convincing appearance of the surface shape is an objective in
itself and biological processes underlying growth are ignored.
This study aims at building an apparatus for testing at the
macroscale hypotheses about processes occurring at the microscale. The proposed method is not a substitute for individual
architectural models or cambial models, but rather offers the
opportunity to integrate them. To achieve this, we recast tree
growth as a surface growth problem, which we address using
a multi-compartment level set method. The level set method
(Osher and Sethian, 1988) offers a modern and robust treatment to simulate interface motion. The method can simulate
the dynamics of surfaces with complex geometry and undertaking topological changes. The present approach allows distinct growth behaviours to be defined in different regions of
the tree surface. The method is not tied to a particular hypothesis of cambial physiology. Any mathematical function of
surface state variables is acceptable to describe growth
speed. While arbitrarily complex shapes can be modelled,
the objective is not to arrive at a predetermined ‘realistic’
tree shape. Instead, shape is seen as a property that emerges
from local growth behaviour. In a manner perhaps closer to
biological reality, we adopt a global-from-local methodology.
It is local growth activity, whether cambial or apical, that generates the global form.
Mann et al. (2007) have previously used a fast marching
method, based on the level set framework, to model tree
surface growth. The present work extends that of Mann et al.
(2007) by coupling primary and secondary growth in one
model, and by considering simulations at the tree scale. We
also generalize the growth model to allow for growth rates
that depend on surface state. These advances allow for the
simulation of potentially more realistic scenarios and the
analysis of more detailed tree structures.
M U LT I - CO M PA R T M E N T L E V E L S E T M O D E L
The level set framework
We define an interface to be a closed surface in threedimensional (3-D) space. Traditionally, interface motion is
studied by explicitly tracking the position of the interface as
it evolves through time. Numerically, a finite number of
marker particles are placed on the interface and the position
of these particles is monitored as a function of time. Several
numerical issues are associated with this approach. Concave
regions can lead to situations where the interfacial position
becomes ambiguous. Issues also arise in convex regions (rarefaction) or contact between interfaces. Although all those
aspects can be mitigated, it comes at the additional complexity
of having to define regularizing behaviours. The level set
method introduced by Osher and Sethian (1988) solves these
issues. Their method relies on a change in perspective,
trading the conventional Lagrangian description of the interface for a Eulerian one. The interface, whether a curve or a
surface, is embedded in a function w so that at any time t,
the w(x, t) ¼ 0 contour defines the interface position. Here, w
is chosen as being the signed distance from the interface
(Osher and Fedkiw, 2003). The signed distance is the
Euclidian distance metric with + or – sign to indicate the
outside and inside of the interface.
The movement of the interface is given by:
∂w/∂t + F|∇w| = 0
(1)
where F is a function defining the growth speed in the normal
direction and where ∇ denotes the differential operator ,∂/∂x,
∂/∂y, ∂/∂z.. This equation corresponds to the classical
relationship where speed is the rate of change of position but
expressed in terms of the embedding function w. From the
fact that the gradient ∇w is zero along contours of w, it can
be observed that growth only happens in the direction
normal to the interface.
Robust numerical schemes exist to solve eqn (1). Here, we
adopt the level set scheme as proposed by Osher and Sethian
(1988). The scheme is a first-order approximation of space
and time derivatives in eqn (1). It has been simplified to
account for the fact that the vascular cambium never grows
inwards, i.e. we always have F ≥ 0. The scheme can be
written:
2
+x
2
n+1
[max(D−x
wijk
− wnijk
ijk , 0) + min(Dijk , 0) +
n
= −Fijk
−y
+y
Dt
max(Dijk , 0)2 + min(Dijk , 0)2 +
2
max(D−z
ijk , 0)
+
(2)
2 1/2
min(D+z
ijk , 0) ]
where w nijk is the discrete value of w at time t ¼ nDt and at i,j,k
grid coordinates. D +xijk is short-hand notation for D +xw nijk
with D +xuijk ¼ +1/Dx(ui+1,j k – ui,j,k) and Dx is the size of
a grid cell. Equation (2) allows the values of w at time step
n + 1 to be expressed as a function of the values at the previous time step n. Given the initial position of the interface
at n ¼ 0, eqn (2) can be used to compute the evolution of w
by iterating through time.
Stability of the scheme is enforced by choosing a time step
that is sufficiently small that the interface does not traverse
more than one cell during an iteration. This is equivalent to
the Courant – Friedrichs – Lewy (CFL) condition, which is satisfied when the time step Dt is chosen so that:
Dt , Dx/max(Fijk ).
(3)
Sellier et al. — A surface growth model of the cambium
Equation (3) is usually satisfied by choosing a CFL number
a ¼ max(Fijk)Dt/Dx so that a , 1. A conservative choice of
a ¼ 0.5 was used throughout this study.
