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Learning Qualitative Models from Physiological Signals by David Tak-Wai Hau S.B., Massachusetts Institute of Technology (1992) Submitted to the Department of Electrical Engineering and Computer Science in partial fulfillment of the requirements for the degree of Master of Science in Electrical Engineering and Computer Science at the MASSACHUSETTS INSTITUTE OF TECHNOLOGY May 1994 ( David Tak-Wai Hau, MCMXCIV. All rights reserved. The author hereby grants to MIT permission to reproduce and distribute publicly paper and electronic copies of this thesis document in whole or in part, and to grant others the right to do so. Author . ........................... . -. Department of Electrical Engineering and Computer Science May 17, 1994 Certified by .................................................. ( Eico Project Manager, Hewlett-P ,~7o----~ Certified by ....................-- ...- ... W. Coiera ard Laboratories Thesis Supervisor ...- Ror G. Mark Grover Hermann Professor in Health Sciences and Technology Cd (x Anen+ [~ ThesisSupervisor W.,n Accepteu Dy............ FredericR. Morgenthaler tee on Graduate Students LIBRARSES Learning Qualitative Models from Physiological Signals by David Tak-Wai Hau Submitted to the Department of Electrical Engineering and Computer Science on May 17, 1994, in partial fulfillment of the requirements for the degree of Master of Science in Electrical Engineering and Computer Science Abstract Physiological models represent a useful form of knowledge, but are both difficult and time consuming to generate by hand. Further, most physiological systems are incompletely understood. This thesis addresses these two issues with a system that learns qualitative models from physiological signals. The qualitative representation of models allows incom- plete knowledgeto be encapsulated, and is based on Kuipers' approach used in his QSIM algorithm. The learning algorithm allows automatic generation of such models, and is based on Coiera's GENMODEL algorithm. We first show that QSIM models are efficiently PAC learnable from positive examples only, and that GENMODEL is an algorithm for efficiently constructing a QSIM model consistent with a given set of examples, if one exists. We then describe the learning system in detail, including the front-end processing and segmenting stages that transform a signal into a set of qualitative states, and GENMODEL that uses these qualitative states as positive examples to learn a QSIM model. Next we report re- sults of experiments using the learning system on data segments obtained from six patients during cardiac bypass surgery. Useful model constraints were obtained, representing both general physiological knowledge and knowledge particular to the patient being monitored. Model variation across time and across different levels of temporal abstraction and fault tolerance is examined. Thesis Supervisor: Enrico W. Coiera Title: Project Manager, Hewlett-Packard Laboratories Thesis Supervisor: Roger G. Mark Title: Grover Hermann Professor in Health Sciences and Technology Acknowledgements First I would like to thank Enrico Coiera, my supervisor at HP Labs, for offering me an extremely interesting project which combines my interests in artificial intelligence, signal processing and medicine, and for being a really great supervisor. I especially thank him for his superb guidance throughout the project, and for his encouragement and patience in the write-up process.. It has been my pleasure to have him as a mentor and a friend. I would also like to thank Dave Reynolds at HP Labs for offering me a lot of help and advice in the project, and for reading part of a draft of the thesis. The overall front-end processing scheme and the idea of using Gaussian filters came from him. I also appreciate his excellent sense of humor. Every discussion with him has been a lot of fun. I am grateful to Professor Roger Mark for taking time out of his busy schedule to supervise my thesis. I thank him for his patience and his comments in the write-up process. I really enjoyed the discussions with him in which I learned a lot about cardiovascular iphysiology. My friend David Maw kindly let me store my enormous number of data files in his server at LCS. I thank him deeply for his help. Most of all, I thank my family for their never-ending love and encouragement. I dedicate this thesis to my mother. This thesis describes work done while the author was a VI-A student at Hewlett-Packard Laboratories, Bristol, U.K. in Spring and Summer 1993. 3 To my mother 4 Contents 1 Introduction 14 2 Qualitative Reasoning 17 2.1 Qualitative Model Constraints ........ 17 2.2 Qualitative System Behavior. 19 2.3 Qualitative Simulation: QSIM......... 21 2.4 Learning Qualitative Models: GENMODEL . 21 2.5 An Example: The U-tube ........... 22 2.5.1 Qualitative Simulation of the U-tube . 24 2.5.2 Learning the U-tube Model ...... 24 3 Learning Qualitative Models 28 ...... .... . .28 3.1 Probably Approximately Correct Learning ........ 3.1.1 Definitions. 3.1.2 PAC Learnability .................. 3.1.3 Proving PAC Learnability ............. 3.1.4 An Occam Algorithm for Learning Conjunctions . ... 3.2 The GENMODEL Algorithm ............... 3.3 3.4 QSIM Models are PAC Learnable ............. 3.3.1 The Class of QSIM Models is Polynomial-Sized . 3.3.2 QSIM Models are Polynomial-Time Identifiable GENMODEL ........................ 3.4.1 Dimensional Analysis. 3.4.2 Performance on Learning the U-Tube Model 3.4.3 Version Space .................... 5 .......... . .28 .......... . .29 .......... . .32 .......... . .33 .......... . .35 .......... . .35 .......... . .35 .......... . .37 .......... . .37 .......... . .38 .......... . .38 . . ... ... . .30 3.5 3.4.4 Fault Tolerance ........................... 39 3.4.5 Comparison of GENMODEL with Other Learning Approaches 40 Applicability of PAC Learning ...................... 4 Physiological Signals and Mode 4.1 4.2 4.1.1 Primary Measurements 4.1.2 Derived Values .... A Qualitative Cardiovascular 5 System Architecture 5.1 Overvievw . 5.2 Front-End System ...... 5.2.1 Artifact Filter..... 5.2.2 Median Filter. 5.2.3 Temporal Abstraction 5.2.4 Gaussian Filter .... 5.3 ...................... ...................... ...................... ...................... Hemodynamic Monitoring 5.2.5 Differentiator. 5.2.6 An Example. Segmenter ........... I .......................... .......................... .......................... .......................... .......................... .......................... .......................... .......................... .......................... 6 Results and Interpretation 6.1 6.2 Inter-Patient Model Comparison: 42 44 44 44 46 49 54 54 54 56 57 57 60 66 68 68 72 Patients 73 1 .5 . . . . . . . . . . . . . . . .. 6.1.1 Patient 1 . . . . . . 73 6.1.2 Patient 2 ......... . . . . . 80 6.1.3 Patient 3 ......... . . . . . 87 6.1.4 Patient 4 ......... . . . . . 94 6.1.5 Patient 5 ......... . . . . . Intra-Patient Model Comparison: Patient ,Sement 1-6.......... 6, 101 108 6.2.1 Segment i ......... . . . . . Se me t . 1-6.. . . . . . . . . . . 108 6.2.2 Segment 2 ......... . . . . . . . . . . . . . . . . . . . . . . 115 6.2.3 Segment 3 ......... . . . . . 122 6.2.4 Segment 4 ......... . . . . . 131 6 6.2.5 Segment 5 ................................. 138 6.2.6 Segment 6 ................................. 145 7 Discussion 7.1 7.2 152 Validity of Models Learned . . . . . . . . . . . . . . . . . ........ 7.1.1 Model Variation Across Time ...................... 153 7.1.2 Model Variation Across Different Levels of Temporal Abstraction . . 153 7.1.3 Model Variation Across Different Levels of Fault Tolerance 154 7.1.4 Sources of Error ..... ............................. Applicability in Diagnostic Patient Monitoring 155 ................ 156 7.2.1 A Learning-Based Approach to Diagnostic Patient Monitoring 7.2.2 Generating Diagnoses Based on Models Learned . .......... 8 Conclusion and Future Work 8.1 152 Future Work 8.1.1 . 156 157 159 ................................... 160 IJsing Corresponding Values in Noisy Learning Data ......... .Bibliography 160 165 7 List of Figures 2-1 Qualitative reasoning is an abstraction of mathematical reasoning with differential equations and continuously differentiable functions 2-2 .......... Qualitative states following addition of an increment of fluid to one arm of a U-tube.................................... 2-3 2-4 18 ... 22 Prolog representation for the U-tube model, initialized for an increment of fluid added to arm a ................................ 23 Qualitative model of a U-tube. 24 .......................... 2-5 Output of the system behavior generated by running QSIM on the U-tube model ................... 2-6 ..................... 25 Qualitative plots for the qualitative states of a U-tube following addition of an increment of fluid to one arm until equilibrium is reached. ........ 2-7 26 Output of the U-tube model generated by GENMODEL on the given behavior. Note that the model learned is the same as the original model shown previously ...................................... 3-1 GENMODEL algorithm 27 .............................. 34 3-2 Using the upper and lower boundary sets to represent the version space... 39 3-3 GENMODEL algorithm with fault tolerance. 40 ................. 4-1 Deriving the systolic, diastolic and mean pressures from the arterial blood pressure waveform ................................. 45 4-2 Deriving the stroke volume from the arterial blood pressure waveform.... 47 4-3 Cardiac output curves for the normal heart and for hypo- and hypereffective hearts................................ ......... 5-1 Overall architecture of the learning system. 8 .................. 50 54 5-2 Architecture used for front-end processing of physiological signals. ..... 55 5-3 An artifact found in blood pressure signals, caused by flushing the blood pressure line with high pressure saline solution 5-4 ................. 56 An artifact found in blood pressure signals, caused by blood sampling from the blood pressure catheter ........................... 5-5 56 A general artifact caused by the transducer being left open to air. ..... 57 5-6 The median filter removes features with durations less than half of its window length, but retains sharp edges of remaining features .............. 5-7 The effect of aliasing in the segmentation of a signal with the temporal ab- straction parameter set to T. g(t) is aliased into h(t) in the qualitative be- havior produced ........... 5-8 58 ....................... Plot of the Hanning window wn]. 61 ........................ 64 5-9 Plots of impulse responses h[n] of Gaussian filters for a = 20,40, 60,80, 100. 64 5-10 Plots of frequency responses H(Q) of Gaussian filters for a = 20, 40, 60, 80, 100. 65 5-11 Impulse response of an FIR bandlimited differentiator ............. 5-12 Frequency response of an FIR bandlimited differentiator ............ 66 67 5-13 Equivalent frequency responses of a Gaussian filter in cascade with a bandlimited differentiator for a = 20, 40, 60, 80, 100 ................. 67 5-14 A segment of the mean arterial blood pressure (ABPM) signal. Note the artifacts at t = 600, 1000, 3400 seconds. ..................... 68 5-15 The ABPM data segment processed by the artifact filter. Note that the artifacts have been filtered but the signal contains many impulsive features. 68 5-16 The ABPM data segment processed by the median filter. Note that the impulsive features have been filtered ....................... 69 5-17 The ABPM data segment processed by the Gaussian filter. Note that the signal has been smoothed ............................ 5-18 The ABPM data segment processed by the differentiator. 69 .......... 69 5-19 Overall scheme of segmentation to produce a qualitative behavior from processed signals and derivatives .......................... 6-1 Patient 70 : Original Signals. Note the relatively constant heart rate due to the effect of beta-blockers ............................ 9 73 6-2 Patient 1: Filtered Signals (L = 61) 74 6-3 Patient 1: Filtered Signals (L = 121) . . . . . . . . . . . . . . . . . . . . . . 75 6-4 Patient 1: Filtered Signals (L = 241) . . . . . . . . . . . .. 76 6-5 Patient 1: Filtered Signals (L = 361) . . . . . . . . . . . . . . . . . . . . . 77 6-6 Patient 1: Filtered Signals (L = 481) . . . . . . . . . . . . . . . . . . . . . 78 6-7 Patient 1: Filtered Signals (L = 601) .............. 79 ...... . . . . . . . ..... . ... 6-8 Patient 2: Original Signals. Note the relatively constant heart rate due to artificial pacing .............. . . . . . . . . . . . . . . . . 80 6-9 Patient 2: Filtered Signals (L = 61) . . . . . . . . . . . . . . . . 81 6-10 Patient 2: Filtered Signals (L = 121) . . . . . . . . . . . . . . . . 82 6-11 Patient 2: Filtered Signals (L = 241) . . . . . . . . . . . . . . . . 83 6-12 Patient 2: Filtered Signals (L = 361) . . . . . . . . . . . . . . . . 6-13 Patient 2: Filtered Signals (L = 481) . . . . . . . . . . . . . . . . 85 6-14 Patient 2: Filtered Signals (L = 601) . . . . . . . . . . . . . . . . . 86 6-15 Patient :3: Original Signals ....... . . . . . . . . . . . . . . . . 6-16 Patient 3: Filtered Signals (L = 61) . . . . . . . . . . . . . . . . 88 6-17 Patient :3: Filtered Signals (L = 121) . . . . . . . . . . . . . . . . 89 6-18 Patient 3: Filtered Signals (L = 241) . . . . . . . . . . . . . . . . 90 6-19 Patient 3: Filtered Signals (L = 361) . . . . . . . . . . . . . . . . 91 6-20 Patient 3: Filtered Signals (L = 481) . . . . . . . . . . . . . . . . 92 . . . . . . . . . . . . . . . . 93 6-21 Patient :3: Filtered Signals (L = 601) . . . . . . .84 .87 6-22 Patient 4: Original Signals. Note the relatively constant heart rate due to the effect of beta-blockers ...... 94 6-23 Patient 4: Filtered Signals (L = 61). Note that the trends of the relatively constant heart rate are amplified. .. . . . . . . . . . . . . . . 95 6-24 Patient 4: Filtered Signals (L = 121) . . . . . . . . . . . . . . 96 6-25 Patient 4: Filtered Signals (L = 241) . . . . . . . . . . . . . . 97 6-26 Patient 4: Filtered Signals (L = 361) . . . . . . . . . . . . . . 98 6-27 Patient 4: Filtered Signals (L = 481) . . . . . . . . . . . . . . 99 6-28 Patient 4: Filtered Signals (L = 601) . . . . . . . . . . . . . . . 100 6-29 Patient 5: Original Signals. Note the relatively constant heart rate due to the effect of beta-blockers ...................... 10 101 6-30 Patient 5: Filtered Signals (L = 61). Note that the trends of the relatively constant heart rate are amplified . . . . . . . . . . . . . . . . . 102 6-31 Patient 5: Filtered Signals (L = 121) . . . . . . . . . . . . . . . . 103 6-32 Patient 5: Filtered Signals (L = 241) . . . . . . . . . . . . . . . .104 6-33 Patient 5: Filtered Signals (L = 361) . . . . . . . . . . . . . . . .105 6-34 Patient 5: Filtered Signals (L = 481) . . . . . . . . . . . . . . . . 106 6-35 Patient 5: Filtered Signals (L = 601) . . . . . . . . . . . . . . . ..107 6-36 Patient 6, Segment 1: Original Signals . . . . . . . . . . . . . . . . . 108 1: Filtered Signals (1 = 61) . . . . . . . . . . . . . . . . 109 6-38 Patient 6, Segment 1: Filtered Signals (1L = 121) . . . . . . . . . . . . . . . ..110 6-39 Patient 6, Segment 1: Filtered Signals (1L = 241) . . . . . . . . . . . . . . . 6-40 Patient 6, Segment 1: Filtered Signals (1L = 361) . . . . . . . . . . . . . . . . 112 6-41 Patient 6, Segment 1: Filtered Signals (1L = 481) . . . . . . . . . . . . . . . . 113 6-42 Patient 6, Segment 1: Filtered Signals (1L = 601) . . . . . . . . . . . . . . . . 114 6-43 Patient 6, Segment 2: Original Signals . . . . . . . . . . . . . . . . . 115 6-44 Patient 6, Segment 2: Filtered Signals (L = 61) . . . . . . . . . . . . . . . . 116 6-45 Patient 6, Segment 2: Filtered Signals (L = 121) . . . . . . . . . . . . . . . ..117 6-46 Patient 6, Segment 2: Filtered Signals (L = 241) . . . . . . . . . . . . . . . . 118 6-47 Patient 6, Segment 2: Filtered Signals (L = 361) . . . . . . . . . . . . . . . . 119 6-48 Patient 6, Segment 2: Filtered Signals (L = 481) . . . . . . . . . . . . . . . ..120 6-49 Patient 6, Segment 2: Filtered Signals (L = 601) . . . . . . . . . . . . . . . . 121 6-50 Patient 6, Segment 3: Original Signals ...... . . . . . . . . . . . . . . . . 122 6-51 Patient 6, Segment 3: 6-37 Patient 6, Segment Filtered Signals (L = 61) l...111 . . . . . . . . . . . . . . . . 123 6, Segment 3: Filtered Signals (L = 121) . . . . . . . . . . . . . . . ..124 6-53 Patient 6, Segment 3: Filtered Signals (L = 241) . . . . . . . . . . . . . . . . 125 6-54 Patient 6, Segment 3: Filtered Signals (L = 361) . . . . . . . . . . . . . . . . 126 6-55 Patient 6, Segment 3: Filtered Signals (L = 481) . . . . . . . . . . . . . . . . 127 6-56 Patient 6, Segment 3: Filtered Signals (L = 601) . . . . . . . . . . . . . . . 6-57 Patient 6, Segment 4: Original Signals ...... . . . . . . . . . . . . . . . . 131 6-58 Patient 6, Segment 4: Filtered Signals (L = 61) . . . . . . . . . . . . . . . . 132 6-59 Patient 6, Segment 4: Filtered Signals (L = 121) . . . . . . . . . . . . . . . . 133 6-52 Patient 6-60 Patient 6, Segment 4: Filtered Signals (L = 241) 11 ... ..... .129 134 6-61 Patient 6, Segment 4: Filtered Signals (L = 361) . . . . . . . . . . . . . . . . 135 6-62 Patient 6, Segment 4: Filtered Signals (L = 481) . . . . . . . . . . . . . . . . 136 6-63 Patient 6, Segment 4: Filtered Signals (L = 601) . . . . . . . . . . . . . . . 6-64 Patient 6, Segment 5: Original Signals ...... 6-65 Patient 6, Segment 5: Filtered Signals (L = 61) 6-66 Patient 6, Segment 5: Filtered Signals (L = 121) 6-67 Patient 6, Segment 5: Filtered Signals (L = 241) 6-68 Patient 6, Segment 5: Filtered Signals (L = 361) . . . . . . . . . . . . . . . . 142 6-69 Patient 6, Segment 5: Filtered Signals (L = 481) . . . . . . . . . . . . . . . .143 6-70 Patient 6, Segment 5: Filtered Signals (L = 601) . . . . . . . . . . . . . . . .144 6-71 Patient 6, Segment 6: Original Signals ...... . . . . . . . . . . . . . . . . 145 6-72 Patient 6, Segment 6: Filtered Signals (L = 61) 6-73 Patient 6, Segment 6: Filtered Signals (L = 121) . . . . . . . . . . . . . . . 6-74 Patient 6, Segment 6: Filtered Signals (L = 241) . . . . . . . . . . . . . . . . 148 6-75 Patient 6, Segment 6: Filtered Signals (L = 361) . . . . . . . . . . . . . . . . 149 6-76 Patient 6, Segment 6: Filtered Signals (L = 481) . . . . . . . . . . . . . . . .137 . . . . . . . . . . . . . . . . 138 . . . . . . . . . . . . . . . .139 ......... ...... .... 141 .................... 140 . . . . . . . . . . . . . . . . 