^{1}

^{*}

^{1}

^{2}

^{*}

Ellipsoid modeling is essential in a variety of fields, ranging from astronomy to medicine. Many response surfaces can be approximated by a hemi-ellipsoid, allowing estimation of shape, magnitude, and orientation via orthogonal vectors. If the shape of the ellipsoid under investigation changes over time, serial estimates of the orthogonal vectors allow time-sequence mapping of these complex response surfaces. We have developed a quantitative, analytic method that evaluates the dynamic changes of a hemi-ellipsoid over time that takes data points from a surface and transforms the data using a kernel function to matrix form. A least square analysis minimizes the difference between actual and calculated values and constructs the corresponding eigenvectors. With this method, it is possible to quantify the shape of a dynamic hemi-ellipsoid over time. Potential applications include modeling pressure surfaces in a variety of applications including medical.

The Cartesian coordinate equation for an ellipsoid is given by the formula:

1)

An alternate form of this equation is the matrix equation given by the formula:

2)

3)

Here, the eigenvalues and eigenvectors of the A matrix give the corresponding squared length of the axis and the direction, respectively. Orthogonal eigenvectors are induced when the matrix is symmetric [

Previous papers have discussed using the method of least squares estimates (LSE) to converge on the ellipsoid best fit for a given set of data [2-5]. LSE analysis minimizes the sum of the distance (or squared difference) between the measured and predicted values of an ellipsoid. We have simplified this concept by implementing existing add-ins in Microsoft Excel such as Solver and Matrix.xla. Additionally, we have automated the process using VBA code so that as the hemi-ellipsoid alters shape and position we can continuously recalculate the new eigenvalues and eigenvectors of the ellipsoid.

Our goal was to create a program that automatically and continuously evaluates data from a pressure map for seated individuals. Currently, most medical institutions evaluate peak pressure when assessing patients for pressure build-up which can lead to pressure ulcer development. However, peak pressure is not the sole determinant in tissue breakdown [

In order to assess the validity of the method, we first tested the ability of the program to predict the values of a known ellipsoid. We chose the simple ellipsoid defined by the following equation:

and considered only the superior half of the shape. We calculated the z values for x and y coordinates known to have corresponding values on the surface of the ellipsoid by solving for the unknown variable z:

The predicted values for z were calculated through minimization via quadratic optimization by gradient descent using the Solver Add-in in Microsoft Excel. Two models were created minimizing the absolute difference and the squared difference between z and

The constraints are summarized here:

Objective Function:

Changing Cells:

k = 0 the A matrix is assumed positive definite

After the optimum predicted values were determined and thus the corresponding

After the validity of the program had been tested using a known ellipsoid, a 36 ´ 36 sample data set of measured values from a pressure map was analyzed. An ellipsoid shape was identified from the data automatically and assessed for characteristics of shape and size. The program was able to identify center values for the ellipsoid and eigenvalues that visually made sense, as shown in

Analysis of the data shows the strength of this program in accurately predicting the ellipsoid shape of the data.

Statistical analysis based on the 13 actual and predicted values of the known ellipsoid showed a very low average error of −0.288% and average absolute error of 0.507%. The center coordinates of the ellipsoid calculated automatically by the program were also extremely accurate, with an average −1.06% error. As discussed above, the

For the measured sample data set, the summed absolute difference between the measured and predicted values ranged from 364.52 to 1070.4 and the average absolute percent error was 20.478%. There are several considerations for the absolute percent error. First, the measured data is an irregular shape and is not a perfect hemi-ellipsoid. Second, while the overall shape may appear hemi-ellipsoid-like, outliers can affect the overall prediction of the model. We determined the values of the mound by eliminating low extreme values, but we did not eliminate high extreme values. Thus, it may be important in future applications to consider removal of these high peak values in order to focus on the values that best fit the shape of the hemi-ellipsoid.

Visualization of the data is an important validation method when testing modeling predictability, which allows interpretation of fit as well location of extreme differences and also verifies that the resulting model is actually ellipsoidal. The Add-in R Excel was used to give 3-D graphical representation, as shown in

To our knowledge, this is the first description of the ischial tuberosities being modeled as hemi-ellipsoids using the matrix equation of an ellipsoid with sum of least squares in a continuous fashion. This model allows for continuous monitoring of data over long periods of sitting time due to its automation.

A major limitation of using a gradient descent method is that it is possible for local maxima or minima to be discovered that are not the true absolute maximum or minimum. In order to circumvent this, we created a loop allowed for different lower bound values for z to be assumed and it initiated the search in different points on the xy grid. We tuned the model to determine the best “center,” represented by a lower bound of “z”. Additionally, the assumption was made that the center of the ellipsoid was on the map. Thus, the parameter k was set equal to 0.

When automating, a trade-off has to be made between accuracy and computation time. Depending on the necessity for precision, a decreased stepping parameter may be desired to ensure the best possible fit occurs. The complexity of the problem and the running time for automation must be taken into consideration when deciding what step should be used and over what range of possible z lower bound values. For example, the running time from 0 to 150 with a step of 30 runs for 13.9 seconds while with a step of 15 runs for 38.8s. If multiple frames of data are collected per second over a period of several minutes to hours, the computation time increases to several hours.

A point of consideration when examining the results of the eigenvalues and eigenvectors is that Matrix.xla offered multiple means of calculating the eigen numbers. For example, the eigenvectors did not have to be assumed to be orthogonal with some methods. However, since we initially based our assumption of the

The method described allows for complex modeling of 3-D data by assuming a hemi-ellipsoid shape. While some of the assumptions and constraints force that data to fit a symmetrical shape, it allows for a means of comparison between constantly evolving shapes. This method is useful in showing trends over time and has a variety of applications in modeling systems with changing hemi-ellipsoid-like shapes.

A.B. would like to thank her faculty advisors and the Colleges of Engineering and Medicine at USF, as well as her patients, family, and Joshua M. Diggs.