Using a level set method, corner formation, rarefaction and
topological changes are handled naturally. No special care is
required to regularize the evolving interface. Because of
these advantages, the level set method has been extensively
used to solve a variety of interface problems such as simulating
tumour expansion, fire propagation, plant colony growth and
shape recognition (Sethian, 1999; Osher and Fedkiw, 2003;
Basse and Plank, 2008). A general introduction to the technique along with the required background can be found in
Sethian (1997). This study demonstrates how the method can
be used to model tree growth.
Growth velocity
The speed at which a curved surface expands can be defined
as a function F that may depend on many factors. We can
assume those factors to be local, global or independent properties of the interface. In the case of tree radial growth, F is what
describes the underlying growth processes. F can potentially
be expressed as a function of any factor impacting cell division, expansion or cell-wall thickening rates.
Local geometrical properties such as normal direction and
curvature are easily accessed using the level set method. The
normal direction is given by n ¼ ∇w/| ∇ w| and the curvature
by k ¼ ∇(∇w/| ∇w|). Numerical approximations to those quantities are available (Sethian, 1999). Global properties of the
surface include its total area or the volume it occupies.
Values can be directly determined from w but at the potential
cost of accuracy (Sethian, 1999). The alternative is to extract
the w ¼ 0 contour as a triangulated surface. Dedicated algorithms like the marching cubes (Lorensen and Cline, 1987)
exist to do this. Once an explicit description has been generated, calculating quantities such as surface area is trivial.
Another advantage in generating a triangulated surface is
that the surface is available for other purposes, e.g. visualization. Independent properties are those that are independent of
the interface shape such as time or position.
Some of the factors impacting on secondary growth rate are
not easily categorized. For example, neither photoassimilate
concentration nor mechanical strain are geometrical quantities.
However, their distribution will depend on the current tree
shape. To account for the spatial distribution of such factors
in the expression of growth rate may lead to complex F formulations. An alternative would be calculating how those factors
are distributed on the tree surface by means of external transport models and expressing F as a function of the local state
variables only.
Another issue with independent or biophysical properties is
they sometimes are meaningful only when defined on the interface itself. Namely, it is unclear what values those properties
should take for level sets other than the zero contour.
However, growth speed F must be known at all points of the
computational domain to update the level set function. A solution is to calculate F at the interface where variables have a
physical meaning and to propagate F from the interface to
all points in the narrow band. This can be achieved by extrapolation (Osher and Fedkiw, 2003) or using the extension
1003
velocity method (Adalsteinsson and Sethian, 1999). Here, we
use the latter approach as it is based on the fast marching
method, which is implemented and employed to re-initialize
narrow bands (see next section).
In most models intended for use in forestry, secondary
growth is not expressed as a radial enlargement rate but as a
total mass of carbohydrates allocated to wood production.
Determining the growth velocity in that case is not straightforward as F must somehow describe how the biomass is
partitioned. Even in the simplest case where available carbohydrates are distributed evenly over the outer xylem surface,
several steps are necessary to formulate F.
First, the rate at which new volume dV/dt is being occupied
by secondary growth can be defined by:
dV 1 dM
=
r dt
dt
(4)
where r is the basic density of the accreted material and dM/dt
is the mass accretion rate. If dM/dt is not exactly known, it can
be obtained by finite difference approximation. The expansion
speed can then be determined from its relationship with the
volume accretion rate:
dV
= Fds
(5)
dt
S
If F does not depend on position, eqn (5) leads to:
F(t) =
1 dV
A(t) dt
(6)
where A(t) is the interface area at time t. In other terms, speed
in the radial direction is equal to the volume enlargement rate
divided by the total surface area. Equation (6) is not valid in
cases where F has a known dependency on position.
The narrow band technique
The main drawback of the level set method is its computational cost. The implicit function that represents the interface
needs to be evaluated at all points of the domain, which is one
dimension higher than the interface. For a closed surface, w is
evaluated on a 3-D grid. For a grid with N points in each
dimension, complexity is O(N 3), per time step. Two major
options exist to alleviate the computational cost issue. The
first is the fast marching method (Sethian, 1996) in which
the surface motion is formulated as a boundary value
problem, thus removing the need to perform time iterations.
Instead, the arrival time of the interface is propagated from
the initial position of the interface through the domain. Each
grid point is traversed only once and, using efficient data structures, the complexity of the method is O(N 3lnN). The method
has already been employed to model tree growth by Mann
et al. (2007). The second option to tackle the computational
cost of the level set method is to reduce the problem size.