146 .147 .150 6-77 Patient 6, Segment 6: Filtered Signals (L = 601) . . . . . . . . . . . . . . . . 151 7-1 Two approaches to diagnostic patient monitoring. In the learning-based approach, models are continually learned from the patient data. In the historybased approach, a hypothesize-test-refine cycle is used to generate models that best match the patient data. based on the current model. In each approach, diagnoses are made ........................... 157 8-1 The two functions a and b shown obey direction-of-change consistency for the qualitative constraint M+ (a, b), but not magnitude consistency...... 162 8-2 Incorporating intervals into corresponding value sets is analogous to adding a bounding envelope to monotonic functions ................. 12 164 List of Tables 5.1 Table showing the orders M and lengths L of the Gaussian filters corresponding to a = 10, 20, 40, 60, 80, 100. . . . . . . . . . . . . . . . 13 . ...... 65 Chapter 1 Introduction Physiological models represent a useful form of knowledge because they encapsulate structural information of the system and allow deep forms of reasoning techniques to be applied. For example, such models are used in many prototype intelligent monitoring systems and medical expert systems. However, generating physiological models by hand is difficult and time consuming. Further, most physiological systems are incompletely understood. These factors have hindered the development of model-based reasoning systems for clinical decision support. Qualitative models permit useful representations of a system to be developed in the absence of extensive knowledge of the system. In the medical domain, such models have been applied to: * diagnostic patient monitoring of acid-base balance [6] * qualitative simulation of the iron metabolism mechanism [14] * qualitative simulation of urea extraction during dialysis [2] * qualitative simulation of the water balance mechanism and its disorders [19] * qualitative simulation of the mechanism for regulation of blood pressure [19] Further, recent developments in the machine learning community have produced methods of automatically inducing qualitative models from system behaviors. Applying such techniques to learning physiological models should not only benefit knowledge acquisition, but also provide a useful tool for physiologists who need to process vast amounts of data and 14 induce useful theoretical models of the systems they study. The learning system could also serve as a tool for model-based diagnosis. For example, it could be incorporated into a diagnostic patient monitoring system to perform adaptive model construction for diagnosis in a dynamic environment. In this thesis, we describe a system for learning qualitative models from physiological signals. The qualitative representation of physiological models used is based on Kuipers' approach, used in his qualitative simulation system QSIM [20]. The learning algorithm adopted is based on Coiera's GENMODEL system described in [5]. There has been much work in the machine learning community on learning qualitative models from system behaviors [4, 38, 28]. However, most of these efforts involve learning from ready-made qualitative behaviors, or at best simulated quantitative data only. Few, if any, address the issues of learning qualitative models from real, noisy data. Yet such domains are precisely ones where incomplete information prevails, and where automatic generation of qualitative models is most useful. This thesis addresses such a need with a learning system that constructs qualitative models from physiological signals, which are often corrupted with artifacts and other kinds of noise. In our system, we use signals derived from hemodynamic measurements, including: * heart rate (HR) * stroke volume (SV) * cardiac output (CO) * mean arterial blood pressure (ABPM) * mean central venous pressure (CVPM) * ventricular contractility (VC) * rate pressure product (RPP) * skin-to-core temperature gradient (AT) The thesis is organized into the following chapters: Chapter 2 provides an overview of qualitative reasoning, with focus on qualitative simulation, learning qualitative models, and the relationship between these two tasks. The classic example of the U-tube model is discussed, showing the sequence of qualitative states obtained by running QSIM on the model, and the result of using GENMODEL to learn the original model from these states. 15 Chapter 3 analyzes the learnability of qualitative models. QSIM models are shown to be PAC learnable from positive examples only, and GENMODEL is shown to be an algorithm for efficiently constructing a QSIM model consistent with a given set of examples, if one exists. The GENMODEL algorithm and the newly added features of dimensional analysis and fault tolerance are discussed in detail. The chapter ends with a comparison of GENMODEL with other approaches of learning qualitative models, and a discussion of difficulties of applying the PAC learning algorithm to learning qualitative models from physiological signals. Chapter 4 describes the various signals from hemodynamic monitoring used for our learning experiments. A qualitative model of the human cardiovascular system using these signals is provided as a 'gold standard' model, representing the author's best estimate of the target concepts learnable from the data. Chapter 5 discusses the architecture of the learning system in detail, with emphasis on the front-end processing stage and the segmenter, and how these two stages provide temporal abstraction. Chapter 6 reports results of using the learning system on data segments obtained from six patients during cardiac bypass surgery. Chapter 7 discusses validity of the models learned, analyzes sources of error, and discusses applicability of the system in diagnostic patient monitoring. Model variation across time and across different levels of temporal abstraction and fault tolerance is also examined. Chapter 8 provides a summary of the main points from the thesis, and discusses promising directions for future work. 16 Chapter 2 Qualitative Reasoning In studying the behavior of dynamic systems, we often model the system with a set of differential equations. The differential equations capture the structure of the system by specifying the relationships that exist among the functions of the system. From the differ- ential equations and an initial state, we can derive a quantitative system behavior using analytical methods or numerical simulation. A qualitative abstraction of the above procedure allows us to work with an incomplete specification of the system. A qualitative model can be represented by a set of qualitative differential equations, or qualitative constraints. From the constraints and an initial state, we can derive a qualitative system behavior using qualitative simulation. Figure 2-1 !illustratesthis abstraction. Different qualitative representations for models and their behavior have resulted from research in different problem domains [7]. In this chapter, we describe Kuipers' representation in [20], used in his qualitative simulation system QSIM. '2.1 Qualitative Model Constraints QSIM represents a model of a system by a set of qualitative constraints applied to the functions of the system. These include arithmetic constraints, which correspond to the basic arithmetic and differential operators, and monotonic function constraints, which correspond to monotonically increasing and decreasing functions that exist between two functions. 17 Perturbation Dynamic System System Behavior Abstraction Abstraction Analytical Methods/Numerical Simulation Quantitative System Behavior (Numerical Data) Differential Equations Statistical Analysis (e.g. RegressionMethods) Abstraction Abstraction Qualitative Simulation (e.g. QSIM) Qualitative Constraints (Qualitative Differential Equations) Qualitative System Behavior (Qualitative States) Inductive Learning (e.g.GENMODEL) Figure 2-1: Qualitative reasoning is an abstraction of mathematical reasoning with differential equations and continuously differentiable functions. Arithmetic Constraints Four arithmetic constraints are included in the QSIM representation: 1. add(f, g,h) 2. mult(f,g,h) f(t) +g(t) = h(t) f(t) x g(t) = h(t) 3. minus(f,g) -f(t) =-g(t) 4. deriv(f, g) a f'(t) = g(t) Monotonic Function Constraints When working with incomplete knowledge of a system, we may need to express a functional relationship that exists between two system functions, without specifying the functional relationship completely. In QSIM, the relationship can be described in terms of regions that are monotonically increasing or decreasing, and landmark values that the functions pass through. The QSIM representation includes two constraints for strictly monotonically 18 increasing and decreasing functional relationships. 1 f(t) = H(g(t)) where H(x) is a strictly monotonically increasing func- 1. M+(f, g) tion 2. M-(f,g) = f(t) = H(g(t)) where H(x) is a strictly monotonically decreasing function Note that since H(x) is a strictly monotonic function in both cases, both M+(f,g) and M-(f,g) require values of f(t) and g(t) to have a one-to-one correspondence, i.e. f(tl) = f(t 2 ) =- g(tl) = g(t2). Also note that the two function constraints M + and M- map onto many different functions including exponential, logarithmic, linear and other monotonically increasing or decreasing functions. This many-to-one mapping enables qualitative models to capture incomplete knowledge of a system. Qualitative constraints can be considered as an abstraction of ordinary differential equations (ODE). Every suitable ODE can be decomposed into a corresponding set of qualitative constraints. For example, Hooke's Law which relates the force of a spring to its displacement with the Hooke's constant k [20]: d2 x dt 2 k m =--x can be decomposed into the following qualitative constraints: deriv(x, v) v= d a d d2 x - dv dt == deriv(v,a) a = -- x == M-(a, x) m 2.2 Qualitative System Behavior A dynamic system is characterized by a number of system functions which vary over time. The system behavior can be described in terms of these functions. In QSIM, system func1 In this thesis, M+ is also referred to as mplus and M- as mminus. 19 tions must be reasonablefunctions f: [a, b] -+ R* where * = [-oo, oo], which satisfy the following criteria: 1. f is continuous on [a, b] 2. f is continuously differentiable on (a, b) 3. f has only finitely many critical points in any bounded interval 4. limt,a+f'(t) and limtb-f'(t) exist in 2*. We define f'(a) and f'(b) to be equal to these limits. Every system function f(t) has associated with it a finite and totally ordered set of landmark values. These include 0, the value of f(t) at each of its critical points, and the value of f(t) at each of the endpoints of its domain. The set of landmark values for a function form its quantity space which includes all the values of interest for the function. A value can be either at a landmark value, or in an interval between two landmark values. The system has associated with it a finite and totally ordered set of distinguished time points. These include all time points at which any of the system functions reaches a land- mark value. The set of distinguished time points form a time space. A qualitative time can be either a distinguished time point, or an interval between two adjacent distinguished time points. The qualitative state of f at t is defined as a pair < qval, qdir >. qval is the value of the function at t, and is either a landmark value or an interval between two landmark values. qdir is the direction of change of the function at t, and is one of inc, std, or dec depending on whether the function is increasing, steady or decreasing at t. Since f is a reasonable function, the qualitative state of f must be constant over intervals between adjacent distinguished time points. Therefore a function can be completely characterized by its qualitative states at all its distinguished time points and at all intervals between adjacent distinguished time points. Such a temporal sequence of qualitative states of f forms the qualitative state history or qualitative behaviorof f. Since a reasonable function f is continuously differentiable, the Intermediate Value Theorem and the Mean Value Theorem restrict the possible transitions from one qualitative state to the next. [20] includes a table listing all possible transitions. 20 2.3 Qualitative Simulation: QSIM QSIM takes a qualitative model and an initial state, and generates all possible behaviors from the initial state consistent with the qualitative constraints in the model. Starting with the initial state, the QSIM algorithm works by repeatedly taking an active state and generating all possible next-state transitions according to the table of possible transitions mentioned in the previous section. These transitions are then filtered according to restrictions posed by the constraints in the system model. Because a model may allow multiple behaviors, QSIM builds a tree of states representing all possible behaviors. Any path across the tree from the given initial state to a final state is a possible behavior of the system. 2.4 Learning Qualitative Models: GENMODEL GENMODEL [5]goes in the opposite direction to QSIM. It takes a system behavior and di- mensional information about the system functions, and generates all qualitative constraints that permits the system behavior. The GENMODEL algorithm works by first generating all possible dimensionally correct qualitative constraints that may exist among the system functions, according to different permutations of the functions. Then it progresses along the state history, successivelypruning all constraints that are inconsistent with each state transition. The set of qualitative constraints remaining at the end represent the most specific model that permits the given behavior. Any subset of this model also permits the given behavior, and therefore is also a possible model of the system. 2.5 An Example: The U-tube The QSIM representation can be illustrated by an example of a U-tube [20]. Figure 2-2 shows a U-tube (two partially filled tanks connected at the bottom by a tube). If an increment of water is added to one arm (A) of the U-tube, the system will undergo three different states before reaching equilibrium again: Initial State The water level in A will rise to a new level with the water level in the other arm (B) unchanged initially. There is a net pressure difference between the two arms, resulting in a net flow of water from A to B. 21 A B I I I I (Normal) t(O) t(O)/t(l) t(l) Time, t Figure 2-2: Qualitative states following addition of an increment of fluid to one arm of a U-tube. ITransitional State As time progresses, the level in A decreases and the level in B increases, while the pressure difference and the water flow diminishes. Equilibrium State When the water in the two arms reach the same level, the pressure difference and the water flow becomes zero and a new equilibrium is reached. A simple model of the U-tube consists of six qualitative constraints, as illustrated in Table 2-3 [6]: 1. The pressure in A increases with the level in A. 2. The pressure in B increases with the level in B. 3. The pressure difference between A and B is the difference in the two pressures in A and B. 4. The flow from A to B increases with the pressure difference between A and B. 5. The level in A is inversely proportional to the derivative of the flow from A to B. 6. The level in B is proportional to the derivative of the flow from A to B. These constraints are represented in a diagram in Figure 2-4. 2.5.1 Qualitative Simulation of the U-tube With this simple model and a partial specification of the initial state, QSIM deduces the remaining states according to the QSIM algorithm mentioned previously. The output of the system behavior generated by running QSIM 2 on the U-tube model is shown in Figure 2-5. 2 We used an implementation of QSIM in UNSW Prolog V4.2 on the HP9000/720 [6]. 22 model(u_tube). % function definitions fn(a). fn(b). % level in arm a % level in arm b fn(pa). fn(pb). fn(pdiff). fn(flow_ab). % pressure in arm a % pressure in arm b % pressure difference % flow rate from arm a to arm b % model constraints mplus(a,pa). mplus(b,pb). add(pb,pdiff,pa). mplus(pdiff,flowab) . inv_deriv(a,flow_ab). deriv(b,flow_ab). % initial conditions of qualitative state at t = t(O) initialize(a,t(0),a(O)/inf,dec). initialize (b,t () ,b(O),_). % ordinal values for function landmarks landmarks(a,[O,a(O),inf]). landmarks(b,[O,b(O),inf]). % definition of function values at normal equilibrium normal( [a/a(O), b/b(O), pa/pa(O), pb/pb(O), pdiff/0, flow_ab/O]). :Figure 2-3: Prolog representation for the U-tube model, initialized for an increment of fluid added to arm a. 23 _1 IT Level A ·_ Pressure A I\ I I~ L1 Pressure Differenct I ~zh M+nA s rluw LauC. -' Figure 2-4: Qualitative model of a U-tube. The qualitative states generated can be presented in qualitative plots as shown in Figure 2-6. In these plots, landmark values are placed on the vertical axis, and distinguished time points and corresponding intervals are placed on the horizontal axis. The only positions that can be taken at each time point are at or between landmark values. 2.5.2 Learning the U-tube Model The system behavior described previously together with dimensional information on the system functions can be given to GENMODEL 3 to generate a set of all qualitative constraints which are consistent with the behavior. Dimensional information on the system functions of the U-tube are specified in Prolog as follows: units(a,mass). units(b,mass). units(pa,pressure). units(pb, pressure). units(pdiff,pressure). units(flow_ab,mass/t). The output of the U-tube model generated by GENMODEL on the given behavior 3 GENMODEL is implemented in UNSW Prolog V4.2 on the HP9000/720 [5]. 24 Simulating u_tube t(O) t(O) / t(1) t(1) simulation completed landmarks used in simulation: landmarks(a, landmarks(b, [0, a(O), a(1), [0, b(O), b(1), inf]). inf]). landmarks(pa, [0, pa(O), pa(1), landmarks(pb, [0, pb(O), pb(1), landmarks(pdiff, [0, inf]). landmarks(flowab, [0, inf]). inf]). inf]). The simulation generated the following qualitative states: t(0) a b a(O) / inf b(O) dec inc pa pb pa(O) / inf pb(O) dec inc pdiff 0 / inf dec flowab 0 / inf dec a b a(O) / inf b(O) / inf dec inc pa pa(O) / inf dec pb pb(O) / inf inc pdiff 0 / inf dec flowab 0 / inf dec a b a(1) b(1) std std pa pa(1) std pb pdiff flowab pb(1) 0 0 std std std t(O) / t(1) t(1) Figure 2-5: Output of the system behavior generated by running QSIM on the U-tube model. 