Nonlinear knowledge in kernel approximation.

Prior knowledge over arbitrary general sets is incorporated into nonlinear kernel approximation problems in the form of linear constraints in a linear program. The key tool in this incorporation is a theorem of the alternative for convex functions that converts nonlinear prior knowledge implications into linear inequalities without the need to kernelize these implications. Effectiveness of the proposed formulation is demonstrated on two synthetic examples and an important lymph node metastasis prediction problem. All these problems exhibit marked improvements upon the introduction of prior knowledge over nonlinear kernel approximation approaches that do not utilize such knowledge.

Importance of nuclear morphology in breast cancer prognosis.

The purpose of this study is to define prognostic relationships between computer-derived nuclear morphological features, lymph node status, and tumor size in breast cancer. Computer-derived nuclear size, shape, and texture features were determined in fine-needle aspirates obtained at the time of diagnosis from 253 consecutive patients with invasive breast cancer. Tumor size and lymph node status were determined at the time of surgery. Median follow-up time was 61.5 months for patients without distant recurrence. In univariate analysis, tumor size, nuclear features, and the number of metastatic nodes were of decreasing significance for distant disease-free survival. Nuclear features, tumor size, and the number of metastatic nodes were of decreasing significance for overall survival. In multivariate analysis, the morphological size feature, largest perimeter, was more predictive of disease-free and overall survival than were either tumor size or the number of axillary lymph node metastases. This morphological feature, when combined with tumor size, identified more patients at both the good and poor ends of the prognostic spectrum than did the combination of tumor size and axillary lymph node status. Our data indicate that computer analysis of nuclear features has the potential to replace axillary lymph node status for staging of breast cancer. If confirmed by others, axillary dissection for breast cancer staging, estimating prognosis, and selecting patients for adjunctive therapy could be eliminated.

Successive overrelaxation for support vector machines.

Successive overrelaxation (SOR) for symmetric linear complementarity problems and quadratic programs is used to train a support vector machine (SVM) for discriminating between the elements of two massive datasets, each with millions of points. Because SOR handles one point at a time, similar to Platt's sequential minimal optimization (SMO) algorithm which handles two constraints at a time and Joachims' SVMlight which handles a small number of points at a time, SOR can process very large datasets that need not reside in memory. The algorithm converges linearly to a solution. Encouraging numerical results are presented on datasets with up to 10,000,000 points. Such massive discrimination problems cannot be processed by conventional linear or quadratic programming methods, and to our knowledge have not been solved by other methods. On smaller problems, SOR was faster than SVMlight and comparable or faster than SMO.

Computer-derived nuclear features compared with axillary lymph node status for breast carcinoma prognosis.

BACKGROUND: Both axillary lymph node involvement and tumor anaplasia, as expressed by visually assessed grade, have been shown to be prognostically important in breast carcinoma outcome. In this study, axillary lymph node involvement was used as the standard against which prognostic estimations based on computer-derived nuclear features were gauged. METHODS: The prognostic significance of nuclear morphometric features determined by computer-based image analysis were analyzed in 198 consecutive preoperative samples obtained by fine-needle aspiration (FNA) from patients with invasive breast carcinoma. A novel multivariate prediction method was used to model the time of distant recurrence as a function of the nuclear features. Prognostic predictions based on the nuclear feature data were cross-validated to avoid overly optimistic conclusions. The estimated accuracy of these prognostic determinations was compared with determinations based on the extent of axillary lymph node involvement. RESULTS: The predicted outcomes based on nuclear features were divided into three groups representing best, intermediate, and worst prognosis, and compared with the traditional TNM lymph node stratification. Nuclear feature stratification better separated the prognostically best from the intermediate group whereas lymph node stratification better separated the prognostically intermediate from the worst group. Prognostic accuracy was not increased by adding lymph node status or tumor size to the nuclear features. CONCLUSIONS: Computer analysis of a preoperative FNA more accurately identified prognostically favorable patients than did pathologic examination of axillary lymph nodes and may obviate the need for routine axillary lymph node dissection.

Indeterminate fine-needle aspiration of the breast. Image analysis-assisted diagnosis.

BACKGROUND: Fine-needle aspiration (FNA) of the breast, although effective for the diagnosis of breast carcinoma, has a significant drawback. A minority of cases cannot be classified as benign or malignant. These FNAs are assigned an inconclusive diagnosis, often prompting surgical biopsy. Surgery is justified in some of these cases, but many of these lesions are benign. If these inconclusive FNAs could be accurately diagnosed as benign or malignant, many of these patients might avoid having to undergo surgical biopsy. METHODS: An image analysis and an automated learning system that was developed at the University of Wisconsin (Xcyt) was used to categorize 56 (37 benign and 19 malignant) breast FNAs diagnosed as "indeterminate" and the computer diagnosis compared with the surgical biopsy. For each case, an operator chose a group of cells within a single field on the FNA slide and digitized this image using a video camera. The outline of each nucleus was manually outlined, and the exact border was delineated by the computer. Based on the analysis of three nuclear features (area, texture, and smoothness), the Xcyt system computed a benign or malignant diagnosis and a corresponding probability of malignancy for each case. RESULTS: Probabilities of malignancy for the respective cases ranged from 0.0-1.0. Benign cases were defined as those having probabilities of malignancy < 0.3; those with probabilities above this limit were considered malignant. Using these criteria, the computer identified 33 cases as benign and 23 cases as malignant. When compared with the surgical biopsy, 42 of the cases (75%) were correctly classified with a sensitivity and specificity of 73.7% and 75.7%, respectively. There were only 5 false-negative cases with a false-negative rate of 13.5% and a predictive value of a negative test of 84.8%. CONCLUSIONS: When faced with inconclusive diagnoses of FNAs of breast masses, the authors believe that image analysis may be used as an aid in the further classification of such lesions, thereby providing a more appropriate triage for surgical biopsy.