This can be achieved by evaluating w on a restricted domain
surrounding the interface. The method, first introduced by
Adalsteinsson and Sethian (1995), is known as the narrow
1004
Sellier et al. — A surface growth model of the cambium
band technique. For a narrow band k-cells wide, complexity is
reduced to O(kN 2), per time step.
One of the key aims of this study is to develop a framework
in which growth hypotheses can be tested. This requires the
ability define general growth functions, which may depend,
for example, on local normal direction and curvature, or
surface state. Because it is not straightforward to compute
these quantities with the fast marching method used by
Mann et al. (2007), the use of the level set method with
narrow banding is preferred to their approach. The methodology employed here to build and update the narrow band is
similar to that documented in Adalsteinsson and Sethian
(1995). A narrow band G is defined as the set of all points
within Euclidian distance w of the interface:
G = {M [ <3 : |d(M)| ≤ w}
(8)
where M is a point of the computational domain <3 and d is
the signed distance function used for constructing G.
Applying eqn (8) would require d to be known at every
point in the domain. However, evaluating d over the entire
grid only to discard points outside the band is wasteful. To
avoid that, we use a marching procedure that computes d by
starting at the immediate vicinity of the interface and yields
increasing d values until the condition d . w is reached. The
algorithm used to create the narrow band, later referred to as
Initialise, is given in the Appendix.
Once the narrow band G has been built, w values are initialized with d values. w is then updated using eqn (1) until the
interface arrives at G boundary, in which case w is not
updated and G is rebuilt. Rebuilding G is performed as
described by the Initialise algorithm. The only difference is
that d values need not to be initially calculated (step 2 of
Initialise) and can be initialized with the latest accepted w
values instead.
To detect when G must be rebuilt, points located at the edges
of G are monitored for a change in the sign of w. A sign change
may also be caused by boundary instabilities. Because of this,
and despite the fact that the cambium grows outwards, the
inner edge also needs monitoring. Edge points are redefined
every time G is (re-)built and consist of all points lying
within one mesh size Dh of the boundaries of G:
+
E = {M [ G : w − d(M) ≤ Dh}
E− = {M [ G : w + d(M) ≤ Dh}
(9a)
(9b)
Here, E + and E 2 represent the outer and inner edge point sets,
respectively.
multiple compartments. To each shoot in the tree is attached
a cambial zone that originates via the procambium from a
different apical meristem. For modelling purposes, it can
also be beneficial to arbitrarily divide a single tissue into multiple regions that follow distinct developmental patterns.
However, modelling the expansion of a multi-compartment
surface is not naturally handled within the level set formalism.
This has already been done by Mann et al. (2007) for the fast
marching method. Their approach was not transposable
directly to the narrow band level set method and has been
adapted to fit in the current framework.
A solution is to use one level set for each surface compartment (Sethian, 1999). Using the narrow band approach mostly
solves the issue of computational overhead. As w is evaluated
within each compartment band, only the grid points that are
located in areas where compartments overlap require several
evaluations (once per compartment they belong to). In the
case of tree shoots, compartments overlap mainly in the
whorl areas. Those regions are spatially very restricted when
compared with the size of the whole structure. Therefore, the
computational overhead is minimal.
A remaining issue is to define how surface compartments
interact with each other. In the case of cambial growth, the
first compartment arriving at one position occupies that point
and no other compartment can grow there. To determine
where xylem is being formed and by which compartment,
we define three internal point sets for each compartment:
Trial, Known and Edge (Fig. 1). Points are allocated to each
set depending on their respective w value. Outside points
(w p . 0) of compartment p are placed in Trial. After an
update of the level set function, all Trial points are monitored
for a sign change from positive to negative in the same way as
Edge points. Any point M where a sign change occurs is added
to the Known point set of the compartment s to which M
belongs. M is then removed from the s Trial set. If point M
belongs to several compartments, one must determine s, the
compartment which is first to occupy M. It is assumed that s
is the compartment with the lowest w(M ) value, i.e.
w s(M ) ¼ minp (w p(M )). M is then handled as a single point
except it is removed and blacklisted from compartments
j
w
w - Dh
Outer Edge
0
Trial
Dh - w
Known
Multi-compartment growth
As initially designed, the level set method applies to problems with a clear distinction between an inside and an
outside (Sethian, 1999). In the simplest model, this is the
case for trees: primary tissues and xylem lie inside and
phloem, bark and surrounding medium are outside. This representation is simplistic in that the cambium is considered
as a single, homogeneous surface. In fact, the vascular
cambium may be better treated as a surface composed of
-w
Inner Edge
Dh
F I G . 1. Zone definition within the narrow band of a surface compartment
based on values of the level set function w on the computational grid. The distance w defines the band half-width. Dh is the size of a grid cell. When the sign
of w changes at any point, the cambial surface has crossed that point and xylem
has formed there. Trial and Known point regions are used to track the advance
of the cambium interface. Values of w in the narrow band Edge are monitored
for sign changes in order to determine when the band needs to be rebuilt.