25 lnf I-lA(A) inf :(I) vrl(B) b(I) b(O) a(O t(0) t(0) t(1) -'n t(1) -inf inf. prss..r(A) inf preun.(B) pb(l)' pb(O) 1 p( ) P(O t(O) t(l) (t() -inf ' t() -inf I t(o) l t (A-B) prrressru diffnce inf t(1) -inf I I Figure 2-6: Qualitative plots for the qualitative states of a U-tube following addition of an increment of fluid to one arm until equilibrium is reached. is shown in Figure 2-7. In the case of the U-tube, the generated model is equivalent to the original model after redundancy elimination by GENMODEL. GENMODEL will be described further in Chapter 3. 26 Generating possible constraints mplus(pdiff, flowab) mplus(pdiff, a) mplus(pdiff, pa) mplus(flowab, pdiff) mplus(flow-ab, a) mplus(flow-ab, pa) mplus(a, pdiff) invderiv(a, flow-ab) mplus(a, flow_ab) mplus(a, pa) deriv(b, flowab) mplus(b, pb) mplus(pa, pdiff) mplus(pa, flowab) mplus(pa, a) mplus(pb, b) add(pdiff, pb, pa) add(pa, pb, pdiff) add(pb, pdiff, pa) add(pb, pa, pdiff) filtering with state t(O) / t(1) filtering with state t(1) filtering : mplLus(pdiff, a) filtering : mp].us(pdiff, pa) filtering : mpllus(flowab, a) filtering : mp].us(flow_ab,pa) filtering : mp]lus(a, pdiff) filtering : mpl.us(a, flowab) filtering : mp]lus(pa, pdiff) filtering : mpllus(pa, flow-ab) filtering : addL(pa, pb, pdiff) filtering : addL(pb, pa, pdiff) Checking for redundancies filtering : mpllus(flowab, pdiff) simple redundancy filtering : mp].us(pa, a) simple redundancy filtering : mpllus(pb, b) simple redundancy filtering: add((pb, pdiff, pa) simple redundancy Model Constraints: deriv(b, flow-ab) invderiv (a, flow-ab) mplus(pdiff, flowab) mplus(a, pa) mplus(b, pb) add(pdiff, pb, pa) Figure 2-7: Output of the U-tube model generated by GENMODEL on the given behavior. Note that the model learned is the same as the original model shown previously. 27 Chapter 3 Learning Qualitative Models In this chapter, we examine the learnability of qualitative models employing the QSIM formalism [20]. We review the Probably Approximately Correct (PAC) model of learning [37], and show that QSIM models are efficiently PAC learnable from positive examples only. The proof is based on the algorithm GENMODEL [5]which can efficiently construct a QSIM model consistent with a given set of examples, if one exists. The chapter continues with a detailed coverage of GENMODEL, including the newly added features of dimensional analysis and fault tolerance, and a comparison of GENMODEL with other approaches of learning qualitative models. We end the chapter with a discussion of the difficulties of applying PAC results to our task of learning QSIM models from physiological signals. 3.1 Probably Approximately Correct Learning A common setting in machine learning is as follows: Given a set of examples, produce a concept consistent with the examples that is likely to correctly classify future instances. We are interested in algorithms that can perform this task efficiently. The Probably Approximately Correct (PAC) model of learning introduced by Valiant [37] is an attempt to make precise the notion of "learnable from examples" in such a setting [17, 30]. 3.1.1 Definitions Let X be the set of encodings of all instances. X is called the instance space. In our learning task, X is the set of all qualitative states within a given set of landmark values and distinguished time points. 28 A concept c is a subset of the instance space, i.e. c C X. In our learning task, c is a QSIM model which can be seen as a concept specifying a subset of X (the set of all qualitative states) as legal states. We can equivalently define a concept c to be a boolean mapping applied to X, i.e. c : X-+ (0, 1}. An example is considered to be a positive example if c(x) = 1, and a negative example if c(x) = 0. A concept class C over X is a collection of concepts over X. In our case, C is the set of all QSIM models with a given number of system functions. The goal of the learning algorithm is to determine which concept in C (the target concept c) is actually being used in classifying the examples seen so far. We define n to be the size parameter of our instance space X. In our case, n is the number of system functions in our system. Note that n affects the number of possible qualitative states and thus the size of the instance space X. The difficulty of learning a concept that has been selected from Cn depends on the size ICn I of Cn. Let 1 An= lg Cl An may be viewed as the minimum number of bits needed to specify an arbitrary element of Cn. If An is polynomial in n, Cn is said to be polynomial-sized. We are interested in characterizing when a class C of concepts is easily learnable from examples. 3.1.2 PAC Learnability Stated informally, PAC learnability is the notion that the concept acquired by the learner should closely approximate the concept being taught, in the sense that the acquired concept should perform well on new data drawn according to the same probability distribution as that according to which the examples used for learning are drawn. We define Pn as the probability distribution defined on the instance space Xn according to which the examples for learning are drawn. The performance of the acquired concept c' is measured by the probability that a new example drawn according to Pn will be incorrectly classified by c'. For any hypothesized concept c' and a target concept c, the error rate of c' is defined as: error(c')= Prepn[c'(x) $ c(x)] llg x = log2 x 29 where the subscript x E P, indicates that x is drawn randomly according to P,. We would like our learning algorithm A to be able to produce a concept whose error rate is less than a given accuracy parameter with 0 < e < 1. However, we cannot expect this to happen always, since the algorithm may suffer from "bad luck" in drawing a reasonably representative set of examples from Pn for learning. Thus we also include a confidence parameter 6 with 0 < 6 < 1 and require that the probability that A produces an accurate answer be at least 1 - 6. As and approach zero, we expect A to require more examples and computation time. We would like the learning algorithm to satisfy the following properties: * The number of examples needed is small. * The amount of computation performed is small. * The concept produced is accurate, i.e. the error rate is arbitrarily small. * The algorithm produces such an accurate concept most of the time, i.e. with a probability of 1 - 6 that is arbitrarily close to 1. The concept class C is said to be efficiently PAC learnable if there exists an algorithm A for learning any concept c E C that satisfies the above criteria. To define PAC learnability formally, we say that a concept class C is efficiently PAC learnable if there exists an algorithm A and a polynomial s(.,., 6, all probability distributions P, on X, ) such that for all n, , and and all concepts c E Cn, A will with probability at least 1 - 6, when given a set of examples of size m = s(n, l, ~) drawn according to P, output a c' E Cn such that error(c') < e. Furthermore, A's running time is polynomially bounded in n and m. The hypothesis c' E C of the learning algorithm is thus approximately correct with high probability, hence the acronym PAC for Probably Approximately Correct. 3.1.3 Proving PAC Learnability One approach of PAC learning due to Blumer et al. [3] is as follows: Draw a large enough set of examples according to P, and find an algorithm which, given the set of examples, outputs any concept c E Cn consistent with all the examples in polynomial time. If there exists such an algorithm for the concept class C, C is said to be polynomial-time identifiable. 30 Formally, we say that C is polynomial-time identifiable if there exists an algorithm A and a polynomial p(A, m) which, when given the integer n and a sample S of size m, produces in time p(A,, m) a concept c E C consistent with S, if such a concept exists. This leads to an interesting interpretation of learning. Learning can be seen as the act of finding a pattern in the given examples that allows compression of the examples. We can measure simplicity by the size of the concept space of the algorithm, or equivalently by An, the minimum number of bits needed to specify a concept c among all the concepts in C,. An algorithm that finds a succinct hypothesis consistent with the given examples is called an Occam algorithm. This formalizes a principle known as Occam's Razor which equates learning with the discovery of a "simple" explanation for the observed examples. A concept that is considerably shorter than the examples is likely to be a good predictor for future data [29]. Next we look at how large a set of examples is "large enough" for PAC learning. [3] includes a key theorem on this issue. Theorem Given a concept c E Cn and a set of examples S of c of size m drawn according to P, the probability is at most ICnl (1- e)m that there exists a hypothesis c' E C, such that: * the error of c' is greater than , and · c' is consistent with S. Proof If c' is a single hypothesis in C, of error greater than , the chance that c' is consistent with a randomly drawn set of examples S of size m is less than (1 - )m . Since C, has ICnI members, the chance that there exists any member of C, satisfying both conditions above is at most - ICE(1 E)m . · Using the inequality -x < e- x we can prove that 1 m >- (ln Cnl+In E 31 1 implies that ICnl(l- E)m < Thus if m satisfies the above lower bound, the probability is less than 6 that a hypothesis in Cn which is consistent with S will turn out to have error greater than e. Therefore to PAC learn a concept, the learning algorithm need only examine m examples where m has a lower-bound as follows: 2 m= Q(-(In lCn + n)) Recall if Cn is polynomial-sized, then An and therefore In ICn is polynomial in n. Therefore m is polynomial in n, and To conclude, an algorithm that draws m examples according to Pn and outputs any concept consistent with all the examples in polynomial time is a PAC learning algorithm. Thus if Cn is polynomial-sized and polynomial-time identifiable, then it is efficiently PAC learnable. 3.1.4 An Occam Algorithm for Learning Conjunctions In [37] Valiant provides an algorithm for PAC learning the set of boolean formulae in conjunctive normal form (CNF) where each clause contains at most k literals. This set of boolean formulae is known as k- CNF. The algorithm is capable of PAC learning from positive examples only. In Section 3.3.2 we will map the problem of identifying a QSIM model consistent with a given set of examples to the problem of identifying a k-CNF consistent with a given set of examples. First we calculate the number of examples needed. The number of conjunctions over the boolean variables x1,..., Xn is 3n since each variable either appears as a positive or negative literal, or is absent entirely. Applying the formula for the lower bound in the previous section, we see that a sample of size O(n + in is sufficient to guarantee that the hypothesis output by our learning algorithm is e accurate with confidence of at least 1 -J. The algorithm starts with the hypothesis conjunction which contains all the literals: c' = X1A 1 A.. 2A = Q(B) denotes B is a lower bound of A. 32 A An Upon each positive example x, the algorithm updates c' by deleting the literal xi if xi = 0 in the example, and deleting the literal xi if xi = 1 in the example. Thus the algorithm deletes any literal that contradicts the data. This can be seen as starting with the most specific concept and successively generalizing the concept upon each positive example given. Since the algorithm takes linear time to process each example, given m examples with m as calculated above, the running time is bounded by mn and hence is bounded by a polynomial in n, 1 and . Therefore this is an efficient PAC learning algorithm for the class of k-CNF. 3.2 The GENMODEL Algorithm GENMODEL is a program for generating qualitative models from example behaviors [5]. Given a set of qualitative states describing a system behavior, GENMODEL outputs all QSIM constraints consistent with the state history. Together, these constraints form the most specific QSIM model given the example behavior. The Algorithm Input: * A set of system functions, Functions. * A set of units for the system functions, Units * A set of landmark lists for the system functions, Landmarks. * A set of qualitative states, States. Output: A qualitative model which consists of all constraints that are consistent with the state history and dimensionally correct, Model. Functions used: search() A function for searching corresponding values from a set of qualitative states. dimcheck() A function for checking dimensional compatibility of functions within a constraint. 33 begin Constraints+-0; Correspondings - search(States); for each f, f2 in Functions such that fl # f2 do for each predicate2 in {inv, deriv, invderiv, M +, M-} do fl, f2, Units) then if dimcheck(predicate2, add predicate2(fl,f 2) to Constraints; for each f, f2, f3 in Functions such that f # f2#f3 do for each predicate3 in {add, mult} do fl, f2, f3, Units) then if dimcheck(predicate3, add predicate3(fl,f2,f3) to Constraints; for each s in States do for each c in Constraints do if not check(c, s, Landmarks, Correspondings) then delete c from Constraints; reduce(Constraints); Model+- Constraints; end Figure 3-1: GENMODEL algorithm. check() A function for checking validity of a constraint given a qualitative state and sets of corresponding values. reduce() A function for removing redundancy from constraints. For example, since M+(A, B) and M+(B, A) specify the same relationship, one of them can be removed. Method: * Search the entire state history States for sets of corresponding values. * Generate the initial search space by constructing all dimensionally correct constraints with different permutations of system functions in Functions. * Successivelyprune inconsistent constraints upon each qualitative state in States. * Remove redundancy from the remaining constraints * Output the result as a qualitative model. The entire algorithm is shown in Figure 3-1. 34 3.3 QSIM Models are PAC Learnable From Section 3.1.3 we conclude that to prove that the concept class of QSIM models is PAC learnable, it suffices to prove that the class is polynomial-sized and that it is polynomial-time identifiable. The following two sections provide these proofs. 3.3.1 The Class of QSIM Models is Polynomial-Sized To show that the concept class of QSIM models is polynomial-sized, we begin by noting that in the QSIM formalism there are five kinds of two-function constraints (inv, deriv, invderiv, M+ and M-), and two kinds of three-function constraints (add and mult). Therefore with n system functions, the number of possible QSIM constraints N is as follows: N = 5n(n - 1) + 2n(n - 1)(n- 2) Therefore, the number of possible QSIM models is: IQSIM-Models(n)l= 2N = 25n(n-1 )+2n(n-1)(n-2) since each QSIM constraint can either be present or absent in the model. This implies that: lg (QSIM-Models(n)I) = N = O(n3 ) Therefore the concept class of QSIM models is polynomial-sized. To PAC learn a QSIM model, we need m examples where m is calculated as follows: m=Q 3.3.2 ((5n(n - 1) + 2n(n - 1)(n - 2))ln2 + In )) QSIM Models are Polynomial-Time Identifiable In this section we show that GENMODEL is an algorithm for efficiently constructing a QSIM model consistent with a given set of examples. We prove this by mapping the problem of identifying a QSIM model consistent with a given set of examples to the problem of identifying a k-CNF consistent with a given set of examples. The algorithm presented in Section 3.1.4 then yields an algorithm for identifying a QSIM model consistent with a 35 given set of examples in polynomial time. We show that this algorithm is in fact identical to GENMODEL. We view each QSIM model as a conjunction of QSIM constraints, and each QSIM constraint as a boolean variable. Then learning QSIM models is equivalent to learning monotone conjunctions 3 with N boolean variables, where N is the number of possible QSIM constraints as calculated in the previous section. Now the algorithm starts with the hypothesis of a monotone conjunction which contains all N of the boolean variables, i.e. all possible QSIM constraints: C'=xlA Ax,n The qualitative states provided for learning constitute the positive examples. Upon each positive example x, the algorithm updates c' by deleting the boolean variable xi if the corresponding QSIM constraint is inconsistent with the example. Since each boolean variable xi corresponds to a QSIM constraint, the algorithm prunes any constraint that is inconsistent with each qualitative state. This can be seen as starting with the most specific model and successively generalizing the model upon each qualitative state given. This is identical to the approach taken by GENMODEL. Now it remains to show that GENMODEL takes polynomial time to process each qualitative state. We review each step taken by GENMODEL in learning a QSIM model: * Search the entire state history for sets of corresponding values. For m qualitative states, there are at most m sets of corresponding values, and the search takes O(m) time. * Generate the initial search space by constructing all constraints with different permutations of system functions. For n system functions, this takes O(n3 ) time as shown in the previous section. * Successivelyprune inconsistent constraints upon each qualitative state. Checking for consistency of a constraint with a qualitative state involves: - Checking landmark values and directions of change. This takes linear time. 3 Monotone conjunctions are ones with positive literals only. 36 - Checking corresponding values. Since there are at most m sets of corresponding values, this takes O(m) time. * Remove redundancy from the remaining constraints. Since we started off with O(n3 ) constraints, there are at most the same number of constraints remaining in the final model. Therefore, removing redundancy takes O(n3 ) time. * Output the result as a qualitative model. Therefore for GENMODEL the processing time of each qualitative state is polynomial in m and n. Since the algorithm takes polynomial time to process each qualitative state, given m states with m as calculated above, the running time is bounded by p(m, n) where p(-, ) is a polynomial in the two arguments. Therefore the running time is bounded by a polynomial in n, and <. Therefore this is an efficient PAC learning algorithm for the concept class of QSIM models. 3.4 GENMODEL In this thesis, the original implementation of the GENMODEL program described in [5] is extended to include dimensional analysis and fault tolerance. 