Using Tactile Images to
Differentiate Breast Tissue Types

Griffin Weber

Advisor : Professor Robert Howe



      Tactile imaging is a medical imaging technique used to improve upon and reduce the subjectivity in clinical palpation. A tactile map is formed using a hand held scanning device equipped with an array of pressure sensors that is stroked along the surface of the body. The goal of this project is to better understand how the features of tactile maps of breast tissue change when a lump with a particular size, depth, and stiffness is embedded between layers of normal glandular tissue and fat. Previous studies have demonstrated that finite element analysis can be used to create tactile maps very similar to those generated by the real device. Therefore, thirty finite element simulations were performed to determine how the tactile sensor behaves given different sets of model "input" parameters. Equations, collectively known as the Forward model, were derived to fit the results of the simulations. These were then used to construct an Inverse model, which when given the simulated tactile maps could predict certain properties of the lump. Finite element analysis showed that with modifications to the tactile sensor to allow for detection of displacement into the breast tissue, a complete Inverse model can accurately predict both the geometry and material properties of the lump.





Contents



Introduction

Background

Tactile imaging is a recently developed medical imaging technique, used primarily in the treatment of breast cancer, to remove the subjectivity from clinical palpation. When a woman visits her physician, part of the physical exam often includes a clinician attempting to palpate the patient for any lumps or changes in the breast tissue that could indicate the presence of a tumor. This method, however, only gives the physician a vague sense of what is actually underneath the skin. Due to the lack of any precise measuring device, if a lump is found through palpation, typically all that can be documented is its general location on the breast and a rough estimate of size. To solve this problem, Assurance Medical developed a Tactile Imaging System consisting of a hand-held device, known as a tactile probe, which replaces the physician's fingertips with an array of pressure sensors (Figure 1a). When this tactile probe is stroked across a patient's skin, the contact pressure between the patient and the tactile probe is recorded by a computer. Simultaneously, a magnetic position tracker measures the location of the probe (Figure 1b).

At any given instant, the pressures recorded by the 16 x 26 array of pressure sensors (the sensors are spaced 1.5 mm apart), forms a Tactile Image. When multiple Tactile Images are arranged spatially by the computer using the position data, they can be combined to form a pressure contour profile known as a Tactile Map. Because lumps are stiffer than the surrounding tissue, they appear as a localized pressure increase in a two-dimensional Tactile Map (Figure 2). A Tactile Map can also be visualized in one-dimension if a line is imagined through the center of the Tactile Map and the variation in pressure along that line is measured. A one-dimensional Tactile Map resembles a Gaussian curve centered above the lump. From the Tactile Maps, it is clear where the lump is positioned. However, it is not obvious from the Tactile Maps if other information about the lump, such as its diameter, depth, or stiffness, can be determined. This data is essential in the diagnosis and treatment of breast cancer. For example, changes in lump diameter observed in sequential Tactile Maps taken each time a women visits her physician can provide important information about the growth of the lump, and knowing the stiffness of the lump can be used to predict whether the lump is benign or actually cancerous. Unfortunately, only limited previous research has been conducted on how to interpret Tactile Maps and extract information about the lump.

Previous Research

Previous research on the use of Tactile Imaging to determine the properties of a lump embedded in breast tissue has been conducted by Dr. Parris Wellman, a graduate of Harvard's Division of Engineering and Applied Sciences. In his PhD thesis (Wellman 1999), he came to three important conclusions. The first is that different types of breast tissues have very different material properties. Suppose fat is assigned a relative elastic modulus, or stiffness, of one. Normal glandular tissue has a higher elastic modulus than fat, and because of the nonlinear material properties of soft tissue, the more force applied to the glandular tissue, the stiffer it will appear. Over the range of forces that is induced by the Tactile Probe in the process of generating a Tactile Map, the average stiffness of normal glandular tissue is six times that of fat. A benign papilloma has an average stiffness of 10 times fat, and a true cancerous tumor such as an infiltrating ductal carcinoma is, on average, thirty times as stiff as fat.