Sellier et al. — A surface growth model of the cambium
other than s. By doing so, we ensure these compartments will
never grow to a region already occupied. A key difference
between Trial and Known sets is that Known points are
unique and immutable whereas Trial points remain tentative
locations for growth and can belong to several compartments.
Xylem characteristics
The proposed model so far only simulates how the tree outer
layer evolves with time. The material inside the surface layer
carries no information about its state except for the identifier
of the compartment that produced it. The geometry of the
growth layer is not sufficient. It is important to be able to
determine additional quantities such as basic density or microfibril angle to characterize the structure of the wood being
produced.
Similarly to growth speed, we assume that xylem characteristics can be expressed as a mathematical function of local,
global and/or independent properties of the growing surface.
Unlike the speed, however, those characteristics need to be
evaluated at any given point of the grid only once during the
entire growth. Evaluation takes place when the cambium first
grows into that point. Determining when this occurs is easily
done thanks to the monitoring already in place. Points entering
a compartment’s Known set during the current time step are
stored. Models of wood properties can then be invoked for
those points using additional information if needed (surface
state variables, position and time of formation). The definition
of such functions is beyond the scope of this study.
Primary growth
three different aspects: the narrow band approach, compartment interactions and non-trivial speed functions.
Table 1 shows results obtained when comparing the seminal
level set technique to the narrow band approach, using either a
large band (w ¼ 7Dh) or a small one (w ¼ 3Dh). The expansion of a sphere expanding at constant speed F ¼ 1 is simulated with three different grid resolutions. We consider a
single-compartment surface only in this test. The main result
is that doubling the resolution halves the numerical error.
This is observed with and without using the narrow band technique. At low resolution, all methods take a similar amount of
time to run. At high resolution, simulations using the narrow
band approach take half to one-third, depending on the band
width, of the computational time without narrow band.
Using the large band, most of the computational time is
spent updating the level set function. In contrast, rebuilding
the band accounts for most of the cost when using the small
band in this test case. Table 1 also shows that band width
impacts the numerical error. The higher the number of
re-initializations, the higher the error. However, the error
induced by re-initializing the band is marginal, amounting to
only a few per cent of the total error.
The second test case is performed to verify that multiple
compartments correctly interact. The test consists of two
spheres S1 and S2 of initial radius R ¼ 0.2 expanding with
an equal, constant speed F ¼ 1. The grid size is nx × ny × nz
with nx ¼ 2(ny – 1) ¼ 2(nz – 1). Sphere centres are located
at grid coordinates (nx/4; (ny + 1)/2; (nz + 1)/2) and (3nx/4;
(ny + 1)/2; (nz + 1)/2), respectively. Band half-width was
chosen equal to w ¼ 4Dh. Table 2 shows that the prediction
error is halved every time the grid resolution is doubled.
Computational times are approximately twice those measured
for a single sphere with a small band (Table 1), which is consistent with a computational grid twice as large. Table 2 also
confirms the spheres are well separated in the middle of the
grid.
Additional tests were carried out on the single sphere
using more complex growth speeds. For that purpose,
four different speed functions were tested: time- or
Shoot elongation and branching are controlled by the apical
meristem. In this study, primary growth is not modelled using
the level set method as it would require that growth occurs in
the direction normal to the interface. The solution retained is to
extend the current tree surface with new surface elements at
each time step. Those elements correspond to the outer layer
of the pith of growth units added to the structure during the
current growth period. Adding surface elements consists only
of local re-initializations; it does not perturb the surface elsewhere. Once inserted, new elements develop according to
growth processes of the compartment they have been attached
to. In the case of a new element that results from branching, a
new compartment is created. Because level sets are updated
using an upwind scheme, pith radius must be chosen as
equal or higher than the elementary mesh size Dh. Any architectural model can be used to determine the elongation and
branching of new growth units. The Grow algorithm describing how primary growth and secondary growth are simulated
in the current work is given in the Appendix.
213
A P P L I CAT I O N TO T R E E GROW T H
413
Numerical tests
Before employing the multi-compartment level set method to
model tree growth and wood formation, it is necessary to
evaluate that it is accurate and efficient. A series of tests
were performed to analyse the error behaviour relative to
1005
TA B L E 1. Modelling a sphere in expansion with and without the
narrow band approach
Grid
size
113
Half-width
L-1 error
(ms)
–
7
3
–
7
3
–
7
3
73.2
73.8
76
36.2
37.2
37.9
18
18.6
18.9
CPU time (s) (update/
re-init.)