3.4.1 Dimensional Analysis Dimensional analysis is an effective way of significantly reducing the size of the initial search space of constraints. Before generating a constraint, GENMODEL checks for units compatibility among functions within the constraint. This is similar to the approach used in several other inductive learning systems, including ABACUS [8], a system for quantitative discovery, and MISQ [28], a system based upon GENMODEL. The dimension of each function is specified at the beginning, usually in terms of the type of quantity the function represents, e.g. 1/time for the heart rate (HR), volume for the stroke volume (SV), and volume/time for the cardiac output (CO). This allows the constraint mult(HR, SV, CO) to be generated since (1/time) x (volume) = volume/time, but does not allow mult(HR, CO, SV) or add(HR, SV, CO) to be generated since they are 37 dimensionally incorrect. Note that the functional constraints M + and M- are not restricted by dimensions. 3.4.2 Performance on Learning the U-Tube Model With dimensional analysis, GENMODEL comes up with exactly the six constraints for the U-tube system described in Section 2.5, given the three qualitative states describing the example behavior. 4 This is a significant improvement to previous results reported in [5] in which 14 constraints were obtained. 3.4.3 Version Space In [24], Mitchell points out that generality forms a partial order among elements in the concept space. This partial order allows the set H of plausible hypotheses at each stage of learning to be represented by its most generalelements (upper bounds) G, and its most specific elements (lower bounds) S, as illustrated in Figure 3-2. A plausible hypothesis is defined as any hypothesis that has not yet been ruled out by the examples seen so far. The set H of all plausible hypotheses is called the version space. Mitchellproposes a learning algorithm, calledthe candidate-eliminationalgorithm,based on this boundary-set representation of H. Initially, the version space contains all possible concepts. As training examples are presented to the learning program, inconsistent concepts are eliminated from the space until only one concept remains - the target concept. Each positive example causes the program to generalize elements in S as little as possible, so that they will cover the example. This is called the Update-S routine. Conversely, each negative example causes the program to specialize elements in G as little as possible, so that they will not cover the example. This is called the Update-G routine. The version space gradually shrinks in this manner until only the target concept remains. In the task of learning QSIM models, the most specific concept at each stage is unique. It is the model which consists of all the QSIM constraints consistent with all the qualitative states presented so far. Since GENMODEL successively updates the most specific model by pruning constraints that are inconsistent with the most current example, it is equivalent to the Update-S routine in Mitchell's Candidate-elimination 4 Learning Algorithm. The U-tube system is a standard reference problem in the qualitative modeling community. 38 more general O ...... -- A model consisting of no constraints (the null description) - the most general concept G S A model consisting of all possible constraints - e InostCaLe morespecific LUILtfL H - the set of all plausible hypotheses (the version space) G - the set of most general plausible hypotheses (the upper boundary set) S - the set of most specific plausible hypotheses (the lower boundary set) Figure 3-2: Using the upper and lower boundary sets to represent the version space. 3.4.4 Fault Tolerance For domains involving noisy learning data, such as noisy signals from hemodynamic monitoring, it is difficult to implement front-end signal processing which filters the noise and restores the signals completely. Part of the difficulty lies in the nature of the noise. Hemodynamic monitoring is vulnerable to a wide variety of artifacts. These include artifacts resulting from various kinds of clinical interventions which we will describe in Section 5.2.1. These artifacts are relatively easy to detect because their values are usually outside the physiologically attainable range. There are also artifacts which do not affect the signals so drastically, and therefore can be hard to detect. For example, patient movements can sometimes affect the operation of transducers and alter the shape of the signals obtained, thereby affecting the qualitative state history subsequently generated. It is hard if not impossible to restore the original signal from a signal heavily distorted by artifacts. To accommodate such difficulties in obtaining a perfectly clean signal for segmentation, we need to incorporate a degree of fault tolerance into GENMODEL. A simple approach is to tag a counter onto every constraint in the initial search space. 39 begin Constraints+- 0; Correspondings+- search(States); for each fl, f2 in Functions such that f 4 f2 do M,+ , M-} do for each predicate2 in {inv, deriv, inv_deriv, M if dimcheck(predicate2, fl, f2, Units) then add < predicate2(fl, f2), 0 > to Constraints; for each fl, f2, f3 in Functions such that fl 0 f2#f3 do for each predicate3 in {add, mult} do if dimcheck(predicate3, fl, f2, f3, Units) then add < predicate3(fl, f2, f3), 0 > to Constraints; for each s in States do begin Remaining e 0 for each < c, i > in Constraints do if check(c, s, Landmarks, Correspondings) then add < c, i > to Remaining else if (i < TOLERANCE) then add < c, (i + 1) > to Remaining Constraints+- Remaining end reduce(Constraints); Model --Constraints; end Figure 3-3: GENMODEL algorithm with fault tolerance. 'This counter keeps track of how many states the constraint has failed in so far. We set a noise level to a fraction of the total number of states in the example behavior. A constraint has to be inconsistent with this many states before it is pruned. The GENMODEL algorithm modified to include fault tolerance is shown in Figure 3-3. 3.4.5 Comparison of GENMODEL with Other Learning Approaches GENMODEL does not require negative examples. The greatest strength of GENMODEL is that it learns from positive examples only. There is no need to generate negative examples as needed in other inductive learning approaches such as GOLEM and genetic algorithms. In [4], Bratko et al. report that learning the U-tube model with GOLEM requires six 40 hand-generated negative example states, in addition to the same positive example behavior we used for GENMODEL which consists of only three states. On each iteration in the GOLEM algorithm, a fixed number of clauses are first generated by Relative Least General Generalization (RLGG) [27]. The clause that covers the most positive examples and none of the negative examples is then chosen for propagation to the next iteration. In [38], Var§ek's genetic algorithm approach requires 17 positive example states and 78 negative example states to learn the U-tube model. In each cycle, candidate solutions are selected for "reproduction" based on a fitness function which is the sum of the fraction of positive and negative examples covered correctly and a "bonus" indicating the size of the solution. In both approaches, it is therefore essential for the user to give the "right" negative examples. Badly chosen negative examples or an inadequate number of them will cause an inappropriate clause to be propagated to the next iteration, which will ultimately affect the concept output in the end. However, there are no existing rules to guide the selection of negative examples. A trial-and-error approach can be tedious, especially in complex domains such as human physiology. GENMODEL does not require ground facts for background knowledge In GENMODEL, the definitions of the QSIM representation are inherent in the check() function used for checking consistency of a constraint with a given qualitative state. There is no need to generate explicit ground facts 5 for this background knowledge, as needed in GOLEM. GOLEM accepts definitions of background predicates in terms of ground facts. In learning QSIM models, explicit ground facts describing QSIM constraint definitions must be generated according to functions and landmark lists relevant to the modeling problem at hand. In [4], Bratko et al. report that learning the U-tube model requires a total of 5408 ground facts as background knowledge. This is already a simplification which excludes rules regarding corresponding values in the M + and M- constraints, rules regarding consistency of infinite values in the add constraint, and rules on the mult constraint. In a more complex domain such as human physiology which potentially involves long landmark lists, the size of the background knowledge required can grow exceedingly large. 5 A clause is said to be ground if it does not contain any variables. 41 GENMODEL is guaranteed to produce a correct model if one exists. Given a set of qualitative states representing a system behavior, GENMODEL successively prunes inconsistent constraints upon each state. The constraints remaining in the end forms the output model. Therefore, GENMODEL is guaranteed to produce a correct model if one exists. On the other hand, GOLEMand genetic algorithms perform heuristic searches across the concept space. GOLEM performs hill climbing with positive and negative example coverage as the heuristic guiding the search. Genetic algorithms similarly perform hill climbing with the fitness function serving as the heuristic. Since neither heuristic is a perfect quality measurement of the current model, GOLEM and genetic algorithms are not guaranteed to produce a correct model even if one exists, unless the search becomes exhaustive. 3.5 Applicability of PAC Learning As discussed in Section 3.3, the following is a PAC learning algorithm for learning a QSIM model from physiological signals: 1. Obtain m qualitative states where m is calculated as follows: m= 1 n 1 ((ln2N + n )) = Q( ((5n(n- 1)+ 2n(n- l)(n - 2))ln2 + n )) 2. Learn a QSIM model from the m qualitative states using GENMODEL. Applying this algorithm to our learning task is difficult for the following reasons: * Qualitative states cannot be modeled as independent examples drawn from an underly- ing probability distribution. Given a reasonable function and a qualitative state, there are only a limited number of possible transitions the system can make, as described in [20]. Further, successive states in signals obtained from hemodynamic monitoring are highly correlated because of physiological constraints limiting for instance the rate of change of signals. * For our experiments, we use 8 different signals. Therefore n = 8. To PAC learn a QSIM model with an accuracy and a confidence level of 80%, i.e. = 6 = 0.2, we need m = 3308 qualitative states. From our experience in segmentation, this translates to 42 about 80-90 hours of patient data. Even to do so with an accuracy and a confidence level of just 50%, i.e. = = 0.5, we still need m = 1322 qualitative states. This translates to about 30-40 hours of patient data. In such a long time span, the patient condition and therefore the corresponding model may have already changed. Signals from hemodynamic monitoring are corrupted by various artifacts and noise. The PAC learning algorithm previously developed assumes learning examples to be noise-free. Therefore, for our learning task, we will apply GENMODEL for polynomial-time identification of a QSIM model from qualitative states only. We will not observe the sample complexity for PAC learning. Even so, as we will see in Chapter 6, we still obtain useful models of reasonable size. 43 Chapter 4 Physiological Signals and Models 4.1 Hemodynamic Monitoring Hemodynamic monitoring provides information on the performance of the cardiovascular (CV) system, and allows the physician to manipulate the CV system with fluids and drugs in the critically ill patient or during surgical procedures. Real time hemodynamic mea- surements cover many aspects of the CV system, including heart rate, blood pressures, temperatures, oxygen supply and others. In this thesis, we use eight signals derived from such measurements for learning a qualitative model that describes the CV system. The basic CV parameters and their characteristics will be reviewed in the following section. 4.1.1 Primary Measurements Heart Rate (HR) The heart rate is determined from the electrocardiogram (ECG) signal as the reciprocal of the interval between two successive QRS complexes. (QRS complexes are the large voltage spikes that correspond to ventricular contraction.) In our experiments, the heart rate signal is sampled at 1 Hz. Arterial Blood Pressure Waveform (ABP) The arterial blood pressure waveform in our data comes from invasive monitoring by a catheter. Invasive monitoring enables recording of rapidly changing arterial pressures, as opposed to noninvasive methods such as the auscultatory detection of Korotkoff sounds. 44 ABP, Systolic pressure (- 120 mmHg) Mean pressure (- 96 mmHg) Diastolic pressure (- 80 mmHg) . 0 0.5 1.0 Time/s Figure 4-1: Deriving the systolic, diastolic and mean pressures from the arterial blood pressure waveform. The signal in our data is sampled at 125 Hz. From the ABP waveform, the following signals can be determined (Figure 4-1): Systolic arterial blood pressure (ABPS) - the value at the height of an ABP pulse. Diastolic arterial blood pressure (ABPD) - the value at the lowest point of an ABP pulse. Mean arterial blood pressure (ABPM) - the mean value of an ABP pulse. This can be calculated by dividing the area under the pulse by the duration of the pulse: ABPM = ABPdt b-a where a is the starting time of the pulse and b is the ending time of the pulse [11]. These signals are derived at 1 Hz. Central Venous Pressure (CVP) Blood from all the systemic veins flows into the right atrium. Therefore, the pressure in the right atrium is called the central venous pressure. Cardiac output curves (from the Frank-Starling law of the heart) can be obtained by plotting the cardiac output versus the CVP [11]. 45 The central venous pressure in our data comes from invasive monitoring by a catheter. The waveform is sampled at 125 Hz. The mean CVP signal (CVPM) at 1 Hz is used in our experiments. Temperature (Tskin, Tcore) The skin temperature and the core temperature are recorded at 1 Hz. From these two signals, a skin-to-core temperature gradient can be determined as described in Section 4.1.2. 4.1.2 Derived Values From the primary measurements, various useful indices of cardiovascular function may be calculated as follows [25, 31]. Stroke Volume (SV) During diastole, the ventricles are filled to about 120-130 ml. This volume is known as the end-diastolic volume. During systole, the ventricles are emptied to a remaining volume of about 50 to 60 ml. This volume is known as the end-systolic volume. The difference between the end-diastolic volume and the end-systolic volume is the volume of blood pumped out of the ventricles during systole. This volume is about 70 ml, and is called the stroke volume [11]. The pulse contour method developed by Wesseling et al. for deriving the stroke volume (SV) and cardiac output (CO) from the arterial pressure waveform has been studied widely in the research literature [40, 15, 34, 10, 36, 39]. Unlike methods such as the thermodilution technique which allows for only intermittent measurements of CO, the pulse contour method enables continuous monitoring of CO, which is important for the management of critically ill patients whose clinical conditions may change rapidly. The pulse contour method estimates the stroke volume from the arterial blood pressure waveform as the systolic ejection area Asys (Figure 4-2) divided by a constant Zao representing the characteristic impedance of the aorta. 1 The systolic ejection area is represented by the integral of the difference between the arterial blood pressure waveform (ABP) and 1The pulse contour method is based on the "Windkessel" model of the systemic circulation which relates the arterial pressure difference to the blood flow via the impedance through which the flow is driven. The stroke volume can be calculated from the pressure as the driving force for flow during the ejection time. 46 ABP/mmHg nichrnfir nntrh ... systolic pressure I 100 nintrairnrocllo --- Systolic Ejection Period I Aortic valve opens An 0 0.5 1.0 Time/s Figure 4-2: Deriving the stroke volume from the arterial blood pressure waveform. the diastolic arterial blood pressure (ABPD) over the systolic ejection period (SEP) 2: SV = Asys_ ftESEp(ABP- ABPD) dt Zao Zao Zao has been shown to be relatively constant over short periods. Since the segments used in our experiments are 16.7 minutes long, the relatively constant Zao does not affect the qualitative behavior of the function and can be disregarded. The results obtained by the pulse contour method and standard techniques such as thermodilution have been shown to correlate well even under dynamic conditions such as in patients undergoing coronary artery bypass surgery [15, 10], patients in surgical intensive care units [36], patients undergoing manipulations of blood pressure [39], and patients undergoing treatments involving vasoactive drugs [34]. 2 The end of the systolic ejection period is marked by the dichrotic notch formed at the closure of the aortic valve. 