The second finding of this work is that Finite Element Analysis can be used to very accurately predict the tactile maps of real breast tissue. Finite Element Analysis is a mathematical approach, frequently used by engineers, to solving complex problems in mechanics. A physical structure is represented as collection of many small, simple geometric entities known as elements. The interactions that exist between these elements can be described in a system of equations. A computer can then be used to solve the equations and predict what will happen when forces are applied to the Finite Element Model. The results of Finite Element Simulations, tests with rubber models, and Tactile Maps generated with a Tactile Probe on real tumors all showed similar patterns when different sizes and types of lumps were used. This is an important result because, for many reasons, controlled tests are extremely difficult to conduct with real tumors. By using Finite Element Simulations, a systematic study of Tactile Imaging is possible without requiring the cooperation of numerous patients, surgeons, and pathologists. Furthermore, noise and other potential sources of error is minimized with Finite Element Analysis. Once Tactile Imaging is understood through Finite Element Analysis, the results can then be extended to real Tactile Maps.

The third finding was a set of mathematical equations, derived using Finite Element Analysis, that could predict the diameter and depth of a tumor using a Tactile Map. Unfortunately, these predictions were only reasonably accurate, and the equations did not account for the stiffness of the lump. Without knowledge of stiffness, the tissue type of the lump can not be determined. An improved set of equations that could use Tactile Imaging to accurately estimate the diameter and depth of a lump, as well as its stiffness, was the goal of my study.

Project Overview

The problem this project set out to answer was how does the diameter, depth, and stiffness of a lump affect a Tactile Map. In order to do this, Finite Element Analysis was used to simulate the generation of Tactile Images and Tactile Maps, a procedure called the Forward Model (Figure 3). In order to analyze the results of the simulation further, a feature extraction step had to be taken. This process involved examining the data collected through the simulations, determining which features were most important in characterizing this data, and designing a method of assigning numeric values to these features. Once this was accomplished, a set of equations, collectively known as an Inverse Model, was developed, which, in contrast to the Forward Model, could use the results of Tactile Imaging to predict the original properties of the lump, including its stiffness. The construction of an Inverse Model for Tactile Imaging is the heart of the design problem of this project. While the development of the Forward Model would require a great deal of analysis and be invaluable in understanding Tactile Maps, the ultimate goal of this project was to produce an Inverse Model that could accurately predict all the properties of the lump.

Finite Element Simulations

Definition of "Input" Parameters

Figure 4 is a diagram showing the anatomy of the breast and the orientation of the different layers of tissue. The uppermost is a thick layer of fat. Beneath the fat is a stiffer layer of normal glandular tissue. The deepest layer is the relatively incompressible chest wall, consisting of muscle and bone. Tumors often form within the glandular layer. They become palpable when they penetrate into the fat. The exact geometry of the layers shown in Figure 4 is quite complex. For the Finite Element Simulations, a simplified model was used. Figure 5 shows how the three layers of tissue were represented as rectangles, with a circular tumor positioned at the base of the layer of fat. Five dimensions of the model were important to this project. These "Input" parameters are the thickness of the layer of fat (t); the diameter of the tumor (d); the thickness of the tissue above the tumor, or the depth of the tumor, in terms of the numbers of tumor diameters (h); the ratio of the stiffness of the tumor to the stiffness of the surrounding fat (E); and the total force applied to the Tactile Probe (F). The parameters t, d, and h are related by the equation t = d*(h + 1/2).

Development of a Finite Element Model

To simulate the pressures and the deformation of the model as the force (F) was increased and the Probe was stroked back and forth over the tumor, a Finite Element Analysis software package ABAQUS was used. FEMAP, a computer aided design (CAD) application was used as a pre- and post-processor for the simulations. The construction of the Finite Element Models was not a trivial task. The simulation of Tactile Imaging is a contact problem involving nonlinear materials experiencing extremely large strains (up to 20%). Unless the Finite Element Model is designed correctly, ABAQUS cannot converge to a solution and will return an error.

Designing the model involved several steps. The first was the basic geometry. Several different shapes for the probe and various dimensions (depth and horizontal distance from the tumor) of the tissue, gland, and chest wall layers were tested (Figure 6). In each iteration of the design process (a few are shown in Figure 6), the complexity of the model was increased to better match the shapes of the real Tactile Probe and breast tissue layers. After the geometry is defined in FEMAP, the model must be "meshed". FEMAP automatically divides the chosen geometry into individual nodes and elements. This process is called meshing. Although rectangular areas are easily meshed, FEMAP often fails to generate acceptable meshes near curved areas such as the surface of the probe or the border of the tumor. This problem was solved by combining FEMAP's automated meshing with manual determination of node and element location. When the first set of completely meshed models were run in ABAQUS, the results were quite poor. There was a lot of "noise" in the output which suggested that the size of the individual elements compared to the size of the entire model was too large. Therefore, the element size was decreased. However, this dramatically increased the time ABAQUS needed to analyze the model, and some noise still remained in the resulting tactile images. Another approach was used where the element size was increased back to its original value, but a thin "skin" of very small elements was placed along the top of the tissue and along the surface of the probe. This model contained about 3000 nodes and over 1000 elements, but finally created smooth tactile images, which ran in ABAQUS in a reasonable amount of time. The final design of the Finite Element Model is shown in Figure 6d.

