Fractal Geometry for Biomimetic Design of Biodevices - Advanced Design and Manufacturing Technologies for Biomedical Devices - page 103

6.3.2 Designs Based on Fractional Brownian Models

Fractional Brownian fractal surfaces have proved to be very adequate for modelling
the micro-textures of natural surfaces (Mandelbrot 1983 ; Falconer 2003 ) and their
application to biodevices, such as scaffolds for promoting cell growth, and the
beginning of tissue formation has been already validated by our team (Díaz Lantada
2012a ).
Further studies linked to its application to the design of alternative biodevices for
several applications, such as promoting osseointegration of implants and especial
contact phenomena, should be addressed and the following explained design proce-
dure may be consequently of help.
The following equation gives the height “ z ” of the mentioned fractional Brownian
fractal surfaces, when assessing the function over a grid of points given by their
( x , y ) coordinates. The model uses several random functions ( A k , B k , C k ), several
control constants (λ, α, m ) and an initial height function “ z 0 ” can also be
introduced:

∞
∑

− a k k

zxyz C (,) =+ • l •sin([•cos() •sin() ]) l x ByBA + +

0 k k k k

k = 1

Figure 6.7 shows the result of evaluating a fractional Brownian fractal function
over a grid of 60 × 60 points (corresponding to a scaffold of 30 mm × 30 mm) and
the influence of introducing changes in the control parameter “α”.
In this example, we use a planar surface ( z 0 = 0) as basis for the fractal, although
multi-scale-based design approaches may wish to combine, for instance, an initial
surface with micrometric features, upon which fractal nanometric details are applied
or even combinations of fractal surfaces with different levels of detail.
Upper image of Fig. 6.7 corresponds to α = 0.8 (fractal dimension around 2.2
with maximum roughness depth reaching 1.2 mm), and lower image of Fig. 6.7 cor-
responds to α = 0.2 (fractal dimension around 2.8 with maximum roughness depth
reaching 2.5 mm). Even though we are referring in this example to a mm × mm × mm
scale, it is important to note that the matrixes obtained give just numerical values
and, depending on final application, we can focus on mm, microns, or nanometres.
In any case, we have to take into account the level of desired detail of the biostruc-
ture being mimicked and the level of detail subsequently attainable with final manu-
facturing technologies.
The calculations have been carried out with help of Matlab software (Mathworks,
version R2009) and the data obtained are stored in three-column matrixes [X, Y, Z].
The command “surf” helps to represent the surfaces linked to the mentioned
matrixes. Some additional details on computation can be found by having a look at
the Matlab (The Mathworks Inc.) code of the different programmes included in the
Annexes of the handbook, hoping it may be of help for carrying out future designs
of biodevices and medical appliances.