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 tree-wide 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.
At the other hand of the spectrum are microscale models of the cambium (Downes et al., 2009
). They capture at various 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.