47 Cardiac Output (CO) The cardiac output is the rate of blood flow from the left ventricle into the aorta. It is related to the heart rate and the stroke volume by the followingequation: CO= HR x SV Ventricular Contractility (VC) VC = d(ABP) dt peak Experimental studies have shown that the rate of rise of arterial blood pressure in general correlates well with the strength of contraction of the ventricle. The peak d(ABP) which occurs at the onset of systole, is often used as an indicator of ventricular contractility [11, 25, 31]. Skin-to-core Temperature Gradient (AT) AT = Tcore- Tskin The difference between the skin and core temperatures of the body is a good indicator of the degree of vasoconstriction, i.e. the state of contraction of the vascular tree. A rise in this differential indicates increasing vasoconstriction while a fall indicates vasodilation. The degree of vasoconstriction in turn reflects cardiac output. Under conditions of poor cardiac output, as in hypovolemia,the body responds by trying to raise the blood pressure by vasoconstriction, at the expense of tissue perfusion [25, 42]. Rate Pressure Product (RPP) RPP = HR x ABPS The rate pressure product (RPP) is the product of the heart rate (HR) and the systolic arterial blood pressure (ABPS). Studies in animals [41] and humans [9] have shown that 48 the RPP correlates well with the myocardial oxygen consumption (mVO 2 ), which is closely related to the work of the heart. mVO 2 depends on several factors, including heart rate, ventricular contractility and ventricular wall tension. RPP is an easy variable to derive, and is clinically useful as an indicator for the appropriate limit of safe ventricular work. 4.2 A Qualitative Cardiovascular Model In this section, we describe a set of possible qualitative constraints that may exist among the different variables of hemodynamic monitoring described in the previous section. These constraints form a "gold-standard" target model of the CV system which allows us to compare our experimental results and evaluate the performance of the learning system. Because of the enormous complexity of the CV system, formulating a model is by no means a simple task. The constraints included in this section are not meant to be a comprehensive coverage of the CV system. For a more detailed coverage, textbooks on medical physiology such as [11] can be consulted. Furthermore, different models may exist for different clinical conditions; a constraint may be valid only under certain conditions. Relationship Among Heart Rate, Stroke Volume and Cardiac Output The heart rate (HR), stroke volume (SV) and cardiac output (CO) are related by the following equation: CO = HR x SV This translates into the followingqualitative constraint: mult(HR, SV, CO) Relationship Among Heart Rate, Arterial Blood Pressure and Rate Pressure Product The heart rate (HR), systolic arterial blood pressure (ABPS) and rate pressure product (RPP) are related by the following equation: RPP = HR x ABPS 49 Cardiac Output (CO) is 5 0 -4 0 +4 + +12 Central VenousPressure(CVP) / mmHg Figure 4-3: Cardiac output curves for the normal heart and for hypo- and hypereffective hearts. Since the behavior of the mean arterial blood pressure (ABPM) approximates that of the systolic arterial blood pressure (ABPS) well in most circumstances, the following qualitative constraint is valid in general: mult(HR, ABPM, RPP) The Frank-StarlingLaw of the Heart The Frank-Starling law states that within physiological limits, the heart pumps all the blood that comes to it without allowing excessivedamming of blood in the veins. In other words, the greater the heart is filled during diastole, the greater will be the quantity of blood pumped into the aorta. The heart automatically adapts itself to the amount of blood that flows into it from the veins, as long as the total quantity does not exceed the physiological limit that the heart can pump. Cardiac output curves (Figure 4-3) illustrate the Frank-Starling law [11]. They show the relationship between the central venous pressure (CVP) (i.e. the right atrial pressure) at the input of the heart and cardiac output (CO) from the left ventricle in different conditions. From the curves, we see that CO increases with CVP until when the venous return overwhelms the heart, in which case CO levels off. Therefore, within the physiological limit of the heart, the followingqualitative constraint holds: M+(CVPM, CO) 50 Heterometric & Homeometric Autoregulation of the Heart When the cardiac muscle becomesstretched an extra amount, as it does when extra amounts of blood enter the heart chambers, the stretched muscle contracts with a greatly increased force, thereby automatically pumping the extra blood into the arteries. This ability of the heart to contract with increased force as its chambers are stretched is sometimes called heterometric autoregulationof the heart. Furthermore, when the heart is stretched, changes in heart metabolism cause an additional increase in contractile strength. It takes approximately 30 seconds for this effect to develop fully, an effect called homeometric autoregulation [11]. Therefore, within the physiological limit of the heart, the ventricular contractility (VC) of the heart increases with the stroke volume (SV): M+(SV, VC) Effect of Heart Rate on Cardiac Output An increase in heart rate can be caused by a higher oxygen demand in tissues and organs, as in physical exercise, or as a compensatory mechanism for a decreased arterial blood pressure, as in hypovolemia. In general, the more times the heart beats per minute, the more blood it can pump, as long as the stroke volume stays roughly the same. This can be seen from the equation CO = HR x SV Therefore, if the stroke volume does not decrease, or if the decrease in stroke volume is more than compensated by the increase in heart rate, the followingqualitative constraint holds: M+(HR, CO) Indeed, a rise in heart rate increases the net influx of calcium ions per minute into the myocardial cells, and enhances ventricular contractility: M + (HR, VC) This increased contractility causes blood to be pumped out faster from the ventricle, 51 shortens the systolic interval, allocates a larger proportion of the cardiac cycle to diastolic ventricular filling, and therefore maintains the stroke volume at a reasonable level. However, once the heart rate exceeds a critical level (150-170 beats per minute in normal individuals), the heart strength itself decreases, presumably because of overutilization of metabolic substrates in the cardiac muscle: M-(HR, VC) This results in a significant decrease in diastolic filling time and consequently a decrease in the stroke volume: M- (HR, SV) The tolerable limits for heart rate decrease with underlying cardiovascular impairment. For example, ventricular tachycardia in a patient with recent myocardial infarction produces a significant reduction in stroke volume, cardiac output, and subsequently arterial blood pressure also [11, 42]. Compensatory Mechanisms for Hypovolemia Hypovolemia refers to an absolute and often sudden reduction in circulating blood volume relative to the capacity of the vascular system [32]. The major homeostatic defense mechanism in hypovolemia is activation of the sympathetic nervous system. The signal for activation of increased adrenergic nervous activity is decreased cardiac output and systemic arterial blood pressure. A decline in systemic arterial blood pressure and/or blood vol- ume activates stretch baroreceptors in the aortic arch, carotid sinus, and splanchnic arterial bed. Volume-sensitive baroreceptors within the heart can also be activated. Increased signals from these baroreceptors to the central nervous system lead to increased sympathetic nerve efferent responses and decreased parasympathetic activity. This triggers a series of compensatory mechanisms including: * arteriolar vasoconstriction with resultant decreased perfusion to skin, skeletal muscle, kidney, and splanchnic organs. This causes an increase in the skin-to-core temperature gradient AT: M-(CO, AT) 52 * tachycardia: M- (CO,HR) * increased rnyocardial contractility: M- (CO, VC) 53 Chapter 5 System Architecture 5.1 Overview The goal of the learning system is to generate qualitative models from physiological signals. The overall architecture of the system is illustrated in Figure 5-1. Figure 5-1: Overall architecture of the learning system. The physiological signal is first processed by a front-end system, which outputs a filtered signal and its derivative. These are entered into the segmenter to produce a qualitative behavior in terms of a set of qualitative states. GENMODEL then uses this qualitative behavior to generate a qualitative model. 1 This chapter describes the front-end system and the segmenter in detail. GENMODEL is described in Chapter 3. 5.2 Front-End System The architecture used for front-end processing of physiological signals is shown in Figure 5-2. The signal first passes through an artifact filter which removes various artifacts and linearly interpolates the intervals of the artifacts removed. The resulting signal is then 1 The front-end system and the segmenter are implemented in Objectworks\Smalltalk on the HP9000/720. 54 Incoming Signal x[n] y[n] y'[n] Processed Signal Derivative Signal Figure 5-2: Architecture used for front-end processing of physiological signals. processed by a median filter which removes impulsive features lasting shorter than half the length of the filter window. A Gaussian filter then smooths the signal to the desired level of temporal abstraction by convolving it with a Gaussian kernel of an appropriate standard deviation a. Finally, this smoothed signal is passed through a differentiator to obtain its derivative. The smoothed signal and its derivative are passed on to the segmenter for seg- mentation, producing a set of qualitative states describing the system behavior represented by the signal. The Gaussian filter and the differentiator are implemented as finite impulse response (FIR) filters. For simplicity, the same length L is used for the median filter window and the impulse responses of the Gaussian filter and the differentiator. Also, since we are interested only in the qualitative behavior represented by the signals, and not in the absolute magnitudes, certain constant factors in the impulse responses have been omitted for simplicity. 2 2 With a filter length of L, the filters require , points before and after the current point to process each point. In processing the first and last L-1 points, we extrapolate the signals by repeating the first and last values beyond the beginning and the end of the signals respectively. This way, the processed signals are of the same length as the original signals. 55 F eEi 20- · o 40 5o I- i . So 70 I 80 . 90 100 - ·'I 110 120 i 30 im·/ Figure 5-3: An artifact found in blood pressure signals, caused by flushing the blood pressure line with high pressure saline solution. f m Tim/. Figure 5-4: An artifact found in blood pressure signals, caused by blood sampling from the blood pressure catheter. 5.2.1 Artifact Filter Physiological signals from hemodynamic monitoring contain various kinds of artifacts which are due to clinical interventions, hardware faults or other causes, rather than real patient conditions. The artifact filter removes these artifacts and linearly interpolates the intervals of the removed artifacts. Figures 5-3, 5-4 and 5-5 show three types of commonly encountered artifacts. These and many other types of artifacts are roughly intervals during which the signal abruptly rises (drops) to an abnormally high (low) value. Therefore we can achieve reasonable performance in our artifact filter by adopting a simple threshold approach. When the magnitude of a signal rises above a certain threshold which cannot be physiologically attained in general, the filter removes the abnormal values and interpolates the interval with the last normal value. This approach of detection and interpolation is relatively simple in nature, and may not work well for signals with complex artifacts or artifacts of long durations. For recognition of complex artifacts, a knowledge-based system can be used. [12] describes a blackboard system for artifact recognition in blood pressure signals. [18] describes 56 N ___ - ___- _ - I1001 i r t F- °-O-.. 6 -I1 O- I s- -1 -- -J50 I L i I I Tim./. Figure 5-5: A general artifact caused by the transducer being left open to air. a neural network approach for the same problem. 5.2.2 Median Filter The median filter operates by centering a window of a given length L at each point of the signal, as shown in Figure 5-6. The output at that point is the median of the values covered by the window. The median filter is therefore a non-lineartime-invariant filter. It removesimpulsive features in the signal with durations less than half of its window length, while retaining sharp edges of remaining features. (Unlike filters based on convolution, the median filter outputs only values existing in the input signal. It does not produce any new averaged values.) This property of the median filter makes it useful for removing short impulsive artifacts in the physiologicalsignals prior to smoothing by the Gaussian filter. This prevents the impulsive artifacts from distorting the smoothed signals. 5.2.3 Temporal Abstraction A complex system such as the human cardiovascular system involves processes operating at different time scales. From the same set of signals, depending on the particular time scale we are interested in, different sets of qualitative states and therefore different models can be obtained. In [21]Kuipers describes a temporal abstraction relation among mechanisms operating at significantly different time scales. Processes that occur significantly faster than the time scale of a model can be considered as instantaneous in the model, while those that occur much slower can be considered as constant. For example, if we look at a system on the order of hours, processes that occur within milliseconds can be considered as instantaneous, 57 Impulsive noise (duration = 2) x[n] r 70 4 l4 I 7 50 4 I Feature with sharp edge (duration = 5) 4I 1 4lII 1 I W 4 ___ _ WW W - - W _ - - _ _ W W 0 10 170170o - - -- - ~_ w w 1 _ n W 1 1 0I 0 o1501505 window length = 5 I I 10i0701701 dian0 70170 00I olsol sorting sorting 0 0 median = 50 median = 0 Features with duration < 3 are filtered. MEDIAN FILTERING y[n] Sharp edge retained 50 I D4 4 4 4I Impulsive noise filtered u h W W M ____ W W W l W W W W W W _____ w W n Figure 5-6: The median filter removes features with durations less than half of its window length, but retains sharp edges of remaining features. 58 while those occurring over days can be viewed as constant. Therefore if we perturb a system by increasing a function A, and observe that another function B responds to this change within milliseconds by increasing its value, then we can still model the relationship between A and B with the functional constraint M+(A, B) even though there is a delay between the perturbation and the response, since the response within milliseconds is seen as occurring instantaneously at this time scale. We incorporate this idea of temporal abstraction into our learning system by the following scheme: 1. First, we use a Gaussian filter to remove changes lasting significantlyshorter than the time scale we are interested in. This is described in Section 5.2.4. 2. Next, we implement the segmenter in such a way that critical points of different functions occurring within r sampling periods are labelled as occurring at the same distinguished time point, where the parameter corresponds to the particular time scale at which we are interested in learning. This is described in Section 5.3. Without the first step, there is a danger of aliasingin the temporal abstraction process. Features lasting for short durations (< -) can be aliased into one lasting for a long duration. This issue is described in the following section. Aliasing We mentioned that temporal abstraction is incorporated into our learning system by means of the Gaussian filter and the segmenter. In particular we described the parameter r of our segmenter which controls the degree of temporal abstraction allowed in the segmentation. Upon reaching a critical point in one of the system functions, the segmenter searches the next r points in the signals of the other functions, looking for critical points. The segmenter then labels the critical point of the initial function together with all other critical points found in the remaining functions as occurring at the same distinguished time point. Therefore in the subsequent stage when GENMODEL learns from the qualitative behavior generated by the segmenter, a function Y which always responds monotonically with the function X within a delay of r sampling periods can be included in the constraint M+(X, Y) or M-(X, Y). In fact, since the segmenter labels the different quantitative times as the same distin- guished time point, the delay has been abstracted from GENMODEL. GENMODEL cannot 59 tell from the qualitative behavior given to it by the segmenter whether a delay has occurred or not. However,if we omit the Gaussian filter and retain features lasting significantly shorter than r sampling periods, there is a danger of aliasingin the temporal abstraction process. Features lasting for short durations (< r) can be aliased into a feature lasting for a long duration. In Figure 5-7 the signal g(t) is distorted into h(t) by aliasing. g(t) contains two impulsive features at t = 100 + 6 and t = 200 + 6. Both of these features begin and end within the temporal abstraction time . Since each feature is associated with a critical point and the segmenter regards all critical points that occur within -r to occur at the same distinguished time point, the feature at t = 100+ and t = 200 + a cause the signal to be labeled as steady at the critical points g(1) and g(2) respectively. The segment between t = 100 + r and t = 200 + r is labeled as decreasing from g(1) to g(2) according to its two endpoints. Therefore g(t) takes on the identity (alias) of h(t) in the qualitative behavior produced. Consequently, the incorrect monotonic constraint: M+(f, g) is learned in the subsequent stage. The aliasing we observe in the segmentation of signals (i.e. a quantitative-to-qualitative conversion) is analogous to the aliasing observed in the sampling of continuous signals (i.e. a continuous-to-discrete conversion). If a continuous signal is sampled at a frequency less than the Nyquist rate (i.e. 2N where LQNis the highest frequency within the signal), without first passing the signal through an anti-aliasingfilter which removesthe Fourier components whose frequencies exceed AQN,these Fourier components will be distorted into signals with frequenciesless than fQN,which will then add to the originally existing components at those lower frequencies and corrupt the signal in a non-recoverable way [26]. This is analogous to the way features with durations significantlyshorter than r corrupt the qualitative behavior produced upon segmentation if we had omitted our anti-aliasing Gaussian filter. 5.2.4 Gaussian Filter The idea of using a Gaussian filter to analyze changes in a signal at different scales is borrowed from the well known technique of scale-space filtering in edge detection. Scale-space 60 <f(O),inc> f(t) <f(O)/f(1),inc> f(300)=(100) --------------- <f(l), std> <f(O)/f(1),dec> <f(1),std> <f(O)/f(1),inc> <f(1),inc> .. - f(0) = f(200) 0 f(l) 10~-, 0160 200 200+ landmark value f() t <g(O),inc> g(t) g(300)="g(100) -- ----- <g(O)/g(1),inc> 1 10aa f <h(O),inc> fl nn n _= | n-lrv nk.I= ntvu = nul g(1), stdi <g(O)/g(1),dec> <h(O)/h(1),inc> t <h(1)i std: 2 <h(O)/h(1),dec> 200+ 11. i00 100+ r-------..