So that the effects of different Input parameters (force, tissue thickness, and tumor diameter, depth, and stiffness) could be studied, thirty variations of the Finite Element Model in Figure 6d were constructed. Linear material properties, which allowed for a constant tumor stiffness, were used in 29 of these variations. The last simulation used nonlinear tissue stiffness to better represent how a real tumor deforms. Each Model was assigned a code in the form dDDhHHeEEfFF, where DD is the tumor diameter, HH is the tissue thickness, EE is the tumor stiffness, and FF is the total force. The Model d10h08e10f10 (d = 1.0 cm, h = 0.8, E = 10, and F = 1.0) was chosen as a reference model. The Tactile Maps of each ABAQUS simulation later would be compared to this reference model. Note that "EE = 1K" means a tumor stiffness of 1000 times that of the surrounding fat, and "EE = NL" indicates nonlinear tissue stiffness. Figure 7 shows the geometries of three of the different Finite Element Models that were used.

Feature Extraction

Formation of Tactile Maps

The ABAQUS output files are enormous. As the Tactile Probe indents into the layers of tissue and moves over the tumor, the contact pressure and model deformation at each Step is saved to disk. Figure 8 shows the deformation and the contact pressure (in the form of a Tactile Image) at four of these Steps. A single simulation to create one tactile map often produces nearly 300 MB of files, too large to analyze in a program such as Excel. To cut the files down to a manageable size, I wrote a computer program (in the Perl language, see Appendix D) to read the files line-by-line and extract only the information relevant to creating the tactile maps: the deformation of the model and the contact pressure between the probe and the tissue. This is saved to a new file only a couple hundred kilobytes in size. A second program was then written to extract, from this subset of the original Finite Element Simulation results, the particular "Output" parameters that were eventually chosen to be used in the Forward and Inverse Models.

Defining a handful of Output parameters that would best characterize the 300 MB of simulation results was the Feature Extraction process of the project. First, a Tactile Map had to be constructed from the Tactile Images. There are a couple of approaches that can be taken when writing the code to accomplish this (Figure 9). In the first method, the individual Tactile Images are arranged spatially; then, the Tactile Images are averaged. In the second method, a particular point is chosen on the Probe and the pressure at that point is traced as the Probe moves over the tumor. Similar traces are found for several other points distributed evenly along the Probe surface. In Method 2, the point traces, rather than the Tactile Images are averaged. In the simulations, the Probe was moved towards the tumor in 2 mm increments. Method 1 is very sensitive to this increment size, so when the Tactile Map was calculated at 1 mm resolution rather than 2 mm, an artificial "lumpiness" was introduced. As can be seen in Figure 9, Method 2 produces essentially the same Tactile Map; but, because of the interpolation that was possible with the point traces, a smooth Tactile Map with a 1 mm resolution could be made. Therefore, Method 2 was chosen to produce all the Tactile Maps in this study.

Definition of "Output" Parameters

Output parameters for the Tactile Maps were then selected (Figure 10). It was previously noted that Tactile Maps resemble a Gaussian curve. The Amplitude (Amp) and the Standard Deviation (Stdev) of the curve were chosen as the first two Output parameters. Also, because there is pressure between the Probe and the tissue even far from the tumor, the entire Tactile Map is shifted upwards by a value defined as a third output parameter called Base. Next, because of the shape of the Tactile Probe, even far from the tumor, the Tactile Images do not show a uniform contact pressure. There is a slight Curvature in the Tactile Images, defined as the pressure at the center of the Probe divided by the average of the pressures at a distance of 8 mm from the center of the Probe (Figure 10).

