Optimization of the step-and-shoot leaf sequence for delivery of intensity modulated radiation therapy using a variable division scheme.

To deliver an intensity modulated radiation therapy (IMRT) plan by multileaf collimator (MLC) it is necessary to convert beam profiles, generated from the inverse treatment planning algorithm, into a series of instructions that the MLC control system can execute. An idealized IMRT beam profile can be regarded as a continuously varying two-dimensional function and is usually represented by an intensity map, i.e., a discretized description in space and in intensity of the beam profile. It is common to assume that the intensity map be defined over a regular grid with N steps and equal increments of intensity levels. In reality, this may not be the optimal representation of the beam profile and may introduce unnecessary discrepancies between the intensity pattern delivered and that ideally required. We have implemented an algorithm capable of minimizing the difference between the two patterns on a beam specific basis. In other words, it can produce optimized intensity maps, individually produced to suit the (continuous function) intensity profile they are intended to approximate. This enhancement in conformation is achieved by allowing variable step size and unconstrained intensity levels in the final leaf sequence.

An investigation into the feasibility of implementing fractal paradigms to simulate cancellous bone structure.

Cancellous bone consists of a framework of solid trabeculae interspersed with bone marrow. The structure of the bone tissue framework is highly convoluted and complex, being fractal and statistically self-similar over a limited range of magnifications. To date, the structure of natural cancellous bone tissue has been defined using 2D and 3D imaging, with no facility to modify and control the structure. The potential of four computer-generated paradigms has been reviewed based upon knowledge of other fractal structures and chaotic systems, namely Diffusion Limited Aggregation (DLA), Percolation and Epidemics, Cellular Automata, and a regular Grid with randomly relocated nodes. The resulting structures were compared for their ability to create realistic structures of cancellous bone rather than reflecting growth and form processes. Although the creation of realistic computer-generated cancellous bone structures is difficult, it should not be impossible. Future work considering the combination of fractal and chaotic paradigms is underway.

Stochastically simulated assessment of anabolic treatment following varying degrees of cancellous bone resorption.

The aim of this study was to investigate the recovery in cancellous bone stiffness resulting from anabolic treatment following varying degrees of resorption, using a stochastic simulation applied to a simplistic structure consisting of five vertical and five horizontal trabeculae. The structure was initially resorbed, and "bone" elements were stochastically removed until nominal resorptions of 10%, 15%, 20%, 25%, and 30% were achieved. A stochastic simulation of anabolic treatment was then applied where bone elements were added, continuing until the original stiffness had been regained, for example, simulating treatment of a patient with an anabolic agent after a period of postmenopausal resorption. The resorption and anabolic simulations were repeated three times for each of the nominal resorptions. The stiffness of the bone structure decreased linearly with resorption, with a slope of approximately -2 and an R(2) of 97.0%; hence, the stiffness fell at approximately twice the rate of the reduction in density. When the various structures regained their original density, the resultant stiffness also had a linear relationship with the original resorption, with a slope of -1 and a lower R(2) of 86.1%. This implies that the reduction in stiffness, when original density was regained, fell proportionately with the degree of initial resorption and, therefore, after a resorption of 30%, when original density was regained, the stiffness of the resultant structure was approximately 30% less than that of the original structure. The density required for the original stiffness to be regained increased linearly with the degree of initial resorption, with a slope of approximately 0.5 and an R(2) of 65.2%, lower than that observed for the previous relationships. This indicates a greater spread of data and suggests greater variability in the formation phase beyond the point of regained original density. Because irreversible connectivity reduction is widely considered to be one of the earliest manifestations of estrogen loss, these findings, although obtained on a simulation of a simplistic cancellous bone structure, support the concept of early intervention to prevent potentially irreversible deterioration of trabecular architecture after menopause.

Finite Element Analysis of Simulations of Cancellous Bone Resorption.

A stochastic simulation of the resorption of cancellous bone has been developed and integrated with a finite element model to predict the resultant change in structural properties of bone as bone density decreases. The resorption represents the net imbalance of osteoclast and osteoblast activity that occurs in osteoporosis. A simple lattice structure of trabecular bone is considered, with an examination of the lattice geometry and discretization indicating that just five trabeculae need to be modelled. The results from the analysis show how the mechanical properties of the cancellous bone degrade with osteoporosis and demonstrate how the method can be used to predict the relationships between stiffness and density or porosity.

Dynamic stochastic simulation of cancellous bone resorption.

A stochastic simulation of cancellous bone resorption was developed and applied to a simple two-dimensional lattice structure representing the vertebral body. The simulation is based upon the concept of a basic multicellular unit (BMU) where net resorption (-deltaB.BMU) is considered at bone/marrow surfaces. The cancellous bone structure is defined as a binary matrix with the size of the pixels corresponding to a square element of approximately 20 microm dimension. The simulation considers both the probability that any surface pixel will be activated into a BMU and, if activated, the length of the resorption cavity. The relationship between relative stiffness and density for the simulation was predicted by finite element analysis. The stochastic simulation was iterated eight times with the mechanical properties assessed after each stage. Perforation of a single trabeculae was first observed at step 2, the structure completely lacking connectivity and mechanical integrity by step 8. The slope of the stiffness-porosity graph was greater than unity for the first five steps, but thereafter approached zero because the structure had lost connectivity and effectively collapsed. The eight-step simulation was repeated five times and demonstrated that, although the stiffness/density relationships were similar at the extremes of density, the dependence of stiffness upon density varied. This clearly demonstrates the stochastic nature of the simulation upon cancellous bone structure, and is probably indicative of a significant dependence of mechanical integrity upon perforation effects.

A comparison of porosity, fabric and fractal dimension as predictors of the Young's modulus of equine cancellous bone.

The purpose of this study was to compare the structural parameters of fabric and fractal dimension as predictors of the Young's modulus of equine cancellous bone. Eight 15 mm cubes of cancellous bone were obtained from three equine third metacarpal bones. Young's modulus was determined for the three orthogonal directions. The fabric and fractal dimension were calculated for each of the six exposed faces of each cube. Fractal dimension plus porosity provided a higher explanatory power for Young's modulus (R2 = 78.7%. P < 0.0001) than fabric plus porosity (R2 = 69.2%, P < 0.0001). Fractal dimension was also significantly correlated with fabric (R2 = 53.8%, P < 0.0001). Although this novel method for combining fractal dimension data into a pseudo-directionally dependent predictor of Young's modulus requires further validation over a greater range of porosities and differing cancellous bone tissues, its potential has been demonstrated.