0.3
0.4
0.4
4.5
3.6
3.0
81.7
33.7
26.6
(0.3/– )
(0.2/0.1)
(0.1/0.2)
(4.1/– )
(2.1/1.3)
(1.1/1.8)
(77.8/–)
(20.3/12.6)
(9.4/16.6)
No.
re-init.
0
2
6
0
4
12
0
8
25
The sphere (S) is centred on the grid origin (0,0,0), has an initial radius
R ¼ 0.5 and expands at speed F ¼ 1. Narrow band half-width is expressed as
a number of grid cells. No. re-init. specifies how many times the narrow band
had to be rebuilt during the simulation. L-1 error is the maximum error on
the predicted time of arrival of (S) over all grid points.
1006
Sellier et al. — A surface growth model of the cambium
TA B L E 2. Two expanding spheres S1 and S2 are modelled as
separate compartments of the same surface: maximum numerical
error on sphere time of arrival, computational cost and
compartment separation are measured
(S1)/(S2) separation
Grid size
L-1 error (ms)
CPU time (s)
∀i ≤ nx/2
∀i . nx/2
84.5
42.9
21.8
0.7
6.4
63.9
(i, j, k) [ S1
(i, j, k) [ S1
(i, j, k) [ S1
(i, j, k) [ S2
(i, j, k) [ S2
(i, j, k) [ S2
20 × 11 × 11
40 × 21 × 21
80 × 41 × 41
As both spheres have the same initial radius and same expansion speed,
they remain contained to their respective half of the grid (see text for
complete test description).
Relative error on arrival time at (1,1,1)
0·06
F=1+t
F=1+t2
F=1+r
F= r2
0·05
0·04
0·03
0·02
0·01
0
11
21
Grid resolution (number of cells)
41
F I G . 2. Relative error on the time-of-arrival at a point of reference for a
sphere of radius r expanding with time- and space-dependent speed functions.
The point of reference is in the corner opposite to the origin of the sphere.
position-dependent speed, with linear or quadratic dependence. For each expansion speed, three different grid
sizes were considered. Prediction error on time of arrival
is evaluated at position (1,1,1), i.e. in the corner of the
grid opposite to the sphere centre. Because the maximal
speed varied with time or band position, the time step
satisfying the CFL condition was re-evaluated at each iteration. Figure 2 shows that the method converges with an
increasing grid size for all the speed definitions trialled.
The degree of dependence on time or location does not
affect the rate of convergence.
Coupling primary growth to secondary growth
Figure 3 shows the evolution of a tree surface over a 4-year
period. Coupling between primary growth and secondary
growth is discrete with a growth cycle equal to 1 year. This
choice is common for models concerned with wood production (Perttunen et al., 1998). Elongation speed is equal to
0.5 m yr21 for the stem and 0.5/p m yr21 for a branch in its
pth year of existence. Branch phyllotactic pattern is 137.5 8.
Distance between two successive branches in a whorl is
equal to 3.5 mm (Pont, 2001). Each whorl has N branches, N
being a random number between 3 and 6. Secondary and
higher order branches are not modelled. Xylem accretion
speed is held constant at 10 mm yr21. Radial expansion
speed of branches is given by F ¼ 5 – 0.5(q – 1) mm yr21
where q is the height index of the branch inside the whorl:
q ¼ 1 denotes the highest branch in the whorl and q ¼ N the
lowest one. This choice of F illustrates intra-whorl competition
with branches positioned higher partly suppressing lower
branches. Here it is only done to show that branches, as distinct
surface compartments, can be assigned different speed functions. The simulation is performed with a mesh size equal to
Dh ¼ 2.5 mm.
A piece-wise tapering is noted at transitions between growth
units with a sudden reduction in shoot cross-sections. A
close-up on the first whorl of the structure reveals more
features (Fig. 4). Overall, the whorl structure is idealistic. No
nodal swelling is observed. Branch – stem junctions are
F I G . 3. Simulated tree surface displayed at times t ¼ 1, t ¼ 2, t ¼ 3 and t ¼ 4 using a 1-D model of primary growth coupled to the level set model of secondary growth.
Sellier et al. — A surface growth model of the cambium
1007
rate, constant with height, as:
A
mx =
1 dM
H dt
(10)
From eqns (4) and (5), we obtain the speed function:
F = rmx /C(z)
B
C
F I G . 4. First whorl of a simulated tree at age 4: (A) growth layers and a
within-whorl view; (B) age of formation; (C) compartment identification.
abrupt whereas they are smoother in nature. The effect of
speed functions defined on a per-branch basis can be observed
in the resulting branch diameter. Figure 4(A) highlights minor
irregularities at the boundaries between surface compartments.