The contact pressure is the only data that is captured by the Tactile Probe. However, the Finite Element Simulations allowed for other potential Output parameters to be considered. The first graph in Figure 11 shows Vertical Probe Displacement. When the Probe was far from the tumor, for a given amount of applied force, it could "sink" several millimeters into the soft tissue. Closer to the stiff tumor, the probe would be forced upwards, unable to deform the tissue as much. The distance that the probe could be pushed into the tissue in the vertical direction was called the Probe Displacement. The Probe Displacement divided by the thickness of the soft tissue in its undeformed state is the nominal strain experienced by the tissue. To characterize the Probe Displacement, the Displacement was recorded when the probe was at a distance of 2.8 cm away from the center of the tumor in the horizontal direction (the distance labeled A in the top graph of Figure 11) and when it was directly above the tumor at x = 0 cm (distance B).

The second graph in Figure 11 shows the point in each Tactile Image where the greatest contact pressure was recorded. For example, when the Probe was 2.8 cm from the tumor, the peak of the Tactile Image was at point A. When the Probe was 2.6 cm from the tumor, the peak pressure was at point B; at 1.4 cm from the tumor, the peak pressure was at C; at 1.2 cm, the peak pressure was at D; and when the probe was directly above the tumor, the peak pressure was at point E. Of interest is the large "jump" between points C and D. Similar jumps in the peak Tactile Image pressures were observed in many of the simulations. However, no clear trends in the length or position of the jumps could be found, so this feature was not chosen to be an Output parameter used in the Forward or Inverse Models. These pressure jumps may be an area of future study.

Forward Model

Effects of Input Parameters on Output Parameters Derived from Pressure

With the Input and Output parameters defined, it was now possible to begin looking for a Forward Model‹a set of equations that could take the five Input parameters (total force F, tissue thickness t, tumor diameter d, tumor depth h, and tumor stiffness E) and transform them to a set of predicted Output parameters (Base, Amplitude, Stdev, Curvature, Displacement at x = 0 cm, and Displacement at x = 2.8 cm). To create the Forward Model, equations would be derived to fit the results of the 29 Finite Element Simulations that used constant tumor stiffnesses. The raw data giving the values of each Input and Output parameter for all thirty simulations are listed in Appendix A. In addition to the absolute values of the simulation Output parameters, relatives values were calculated. The relative values are the ratios of the absolute parameters of a given simulation and those of a reference simulation based on Model d10h08e10f10 (d = 1.0 cm, h = 0.8, E = 10, and F = 1.0). Although explicit mention of it is often omitted later in this report, the relative values are the only ones used in the Forward and Inverse Models.

The first step in creating the Forward Model was to change one Input parameter at a time in the Finite Element Simulation and observe the changes to the Output parameters. Figure 12 shows the Tactile Maps of these simulations. These Tactile Maps have been grouped into six separate graphs. The first illustrates changes in the total force F. Note that as the force F was increased, the Base, Amplitude, and Stdev of the Tactile Map increased as well. The second graph (moving downward in the figure) shows the effects of changing tumor stiffness E, and the third demonstrates how the tactile maps change when varying both force and stiffness. The next three graphs similarly show the effects of tumor diameter d and tissue thickness h, both independently and combined.

By examining these six graphs, several trends are clearly visible, others are more subtle. To numerically quantify these trends, graphs were created to show how one simulation Output parameter changes when modifying one Input parameter of Model d10h08e10f10 and keeping all others constant. Twelve of these graphs are shown in Figure 13. The first shows how the relative Base value increases linearly with the total force F. No other input parameters had any significant effects on the Base value. Therefore, a Forward model for the Base value can immediately be determined: the Output parameter Base is directly proportional to the total force F.

The next two graphs (moving downward in the figure) show that the relationship between relative Curvature and the Input parameters is more complex. Parameters d, h, and F all had nonlinear effects on the Curvature. Since the Curvature is measured far away from the tumor, it makes more sense to use the Input parameter t, the total thickness of the fat, equal to d*(h + 1/2), rather than d and h separately. The next graph, of Curvature and stiffness, shows that the elastic modulus of the tumor does not affect the Curvature, which was expected. Far from the tumor, the material properties of the tumor should not have any effects. The only Output parameter derived from contact pressure that tumor stiffness did have a significant effect on was the relative Amplitude of the Tactile Map. As the stiffness increased, the relative Amplitude increased as well. However, this value quickly approached a maximum relative Amplitude with moderately large stiffnesses. Best fit curves are shown for each graph in Figure 13.