Computer-derived nuclear "grade" and breast cancer prognosis.

Visual assessments of nuclear grade are subjective yet still prognostically important. Now, computer-based analytical techniques can objectively and accurately measure size, shape and texture features, which constitute nuclear grade. The cell samples used in this study were obtained by fine needle aspiration (FNA) during the diagnosis of 187 consecutive patients with invasive breast cancer. Regions of FNA preparations to be analyzed were digitized and displayed on a computer monitor. Nuclei to be analyzed were roughly outlined by an operator using a mouse. Next, the computer generated a "snake" that precisely enclosed each designated nucleus. Ten nuclear features were then calculated for each nucleus based on these snakes. These results were analyzed statistically and by an inductive machine learning technique that we developed and call "recurrence surface approximation" (RSA). Both the statistical and RSA machine learning analyses demonstrated that computer-derived nuclear features are prognostically more important than are the classic prognostic features, tumor size and lymph node status.

Computer-derived nuclear features distinguish malignant from benign breast cytology.

This article describes the use of computer-based analytical techniques to define nuclear size, shape, and texture features. These features are then used to distinguish between benign and malignant breast cytology. The benign and malignant cell samples used in this study were obtained by fine needle aspiration (FNA) from a consecutive series of 569 patients: 212 with cancer and 357 with fibrocystic breast masses. Regions of FNA preparations to be analyzed were converted by a video camera to computer files that were displayed on a computer monitor. Nuclei to be analyzed were roughly outlined by an operator using a mouse. Next, the computer generated a "snake" that precisely enclosed each designated nucleus. The computer calculated 10 features for each nucleus. The ability to correctly classify samples as benign or malignant on the basis of these features was determined by inductive machine learning and logistic regression. Cross-validation was used to test the validity of the predicted diagnosis. The logistic regression cross validated classification accuracy was 96.2% and the inductive machine learning cross-validated classification accuracy was 97.5%. Our computerized system provides a probability that a sample is malignant. Should this probability fall between 30% and 70%, the sample is considered "suspicious," in the same way a visually graded FNA may be termed suspicious. All of the 128 consecutive cases obtained since the introduction of this system were correctly diagnosed, but nine benign aspirates fell into the suspicious category.(ABSTRACT TRUNCATED AT 250 WORDS)

Computerized breast cancer diagnosis and prognosis from fine-needle aspirates.

OBJECTIVE: To use digital image analysis and machine learning to (1) improve breast mass diagnosis based on fine-needle aspirates and (2) improve breast cancer prognostic estimations. DESIGN: An interactive computer system evaluates, diagnoses, and determines prognosis based on cytologic features derived from a digital scan of fine-needle aspirate slides. SETTING: The University of Wisconsin (Madison) Departments of Computer Science and Surgery and the University of Wisconsin Hospital and Clinics. PATIENTS: Five hundred sixty-nine consecutive patients (212 with cancer and 357 with benign masses) provided the data for the diagnostic algorithm, and an additional 118 (31 with malignant masses and 87 with benign masses) consecutive, new patients tested the algorithm. One hundred ninety of these patients with invasive cancer and without distant metastases were used for prognosis. INTERVENTIONS: Surgical biopsy specimens were taken from all cancers and some benign masses. The remaining cytologically benign masses were followed up for a year and surgical biopsy specimens were taken if they changed in size or character. Patients with cancer received standard treatment. OUTCOME MEASURES: Cross validation was used to project the accuracy of the diagnostic algorithm and to determine the importance of prognostic features. In addition, the mean errors were calculated between the actual times of distant disease occurrence and the times predicted using various prognostic features. Statistical analyses were also done. RESULTS: The predicted diagnostic accuracy was 97% and the actual diagnostic accuracy on 118 new samples was 100%. Tumor size and lymph node status were weak prognosticators compared with nuclear features, in particular those measuring nuclear size. Compared with the actual time for recurrence, the mean error of predicted times for recurrence with the nuclear features was 17.9 months and was 20.1 months with tumor size and lymph node status (P = .11). CONCLUSION: Computer technology will improve breast fine-needle aspirate accuracy and prognostic estimations.

Image analysis and machine learning applied to breast cancer diagnosis and prognosis.