Those irregularities are contained within one mesh size and do
not propagate with time. Figure 4(B, C) show that xylem age
and its cambial origin are characteristics that are available at
high resolution inside the tree surface.
(11)
where C(z) is the cross-sectional circumference at discrete
height z ¼ kDh. Equation (11) introduces an approximation
but the error remains small provided that the variation of
C(z) over Dh is small. C(z) is computed after extracting the triangulated surface.
In Fig. 5(A), the tapering of the stem developing with a
homogenous growth rate is distinctly conical. In contrast, the
stem expanding under F as defined in eqn (11) has a less pronounced taper (Fig. 5B). The outer shape at the end of the
growth period appears to be parabolic. For both speed functions, the width of the latest annual ring (measured at breast
height) decreases with age (Fig. 5C). Ring width is relatively
higher when growth speed is homogeneously distributed over
the stem surface.
Figure 6(A) shows how simulated volume varies as a function of time. The volume error measures the difference
between the volume measured in the simulations and the original tree volume given by CABALA that is used to determine
the enlargement rates dV/dt in order to derive F. Wood volume
is underestimated in both simulations although the error
depends on the choice of speed function. After 7 years, the
relative error remains less than 10 and 5 % for homogeneous
and section-dependent speed functions, respectively (Fig. 6B).
DISCUSSION
Distribution of cambium activity
Numerical considerations
Simulations of tree growth were performed using speed
functions that encapsulate how cambial activity is distributed
over the stem surface. These simulations illustrate how to formulate a local growth rate from a global value indicating how
much biomass is allocated to stem wood production. The modelled tree corresponds to a eucalypt (Eucalyptus globulus)
grown for 7 years in a high-productivity site (Northcliffe site
in Hingston et al., 1998). The presence of branches is neglected. Monthly elongation rate and stem biomass increments
are controlled by CABALA (Battaglia et al., 2004). We consider two simple allocation patterns to illustrate the relationship between growth function and stem shape. Those cases
are intended to test growth rates that depend on surface state,
an aspect not considered previously (Mann et al., 2007).
They are idealized mechanisms and are by no means an
attempt at depicting biological reality. In the first case, we
assume that the available biomass is distributed homogeneously over the cambial layer. F is obtained using eqn (6)
and is constant over the entire surface. In the second case,
we introduce a two-step partitioning: the available carbon is
first distributed equally along stem length then over the crosssection circumference. We define the linear mass accretion
Level set methods have been successfully applied to many
interface problems. In this study, we show that with the necessary adaptations, the level set approach can be applied to modelling tree growth and wood formation. The tree is a branched
and highly anisotropic structure that sparsely occupies its
bounding volume. Use of the narrow band technique makes
this problem computationally tractable. In simple problems,
this resulted in two- to three-fold gains in efficiency. This is
less than the 10-fold gain reported by Adalsteinsson and
Sethian (1995), although would be expected to increase for a
larger problem. In addition, there is currently no established
method for choosing the optimal width of the narrow band,
which involves a trade-off between update speed and
re-initialization frequency (in the simulations presented here,
the narrow band width was selected by trial and error).
Alternative narrow band techniques where points are added
and removed dynamically (Desbrun and Cani-Gascuel, 1998)
may enable further efficiency gains. In cases where the
speed function is only required to depend on
surface-independent properties, using the fast marching
method (Mann et al., 2007) may be computationally more
advantageous than an adaptive level set method.
1008
Sellier et al. — A surface growth model of the cambium
A
B
Ring width (mm)
C
30
Growth
25
Homogeneous
Section-dependent
20
15
10
5
0
1
2
3
4
Year
5
6
7
F I G . 5. Stem shapes obtained with speed functions representing two simple carbon partitioning hypotheses: (A) uniform distribution of growth activity over the
cambial layer; (B) uniform distribution over stem length and then distributed over cross-sectional circumference; (C) width of the latest annual ring at breast
height. Height is reduced by a factor of 100.
0·30
A
Volume (m3)
0·25
CABALA
Section-dependent
Homogenous
0·20
0·15
0·10
0·05
0·00
Relative error
0·05
B
0·00
Section-dependent
–0·05
Homogenous
–0·10
–0·15
0
2
4
Time (years)
6
8
F I G . 6. (A) Stem volume as a function of time for the initial CABALA input and for simulations with both homogeneous and section-dependent allocations. (B)
The CABALA model also provides a reference against which the relative amount of volume numerically dissipated can be measured.
Sellier et al. — A surface growth model of the cambium
The narrow band technique combines very well with the
multi-compartment approach. The double sphere test shows
that compartmentalizing the surface incurs little penalty.