Effects of Input Parameters on Probe Displacement

The effects of modifying one Input parameter at a time on the Probe Displacement was also observed. The first graph in Figure 14 shows that the Displacement increases as the diameter d is increased while the tumor depth h is held constant. The Displacement at 2.8 cm increases faster, though, than the Displacement at 0 cm. As the tumor depth h is increased and the diameter d remains constant, the Displacement increases at approximately the same rate at all distances from the tumor. The fourth graph in Figure 14 shows that the force F changes the Displacement at all distances from the tumor; however, varying tumor stiffness E only changes the Displacement above the tumor. Note that far from the tumor, the curve representing the Displacement of the model with nonlinear tumor stiffness (d10h08eNLf10) resembles a linear model with a tumor stiffness of E = 2. Though, as the Probe approaches the tumor, the nonlinear model shows a Displacement gradually approaching that of a tumor stiffness of E = 10.

The graphs in Figure 15 show how Probe Displacement changes as E and F are varied. In these graphs, Displacement is divided by the undeformed tissue thickness to give Nominal Strain. The results of simulations with similar tumor stiffnesses are connected with a curve of the form A*Strain^B. These are the Strain-Force curves in the graphs in Figure 15. The nonlinearity of the Strain-Force curves, present despite the fact that only materials with linear elasticity were used, is a result of the large deformations in the simulation model. The two curves fitting E = 1 and E = 1K (1000) form bounds within which the remaining Strain-Force curves lie. As the stiffness increases, the slope of the curve increases from that of the E = 1 curve to that of the E = 1K curve. Note that with a larger and larger stiffness, it becomes harder to see the increase of slope. Note further that in the range of F values that were tested (0.2 - 1.0), none of the curves intersect. In the second graph in Figure 15, this is no longer the case. Superimposed on the curves of the first graph is now the Strain-Force curve when a nonlinear lump stiffness was used. At a low total force it behaves like a E = 1x or 2x lump, but at higher values of F the Strain resembles that of a E = 10x or greater lump. The graphs in Figure 16 show that changing d and h can also influence the Strain-Force curves. However, the effects of modifying d or h are less drastic than those which result from changing the stiffness. This will eventually allow for the development of an accurate Inverse Model for tumor stiffness E.

Combining Parameters

Once it was known how individual Input parameters influenced the Output parameters, in order to complete the Forward model, these individual effects would have to be combined. A first attempt at combining the Input parameters involved simply multiplying the best fit curves found in Figure 13. For example, it was found that if all Input parameters are held constant except F, then the Curvature is proportional to F^-1.1. If F is held constant and tissue thickness t is changed, then the Curvature is proportional to t^-0.55. Therefore, a Forward Model was constructed stating that Curvature is proportional to F^-1.1*t^-0.55. The assumption that the individual effects could be multiplied actually worked very well. The Forward Model could predict the Finite Element Simulation Output parameters almost exactly. However, some of the equations, particularly for the Output parameter Stdev contained many terms. The simpler the Forward Models equations, the easier it would be to create an Inverse Model later. To see if any of these terms could be dropped and to ensure that their coefficients in the equations were set to minimize the error of the Forward Model, multiple regression analysis was used. To determine which terms to remove, several regression equations were found, each using different subsets of terms. The t-stats and P-values of the coefficients in each equation were examined, and the effects on the R^2 and Adjusted-R^2 values were considered. This is a standard statistical technique used to select the most appropriate form for the regression equation.

Appendix B shows the regression analysis results for each of the final Forward Model equations. Note that in all cases, the R^2 was almost 1, indicating that the Output parameters can be very accurately predicted. In Figure 17, the validity of the Forward Model is illustrated. Each graph shows the predicted value of one of the Output parameters against the actual Output parameter that was obtained through the simulation. Note that in the simulations on which the Forward Model is based, the Input parameters were varied over a relatively large ranges of values. One limitation of multiple regression is that caution must be taken when attempting to extrapolate outside of the range of data used in deriving the equations. By choosing a large range of different Input parameters, any real breast tissue geometries should fall within these parameter ranges. This should minimize any "misuse" of the Forward model that could arise from extrapolation.