------- 200 I t(1) ------------------- I .LI") 300 "tI 200+% 20048 100+8 t(O)/t(1) g(O) ch(1)j std> <h(O)/h(1),inc> <h(l), inc> ---------------------- ----------------------------------------- 0 g(l) g _ --- t(O) std> <g(O)/g(l), inc> <g(l), inc> .g(1), - g(0) = "g(200)" h(t) , t t(2) t(1)/t(2) t(2) t(2)/t(3) t(3) distinguished time point Figure 5-7: The effect of aliasing in the segmentation of a signal with the temporal abstrac- tion parameter set to r. g(t) is aliased into h(t) in the qualitative behavior produced. 61 filtering constructs hierarchic symbolic signal descriptions by transforming the signal into a continuum of versions of the original signal convolved with a kernel containing a scale parameter. In an image, changes of intensity take place at many spatial scales depending on their physical origin. Marr and Hildreth [23] observed that detecting zero crossings in the Laplacian of the intensity values across different scales enables a system to distinguish between a physical edge from surface markings or shadows. They suggested that the original image be bandlimited at several different cutoff frequencies and that an edge detection algorithm be applied to each of the images. The resulting edge maps have edges corresponding to different scales. In our learning system, we need to segment a set of signals at different time scales. We can do so by bandlimiting our original signals at several different cutoff frequencies and segmenting the signals by detecting zero crossings in the first derivative of the signals at different scales. The segmentation then produces a set of qualitative behaviors at different time scales which can be given to GENMODEL to produce qualitative models at different scales. To bandlimit an image at different cutoff frequencies, the impulse response of the lowpass filter proposed by Marr and Hildreth is Gaussian shaped. This choice is motivated by the fact that the Gaussian function is smooth and localized in both the spatial and frequency domains. 3 A smooth impulse response is less likely to introduce any changes that are not present in the original shape. A localized impulse response is less likely to shift the location of edges. Further, Yuille and Poggio [43] and Babaud et al. [1] have separately shown that the Gaussian filter has a unique property concerning zero crossings of the first derivative of the filtered signal: 4 moving from coarse to fine scale, new zero crossings appear, but existing ones never disappear. Consequently, the extrema can be used to construct a tree describing the successive partitioning of the signal into finer subintervals as new zero crossings appear at finer scales. This partitioning of the signal by extrema moving from coarse to fine scale forms a strict hierarchy. Scale-space filtering in edge detection can be seen as a form of the more general technique of wavelet transforms in multiresolution signal analysis, with the wavelets here being Laplacians of shifted Gaussians, and signal edges located by zero 3 The Gaussian function has the smallest duration-bandwidth product with duration and bandwidth as defined in [33], and is therefore optimally localized in both the time and frequency domains. 4In general, this property holds true for zero crossings obtained by applying any linear differential operator (including the Laplacian and the first derivative) to the filtered signal. 62 crossings of the wavelet transform [35]. We adopt a similar approach for segmenting our signals. The impulse response of the lowpass filter used is based on the Gaussian function: g(t) = 1 t2 e 2 for -oo < t < oo and a > 0. The standard deviation a determines the cutoff frequency with a larger a corresponding to a lower cutoff frequency. a therefore determines the time scale we are operating at, with a smaller a corresponding to a finer time scale and a larger a corresponding to a coarser scale. The frequency response of the lowpass filter is the Fourier transform of g(t) which is also Gaussian shaped: G(Q) e n2,2 2 To translate this into a discrete-time filter, we simply replace t with n, yielding g[n] as follows: g[n]= 1 n2 ,e 2 for -oo < n < oo and a > 0. To obtain a finite impulse response (FIR) h[n] from the infinite impulse response g[n], we multiply g[n] by the Hanning window w[n]: h[n] = g[n]w[n] where [] 0.5 - .cos( 2r(n+2) M << M otherwise 0 This avoids the nonuniform convergence (the Gibbs phenomenon) associated with abrupt truncation of an infinite impulse response [26]. The Hanning window w[n] is plotted in Figure 5-8. In our experiments, we used values of a at 10, 20, 40, 60, 80 and 100. h[n] is plotted as shown in Figure 5-9 for these values of a. M is the order of the FIR filter. Therefore M + 1 is the length of the impulse response: L=M+1 63 n Figure 5-8: Plot of the Hanning window w[n]. 0.02 A 0.018 [ .:a=20 .. :a=40 -. :a=60 -:=80O -: a= 100 1. : 0.016 0.014 0.012 0.01 0.008 0.006 1,' 0.004 0.002 -00 -200 -100 0 100 200 300 n Figure 5-9: Plots of impulse responses h[n] of Gaussian filters for a = 20, 40, 60, 80, 100. 64 Ia M(=6a) 10 20 40 60 80 100 60 120 240 360 480 600 L(=M+ 1) 61 121 241 361 481 601 Table 5.1: Table showing the orders M and lengths L of the Gaussian filters corresponding to a = 10,20,40,60,80,100. .:=20 :.~ 0.9 -. :a=60hO !,* 0.6 ' ,,-: = 100 0.7 0.6 '0.5 0.4 :I't 0.3 ·I 0.2 i .'''I .. 0.1 -. 08 -0.06 -0.04 -0.02 ' 0 0.02 0.04 0.06 0.08 A1. Figure 5-10: Plots of frequency responses H(Q) of Gaussian filters for a = 20,40,60,80, 100. In our learning system, M is set so that the finite impulse response extends to three standard deviations from the origin: M -= 3a 2 which yields: M =6 Table 5.1 shows the lengths and the orders of the filters that correspond to the six values of a we used in our experiments: The frequency response H(Q) is Gaussian shaped as shown in Figure 5-10 for the six values of a used in our experiments. 65 0.8 0.8 0.4 0.2 -0.2 -0.4 -0.8 -0.8 -300 i -100 -200 0 100 200 300 Figure 5-11: Impulse response of an FIR bandlimited differentiator. 5.2.5 Differentiator The differentiator is implemented as an FIR filter based on the frequency response of a bandlimited differentiator [26]: H(e- ' ) = jw -7r < w < r The corresponding impulse response is: h[n]= n o rnosrn-sinrn _ 00 <n < 7rn7--~--- -Cx < < cx =0 fn 0O h[n]= cosrn o n540 n This infinite impulse response is multiplied by the Hanning window to obtain a finite impulse response hd[n] as shown in Figure 5-11. The frequency response of this FIR bandlimited differentiator is shown in Figure 5-12. It is interesting to note that the lowpass filtering operation of the Gaussian filter and the derivative operation of the differentiator may be combined to obtain a single filter with the derivative of the Gaussian function as its impulse response: t vJ27r , 66 t2 WY~ Figure 5-12: Frequency response of an FIR bandlimited differentiator. 0.035 .: o = 20 .. :=40 -. :a=60 0.03 -:a=0 0.025 ._0.02 -0.015 0.01 0.005." . ' '' ' " Figure 5-13: Equivalent frequency responses of a Gaussian filter in cascade with a bandlim- ited differentiator for a = 20,40,60,80, 100. The corresponding frequency response is as follows: G'(Q) = je 02,2 2 This frequency response is plotted in Figure 5-13. From the frequency response, we note that the combined operation is equivalent to bandpass filtering where a controls the bandwidth of the bandpass filter. Bandlimiting the signals tends to reduce noise, thus reducing the noise sensitivity problem associated with detecting zero crossing points. With increasing values of a, the bandwidth of the bandpass filter decreases and therefore more noise rejection is achieved. This agrees with our expectation since larger values of a correspond to coarser time scales. 67 2E To III .r i I" AI Figure 5-14: A segment of the mean arterial blood pressure (ABPM) signal. Note the artifacts at t = 600, 1000, 3400 seconds. So Time/s Figure 5-15: The ABPM data segment processed by the artifact filter. Note that the artifacts have been filtered but the signal contains many impulsive features. 5.2.6 An Example Here we show a segment of mean arterial blood pressure signal (Figure 5-14) and show the signal as it passes through the 1 artifact filter (Figure 5-15), the median filter (Figure 5-16), the Gaussian filter (Figure 5-17), and the differentiator (Figure 5-18). In this example, the length of the filters used is 61, i.e. L = 61. 5.3 Segmenter The segmenter consists of two parts: a function segmenter for each function of the system, and a qualitative behaviorgenerator to coordinate the whole segmentation process. The overall scheme is illustrated in Figure 5-19. The function segmenter segments each signal at zero crossings of its derivative obtained 68 E E I I a r- I ... .. I . ... .... . 500 I .... . 1000 . I 1500 2000 2500 l 3000 .. - 3500 Time/$ Figure 5-16: The ABPM data segment processed by the median filter. Note that the impulsive features have been filtered. E E 2I I I I .9 -6 .r i I 0 500 1000 1500 2000 2500 3000 3500 Time/s Figure 5-17: The ABPM data segment processed by the Gaussian filter. Note that the signal has been smoothed. I, E E :8 It II E RI I I 2 -6 .C 2 -k I E Time/s Figure 5-18: The ABPM data segment processed by the differentiator. 69 Distinguished Time Point Qualitative Function State t(7) x(t(7),x(4)/x(l),inc) Qualitative Behavior Generator Instance Variables: Disting. Time Point List Temporal Abstraction QualitativeSystemState (x(t(7),x(4)/x(1),inc),y(t(7),y(7),std),z(t(7),z(4),dec)) Figure 5-19: Overall scheme of segmentation to produce a qualitative behavior from processed signals and derivatives. 70 from the differentiator. It then looks up its local landmark list to see if there is any existing landmark within a tolerance from the current value of the function. If so, the existing landmark becomes the qualitative value of the function in this state. If not, the segmenter creates a new landmark corresponding to the current value of the function, returns this landmark as the qualitative value of the function in this state, and stores the new landmark in the local landmark list. The direction of change of the function in the current state is obtained by observing the sign of the derivative. A positive derivative corresponds to inc (increasing). A negative derivative corresponds to dec (decreasing). A derivative within a tolerance from zero corresponds to std (steady). The qualitative value and the direction of change together form a qualitative state of the function. The qualitative behavior generator keeps track of distinguished time points and coordinates the entire segmentation process. When any one or more of the function segmenters detects a zero crossing in their derivatives, the generator waits for r sampling periods to see if any other segmenters also detect a zero crossing in their derivatives. The parameter - therefore determines the level of temporal abstraction, as discussed in Section 5.2.3. 5 The generator labels all times within these r sampling periods as the same distinguished time point. It then signals all segmenters to segment their signals at this time point. The generator then collects a qualitative state of each function from its segmenter, and combines the qualitative states of all the functions of the system into a qualitative state of the system at the current distinguished time point. A series of such qualitative states form a qualitative behavior of the system. This is written into a text file for subsequent input into GENMODEL. 5r is set to M = 3 so that the corresponding frequency Q2= is well within the cutoff region of the lowpass Gaussian filter. This avoids the aliasing problem discussed in Section 5.2.3. 71 Chapter 6 Results and Interpretation The learning system was applied to data segments obtained from six patients during cardiac bypass surgery. 1 A data segment from each of the first five patients was used to study how qualitative models learned vary across patients. Six data segments from the same patient (Patient 6) during the same surgery were used to study how qualitative models learned vary within a patient. The models learned are compared with the cardiovascular model described in Section 4.2. Each data segment was 1000 seconds (16.7 minutes) long, sampled at 1 Hz. The fault tolerance level in GENMODEL was set at 20% of the total number of qualitative states in each data segment. The operation performed in each case was to insert coronary artery bypass grafts, except in the case of Patient 2 which was to replace the aortic valve. Models were learned from the data segments at six different levels of temporal abstraction, represented by the six different values of L as shown in Table 5.1 in Section 5.2.4. The results are described below. For each data segment, a brief overview of the patient's condition is given, followed by a plot of the original signals. Then the filtered signals at each of the six levels of temporal abstraction is shown followed by the model learned and an interpretation of each of the model constraints. 'Raw data was recorded from the HP Component Monitoring System at a local hospital. The eight parameters used for the experiments were derived from these primary measurements as described in Section 4.1. 72 6.1 Inter-Patient Model Comparison: Patients 1-5 6.1.1 Patient 1 The patient was a 39-year-old gentleman with familial hyperlipidemia syndrome and a 3-4 year history of angina. Upon cardiac catheterization, he was found to have widespread triple vessel coronary artery disease but a well functioning left ventricle. He was on calcium channel blockers, beta-blockers, aspirin, GTN (glyceryl trinitrate or nitroglycerin, a vasodilator) and a drug to control his hyperlipidemia. Prior to the interval corresponding to the data segment, lightness in anesthesia caused a rise in ABP following skin incision from 109 mmHg systolic to a peak of 137 mmHg systolic. During the interval, the anesthetic (Enflurane) was increased and a bolus of analgesic (2 ml of Alfentanil) was administered to dampen the response to surgical insult and avoid myocardial ischemia. The ABP was subsequently brought down. . 4nn i 4n_ i i lVV r m 80 - - ____III I 5 r Fill i o00 I- I 1000 1500 2000 goo 1000 t/sec 1000 - UU I 100 oo 1500 2000 t/sec VI II 1000 1500 C> 500 III 00 2000 . . rrrmarrmrru 1000 t/sec 1500 2000 1500 2000 t/sec X 10 5 5 01 O~>a T-If LA&---------1'11 o00 1500 1000 -500 2000 t/sec t/sec X 10 4 a- -1 kal - u 500 I _'.I j -A- IIaJ w0 - I 1000 1000 1500 2000 90 t/sec t/sec Figure 6-1: Patient 1: Original Signals. Note the relatively constant heart rate due to the effect of beta-blockers. 73 -7 / Il co 60 C-· A C 2a- 6 Q. L 00oo Jl 1000 1500 2000 - IL 500 goo 1000 t/sec 100 I __ · 1500 500 - · 80 - > 400 T' "' 0--41! "~oVo ' 1500 1000 '00 2000 t/sec 2000 1000 1500 2000 1500 2000 1500 2000 t/sec t/sec x 1)04 5 - 0 o C) oo00 1000 1500 1 g0 2000 1000 t/sec t/sec nAnn I i lf " " lI- I- a a. 8000n' 6000 - 500 1. ·a I . I-J 1.4 LU 0 1500 1000 2000 i 4 5'00 1000 t/sec t/sec Figure 6-2: Patient 1: Filtered Signals (L = 61) L=61 .No constraints were obtained. 74 70- ! 3,4 ~~~~ ~ ~ >L 6 60 z[ 50o)0 1000 1500 I 500 2000 1000 t/sec 80 ·- .51 1000 1500 1000 2000 1500 2000 t/sec x 104 4- jiu .II 1000 1500 _ oo 2000 --------i 1000 t/sec 10000 v500 500 ..... t/sec . . - a0 . 8000 cc .. 1500 IIB "oo 2000 /sec 0( 2000 Cn400 'goo 4.5 1500 .1 II I I IIvv -· 2000 rnn_ nn _ I 1500 t/sec Q- 1.6 - i w 1000 1500 2( )00 t/sec 1.4 1.2 500 1000 t/sec Figure 6-3: Patient 1: Filtered Signals (L = 121) L=121 mult(HR, ABPM, RPP) (Correct) mult(HR, SV, CO) (Correct) 75 iB "I · it 7 0 1000 1500 2000 oo00 1000 t/sec CTI 1500 2000 1500 2000 t/sec Arn 100 > 400 r--n an ""' -'hql VToo 1000 1500 l' 2000 1000 t/sec 84 4 3.5 3.&I II . cr I 1 II 2 I .. .. · 1000 1500 · ii i· 1000 1500 !00 2000 t/sec 2000 t/sec 4 .. L- I.V . 8000 1 · o1 0 00 a t/sec - 1 .4 500 o 1000 1500 0 I 2( )00 I 5000 t/sec 1000 1500 2( 100 t/sec Figure 6-4: Patient 1: Filtered Signals (L = 241) L=241 M+(ABPM, RPP) Correct given that HR was constant due to beta-blockers. increased due to increased Enflurane dosage. mult(HR, ABPM, RPP) (Correct) mnult(HR, SV, CO) (Correct) 76 ABPM 71n IIIX <a 7 Iv 60 0 u - /--/- I -1 'goo 1000 1500 2000 1000 t/sec 100 I-r 13: Jl 80 rn 1 . 1500 2000 2000 1500 2000 1500 2000 - 1000 -. goo t/sec t/sec 4.5rX 104 2 4. 01. 4 I ' o0 1000 1500 2000 1000 t/sec t/sec x 1500 _ n=^ .s 2000 >3 400 1000 00 1500 t/sec nnnn I, I UUUU a - 8000 E: ]f · 1 Ln[ Jli[ w 500 1.4 0 - I 1000 1500 2000 t/sec 1. J -- .0 50 500 1000 t/sec Figure 6-5: Patient 1: Filtered Signals (L = 361) L=361 M- (ABPM, VC) VC increased to compensate for decreasing ABPM due to increased Enflurane dosage. mult(HR, ABPM, RPP) (Correct) 77 "7 !7 Iu E-6 60 oI_ " 1000 1500 2000 1000 goo t/sec 80 _ - 1500 2000 1500 2000 1500 2000 1500 2000 t/sec · la Arn ~1 a_ I.v I tr m 70 400 =n _ -goo · 1000 65 0 tsec 1500 2000 1000 Vsec 4 y 1,v 4.2 1 ' 0o z 4 1- &R~-0 A 1000 1500 . 2000 1000 t/sec t/sec I----n_ IllllI Ill I- 1.6 I UUVVVV a- a. Cc 8000 5 1.4 w Ernnn .1 II BI 500 1000 1500 . 4 ni - I 2000 t/sec _ 500 1000 t/sec Figure 6-6: Patient 1: Filtered Signals (L = 481) L=481 M+(ABPM, RPP) Correct given that HR was constant due to beta-blockers. increased due to increased Enflurane dosage. M+(CVPM, VC) Both increased - Frank-Starling Law of the Heart. M-(ABPM, SV) (Spurious) M- (SV, RPP) (Spurious) mult(HR, SV, CO) (Correct) 78 ABPM 70 7_ f- - n 60 a 6 500 1000 1500 "8 2000 1000 o00 t/sec z 70 -I- ""' :1!'1 1000 1500 L -,' 0o 1000 x1 I nI oo 2000 1500 2000 i~~~~~~ II 1000 1500 . !00 2000 1000 t/sec t/sec xrnnrn_ I BB BI BB] 1 1 IVVI t- 13 8000 n l1a Ern v.vgoo .B . 1500 Vsec 4 ." a 2000 I' 2000 t/sec " 1500 > 400 --- ___ o.o 2000 . ApJ' .LL z . 1500 t/sec .. W .. .I 1000 1500 I .u 1.4~ I 2000 t/sec . ' 500 1000 t/sec Figure 6-7: Patient 1: Filtered Signals (L = 601) L=601 M+(ABPM,RPP) Correct given that HR was constant due to beta-blockers. increased due to increased Enflurane dosage. M+(CVPM, VC) Both increased - Frank-Starling Law of the Heart. M+ (HR, CO) Both decreased because of increased Enflurane dosage. invderiv(ABPM, RPP) (Spurious) inv_deriv(ABPM, VC) (Spurious) mult(HR,ABPM, RPP) (Correct) mult(HR, SV, CO) (Correct) 79 ABPM 6.1.2 Patient 2 The patient was a 65-year-oldblind and partially deaf lady who was admitted in acute left ventricular failure and found to have significant aortic valve regurgitation. The operation performed was to replace the patient's aortic valve and to check a mitral valve installed approximately 27 years ago. The data segment was taken when the patient was just coming off bypass and was artificially paced. 