This is a consequence of the fact that each compartment has
little spatial overlap with others. Some irregularities appeared
at compartment boundaries during tree growth simulations.
However, those irregularities remain within mesh size and
their impact can be lessened by increasing grid resolution.
More importantly, those irregularities remain at boundaries
and do not degenerate into numerical instabilities.
In the last tree growth application, both simulated trees have a
lower volume than the one prescribed (Fig. 6). Error at the end of
growth is at worst 10 %. Results have shown the error on volume
does not hide important aspects such as seasonal dynamics of
growth, which are predicted by CABALA and observed as
small-scale oscillations in the volume variations (Fig. 6a).
Error on interface position and thus occupied volume can be
reduced by increasing grid resolution. Results with time- and
space-dependent speed functions have shown that the model
converges towards theoretical interface position as resolution
increases. In cases where tree size might limit how finely the
surface is resolved, improving accuracy through the use of
second-order schemes may prove to be more advantageous. If
the error remains unacceptable after taking those additional
steps, an option is to measure the amount of dissipated
volume and re-inject it at each step. Alternatively, particle
level set methods (Osher and Fedkiw, 2003), developed
especially to alleviate the dissipative nature of level set
methods, could be employed. Results have also shown that the
error in the volume is also related to the growth cycles and
not to the level set method itself. Error arises from numerically
approximating the volume enlargement rates from discrete
volume time series. Although the error is not biased towards
volume loss, it can cause strong fluctuations in the predicted
volume and therefore should be mitigated. This can be done
by using higher order finite differences whenever possible.
Modelling wood formation at tree scale
For many tree modelling applications, both apical growth
and cambial growth are desirable. The feasibility of coupling
these two mechanisms has been demonstrated in this paper.
However, level set methods can only model movement in
the direction normal to the interface. Normal growth is a
poor choice of idealization for primary growth, at least at
tree scale. In this model, we choose to initialize the surface
elements corresponding to the new shoots created during
each growth cycle independently of the secondary growth
process. This is a reasonable assumption because vascular
cambium derives from the procambium. While the procambium exists in the embryo and is perpetuated by primary meristems, the transition to cambium generally occurs in
internodes that have just ceased elongating (Larson, 1994).
Separating the two growth processes allows apical growth to
be controlled by an arbitrary architectural model. Once
growth units are inserted as implicit surfaces, their radial
growth is entirely controlled by the speed function attached
to them. Feedback between elongation and enlargement can
be desirable as primary and secondary meristems are functionally linked (Vaganov et al., 2006). With the present approach,
1009
feedback is possible but, because the coupling is discrete in
time, it can occur at the end of the growth period only. As
the growth period is a model parameter that is user-defined,
time resolution is easily refined and adapted to the time
scales of the modelled processes. In the simulations presented,
simple assumptions were made for the parameterization of
primary and secondary growth and this leads to an idealized
tree. With yearly growth cycles, the output does not appear
very realistic. In particular, the external surface of the whorl
does not compare well with real patterns. Shoot sections
remain cylindrical with circular sections as a result of the constant growth rates. To simulate more realistic whorls, showing
nodal swelling for example, it would be necessary to account
for photoassimilate flow, hormone concentration (Kramer,
2002) or biomechanical adaptations (Müller et al., 2006).
The simplification is also obvious in the stem shapes of the
modelled eucalypt trees. Monthly growth cycles led to a
smoother stem surface length-wise. Including the crosssectional circumference for carbon partitioning improved the
overall shape. However, both tapering modes are unrealistic
in that they show neither a stem enlargement near tree base
nor a bi-linear taper with different slopes below and above
crown base. As in the previous application, a more thorough
description of the growth processes is critical to producing
shapes that match biological reality.
Conceptually, the present model generates a global quantity
(tree shape) as a result of a local process (cambial growth)
being integrated in time and space. Growth rate may depend
on a set of variables that define the local operating environment such as temperature or water potential. Predicting how
these variables are distributed over the tree surface can be
done by dedicated transport models. For instance, Mann
et al. (2007) have demonstrated how a flow model can be
applied to the outer surface to predict grain orientation. The
same type of model can also be used to simulate auxin flow
(Kramer, 2002). The next step towards a more realistic secondary growth outcome is formulating a speed function F that
depends on the quantities predicted by these transport
models. Linking growth activity and external models has not
been attempted thus far.
