It is now time to talk about the concept of a fractal dimension. Most people are familiar with the three spatial dimensions that we live in. These dimensions are known as topological dimensions, and have been used for many years to describe the shape and position of objects. Benoit Mandelbrot, however, found that certain geometrical objects couldn't be described well with the usual topological dimensions, and formulated the idea of a fractional or fractal dimension, existing somewhere between the usual topological dimensions.

Each of the following objects has a topological dimension of 1. However, the more complex they become, the more they tend to fill the space about them. The amount of space filled by one of these objects is represented by the fractal dimension or index (D), which can be thought of as a "filling factor".

For the line and circle, D is equal to 1.0. The snowflake pattern on the right fills more space than the line or circle, and its D is 1.26.

The Koch fractal above fills even more space than the snowflake pattern, and has a D of 1.5.

The line drawing of pulmonary parenchyma above is an even busier pattern, and fills even more space. Its D is about 1.82.

The fractal dimension ranges between 1 and 2. It is 1 where the structure is a simple straight or curved line, and fills almost no space. When a structure fills all available space, such as the yellow square below, its fractal dimension is 2.

Many past models of bone have confined themselves mostly with modelling breaking strength as a simple function of bone mineral density. Bone density is certainly strongly related to breaking strength. However, there are wide biological variations seen in breaking strength among patients with the same bone density. Therefore, investigators have sought to create more refined models of bone strength.

Some of these models are based on the idea that it's not just how much mineral you have in a given bone, but also how it's arranged within that bone that determines bone strength. Indeed, finite element analysis models of bone strength have been fairly successful in predicting bone strength. However, these models are quite complex, and are unlikely to become clinically useful for a given patient in the near future. The idea that the fractal index of trabecular bone might be related to bone strength is an appealing one, since the fractal index is simple to calculate from clinical CT images of a given bone.

The image below shows two possible models for bone. Normal bone lies somewhere between the hollow cylinder of bone on the left and the solid cylinder on the right. If one analyzes CT slices of these two bones, one would measure a fractal index of 1.0 on the hollow bone and an index of 2.0 for the solid bone. Measurements of normal bone fall somewhere between these two extremes (about 1.7 - 1.8), and it is plausible to hope that the fractal index may prove useful in estimating bone strength.

Bone with no trabeculae ("osteoporosis") -- D = 1.0		Bone with solid center ("osteopetrosis") -- D = 2.0