Inverse Model

Total Applied Force and Tissue Thickness

The Inverse Models were created using an approach similar to that used in deriving the Forward Models. However, unlike before, when Finite Element Analysis could provide a set of Output parameters given a set of Input parameters, it was not possible to begin with any desired combination of Output parameters and run the Finite Element Analysis backwards to obtain the Input parameters. The only pairs of Output-Input parameters that could be used were those that had already been found in the generation of the Forward Model. This presented a problem. Before, it was very useful to hold all but one Input parameter constant and observe the effects on the Output parameters. This systematic method could not be used to determine the "effects" of the individual Output parameters on the Input parameters because in each simulation every Output parameter was different. In other words, no Output parameters could be held constant. However, the forms of the Forward Model equations could be used as a starting point for designing the Inverse Model equations.

For example, previously it was found that the Base value is directly related to the total force F applied to the Probe. Therefore, it would be logical that F is directly proportional to the Tactile Map Base. Indeed, the Base can be used to predict F with R^2 = 0.999. Similarly, the thickness of the tissue t can be predicted using the Curvature and the Base value. However, this only gives an R^2 = 0.810. The Curvature and the Base values are both derived from the contact pressure recorded by the Tactile Probe. If modifications were made to the Tactile Probe that could allow it to measure the Displacement, then an Inverse Model for tissue thickness t with R^2 = 0.990 can be found using the Displacement at 2.8 cm from the tumor.

Lump Diameter and Depth

Forming the remaining Inverse Model equations involved much trial-and-error work to determine possible terms to include in the regression analysis. The earliest forms of the Inverse Model equations were found by attempting to directly invert the Forward Model equations mathematically. This, however, resulted in very poor results. The Forward Models were designed to minimize the error in predicting the Output parameters. In no way do they minimize the error when using the Output parameters to minimize the Input parameters. After experimenting with many versions of Inverse Models, a final set of equations, listed in Appendix C, were chosen.

The R^2 values of the Inverse Model for d, h, and E require some explanation. Using pressure only, tumor diameter d could only be predicted with R^2 = 0.790. The lower R^2 was due largely to cases where the tumor stiffness E was less than 10 times the surrounding fat. If the clinician performing the Tactile Imaging has prior knowledge that the lump is indeed cancerous, in which case it can be assumed that E is at least 10, the R^2 for the Inverse Model increases to 0.920. On the other hand, using Probe Displacement, tumor diameter d for all E can be predicted with R^2 = 0.925. For stiffness E at least 10, Displacement can be used to predict d with R^2 = 0.966. Thus, Displacement is clearly a better predictor of tumor diameter than the Tactile Maps alone.

A somewhat similar pattern is found with tumor depth h. Using pressure, the Inverse equation for h has an R^2 = 0.824; however, when the equation is fit through models where E was at least 10, R^2 = 0.957. Unlike the Inverse Model for d, using Displacement values rather than the Tactile Map results in a less accurate Inverse Model, with R^2 = 0.800 for all E and R^2 = 0.882 for E at least 10.

Lump Stiffness and Differentiating Tissue Types

The values for tumor stiffness E that were simulated are 2, 5, 10, 50, and 1000 times as stiff as the surrounding fat. Highly skewed data like this causes problems in regression analysis because the few simulations using E = 50 and particularly E = 1000 will result in excessively high leverage. The least-squares method will pull the "best-fit" equation towards these few data points, without considering trends that occur at low stiffnesses. Therefore, the regression was performed not using E, but rather using E^-2/3, which has the effect of evenly spacing the data. The exponent was chosen based on the results of the Forward Model. Previously, it had been found (see Figure 13) that features of the Tactile Maps were most closely associated with E^-2/3, not a simpler transformation that would have had the same effect of evening the data such as 1/E or log(E).

Using pressure, E^-2/3 could be predicted with R^2 = 0.649. The reason for the low R^2 is that the only feature of the Tactile Maps that was significantly influenced by tumor stiffness was the Amplitude. The total force applied to the probe, the tumor depth, and the tumor diameter all had effects on the Amplitude also. Therefore, it is difficult to "see" what part of the Amplitude is due to tumor stiffness. Furthermore, any tumors that are 10 or more times as stiff as the surrounding tissue behave as if they were all essentially incompressible. As a result, the difference between the Amplitude caused by a tumor with E = 10 and one with E = 1000 is nearly zero. However, softer lumps with stiffnesses less than E = 10 have a much greater impact on the Amplitude of the Tactile Map. The difference in Amplitudes between a lump with E = 2 and E = 5 is many times greater than the difference between a lump with E = 50 and E = 1000. If the clinician has prior knowledge that the lump is not a cancer and therefore the stiffness is likely to be less than E = 10, then the pressure data can be used to predict E^-2/3 with R^2 = 0.960.