The need for the level set representation may not be obvious
with our simple case studies and there exist more efficient
ways to simulate realistic tree surface shapes. However,
realism at the expense of underlying mechanisms is not the
main purpose of the current study. The model is causal in
that tree shape emerges from the function that describes
local growth behaviour. Both tree growth applications presented in this study have shown that a simplified description
of growth processes gives rise to a simplified shape but a
shape that is entirely the consequence of the hypothesized
growth or speed function. This trait can be used to investigate
more complex hypotheses. What the simulations demonstrate
is (1) that the model produces a shape corresponding to an
arbitrary growth function and (2) that an individual tree can
be simulated as a surface in expansion at the organism level.
Another advantage offered by the level set method is the
ability to explore ring structure and wood anatomy. Indeed,
wood layering is simply the succession of past shapes. For
instance, the model can be used to investigate whorls, a
region where layering is notoriously complex. Cambial activity
1010
Sellier et al. — A surface growth model of the cambium
depends strongly on the position on the surface. Because of
developmental constraints and environmental influences, a
high degree of longitudinal and circumferential variation
arises. The heterogeneity in wood formation is obvious when
looking at the resulting material and its structure. Variability
also exists temporally and it may be necessary to account for
kinetics of cell formation (Dünisch and Rühmann, 2006) in
order to depict some growth patterns. Although this aspect
has not been developed during the study, the present framework permits characteristics other than position to be computed and attached to the cambial layer as it grows. In the
simulations presented, time of formation and cambial origin
information are determined. In conjunction with wood property models, the surface growth approach may prove useful
in creating high-resolution maps of wood quality.
CO NCL USI ON S
A framework based on a level set method for modelling
cambium development as a surface growth problem has been
presented. The model avoids the traditional problems encountered when modelling interface motion using, for example,
methods based on marker particles. It places the complexity
in formulating a mathematical function that can represent the
cambium rate of expansion and its dependencies on geometrical and biophysical quantities. The model proposed is a framework that is not tied to a particular function to express the
growth rate and its dependencies. The only limitation is that
cambial growth always occurs outwards, which is in accord
with biological reality. Furthermore, the model offers the
possibility of dividing the cambial layer into multiple compartments. Each compartment can then be assigned its own mechanism of expansion and will compete for available space with
other compartments. This is useful for dealing with tissues of
different origin, when differences in growth behaviour have
not been linked to differences in the local growth environment,
or when the local environment cannot be determined. Despite
some precautions inherent to employing a numerical method,
tests and applications show that the multi-compartment level
set method provides a sound framework to analyse the development of the vascular cambium.
We have demonstrated that the level set method can be
coupled to a traditional architectural model of tree growth,
thus giving explicit transverse dimensions and an internal
structure to the entire tree. We have also shown how the
surface representation can be tied to the type of physiological
model encountered in forestry to predict characteristics of individual trees. We plan to develop models to predict the physiological and mechanical variables that define the local
environment in which the cambium operates. Doing so will
allow us to tie the current level set model to existing, fine-scale
models of cambial growth and investigate the mechanisms that
control wood formation.
AC KN OW LED GEMEN T S
The Mathematics-in-Industry group is gratefully acknowledged for its role in initiating this research on tree surface
growth. This research is supported by the New Zealand
Foundation for Research, Science and Technology and by
the Future Forest Research initiative.
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APPENDIX
The Initialise algorithm used to create and rebuild a narrow
band is as follows:
(1) Mark all points surrounding the interface as the tentative
set L.
(2) Compute the signed distance function d for all points in L.
(3) Remove the point P [ L with the smallest d value.
(4) Add P to the accepted set G.
(5) Compute d for all neighbours of P that are not already in
G; add neighbours to L.
(6) Go to 3 if d(P) ≤ w.
(7) Initialize the level set function w with d.
At step 2, d values must be geometrically constructed. At step
4, d may also be computed accurately by geometrical construction if the interface has a simple shape. Alternatively, d can be
numerically constructed using the Godunov solver (Rouy and
Tourin, 1992). It introduces an approximation but can be performed for arbitrary shapes.
The Grow algorithm used to simulate tree surface development is given below. A growth unit (GU) is a shoot segment
created during one growth period. Steps 1– 3 relate to
primary growth, steps 4 – 8 to secondary growth.
(1) Determine position of new growth units (GU).
(2) Initialize an implicit surface for each GU.
(3) Append GU to an existing compartment (elongation) or
create a new compartment (branching).
(4) Create/update internal zones in each compartment (Edges,
Trial).
(5) For each compartment:
(a) Compute the level set update.
(b) Monitor edges. If a sign change occurs:
(i) Re-initialize the compartment’s narrow band,
(ii) Go to (5).
(6) Carry out level set update for all compartments.
(7) Monitor Trial. For all points where a sign change occurs:
(a) Add point to Known.
(b) Determine compartment s crossing point first.
(c) Remove point from s Trial.
(d) Remove point from other compartments altogether.
(8) Increment time step. Go to 1.