Fractal Analysis of Trabecular Bone
Fractals are unusual geometric structures that can be used to analyze many biologic structures not amenable to conventional analysis. The purpose of this exhibit is to teach some of the fundamentals of fractal analysis, and to show how they can be applied to analysis of trabecular bone.
Simple models of biologic structures can be constructed with simple shapes such as lines, circles, spheres, and simple polygons. One can then estimate properties of the structure, such as length, area, volume, strength, etc.
The earth could be modeled as a sphere, while an artery can be modeled as a hollow cylinder.
However, there are many complex biologic structures that cannot be easily modelled by simple shapes such as these. One of the most common such patterns is the branching structure of many biological structures. Branching structures in the human body include the arteries, veins, nerves, the Bundle of His, parotid gland ducts, and the bronchial tree.
Other Fractal Processes in Nature
- Regional distribution of pulmonary blood flow
- Pulmonary alveolar structure
- Mammographic parenchymal pattern as a risk for breast cancer
- Regional myocardial blood flow heterogeneity
- Fractal surfaces of proteins
- Distribution of arthropod body lengths
Fractal objects have several interesting properties. One of the most interesting is self-similarity. The Sierpinski triangle below is a good example of this. The big triangle is composed of four smaller triangles, each of which are composed of four even smaller triangles, ad infinitum. A fractal object such as this exhibits this self-similarity over many scales of observation, from the full-sized object down to the microscopic level.
Another property of a fractal object is the lack of a well-defined scale. A good example of this is seen in clouds, which tend to look very similar no matter what their size. Another way to state this is that it is difficult to tell just how big a cloud is without some external reference.
Trabecular bone has a branching pattern, as seen in this vertebral specimen. One can also see that it exhibits self-similarity. That is, the trabeculae and the marrow spaces between them look very similar no matter what their size.
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
How then, does one estimate the fractal index of a complex fractal structure like the trabecular bone in the vertebra above?
We first need to tell our image analysis program which tissue is bone and which is not. This is generally simple since bone is far denser than soft tissue. In our image analysis program, we adjusted the window and level controls so that only tissue with CT density greater than a certain level would be included in the analysis. This can be done visually in our software by adjusting the window and level controls until only the tissue of interest is shown in red, as below.
Once these threshold levels have been set, the imaging software now knows what is bone
and what is not. One can now use the region of interest cursor to select an
appropriate section of trabecular bone to analyze, as shown below.
Algorithms for Estimating the Fractal Index
There are a wide variety of computer algorithms for estimating the fractal index of a structure, such as the box-counting algorithm, the pixel dilation method, the calipher method, the radial power spectrum method, and others. We will demonstrate the box-counting algorithm, because of its simplicity.
This algorithm estimates how much of the available space is taken up by the fractal structure. First, an arbitrary grid is placed over the structure to be measured. Then, one counts how many boxes in the grid are filled by the fractal structure. For the grid size above, the structure fills 6 boxes.
The process is then repeated with a grid half the size of the previous one. With the grid below, the structure fills 9 boxes.
The process is again repeated with a grid half the size of the previous one. With the grid below, the structure fills 18 boxes.
This process can be carried on indefinitely, using smaller and smaller grids. For this demonstration, we will only count the boxes one more time. With the final grid below, the structure fills 49 boxes.
So, our box-counting data (omitting the first count) is summarized in the table below:
The data from these three countings are tabulated and plotted on a log-log plot as shown below. A linear regression is done to find the best fit. The slope of this line is used to calculate the fractal index. The best fit equation is y = 1.2224 x + 1.2985
The fractal index can be estimated as the slope of this line. Thus, for this fractal object, the estimated fractal index D = 1.2224. This approximation, using only 3 steps of box counting, slightly underestimates the actual fractal index of this figure, which is ln(4)/ln(3) = 1.261860. With the addition of multiple successive steps of box counting, one should be able to get an estimate much closer to the theoretical fractal index.
This concludes our exhibit on fractal analysis. We hope that you have found it of interest.