With Probe Displacement, the Inverse Model for tumor stiffness can be further improved so that for all stiffnesses, E^-2/3 can be predicted with R^2 = 0.724, and for lumps with E less than 10, R^2 = 0.981. To summarize, in general, lump stiffness can only be roughly categorized as being either hard (E >= 10) or soft (E < 10). The Inverse Model is not good enough to predict an exact value for stiffness. However, whether the lump is simply hard or soft provides much information that can aid in the differentiation of breast tissue types. As was previously mentioned, lumps of normal glandular tissue have a stiffness of less than E = 10, and the only tissue types that have stiffnesses larger than E = 10 are potential cancers. While some tumors are in fact softer than glandular tissue, Tactile Imaging can be used to detect and monitor small changes in the stiffness of these lumps because, as was seen in the regression analysis, an accurate value for stiffness can be made if it has been determined that E < 10.

The Complete Inverse Model

Figure 18 and Figure 19 illustrate the validity of the complete Inverse Model. Figure 18 shows how well the Inverse Model can be used predict the original Finite Element Model Input parameters using only Output parameters derived from pressure data (the Curvature and the Tactile Map Base, Amplitude, and Stdev). Figure 19 shows the improvements to the Inverse Model that can be made in many cases if Probe Displacement is considered. A correlation matrix for the complete Inverse Model is shown in Table 1. From Table 1, it can be seen that pressure data, in particular the Base value of the Tactile Map must be used to predict the total applied force F. Pressure can also be used to predict tissue thickness t and tumor diameter d; however, a much more accurate prediction can be made if somehow Probe Displacement could be measured and incorporated into the Inverse Model. Tumor depth, or tissue thickness expressed as the Input parameter h can best be predicted using the Tactile Maps. Finally, while Displacement should be used to predict tumor stiffness E, in general only a rough estimate can be made. However, this is enough to differentiate many true cancers from lumps of normal glandular tissue; and, an accurate prediction of stiffness E, important in the monitoring of changes in breast tissue, is possible in cases where the stiffness can be assumed to be less than ten times that of the surrounding fat.

ParameterPressureDisplacementn
F R^2 = 0.999 * - 29
t 0.810    0.990 * 29
d 0.790    0.925 * 29
d (E >= 10) 0.920    0.966 * 22
h    0.824 * 0.800 29
h (E >= 10)    0.957 * 0.882 22
E^-2/3 0.649    0.724 * 29
E^-2/3 (E < 10) 0.960    0.981 * 7
Table 1: Summary of the complete Inverse Model. A (*) indicates the best Inverse Model equation to use for predicting the corresponding Input parameter.

Considerations of Real Tactile Imaging

Although the Forward and Inverse Models were shown to work extremely well for the Finite Element Simulations, to be of any use in the diagnosis and treatment of breast cancer, the results of this project must also be applicable to real Tactile Maps. Figure 20 shows a real Tactile Map (the circles in the graph) superimposed with a Gaussian curve similar to those predicted by the Finite Element Simulations. There is very little difference between the curves; an Amplitude, Stdev, and Base value can be assigned to the real Tactile Maps just as they could to the output of the ABAQUS simulations. Most of the deviation in the real data from the Gaussian curve is due to the fact that real tumors are not perfect spheres. If one side of the tumor extends further than the other, then the Tactile Map will be slightly skewed. Previous research (Wellman 1999) has shown that Finite Element Simulations do produce similar Tactile Maps to tests with rubber models and real tumors.

To determine whether Probe Displacement could indeed distinguish between soft and hard lumps, I asked Assurance Medical, the company which developed the Tactile Imaging System, to perform a Displacement experiment using an actual Tactile Probe. In this experiment, weight was applied to a Tactile Probe and the Displacement of the Probe into a rubber model of breast tissue was measured. Because the Tactile Probe cannot record Displacement by itself, a separate caliper was used for the measurements. From these tests, Force-Strain curves (Figure 21), similar to those shown in Figure 15 and Figure 16 were generated. Four different lump stiffnesses are shown. Lump stiffnesses of E = 1 and E = 4.84 had nearly the same Force-Strain curves. Similarly, there was very little difference between lumps with siffnesses of E = 8.93 and E = 17.9. However, there was a clear distinction between the softer stiffnesses and the two lumps that had a higher stiffness. Also, the general trend of increasing slope with increasing stiffness followed exactly what was predicted by the Finite Element Simulations (Figure 15). Thus, it does seem possible that useful information could be obtained from the Displacement of a real Probe if Assurance Medical modified their Tactile Probe to capture this data.