4 nn I I V IE 2 50 CD 0 .I I 9000 9500 10000 10500 9o600 _ k_ 9500 i/sec Jl M 100 -Iz zi r I LII Trrr-L n · -I 9000 9500 10000 9000 10500 9500 i/sec x 106 10000 · 10 IL JL 0 E n-'Y·1 · YU~y-. i -=h 9500 · ~·YI· i 10000 -900 10500 9500 t/sec 10000 10500 i/sec -x 104 'V 4I a JI-- Er -J w 0 9500 10500 t/sec · -1 9000 10500 x 104 -n Cv 10000 i/sec 10000 In "6oo 10500 9500 10000 10500 t/sec i/sec Figure 6-8: Patient 2: Original Signals. Note the relatively constant heart rate due to artificial pacing. 80 ^A "All Jl c. _ > 10t m 0 r 9ooo00 500 · 9500 n, Y..- 1000_ 7$y-- ,.. 9500 10000 ... 9'0o 900 10500 t/sec 00 X 104 i- rv _ 9500 10500 10000 10500 10000 10500 _ 10000 10500 9ooo00 9500 t/sec t/sec "IB " n I- " . i L n~ J.u ' m I-; 0.6 nE 8000 -J 0 nA [ I 6000- 10000 t/sec IB 9500 9ooo00 nn I II · 0 0.5 5 I I U, 500 i75 10 10500 t/sec Vsec '9o00 10000 9500 10000 10500 9600 inn 9500 t/sec t/sec Figure 6-9: Patient 2: Filtered Signals (L = 61) L=61 M+(SV, CO) Correct given that HR was constant due to artificial pacing. mult(HR, SV, CO) (Correct) 81 /~s Ar, 35 50 i- m3 000oo 9500 10000 n_ 0 5j9000 10500 9500 10000 10500 10000 10500 10000 10500 10000 10500 t/sec t/sec 800 I-r- 78 > 600 (0 -o JII·II BI 900o 9500 10000 '100oo 10500 t/sec 9500 t/sec 4~~o° x 104 r 0o 6 U> 0.5 9000 -vv' i iq 9500 10000 10500 9'000 9500 t/sec 9000 a. n n- - t/sec · -0.8 0.6 8000 0 7000 9000 9500 10000 10500 0.4 9000 t/sec 9500 t/sec Figure 6-10: Patient 2: Filtered Signals (L = 121) L=121 M+(SV, CO) Correct given that HR was constant due to artificial pacing. 82 55 · · Jr., _ m rn 50 > 4_ 0 ,-I 4! 93000 ^ /_// · · 9500 10000 105500 9500 t/sec 10000 10500 t/sec 0OU 800_ 78 60 0 II . . - OO_ ·· _ I7'9 00 Jl III 9500 10000 I I 9000 10500 9500 Vsec 10000 10500 10000 10500 Vsec 4 ,X 10 4i . II 0 0.5 06 AII _ ! -1 9000 9500 10000 10500 90ooo 9500 t/sec 9000 - t/sec 0.8 . . a. a 8000 0.6 _VO i I VVV I....... . . - 9000 w III ~rll 9500 10000 _ 9U i00 10500 t/sec 9500 10000 t/sec Figure 6-11: Patient 2: Filtered Signals (L = 241) L=241 M+(SV, CO) Correct given that HR was constant due to artificial pacing. 83 10500 _' -i - 50 aC2 t 4 1j. §-oo 9500 10000 _ 9000 10!500 9500 t/sec 80 700 . . 78 '9'600 9500 10000 1. . . 9500 10000 10500 10000 10500 10000 10500 t/sec I _ 9%00 0 0.5 .l 9500 10000 9'00 10500 9500 t/sec t/sec .. | .no V...- 8000 AL a- 7nA . . 900 ,x 104 o6 - 500 10500 t/sec I 10500 i 600 --lm J 10000 t/sec 0-J " 'A _ , vv 9000 0.6 9500 10000 10500 IL ''6oo t/sec 9500 t/sec Figure 6-12: Patient 2: Filtered Signals (L = 361) L=361 No constraints were obtained. 84 A_ U 2 50 4 , j, 900 9500 10000 10500 900oo t/sec m 9500 9EvD fUU I 78 10000 10500 /sec . . j 650 9000 9500 10000 l I I t w9ooO 10500 . 9500 t/sec 2 . . 10500 t/sec 104 7x . 10000 . . C I6 ,1 Il 9o00 9500 10000 10500 9o00 9500 t/sec n-OUUU 10500 10000 10500 0.8 0- a. Ir 10000 t/sec 0.6 nn J ~ IIB Wa 0C.n __ 9000 9500 10000 10500 i JA _ _ Vo00 t/sec 9500 t/sec Figure 6-13: Patient 2: Filtered Signals (L = 481) L=481 No constraints were obtained. 85 · 0 - 2 2 50 6o o000 9500 10000 9'00 10500 9500 t/sec 10000 10500 10000 10500 t/sec 7n.. III 80 78 Cn650. '9000 9500 10000 ,n.^ 9VooO 10500 9500 t/sec t/sec 14 I · . . o6 I 9000 9500 10000 I 10500 I 9000 9500 t/sec 10500 10000 10500 t/sec nnn II ·i · I ... aa- 10000 OvVV i w 0.6 a: -Inn a I 9000 9500 10000 ..n .A _ 9000 10500 t/sec 9500 t/sec Figure 6-14: Patient 2: Filtered Signals (L = 601) L=601 M- (ABPM, SV) (Spurious) 86 6.1.3 Patient 3 The patient was a 47-year-old gentleman who had a recent large myocardial infarction and recurrent episodes of left ventricular failure since then. His exercise tolerance was very poor and he developed a left ventricular aneurism following the myocardial infarction. The operation performed was insertion of coronary artery bypass grafts and excision of the ventricular aneurism. The data segment was taken when the surgery was just starting. 4n_ lBtI. n 80 0- 0 o 1000 1500 LLSU-·Y" · 1000 2000 i/sec I WU - aI 0 1 1Eli _UU9660 rY 1000 1500 2000 1500 2000 i/sec 2000 · - 0- , W.'pi IL. ---- '....... 1500 'WVVKo 0 2000 1000 t/sec /sec x 105 d~ I E B IV ILI. ,..L-- 0-500 1000 1500 0 0 O.1 2000 - - I 'oo00 1000 t/sec 1500 2000 1500 2000 MJ, 0 11.5i 04 ! 1500 II t/sec x 10 4 1000 I W l-1 I -- / L- J i 500 2000 1000 t/sec t/sec Figure 6-15: Patient 3: Original Signals 87 7r I - to, m 60 500 50; 1000 1500 2000 t/sec 4 2 500 1000 1 . > 300 '1 200 500 t/sec L x 134 ._ 113~~~~~~ sX 1500 2000 1500 2000 1 ° 0.5- n v __ I 500 · 1000 1500 'iSy I~~~~~~ 2000 500 t/sec H 2 5000 5000 1 . 1000 1000 t/sec 10000 II lJ 1000 t/sec 02 o 0- 2000 400 I a 1500 t/sec 1500 2000 t/sec O_ 500 . . 1000 1500 t/sec Figure 6-16: Patient 3: Filtered Signals (L = 61) L=61 No constraints were obtained. 88 2000 70II I co 60 · 2a. 0 _ 4 i/ n- 9 Ern 500oo 1000 1500 2000 500 1000 t/sec 400 - 100 IX 2000 . . > 300 0---JII 1000 1500 BI '"oo 2000 1000 t/sec 4 x 104 1500 t/sec 1000 t/sec 11 " aaa: 5000 1 uJ ! 1000 1500 2000 I ! 00oo I I 1000 1500 t/sec t/sec Figure 6-17: Patient 3: Filtered Signals (L = 121) L=121 M + (HR, RPP) 2000 Bi oo 2000 nnnr 0- 1500 0 0.5 i 500 2000 _ I 1000 BlBB "I 1500 t/sec 32 (D B 1500 t/sec Correct due to slowly rising ABPM. mult(HR,ABPM,RPP) (Correct) mult(HR, SV, CO) (Correct) 89 2000 70 - >4 a- 60 EL co I O rn. 1000 oo00 1500 2000 500 1000 t/sec 1500 2000 1500 2000 1500 2000 t/sec 70 barn 6JDU I 60- > 300 T r- n -- 500 1500 1000 2000 1000 t/sec I I 1 t/sec X 1 ()4 4_ ]r (>)0 00 23 yo1 - 12 1000 5oo 1500 2000 o00 1000 t/sec Onnf J uv- ]] ~l t/sec _ .¢' a_ a- 6000 n' 4000 - 500 5i-- · · 1000 1500 0 · 2000 500 t/sec 1000 1500 t/sec Figure 6-18: Patient 3: Filtered Signals (L = 241) L=241 AM+(HR, RPP) Correct due to slowly rising ABPM. mult(HR, ABPM, RPP) (Correct) nmult(HR,CVPM, RPP) (Spurious) ,mult(HR, SV, CO) (Correct) 90 2000 70 a. 4 60 L rEn -oo 0 . . 1000 1500 0 1000 0oo 2000 ,, /U 400 3 60 > 300 [ 500oo 1000 1500 2000 1500 2000 1500 2000 t/sec t/sec 1500 ok4 )O sV0 2000 ------/ 1000 t/sec t/sec x 104 1 0 >0.5 O 2 t _ II oo 1000 1500 goo 2000 1000 t/sec annnII I t/sec eII I a L 6000 4000 500 - 1 0 I 1000 1500 ! 500 2000 1000 1500 t/sec t/sec Figure 6-19: Patient 3: Filtered Signals (L = 361) L=361 mult(HR,ABPM,RPP) (Correct) mult(HR, SV, CO) (Correct) 91 2000 '= · /U 6 2 · a-4 60 O i ... -~l! | ffi 1000 1500 2000 1000 t/sec 7^ 2000 1500 2000 400 - /U C> 300 60 cn Inn 'oo _ -III 5'oo 1000 1500 2000 I~ I 1000 /sec ex t/sec 104 1 0 0.5 02 i 1500 t/sec I-- i 500 1000 1500 OL 0 2000 1000 t/sec I 1500 2000 1500 2000 i/sec 8000 H 1.5 f- 6000A1~1 ^Ill LU a 1 500V 500 1000 1500 2000 iJ i r, _ 500 t/sec 1000 t/sec Figure 6-20: Patient 3: Filtered Signals (L = 481) L=481 M+(HR, RPP) Correct due to slowly rising ABPM. 92 9 6 aa. 60 a 0 o00 1000 1500 4 -- ( 1000 2000 t/sec 70 400 - 1500 2000 t/sec . . >c 300 I JII 1000 1500 2000 ' 1000 oo t/sec 10)4 ,x Li 0 0.5 02 I 1000 1500 2000 1000 t/sec 1.EIr · 0 6000 1 C \r I V.0 5 AAlto II IB 1000 1500 2000 2000 _ 1000 t/sec i/sec Figure 6-21: Patient 3: Filtered Signals (L = 601) L=601 M + (HR, RPP) 1500 f I 0- 500 2000 t/sec c 8000 - III 1500 t I I go 0 2000 1 . . 1500 /sec Correct due to slowly rising ABPM. mult(HR,ABPM, RPP) (Correct) 93 6.1.4 Patient 4 The patient was a 60-year-old gentleman with a history of angina and breathlessness on exercise. He had been a non-insulin dependent diabetic for many years. His coronary angiography showed severe triple vessel disease with an occluded right coronary artery and severely impaired left ventricular function. He was on beta-blockers (Atenolol) and calcium channel blockers (Nicardipine). The data segment was taken when the operation had just started, and the period was relatively uneventful. 1 \~m~ · aco 70 8 0 IwFn#ljvI cJ 1500 1000 2000 2500 1500 1'600 t/sec -1· rn 2500 nrnr 'JII - 100 L,,,,1, ,,,, , n 100oo I 2000 t/sec I· Al lf00 , A 111TAP" BI 1500 tsec 2000 1000 2500 1500 2000 2500 t/sec ,,x 105 o_ .I- -kill. Ii IAMI ULWL 'I'T'F7TrIq7rrI1TWi"7 1500 1000 J L&dIdlILL.. 2000 · _c. 2500 1000 1500 t/sec Ax z tsec 2000 2500 104 a 1- 0- II. 1.-L I 1000 nFTL . 4~u1LLJX 1500 2000 a 1.5 10002500 1000 t/sec 1500 2000 2500 t/sec Figure 6-22: Patient 4: Original Signals. Note the relatively constant heart rate due to the effect of beta-blockers. 94 11 a eo 100 1500 2000 0 6_ 2500 o000 1500 t/sec 4hi Or -44 * , . 800 . 1500 2000 8000 2500 1500 t/sec x 10 0 0 2000 2500 900 c) A- 2500 I UUU- : I 1000oo 2000 t/sec t/sec 4 1 . . J.i 0>0.5 AI fl 1000 1500 2000 2500 1ib00 1500 t/sec 2000 2500 2000 2500 t/sec ""' rcnn _ I 4· a. a- 5000n 4500 1000 w 0 1500 2000 2500 J q B _ 1 boo t/sec 1500 t/sec Figure 6-23: Patient 4: Filtered Signals (L = 61). Note that the trends of the relatively constant heart rate are amplified. L=61 No constraints were obtained. 95 i/\i n. M · 6CI.I a: "` 'i000 1500 2000 44 l G" A,1 ZL.4 __ 1o00 1500 2000 ~~~~ ~ ~ 6 C A 2500 L 10 0 t/sec AC ··- ! a n'n iii 900 M 2500 I... 2000 2500 2000 2500 t/sec vJ VVI .B. (n 1500 1500 100 t/sec t/sec x 104 0 0.5 00 5 · Z 1000 1500 2000 25 00 1000 1500 t/sec PA I II 1 .,II 11 2500 So w 0 ... 1000 2000 I I- 5000 I~. 2500 t/sec rRt5 O--IL 2000 1500 2000 2500 4 I I b 00 110 t/sec 1500 t/sec Figure 6-24: Patient 4: Filtered Signals (L = 121) L=121 M+(SV, CO) Correct given that HR was constant due to beta-blockers. M+(CVPM, /T) (Spurious) mult(HR, SV, CO) (Correct) 96 6. oh><. i " 56 0 ED c·· 1500 0boo 2000 2500 ,_ . 1000 150 1500 o- 44 i/V & Z3." i000 1UUU | w l ! 2500 2000 · 900 ( nr-- ! 1500 2000 t/sec t/sec III nl 'too 2500 t/sec 1500 2000 2500 2000 2500 2000 2500 t/sec x 104 U 0C 5 0 0.5 it nA!I 1oo00 1500 2000 2500 1000 1500 t/sec t/sec -- nn c' [.013.5000 I[ w 1.· .J.IJVV 0 1 ~1 1"00 1500 2000 2500 1 boo 1500 t/sec t/sec Figure 6-25: Patient 4: Filtered Signals (L = 241) L=241 M+(CO,RPP) (Spurious) M+ (SV, CO) Correct given that HR was constant due to beta-blockers. M+ (SV, RPP) (Spurious) mult(HR, SV, CO) (Correct) 97 6.5 1 Q- 57 a 6 13 C . 100 1500 2000 2500 00 5 . 1500 t/sec 2000 2500 t/sec non _u ..IVVV ...... I C0 900 Inn ... ii. u1oo A1 00oo 1500 2000 2500 1500 2000 2500 t/sec t/sec x 10 (. 0.5 85 0 ! 100 1500 2000 1000 2500 1500 t/sec 2000 2500 t/sec ,, ac"""" 1. . . DUO L 5000 I-- - uJ w 0 _ 1000 I 1500 2000 2500 i t/sec b000 1500 2000 t/sec Figure 6-26: Patient 4: Filtered Signals (L = 361) L=361 M + (SV, CO) Correct given that HR was constant due to beta-blockers. mult(HR, SV, CO) (Correct) 98 2500 6.5 r E m 57- CC7 _ 1000 / 1500 2000 . . 6 I 1i000 2500 1500 t/sec 2000 2500 2000 2500 t/sec xrnnn I I III I vIU I 44.5 >) 900 arn_ AA , ~1 1500 1000 2000 2500 III. 1000 1500 t/sec t;c t/sec x 104 0 0.5 00- 5 4 - 1000 _- . · 1500 2000 2500 1000 t/sec rri Ir 5000 A-n Jl "l 2000 2500 2000 2500 t/sec rnn JJUU a0 . 1500 I- 1.5 _ 1000 1000 uJ w 0 1 I 1500 2000 2500 I __ Iboo 10 00 t/sec 1500 t/sec Figure 6-27: Patient 4: Filtered Signals (L = 481) L=481 rmult(HR, SV, CO) (Correct) 99 57 a-) O Gc 1000 1500 2000 2500 6 L 1 000 1500 t/sec 2500 .nn I UUU AZI > 900 r44.5 no -- AA · 0)~r!I I 1000 1500 2000 1000 2500 1500 t/sec 2000 2500 2000 2500 2000 2500 t/sec 1 x 104 O 0.5 0) 5 Jl 2000 t/sec AI 1000 1500 2000 1000 2500 1500 t/sec t/sec r Ann 0Q 0- 5000 aI A Jl "1 UJ 0 An 1000 1500 2000 2500 4 i t/sec B boo 1500 t/sec Figure 6-28: Patient 4: Filtered Signals (L = 601) L=601 mult(HR, ABPIM, RPP) (Correct) mnult(HR, SV, CO) (Correct) 100 6.1.5 Patient 5 The patient was a 66-year-oldgentleman with a fairly long history of angina and a proven inferior infarct 3 months before the operation. He had very poor exercise tolerance, developing severe ischemia after very moderate exercise. His catheterization showed severe triple vessel disease with reasonably good left ventricular function. He was hypertensive and was treated with beta-blockers (Atenolol). He was also a non-insulin dependent diabetic. The data segment was taken quite some time after the surgery had started. Before the period, the initial lightness of anesthesia caused a sharp rise in ABP from 90 mmHg systolic up to 160 mmHg systolic and that was sustained for several minutes. During the period, the depth of anesthesia (Enflurane) and analgesia (Alfentanil) and the dosage of GTN were increased to bring the ABP back down. 80 · - 1 2a. m 60 C) ": An gl Ill _ 'o0( 3500 0 4000 4500 30 t/sec I'll Jtc-- r I. I III I i/sec _ et3 :s0oo0 I 4000 ! ! - -;·C I 3500 X 104 3'000 4500 · 3500 t/sec 106 -x I · · I 3000 3500 -B 4000 4500 4000 4500 t/sec · 5 >r, S 4000 4500 3000 t/sec 3500 t/sec x 104 a Ec 0 - -r -J _,^ C3 ! 3000 3500 4000 4500 30 t/sec t/sec Figure 6-29: Patient 5: Original Signals. Note the relatively constant heart rate due to the effect of beta-blockers. 101 2a- A 60 50 3 0 40 300 3500 4000 4500 3'boo t/sec zir 1 3500 '1I IIIBI nn 7: .hh1 300 __ I · 3500 4000 · II 4500 3ooo00 3500 t/sec x 104 1 o0 4500 . . > 0.5 ll I 30ooo 3500 4000 _ 3600 4500 3500 t/sec ---I I VVV . a- 6000 rnn _ ... 4000 t/sec -1 cc 4500 > 500 iiri--? Jl 4000 t/sec 3--0 II is-\ 300( 0 3500 4000 4500 4000 4500 t/sec 4000 2.2 4500 3000 t/sec 3500 t/sec Figure 6-30: Patient 5: Filtered Signals (L = 61). Note that the trends of the relatively constant heart rate are amplified. L=61 M-(CVPM, AT) (Spurious) M+(SV, CO) Correct given that HR was constant due to beta-blockers. M+(CVPM, AT) (Spurious) 102 "I n.l 50 a0 n A Jlm Fr : a;3- C - | 3500 -3o00 I I 3500 4000 4500 4000 4500 4000 4500 4000 4500 ^~~~~~ 4000 3000 4500 t/sec ._ __ bV I t/sec 600 · I 300(0 -\ > 400 -ii7 "" l nnhl_ 3500 4000 -Jl I 1 LUUv 4500 3000 3500 t/sec t/sec x10 4 1 t. oO 4 > 0.5 , A J 3000 3500 4000 4500 3v00 3500 t/sec 7000 0- - 6000 mnn .. - | t/sec w \N \ L 2.2 UJ JJ..J 300JU_ 300( 0 3500 4000 4500 3000 t/sec 3500 t/sec Figure 6-31: Patient 5: Filtered Signals (L = 121) :L=121 .No constraints were obtained. 103 0u - · · · · I Im an 50 co 0 3000 3500 0 4000 4500 - - 3500 3'b00 I.3000 > 400 4000 3500 ---V onn I 4500 3oo000 · · 3500 4000 tsec x104 1 · 0 0.5 0 I 3000 3500 4000 3'o00 4500 ! 3500 4000 4500 4000 4500 Vsec t/sec · - - 6000 w (r --I 4500 t/sec 04 7000 4500 600 60 a5 I 58 6 4000 t/sec t/sec 2.2 0 II VVUV 3000 3500 4000 4500 3000 3500 t/sec t/sec Figure 6-32: Patient 5: Filtered Signals (L = 241) L=241 M+(SV, CO) Correct given that HR was constant due to beta-blockers. mult(HR, SV, CO) (Correct) 104 I f! <I 4 · · "3. CL- 50 30( · al 3500 oo 4000 4500 i )0 3500 t/sec ,, 600 · · '30000 3500 4000 40030( )0 4500 I 3500 Vsec 104 0.5 A 901DO 3500 4000 4500 _ 3vo00o 0 ! 3500 t/sec - a- 4500 1 0 35 4000 t/sec 4 7000 · > 500 U58 0 4500 t/sec · 4.5 4000 4000 4500 t/sec · · · · - I-r 2.2 - 6000 w 0 5000 300C 3500 4000 4500 . 3000 t/sec . 3500 4000 4500 t/sec Figure 6-33: Patient 5: Filtered Signals (L = 361) L=361 M+(ABPM, HR) (Spurious) M+(ABPM, RPP) Correct given that HR was constant due to beta-blockers. dropped because of increased depth of anesthesia. M+ (HR, RPP) (Spurious) M+(SV, CO) Correct given that HR was constant due to beta-blockers. rmult(HR,ABPM, RPP) (Correct) rmult(HR,CVPM, RPP) (Spurious) mult(HR, SV, CO) (Correct) 105 ABPM 60 4- - m 55. · · 3 - ^ S 3500 5300 4000 300 4500 · · 3500 4000 60 8'g 4500 t/sec t/sec 600 - E · · > 500 1 58 "'A^O 3000 3500 4000 4500 "o00 3500 t/sec x 104 I' I. 4500 4000 4500 1 _ 00 t4. ', 4000 Vsec 0 0.5 iI o7000 3500 4000 4500 3'000 3500 t/sec t/sec 7000 - · - 2.2 w LJ a. 6000 5000 · - 3000 · 3500 4000 4500 3000 t/sec · 3500 4000 4500 t/sec Figure 6-34: Patient 5: Filtered Signals (L = 481) L=481 A+ (ABPM, RPP) Correct given that HR was constant due to beta-blockers. ABPM dropped because of increased depth of anesthesia. M-(ABPM, VC) VC increased to compensate for decreasing ABPM due to increased dosage of anesthetic, analgesia and GTN. M- (RPP, VC) (Spurious) mult(HR, ABPM, RPP) (Correct) 106 "I A 11 r u_ rm 55 n I ~1 h1UC >C I 3500 30003 4000 3'00 4500 3500 t/sec ^A I Lred, nlJ 58 N 58 4000 4500 4000 4500 > 5000 300o 3500 4000 4500 /11 nn I 3000 3500 t/sec X 10 4500 600- · _ l 4000 t/sec t/sec 4 ,, 0o (> 0.5 4 O or soc iii 3500 4000 4500 3000 3500 t/sec 7nrnn_ I t/sec I- a- 6000 2.2 5000 300( 0 0 I 3500 4000 4500 I 3000 t/sec 3500 4000 4500 t/sec Figure 6-35: Patient 5: Filtered Signals (L = 601) L=-601 M+(ABPM,HR) (Spurious) M+(ABPM, RPP) Correct given that HR was constant due to beta-blockers. dropped because of increased depth of anesthesia. .M(CVPM, CO) Frank-Starling Law of the Heart. .M4(HR, RPP) (Spurious) nmult(HR,ABPMI, RPP) (Correct) mrult(HR, CVPA1, RPP) (Spurious) mnult(HR, SV, CO) (Correct) 107 ABPM 6.2 Intra-Patient Model Comparison: Patient 6, Segments 1-6 The patient was a 63-year-old gentleman having 1 internal mammary artery and 3 coronary artery grafts. He had a history of 6 years of hypertension, 4 years of angina and 20 years of chronic bronchitis. His angiogram showed well presented left ventricle, totally occluded right ventricle and severe disease at the origin of all left sided vessels. He was not on beta-blockers. 6.2.1 Segment 1 Previous to this segment, lightness in anesthesia caused rises in ABP (up to 180 mmHg systolic) at leg surgery, chest incision and sternotomy. The patient then developed myocardial ischemia. In response to this, the GTN dosage was increased, which along with hypovolemia caused the ABP to drop, with the result that ischemia improved at the expense of blood pressure. The depth of anesthesia was also increased. 150 4 · I a. . 100. 0 G-- 1000 55oz 0 1500 2000 n 1v; I _ '0 -100 _ I 1000 500 i/sec 1500 cnn III .1 5000 - I.1VV 0 cr X -111\ l .. -5000 - . . 1000 -"oo00 1500 --1W.,.III, I 6-1- 0- cn -- 2000 500 1000 t/sec 0 2000 i/sec 4x r | 106 2000 5_LL..iL.L ( 0 1500 /sec i...-- alh. Ir1- lr nr' -- mI·- -oo 1000 1500 2000 1000 t/sec 104 Px J.~ a.e 0 2 aU in3 UIUIIIYrUIJIU*; _-11l 500 2000 1500 2000 1-4 w: - I I 1500 t/sec I 1000 1500 2000 oo00 t/sec - I. 1000 t/sec Figure 6-36: Patient 6, Segment 1: Original Signals 108 nn_ I r I III iVV CL M: 50 -) O L A 1500 1000 0oo 0okO -o00 2000 71 1| 1000 2000 1500 2000 nnn 100 III I VUv > 500 50 l' - I goo | | 1000 1500 \· goo 2000 1000 t/sec t/sec 10 1500 t/sec t/sec x 10 14 ie I 1000 goo 1500 1000 2000 1500 2000 x 104 \ 500 0 2000 t/sec t/sec 2 1500 500 _ U3 1000 1500 wD 1 2000 500 1000 t/sec t/sec Figure 6-37: Patient 6, Segment 1: Filtered Signals (L = 61) L=61 mult(HR, ABPM, RPP) (Correct) 109 5 100 a . . a- 0 O3 .) '~ 1000 "o00 1500 2000 _ -; 500 . 2000 1500 2000 1500 2000 1500 2000 Ir-nr III I. II I I 1UU cc 50 I C> L 500 iii goo 1000 1500 2000 r goo 1000 t/sec t/sec x 104 10 0o 1500 t/sec t/sec rrrr II , IUV . 