- Microvascular networks: fractal structure, hemodynamic properties and physiological functions. A satellite symposium of the XXXI International Congress of Physiological Sciences. July 16-19, 1989. Int J Microcirc Clin Exp, 1989.
- Akbarieh M, Dubuc B, Tawashi R. Surface studies of calcium oxalate dihydrate single crystals during dissolution in the presence of urine. Scanning Microsc 1987;1(3):1397-403.
- Akbarieh M, Tawashi R. Surface studies of calcium oxalate dihydrate single crystals during dissolution in the presence of stone-formers' urine. Scanning Microsc 1989;3(1):139-45.
- Antognetti PF, Dellepiane S, Serpico SB, Vernazza G. [The application of fractal technics for the enhancement of radiographic images]. Radiol Med (Torino) 1989;77(5):535-9.
- Arle JE, Simon RH. An application of fractal dimension to the detection of transients in the electroencephalogram. Electroencephalogr Clin Neurophysiol 1990;75(4):296-305.
- Barnsley MF, Massopust P, Strickland H, Sloan AD. Fractal modeling of biological structures. Ann N Y Acad Sci 1987;504(4):179-94.
- Bassingthwaighte JB, King RB, Sambrook JE, van SB. Fractal analysis of blood-tissue exchange kinetics. Adv Exp Med Biol 1988;222(1):15-23.
- Bassingthwaighte JB, King RB, Roger SA. Fractal nature of regional myocardial blood flow heterogeneity. Circ Res 1989;65(3):578-90.
- Blinc A, Lahajnar G, Blinc R, Zidansek A, Sepe A. Proton NMR study of the state of water in fibrin gels, plasma, and blood clots. Magn Reson Med 1990;14(1):105-22.
- Bryant SH, Islam SA, Weaver DL. The surface area of monomeric proteins: significance of power law behavior. Proteins 1989;6(4):418-23.
- Caldwell CB, Stapleton SJ, Holdsworth DW, et al. Characterisation of mammographic parenchymal pattern by fractal dimension. Phys Med Biol 1990;35(2):235-47.
- Cerny LC, Cerny EL, Granley CR, Compolo F, Vogels M. The erythrocyte sedimentation rates: some model experiments. Biorheology 1988;25(1-2):85-94.
- Cutting JE, Garvin JJ. Fractal curves and complexity. Percept Psychophys 1987;42(4):365-70.
- Dewey TG, Datta MM. Determination of the fractal dimension of membrane protein aggregates using fluorescence energy transfer. Biophys J 1989;56(2):415-20.
- Diaz G, Quacci D, Dell'Orbo C. Recognition of cell surface modulation by elliptic Fourier analysis. Comput Methods Programs Biomed 1990;31(1):57-62.
- Dieckert JW, Dieckert MC, Creger CR. Calcium reserve assembly: a basic structural unit of the calcium reserve system of the hen egg shell. Poult Sci 1989;68(11):1569-84.
- French AS, Stockbridge LL. Fractal and Markov behavior in ion channel kinetics. Can J Physiol Pharmacol 1988;66(7):967-70.
- Gates MA. A simple way to look at DNA. J Theor Biol 1986;119(3):319-28.
- Goldberger AL, West BJ. Applications of nonlinear dynamics to clinical cardiology. Ann N Y Acad Sci 1987;504(4):195-213.
- Goldberger AL, West BJ. Fractals in physiology and medicine. Yale J Biol Med 1987;60(5):421-35.
- Grosberg A, Shakhnovich EI. [Solid interactions in statistical physics of biopolymers]. Biofizika 1987;32(6):949-59.
- Grosberg A, Nechaev SK, Shakhnovich EI. [The role of topological limitations in the kinetics of homopolymer collapse and self-assembly of biopolymers]. Biofizika 1988;33(2):247-53.
- Guo XH, Zhao NM, Chen SH, Teixeira J. Small-angle neutron scattering study of the structure of protein/detergent complexes. Biopolymers 1990;29(2):335-46.
- Horsfield K. Diameters, generations, and orders of branches in the bronchial tree. J Appl Physiol 1990;68(2):457-61.