Fine needle aspiration (FNA) accuracy is limited by, among other factors, the subjective interpretation of the aspirate. We have increased breast FNA accuracy by coupling digital image analysis methods with machine learning techniques. Additionally, our mathematical approach captures nuclear features ("grade") that are prognostically more accurate than are estimates based on tumor size and lymph node status. An interactive computer system evaluates, diagnoses and determines prognosis based on nuclear features derived directly from a digital scan of FNA slides. A consecutive series of 569 patients provided the data for the diagnostic study. A 166-patient subset provided the data for the prognostic study. An additional 75 consecutive, new patients provided samples to test the diagnostic system. The projected prospective accuracy of the diagnostic system was estimated to be 97% by 10-fold cross-validation, and the actual accuracy on 75 new samples was 100%. The projected prospective accuracy of the prognostic system was estimated to be 86% by leave-one-out testing.

Machine learning techniques to diagnose breast cancer from image-processed nuclear features of fine needle aspirates.

An interactive computer system evaluates and diagnoses based on cytologic features derived directly from a digital scan of fine-needle aspirate (FNA) slides. A consecutive series of 569 patients provided the data to develop the system and an additional 54 consecutive, new patients provided samples to test the system. The projected prospective accuracy of the system estimated by tenfold cross validation was 97%. The actual accuracy on 54 new samples (36 benign, 1 atypia, and 17 malignant) was 100%. Digital image analysis coupled with machine learning techniques will improve diagnostic accuracy of breast fine needle aspirates.

Breast cytology diagnosis with digital image analysis.

An interactive computer system has been developed for evaluating cytologic features derived directly from a digital scan of breast fine needle aspirate slides. The system uses computer vision techniques to analyze cell nuclei and classifies them using an inductive method based on linear programming. A digital scan of selected areas of the aspirate slide is done by a trained observer, while the analysis of the digitized image is done by an untrained observer. When trained and tested on 119 breast fine needle aspirates (68 benign and 51 malignant) using leave-one-out testing, 90% correctness was achieved. These results indicate that the method is accurate (good intraobserver and interobserver reproducibility) and that an untrained operator can obtain diagnostic results comparable to those achieved visually by experienced observers.

Computer-designed expert systems for breast cytology diagnosis.

Three expert systems have been developed to diagnose from nine scalar values visually assigned to epithelial cells obtained by breast fine needle aspiration. These expert systems achieved up to 0.98 sensitivity and 0.97 specificity. When applied to 804 breast masses, the clinical sensitivity was 0.98 and specificity was 0.93 (exclusive of the 0.04 unsatisfactory aspirates). Cancers can be missed physically during the aspiration process; thus, some clinically suspicious masses were biopsied despite benign cytology. This contributed to the difference between the expert system and clinical specificities.

Multisurface method of pattern separation for medical diagnosis applied to breast cytology.

Multisurface pattern separation is a mathematical method for distinguishing between elements of two pattern sets. Each element of the pattern sets is comprised of various scalar observations. In this paper, we use the diagnosis of breast cytology to demonstrate the applicability of this method to medical diagnosis and decision making. Each of 11 cytological characteristics of breast fine-needle aspirates reported to differ between benign and malignant samples was graded 1 to 10 at the time of sample collection. Nine characteristics were found to differ significantly between benign and malignant samples. Mathematically, these values for each sample were represented by a point in a nine-dimensional space of real variables. Benign points were separated from malignant ones by planes determined by linear programming. Correct separation was accomplished in 369 of 370 samples (201 benign and 169 malignant). In the one misclassified malignant case, the fine-needle aspirate cytology was so definitely benign and the cytology of the excised cancer so definitely malignant that we believe the tumor was missed on aspiration. Our mathematical method is applicable to other medical diagnostic and decision-making problems.

Computer-aided diagnosis of breast aspirates via expert systems.

Two computer-driven expert systems trained to correctly diagnose 369 fine needle aspirates of the breast on the basis of nine cytologic descriptive parameters were tested on 70 newly obtained aspirates (57 benign and 13 malignant). The system generated by multisurface pattern separation misclassified one malignant test sample (i.e., one false negative) while the system generated by a connectionist algorithm (neural network) misclassified two benign test samples (i.e., two false positives). A decision tree misclassified three of the benign test samples (i.e., three false positives). These expert systems aid in the cytologic diagnosis of breast aspirates and can serve as models for other applications.

A conjugate decomposition of the Euclidean space.

Given a closed convex cone K in the n-dimensional real Euclidean space R(n) and an nxn real matrix A that is positive definite on K, we show that each vector in R(n) can be decomposed into a component that lies in K and another that lies in the conjugate cone induced by A and such that the two vectors are conjugate to each other with respect to A + A(T). As a consequence of this decomposition we establish the following characterization of positive definite matrices: An nxn real matrix A is positive definite if and only if it is positive definite on some closed convex cone K in R(n) and (A + A(T))(-1) exists and is positive semidefinite on the polar cone K(0). If K is a subspace of R(n), then K(0) is its orthogonal complement K[unk]. Other applications include local duality results for nonlinear programs and other characterizations of positive definite and semidefinite matrices.