Conclusions and Further Development

The conclusions of this study are as follows. A Forward Model can be constructed to very accurately predict the Tactile Maps, Tactile Images, and Probe Displacement of the Finite Element Simulations. Using the current design of the Tactile Probe, which only captures pressure data, an Inverse Model can be used to accurately predict the total force applied to the Probe, the thickness of the breast tissue, the diameter of a tumor, and the depth of a tumor. By incorporating Probe Displacement data into the Inverse Model, the tissue thickness and tumor diameter can be predicted with much more accuracy. The stiffness of lumps that are relatively soft can easily be determined; and, in general for all lumps, the stiffness can be estimated well enough to monitor changes in breast tissue and in many cases differentiate breast tissue types. Therefore, with modifications to the Tactile Probe that would enable it to measure Displacement, a complete Inverse Model is possible. Further research could explore different methods of implementing this feature into the Tactile Imaging System.


References

Akin, J. ED. Finite Element Analysis for Undergraduates. New York: Academic Press. 1986.

Engel, June. The Complete Breast Book. Canada: Key Porter Books Limited. 1996.

Fung, Y.C. Biomechanics: Mechanical Properties of Living Tissues Second Edition. New York: Springer. 1993.

Gallagher, R. H. Finite Elements in Biomechanics. New York: John Wiley & Sons. 1982.

Pederson, Licille M. and Janet M. Trigg. Breast Cancer : A Family Survival Guide. London: Bergin & Garvey. 1995.

Ruddon, Raymond W. Cancer Biology. Third Edition. New York: Oxford University Press. 1995.

Siever, Ellen. Perl in a Nutshell. O'Reilly & Associates, Inc. January 1999.

Wellman, Parris Saxon. "Tactile Imaging", a PhD thesis presented to Harvard Universitiy's Division of Engineering and Applied Sciences. 1999.

and

Assurance Medical
103 South Street
Hopkinton, MA 01748
Tel: 888-470-1842
Fax: 508-435-5166
www.assurancemed.com


Acknowledgments

The following people provided me with assistance in this project:

Professor Robert Howe
Dr. Parris Wellman
The employees of Assurance Medical
Professor John Hutchinson
Professor Joseph Harrington
Anna Galea
Heather Gunter


List of Figures

Figure 1: The Tactile Probe and the Formation of Tactile Maps
Figure 2: A Two-Dimensional Tactile Map of a Real Tumor
Figure 3: Design Flow of the Project
Figure 4: Breast Anatomy - Three Layers of Breast Tissue
Figure 5: Breast Tissue Geometry Used in the Finite Element Simulations
Figure 6: Development of the Finite Element Model
Figure 7: Finite Element Models for Selected Tissue Geometries
Figure 8: Tissue Deformation and Tactile Images
Figure 9: Formation of Tactile Maps from Contact Pressure Data
Figure 10: Feature Extraction (Tactile Maps and Tactile Images)
Figure 11: Feature Extraction (Displacement and Pressure "Jumps")
Figure 12: The Effects of Input Parameters on Tactile Maps
Figure 13: The Relationship Between Input and Output Parameters
Figure 14: The Effects of Input Parameters on Probe Displacement
Figure 15: Strain-Force Curves (Different E)
Figure 16: Strain-Force Curves (Different d and h)
Figure 17: Validity of the Forward Models
Figure 18: Validity of the Inverse Models (Using Pressure Only)
Figure 19: Validity of the Inverse Models (Using Pressure and Displacement)
Figure 20: Gaussian Fit to a Real Tactile Map
Figure 21: Strain-Force Curves Generated Using a Real Tactile Probe


Appendices

Appendix A: Raw Data from Simulations
Appendix B: Regression Analysis Results for the Forward Model
Appendix C: Regression Analysis Results for the Inverse Model
Appendix D: Computer Code

 
Copyright 2000    Griffin Weber    http://www.griffinweber.com