- Iannaccone PM. Fractal geometry in mosaic organs: a new interpretation of mosaic pattern. Faseb J 1990;4(5):1508-12.
- Ishibashi A, Aihara K, Kotani M. [Chaos in brain and neurons and an analysis on the fractal dimensions]. Iyodenshi To Seitai Kogaku 1988;26(1):57-61.
- James TN. The spectrum of diseases of small coronary arteries and their physiologic consequences. J Am Coll Cardiol 1990;15(4):763-74.
- Javanaud C. The application of a fractal model to the scattering of ultrasound in biological media. J Acoust Soc Am 1989;86(2):493-6.
- Katz MJ. Fractals and the analysis of waveforms [published erratum appears in Comput Biol Med 1989;19(4):291]. Comput Biol Med 1988;18(3):145-56.
- King RB, Weissman LJ, Bassingthwaighte JB. Fractal descriptions for spatial statistics. Ann Biomed Eng 1990;18(2):111-21.
- Korn SJ, Horn R. Statistical discrimination of fractal and Markov models of single-channel gating. Biophys J 1988;54(5):871-7.
- Lew RR, Schauf CL. Fractal filtering of channel data. Biochim Biophys Acta 1990;1023(2):305-11.
- Liebovitch LS, Sullivan JM. Fractal analysis of a voltage-dependent potassium channel from cultured mouse hippocampal neurons. Biophys J 1987;52(6):979-88.
- Liebovitch LS, Fischbarg J, Koniarek JP, Todorova I, Wang M. Fractal model of ion-channel kinetics. Biochim Biophys Acta 1987;896(2):173-80.
- Liebovitch LS. The fractal random telegraph signal: signal analysis and applications. Ann Biomed Eng 1988;16(5):483-94.
- Liebovitch LS. Testing fractal and Markov models of ion channel kinetics. Biophys J 1989;55(2):373-7.
- Liebovitch LS, Toth TI. Using fractals to understand the opening and closing of ion channels. Ann Biomed Eng 1990;18(2):177-94.
- Mainster MA. The fractal properties of retinal vessels: embryological and clinical implications. Eye 1990;.
- Marchais P. ["Fractalì forms and developments in mental disorders]. Ann Med Psychol (Paris) 1988;146(10):966-73.
- Marsh DJ, Osborn JL, Cowley AJ. 1/f fluctuations in arterial pressure and regulation of renal blood flow in dogs. Am J Physiol 1990;.
- Matsuyama T, Sogawa M, Nakagawa Y. Fractal spreading growth of Serratia marcescens which produces surface active exolipids. Fems Microbiol Lett 1989;52(3):243-6.
- McManus OB, Weiss DS, Spivak CE, Blatz AL, Magleby KL. Fractal models are inadequate for the kinetics of four different ion channels. Biophys J 1988;54(5):859-70.
- McManus OB, Spivak CE, Blatz AL, Weiss DS, Magleby KL. Fractal models, Markov models, and channel kinetics. Biophys J 1989;55(2):383-5.
- McNamee JE. Fractal character of pulmonary microvascular permeability. Ann Biomed Eng 1990;18(2):123-33.
- Meakin P. A new model for biological pattern formation. J Theor Biol 1986;118(1):101-13.
- Miyashita Y. Neuronal correlate of visual associative long-term memory in the primate temporal cortex. Nature 1988;335(6193):817-20.
- Montague PR, Friedlander MJ. Expression of an intrinsic growth strategy by mammalian retinal neurons. Proc Natl Acad Sci U S A 1989;86(18):7223-7.
- Morigiwa K, Tauchi M, Fukuda Y. Fractal analysis of ganglion cell dendritic branching patterns of the rat and cat retinae. Neurosci Res Suppl 1989;10(43).
- Nelson TR, West BJ, Goldberger AL. The fractal lung: universal and species-related scaling patterns. Experientia 1990;46(3):251-4.
- Nonnenmacher TF. Fractal scaling mechanisms in biomembranes. Oscillations in the lateral diffusion coefficient. Eur Biophys J 1989;16(6):375-9.
- Obert M, Pfeifer P, Sernetz M. Microbial growth patterns described by fractal geometry. J Bacteriol 1990;172(3):1180-5.
- Onaral B, Tsao YY. Fractal dynamics of polarized bioelectrodes. Ann Biomed Eng 1990;18(2):151-76.
- Purugganan MD. The fractal nature of RNA secondary structure. Naturwissenschaften 1989;76(10):471-3.
- Rahamimoff R, DeRiemer SA, Ginsburg S, et al. Ionic channels in synaptic vesicles: are they involved in transmitter release? Q J Exp Physiol 1989;74(7):1019-31.
- Rigaut JP. Automated image segmentation by fractal grey tone functions. Gegenbaurs Morphol Jahrb 1989;135(1):77-82.
- Risset JC. Pitch and rhythm paradoxes: comments on "Auditory paradox based on fractal waveform. J Acoust Soc Am 1986;80(3):961-2.
- Sansom MS, Ball FG, Kerry CJ, McGee R, Ramsey RL, Usherwood PN. Markov, fractal, diffusion, and related models of ion channel gating. A comparison with experimental data from two ion channels. Biophys J 1989;56(6):1229-43.
- Schlesinger MF. Fractal time and 1/f noise in complex systems. Ann N Y Acad Sci 1987;504(4):214-28.
- Schroeder MR. Auditory paradox based on fractal waveform. J Acoust Soc Am 1986;79(1):186-9.
- Smith TJ, Marks WB, Lange GD, Sheriff WJ, Neale EA. A fractal analysis of cell images. J Neurosci Methods 1989;27(2):173-80.
- Stein KM, Kligfield P. Fractal clustering of ventricular ectopy in dilated cardiomyopathy. Am J Cardiol 1990;65(22):1512-5.
- Stockbridge LL, French AS. Characterization of a calcium-activated potassium channel in human fibroblasts. Can J Physiol Pharmacol 1989;67(10):1300-7.
- Sun HH, Wang X, Onaral B. Onset of nonlinearity in fractal dimension systems: an application to polarized bioelectrode interfaces. Ann Biomed Eng 1988;16(1):111-21.
- Takahashi M. A fractal model of chromosomes and chromosomal DNA replication. J Theor Biol 1989;141(1):117-36.
- Teich MC. Fractal character of the auditory neural spike train. Ieee Trans Biomed Eng 1989;36(1):150-60.
- Ten DH, Klasen HJ, Nijsten MW, Pietronero L. Superficial lightning injuries--their "fractal shape and origin. Burns Incl Therm Inj 1987;13(2):141-6.
- Thibert R, Akbarieh M, Tawashi R. Application of fractal dimension to the study of the surface ruggedness of granular solids and excipients. J Pharm Sci 1988;77(8):724-6.
- Tsuchiya T, Ichimura A, Nagai Y. Correlations in one-dimensional fully developed chaos. Cell Biophys 1987;11(52):109-22.
- Van BJ, Roger SA, Bassingthwaighte JB. Regional myocardial flow heterogeneity explained with fractal networks. Am J Physiol 1989;.
- Van BJ, Bassingthwaighte JB, Roger SA. Fractal networks explain regional myocardial flow heterogeneity. Adv Exp Med Biol 1989;248(1):249-57.
- Victor JD. The fractal dimension of a test signal: implications for system identification procedures. Biol Cybern 1987;57(6):421-6.
- West BJ. Physiology in fractal dimensions: error tolerance. Ann Biomed Eng 1990;18(2):135-49.
- Xu N, Xu JH. The fractal dimension of EEG as a physical measure of conscious human brain activities. Bull Math Biol 1988;50(5):559-65.
- Zbilut JP, Mayer KG, Sobotka PA, O'Toole M, Thomas JJ. Bifurcations and intrinsic chaotic and 1/f dynamics in an isolated perfused rat heart. Biol Cybern 1989;61(5):371-8.