1000 14 i r , 0.5 5 _ II n _ I I, 0oo 1000 1500 2000 goo 1000 t/sec j-I- t/sec x 104 aa1 Cn 00-500 1) 50 2- iri1000 1500 2000 500 t/sec 1000 t/sec Figure 6-38: Patient 6, Segment 1: Filtered Signals (L = 121) L=121 'rlult(HR, ABPllI, RPP) (Correct) mult(HR, SV, CO) (Correct) 110 100 > r 0 0 I_ 50s0 1000 1500 2000 -o00 Iv 1500 2000 t/sec -n ·· · 1000 t/sec 1000 > 500 50 A 00oo n 5o 0 · 1000 1500 2000 · 1000 2000 1500 2000 1500 2000 tsec Vsec 4 1500 104 1 0.5 2 I 0oo ! 0 1500 1000 2000 1000 t/sec t/sec _ n'' 1·04 a a13 1r- 2 0. LU . I vU 500 1000 1500 2000 I _ goo 1000 t/sec /sec Figure 6-39: Patient 6, Segment 1: Filtered Signals (L = 241) L=241 M-(HR, SV) (Spurious) inv-deriv(ABPM, RPP) (Spurious) invderiv(ABPM, VC) (Spurious) 111 _nr ,.... Ivv 0 a. U a- m _ c,\ 'goo .. 1000 1500 500 2000 1000 1500 2000 1500 2000 1500 2000 1500 2000 t/sec t/sec 100 4 ffnnn I .... Ivvv 50 u) 500 l- I 1000 goo 1500 500 2000 1000 t/sec t/sec X 104 4 1 0(D o 0.5 n 0oo 1000 1500 2000 00oo 1000 t/sec fx t/sec 10 4 e· a a- 1 n 2- A1L 500 q 1000 1500 2000 I _ goo t/sec 1000 t/sec Figure 6-40: Patient 6, Segment 1: Filtered Signals (L = 361) L=361 M + (HR, RPP) Both dropped because of increased depth of anesthesia. mult(HR, ABPM, RPP) (Correct) mult(HR, SV, CO) (Correct) 112 A ·4El nnlI I UV a. >2 M n, ... 0 __ - oo 1000 1500 2000 0 1000 t/sec 1500 2000 t/sec 100 - 1000 I 50 > 500 I 1000 o60 1500 1000 2000 Vsec 1500 2000 Vsec 4 10 I! 0 82 >0 0.5 III II oo00 1000 1500 2000 500 5 I I 1000 1500 20 00 1500 2000 t/sec t/sec 4 x o10 I J a. 1 0 I. C.,4·- . 500 I~~~~~~~~~~~~~~~~ 1000 1500 oo 2000 t/sec 1000 i/sec Figure 6-41: Patient 6, Segment 1: Filtered Signals (L = 481) L=481 M + (HR, RPP) Both dropped because of increased depth of anesthesia. M+(ABPM, AT) Both dropped because of vasodilating effect of GTN. invderiv(ABPM, RPP) (Spurious) inv deriv(ABPM, VC) (Spurious) mult(HR, ABPM, RPP) (Correct) mult(HR, ABPM, VC) (Spurious) mult(HR,CVPM,RPP) (Spurious) mult(HR, SV, CO) (Correct) 113 11Il1 4- _rr I UU L a- 2 m 1 n I, IJ90) I 1000 1500 2000 o00 1000 100 2000 1500 2000 1500 2000 1500 2000 UU - a: I 50 I! 1500 t/sec t/sec > 400 206- I' 1500 1000 2000 1000 t/sec t/sec 4 n x 10 4_ I- o 0.5t II I I III g00 1000 1500 00 2000 1000 t/sec t/sec ,x 104 m _ !a1 1 '- , I 500 1000 1500 2000 500oo t/sec 1000 t/sec Figure 6-42: Patient 6, Segment 1: Filtered Signals (L = 601) L=601 t+ (ABPM, CO) Both dropped because of vasodilation and increased venous tone caused by increased GTN dosage. M + (HR, RPP) Both dropped because of increased depth of anesthesia. M + (ABPM, AT) Both dropped because of vasodilating effect of GTN. IM+(CO,AT) Both dropped because of vasodilation and increased venous tone caused by increased GTN dosage. invderiv(ABPM, RPP) (Spurious) invderiv(ABPM, VC) (Spurious) mult(HR,ABPM, RPP) (Correct) mult(HR,CVPM,RPP) (Spurious) mult(HR, SV, CO) (Correct) 114 6.2.2 Segment 2 The increased GTN dosage successfully lowered ABP and improved ischemia. In this segment, GTN dosage was reduced. 80 - r > a211*uun~~unrIII ~~~~~~ m 60 rn IL 4 L--- I 2000oo 2500 "'fij 0D 3000 V~ t/sec An- ii I i 500 An JfiJ ZUL I 10( I Fiji .1 1,.. ) APW U .. .. . 2500 I,.,~.. 2000 3000 10 .1 . 3 0 >C - 2oo00 2500 - ri -r 3000 'ooo t/sec x 10 nI C 2(500 - - - - ,7 Onr11 2500 t/sec 30100 C 1. ,4TJ II 3000 2500 t/sec t/sec x ,L111~1 _- 200UU .. 2( 000 00 3000 2500 b0oo t/sec LL. 2500 t/sec 1. Ff 1 3000 4;nl 200 i 2500 t/sec Figure 6-43: Patient 6, Segment 2: Original Signals 115 30(30 60 c 50 > AN 0 C- illI ,l'l I, .... 2500 t/sec 3000 o00 2500 3000 t/sec 1000 I ,,vv szj7C I 50 ,I '" 2v00 2000 Il 2 15 _ 2000 2.5 > 500 r A'00 % 2500 t/sec ! 3000 -x 1(4 2500 t/sec 3000 2500 t/sec 3000 2500 3000 2 0U I 1 2000 I "v A 2500 t/sec ---A ,I~1111E II II III Q. 5000 2000 20C00 I-- 1.2 r iii [ 2v- 3000 L- 1.1 w --- 0 · 2500 t/sec 3000 4i _ 2b00 t/sec Figure 6-44: Patient 6, Segment 2: Filtered Signals (L = 61) L=61 mult(HR, SV, CO) (Correct) 116 vuv ... __ · ~~~~~~~~~~~~~~~~~2.5 0EL - 50 2 O An -- 2000 .~~~~~~~~~~~ 2500 t/sec 15 3000 ZO00 n, I I~ IUU 1000 50 > 500 '1 2500 t/sec 3000 2500 t/sec 3000 2500 3000 A 00 2500 t/sec 2O 90 20( 3000 Ax 104 2 4 r 01 I- o O22 i i ! II 2d00 2500 2000 30i 00 t/sec t/sec 100( U cc | 5 00 - , 1 .0 - 1.1 \ 2000 2500 30100 12_ 20( 0O t/sec 2500 t/sec Figure 6-45: Patient 6, Segment 2: Filtered Signals (L = 121) L=121 M+ (HR, RPP) Correct given relatively constant ABPM. mult(HR, ABPM, RPP) (Correct) 117 3000 hi 2.5 f a- 50 <c 20oo 2500 t/sec 3000 2 0 IE- = '' oo 2500 t/sec 3000 2500 30 00 1000 uv Ic T 40 > 500 nIl _ '200oo 2500 3000 2600 /sec A /sec x 10 4 _.-__ i 1.5 02 0 c 1 n l I! 2500 t/sec 2000 I-_- 3UUU 3000 t/sec I . L 0r w 2000 2500 t/sec ~~~~~~~~~~~~~~~~~~~~~ 2500 30 00 ooo 3000 \-1- 1.1 1! -- 2b00 . 2500 t/sec Figure 6-46: Patient 6, Segment 2: Filtered Signals (L = 241) L=241 deriv(SV, CO) (Spurious) 118 3000 > 'L Cl 2.5 2 50 O 2500 t/sec 2000 3000 I 5, '00oo 3000 t/sec hl\h 60 I UUU I 40\ > 500 r% ~Ct 2500 t/sec 3000 20(30 00 2 0 | 2i 00 600( II= 1 n -1 , . 2500 t/sec :oo 3000 2500 - 1.2 0 400( r -* 1.1 - - - w 0 2500 3000 t/sec - 200C I 2( 00 3000 m · .1 1 _ _ 2500 t/sec X 10 4 ,__ _ ., 2500 3000 1 ! 2b00 t/sec - ~ 2500 t/sec Figure 6-47: Patient 6, Segment 2: Filtered Signals (L = 361) L=361 'mult(HR, SV, CO) (Correct) 119 - 3000 I J > a- co 50 It 2 0o c-l _, B 0oo 2500 t/sec 'oo 3000 .4I nfrf ..... n- 40 T' c 500 _ 2500 t/sec n 2v0(00 3000 21 1 ., 2500 t/sec 2C)00 I-- n0- 4000 n l~~~~~~~~~~~~~~~~~~~ 2500 t/sec 3000 I" oo u2U000 3000 l~~~~~~~~~~~~~~~~ 2000 3000 1.5 0 o 2 2000 - 2500 t/sec x 104 6000 - 3000 l B B 2VOI 00 2500 t/sec 60 on t ! 4 i 2500 t/sec 3000 2500 t/sec 3000 I 1.1 .4 B 2b00 Figure 6-48: Patient 6, Segment 2: Filtered Signals (L = 481) L=481 nM+ (HR, VC) Both were relatively steady. deriv(SV, CO) (Spurious) 120 rr 55 ~~ ~ ~ ~ ~ ~ ~ ~ ~ ~ 2. am 50 2 E > _ AC 2000 _ _ _ _ _ _ _ 0 4 _ 2500 t/sec I__ 3000 '200 2500 t/sec 3000 2500 t/sec 3000 2500 t/sec 3000 2500 t/sec 3000 f'n -- Du I UW0 > 500 :40 on 00 2500 t/sec n 2'000 3000 1 x 104 j2 O _. co2 A>r 2000 2500 t/sec u60o 3000 ---- nx a. a 4000 - 1= n 1.1 LU 2500 t/sec 2000 3000 2b00 Figure 6-49: Patient 6, Segment 2: Filtered Signals (L = 601) L=601 M+ (HR, RPP) Both continued to drop because of increased depth of anesthesia. 1M-(ABPM, HR) ABPM started to rise moderately following decreased GTN dosage. M- (ABPM, RPP) ABPM started to rise moderately following decreased GTN dosage. deriv(SV, CO) (Spurious) mult(HR, ABPM, RPP) (Correct) mult(HR, ABPM, VC) (Spurious) mult(HR,CVPM,RPP) (Spurious) mult(HR, CVPM, VC) (Spurious) 121 6.2.3 Segment 3 The patient condition was stabilized. Depth of anesthesia was decreased. This period was uneventful. QU a. 0 4 1 . a. 70 0 ,imnflM]bhII iJlnlyrJlj\/i*N*-J/JC/C 3500 i/sec 0ooo 4000 3500 3bl00 4000 t/sec Annr. 20 UUUVV 100 > 2000 ~.L. _..L...,. =.. _ n II,' n1111 3500 i/sec 3V00 4000 3'600 . AL. J AM . -~,~. ~.A. 3500 t/sec 40100 x 105 4I 0 02 A, tr~~~~ulr~~A11 0 b- 3500 4000 3'000 t/sec 4 I' 4 4 I. - a. rI 3ooo00 I. I I- 6.1 w0 rd 3500 3500 40100 t/sec X 104 n Ir .I.. - 3600 a- 1I I 4000 I '3600 t/sec 3500 t/sec Figure 6-50: Patient 6, Segment 3: Original Signals 122 40100 r ml I li 2. r Z.b 2 > 2 _ 4 3500 t/sec '300 4000 1I nn 3500 t/sec 4000 3500 t/sec 4000 3500 4000 nnnrr I UU I-v I 50T"', > 500 n I m _ I! I' 3500 t/sec 3v000 4000 a300 104 1U I 0o- 1 _- 5 o 0.5 - Il I 30(00 II ll l VVVV c 5000 CC Ai S~~~~~~~~~~ 3500 t/sec II 4000 t/sec -r 3r00 3v000 1. I r -H I- 0 3500 t/sec 4000 1 1 = 3000 3500 t/sec Figure 6-51: Patient 6, Segment 3: Filtered Signals (L = 61) L=61 mult(HR, ABPjM, RPP) (Correct) rmult(HR, SV, CO) (Correct) 123 4000 Fij 20 I a 54 &J in, 2 01I ! 3500 0oo 4000 . '3000 4 100 V > 500 n1 I! 3ooo00 *.l lI 3500 3oo00 4000 tsec 4000 1 o 0.5 5 n I remN I I! 3Vooo 3500 4000 3oo00 t/sec 3500 4000 t/sec Iv 100CEn a. a 3500 t/sec x 104 1u I o 4000 nn 1II I 50 0 3500 t/sec t/sec 4,.. ro I- 500)0 1 -L 3'6000 3500 4000 t/sec 1.1 = 3000 3500 t/sec Figure 6-52: Patient 6, Segment 3: Filtered Signals (L = 121) L=121 M- (CVPM, AT) (Spurious) M + (HR, RPP) Both rose because of decreased depth of anesthesia. mult(HR,ABPM, RPP) (Correct) mult(HR, CVPM, RPP) (Spurious) mult(HR, SV, CO) (Correct) 124 4000 11~~~~~~~~nL "I 00 - ---p t.0 54 A6 '300 4000 '3000 i/sec 3500 4000 i/sec -~r I--1uuu u 50 - I 1 ri 3500 > 500 A n 3'000 3500 4000 300 t/sec 3500 4000 t/sec x 104 4 IV o 0.5 05 'U n n 3000 3500 4000 3V000 t/sec 3500 I 4000 t/sec 4I AAAA - 0V I- f- 5000 A A 2 0 3000 3500 4000 3000 t/sec 3500 t/sec Figure 6-53: Patient 6, Segment 3: Filtered Signals (L = 241) L=241 M + (HR, CO) Both rose because of decreased depth of anesthesia. M+(RPP, VC) Both rose because of decreased depth of anesthesia. mult(HR, ABPM, RPP) (Correct) mult(HR, CVPM, RPP) (Spurious) mnult(HR, CVPM, VC) (Spurious) mult(HR, SV, CO) (Correct) 125 4000 _IIA r-.I. 53 ~- >. '- rI '3000 1 no u C, Ui n X 10 4000 _ *Il 3500 t/sec 3500 t/sec 50, I · 4000 o n rooo - 2 0 3500 t/sec 3000 - I -I r I3 I 50 D 40100 3 30oo 3500 /sec 4000 4 1 0 >0.5 (D 5 I!I 300oo 3500 t/sec 40100 i 3'000 3500 t/sec 4000 3500 4000 II nnn on 1I Ilil uuuq a. 0 ) I2r 0/l 500( D III i I ri i1 33000 3500 40 DO 3000 t/sec t/sec Figure 6-54: Patient 6, Segment 3: Filtered Signals (L = 361) L=361 M+(CO, RPP) Both rose because of decreased depth of anesthesia. AM+(HR, CO) Both rose because of decreased depth of anesthesia. M + (HR, RPP) Both rose because of decreased depth of anesthesia. mult(HR, ABPM, RPP) (Correct) mult(HR, ABPM, VC) (Spurious) mult(HR, CVPM, RPP) (Spurious) mult(HR, CVPM, VC) (Spurious) mult(HR, SV, CO) (Correct) 126 r 2.EIh f a ll. r m 53 I _ ob00 rrrI I 2 i 3500 t/sec 4000 U '3000 > 500 50 A\ _ 300 3500 4000 03_ '00 tsec I 3500 40 00 t/sec .n_ 0(D 4000 1000 IrV ··· .11 i 3500 t/sec I JUU I 02a. 1 r, _ X 104 1 0 0.5 0 n i I 300oo 3500 40100 3ooo00 3500 4000 t/sec t/sec 10000 a 5000 0. 5000 I= nIl _ 3000 w2 3500 ] 4C100 o 3000 t/sec 3500 t/sec Figure 6-55: Patient 6, Segment 3: Filtered Signals (L = 481) L=481 M+(HR, CO) Both rose because of decreased depth of anesthesia. AlM+ (HR, RPP) M+(HR, SV) M+(HR, VC) M+(CO,RPP) M+(CO,VC) M+(RPP, VC) M+(SV, CO) M+(SV,RPP) M+(SV, VC) mult(HR,ABPM, RPP) (Correct) mult(HR, ABPM, VC) (Spurious) mult(HR,CVPM,RPP) (Spurious) mult(HR, CVPM, VC) (Spurious) 127 4000 rnult(HR, SV, CO) (Correct) 128 ,Fat| I W 11 - a. 53 0 -;R 4 L %3oo l 3500 f 4000 3500 t/sec 4 IIIIAnn 100 I-r 50 _ C,, 0o 30 00 3500 500 I n 30oo 4000 t/sec 3500 4000 /sec 0I X 104 1 - JrI o 4000 t/sec 0. 5 O- 3000 A 3500 3'00 40(20 t/sec 3500 4000 t/sec 4I nnnn .. .. .I. Jl a aC 5000 2I D n' ..v I I I= 3o00 3500 3000 40( 20 t/sec Figure 6-56: Patient 6, Segment 3: 3500 t/sec Filtered Signals (L = 601) L=601 A+ (CO, RPP) Both rose because of decreased depth of anesthesia. Ml+(CO, VC) ATM+(HR,CO) M+(HR,RPP) MI+(HR, SV) M+(HR, VC) M+(RPP, VC) M+(SV, CO) M+(SV, RPP) M+(SV, VC) mult(HR, ABPM, RPP) (Correct) mult(HR, ABPM, VC) (Spurious) mult(HR, CVPM, RPP) (Spurious) mult(HR, CVPM, VC) (Spurious) 129 4000 rnult(HR, SV, C'O) (Correct) 130 6.2.4 Segment 4 The patient condition was steady. This period was uneventful. 200 10C I I am l- > 100 50 0, I J 4000 4500 5000 4o00 ,n J , t/sec A I 4500 -. . 5000 rnn ·. ' I -4V00 Ix 5000 4500 /sec /sec 105 J 0C. 0 .4 _c, 40oo 0 o or - .r - rlr -. -w .-n . I A I 5000 I---r I500 rr t.~'~ ·- · ·· vUUU I 10o nL a- 1 ,,, t/sec 20CII Z A R 4500 4500 t/sec 5000 4oo 4500 5000 i/sec x 1o04 I W1~tZ 4000 4500 I- 5 1 w 0 A VUoo 5000 4500 i/sec i/sec Figure 6-57: Patient 6, Segment 4: Original Signals 131 5000 60I nI IF - E 5 2C 4( I --~0boo 10TV · [ · Jt I l - 4500 t/sec 5000 4000 --- 1UUU | 500 A 5000 4u00v 4000 I 4500 5000 t/sec I Iv o 4500 t/sec X 104 I 0 5000 t/sec 50 >1' 4000 4500 1 r - 5 I'O. n u _ 4000 x rr 0- mr 4500 t/sec 4000 5000 nn III II 5000 4500 t/sec 5000 4500 t/sec 5000 J V -i1 w a I I. 40000 4500 t/sec 5000 4000 Figure 6-58: Patient 6, Segment 4: Filtered Signals (L = 61) L=61 nmult(HR, ABP, RPP) (Correct) mult(HR, SV, CO) (Correct) 132 In nJ 10 .1 X 0 Q- 40 m L- I'l '4000 ,..l 5 o 4500 5000 4'000 t/sec 5000 t/sec 1000 10 I I JV a Lo 500 VIj ,I ·I 4500 '4.000 5000 4 00 4000 t/sec x10 0o 1 0 0.5 5 nJJ J 4000 4500 5000 W l_ - 4Vo00 t/sec nnn JJ JJ 5000 4 f .J 4500 t/sec lU Er a: 4500 4500 5000 t/sec _ - 2 6000 A ~J nn JJ JJ 0 I. 4000 4500 t/sec n 4000 5000 4500 t/sec Figure 6-59: Patient 6, Segment 4: Filtered Signals (L = 121) L=121 mult(HR, ABP.M, RPP) (Correct) mult(HR, SV, CO) (Correct) 133 5000 nf, ~E 1 a 03 50 0 Ar! oo00 4500 5000 5v 5 n 4v000 t/sec j OU r70 70 4000 4500 5000 A,f~. 0 50 0 Ai---1 4500 4000 5000 t/sec t/sec 4 x 1C0 I 5000 100 lI c,\ I,, UV 4500 t/sec h'1 1 I I 0 ! Q A 'L -L_~ · v 4000 4500 -· - i4 _ 5000 4000 -boo t/sec anrar _ III II AAn 4000 5000 4500 5000 2 ~I a 6000 4500 t/sec U 4500 4000 5000 4000 t/sec t/sec Figure 6-60: Patient 6, Segment 4: Filtered Signals (L = 241) L=241 invderiv(ABPM, RPP) (Spurious) mnult(HR, SV, CO) (Correct) 134 55 - 2a rn50 4 0 Ar DO 401 Gu 5000 4500 t/sec 9 4000 5000 t/sec 1000 - 500 > 70 ni 40400 4500 5000 I 4600 t/sec b 4500 4500 5000 /sec x 104 m 1 _ 0 Or n_ -· 4 i 4vo00 4500 5000 _ 4000 t/sec ---- cc 1.'rI I 6000 Ann · n· · 5000 t/sec bUUU L 4500 II-w a _ 4000 4500 5000 t/sec 1 A r, 400ooo 4500 t/sec Figure 6-61: Patient 6, Segment 4: Filtered Signals (L = 361) L=361 No constraints were obtained. 135 5000 6r -1-1 f Cn 50 m o 73I 4[- 4500 t/sec 4o000 M . - f | 400 50)00 4500 t/sec 5000 4500 tsec 5000 4500 t/sec 5000 4500 5000 1000 _ I:3 70 500O- 4500 4')00 tsec 50)00 4000 x 104 1 0 . I! 4000 -· -Iboo .~~~~~~~~~~~~ 4500 t/sec __ 5000 4000 brAA I 4 a1 a 7000 ccN E fiAnn v0vv 4000 w 0 - 4500 t/sec II 5000 _ "4boo t/sec Figure 6-62: Patient 6, Segment 4: Filtered Signals (L = 481) L=481 M+(ABPM, SV) Both decreased overall. M+(RPP, VC) (Spurious) 136 rr 50 0 > 4I Ar 4500 t/sec 400 no 5000 400 4500 t/sec 5000 4500 5000 rIU --- I VUV ·- > 500 M70 an 4000 0 4500 t/sec x 1C)04 0 A - - 4000 n 4'000 5000 >"' " 4500 t/sec t/sec o _ ___________________________ _. 5000 4b00 4500 t/sec 5000 n 7000 nfAn oouu - 4000 4500 Au, r4 --- 4000 5000 -; 4500 t/sec t/sec Figure 6-63: Patient 6, Segment 4: Filtered Signals (L = 601) L=601 M- (ABPM, CO) Both decreased overall. AI (ABPM, SV) Both decreased overall. M+ (SV, CO) Both decreased overall. mult(HR, ABPM, RPP) (Correct) mult(HR, SV, CO) (Correct) 137 5000 6.2.5 Segment 5 The patient experienced low ABP post bypass due to poor cardiac performance secondary to a technically poor graft and possibly hypovolemia. Inotropic therapy (Dobutamine) was given which potentially caused the patient to develop myocardial ischemia. Blood infusion was performed to bring ABP back up. 100 uu Il 60 n ,:C 1 -- > 50 s-JJ 1000 1500 0 10(D0 2000 ~~~l~~~n~~. t/sec ~ J 2000 -TU r 100 .l I Z - II II '~UUU | 1500 2000 1000 1500 t/sec t/sec 00 x 105 , C) 0 ......... > 9 .1o000 L-L 1500 2000 1000 t/sec ' G 1. 0- Cr x 104 '.*W. r/ hI'I"f -E 2000 . TT .-" " -. 1500 t/sec 20 00 1500 20 00 -j r- nI _ 1.,,0 1000 I. W- nmnrrrr (c 1000 2 2000 1500 t/sec I- 31.5 ·· · 1500 1 1000 2000 t/sec t/sec Figure 6-64: Patient 6, Segment 5: Original Signals 138 10 50 -- a. 40 co 0 T- U 5 A .5{ _ I'l 1000 1 00 1500 2000 It 10oo0 t/sec 100 1000 I rr 90 T lrl l 1500 2000 i'6oo n x 104 l[ I 5 ·-z Ai i'66 1000 0o .5 n 1500 2000 1000 ~~n III 1· I 0a 8000 "1 II vv00 I 1000 201 O0 'A 5 1.2 0 1500 t/sec 1500 t/sec t/sec a- 2000 4I- 11 5 1500 t/sec t/sec 0o · > 500 1000 Iv 2000 t/sec B - 1500 2000 1 1000 A A~~~~~ 1500 t/sec Figure 6-65: Patient 6, Segment 5: Filtered Signals (L = 61) L=61 mnult(HR, ABPM, RPP) (Correct) mult(HR, SV, CO) (Correct) 139 20100 50 10 a- 40 m 0 nn L - - 1000 | 1500 2000 5 t n 1000oo t/sec 1000 90 > 500 __ 000 1000 2000 t/sec 100 IT 1500 orb Y· ! A\ I 1500 t/sec U-o 1000 2000 1500 2000 t/sec x 104 [ [ ,, 0 1 r 5 o > 0.5 n1__ 1000 *1' rs I [ [ 1500 2000 1000 t/sec 10000 .l F- f vvv 1000 I 1.4 2000 | .l L 1.2 D 1000 1 - 1500 t/sec 2000 t/sec aa- 8000 cc -n---II 1500 [I 1500 t/sec Figure 6-66: Patient 6, Segment 5: Filtered Signals (L = 121) LJ=121 mult(HR, SV, CO) (Correct) 140 2000 Iou lU 40 5 cn 1000 1500 t/sec 2000 > A1 1o00o 1500 t/sec 2000 1500 2000 cnn~~~~~~~~~~~-. DUU nn I'vv 9 90 an v00oo 1500 2000 1oo00 t/sec lU t/sec x 10 4 0 0.5 A5 7 n 100oo 1500 t/sec 2000 a 1500 t/sec 2000 1500 t/sec 2000 I1.L I UUUU a. 1o00o 8000 annn_ 1000 1.2 Jw . J 0 .44 1500 t/sec 2000 i000 Figure 6-67: Patient 6, Segment 5: Filtered Signals (L = 241) L=241 M + (ABPM, CO) Both dropped initially because of poor cardiac performance and hypovolemia, and started to rise following blood infusion. M+(ABPM, SV) Both dropped initially because of poor cardiac performance and hypovolemia, and started to rise following blood infusion. M+(SV, CO) mult(HR, ABPM, RPP) (Correct) mult(HR, SV, CO) (Correct) 141 50 - * 8 | 2 00 6 m 45 40* 1 0(30 1500 2000 1000 t/sec r-· II VVUU I 95 C) ,_ 1000 n E 1500 _ _ 1000 2000 t/sec 0 0.5 5 _ J 1500 t/sec 0 2000 10(00 4 [[ r · H. 1 H 1.2 -J QU _. 1000 I 1500 t/sec 1500 2C)00 t/sec 0_ Q 8000 mflnnn 2000 1 , n1 1i000 1500 t/sec x 10 4 [ I It ..nnnn I I [[II 2000 t/sec 100 cD 1500 2000 __ t 11 1000 1500 2C)00 t/sec Figure 6-68: Patient 6, Segment 5: Filtered Signals (L = 361) L=361 M+(ABPM, CO() Both dropped initially because of poor cardiac performance and hypov- olemia, and started to rise following blood infusion. 142 Q1 J'~lI 0(I 6 45 An 10400 A 1500 2000 1000 t/sec 2000 t/sec 50s )I lUU I I 1500 95 u, 9C600 1500 tsec 2000 A 1V00 1500 tsec 2000 x 10 4 .1 1 _ O o 0.5 0 w4 i _ nI __ i 1010o 1500 2000 1000 t/sec ---- ~ 1.3 X 8000 I-U 1.2 0 11 7nfn 1500 2000 t/sec YUUU l10o 0o 1500 - 'iboo 2000 t/sec 1500 t/sec Figure 6-69: Patient 6, Segment 5: Filtered Signals (L = 481) L=481 No constraints were obtained. 143 2000 rn n oU 0 E 45 An 02 ~~A I -- 1000 1500 _ 2000 _ 1000 _ _ _ _ 1500 _ 2000 t/sec t/sec 100 G _ 500 95 03 1000 1500 2000 1000 t/sec 1500 2000 t/sec x104 O O 0.5 n > A _ _ 1000 _ _ _ 1500 t/sec _ ----\ 2000 1000 1500 t/sec 2000 1500 t/sec 2000 nnn-.G 8000 7nAn I uuu 5 1.2 - 01 W -- 1000 1500 t/sec 2000 1 , . I 1000 Figure 6-70: Patient 6, Segment 5: Filtered Signals (L = 601) L=601 A/+ (SV, CO) Both dropped because of hypovolemia. MA--(HR, CO) HR increased both as a compensatory response to decreasing CO due to hypovolemia, and as a response to inotropic therapy. A1--(HR, SV) invderiv(SV, CO) (Spurious) mult(HR, ABPiM, RPP) (Correct) mult(HR, SV, CO) (Correct) 144 6.2.6 Segment 6 ABP gradually rose as a result of switching to another inotrope (Adrenaline) and blood infusion. 80 o m 60 10 I 3000 2500 2ooo00 3000 200 VIII L Cn 100 0 I A 2500 t/sec 2000 5 2500 t/sec 50CIvf- "1 II u2000 I 11 III t/sec 0 11 8 0 ArI -200o I11 1,10111. 11,II'' 00 2000 3000 'MMM1M1,p4pWiJU_,; 1111 11 HIM 2500 3000 t/sec x 1C5 .. 0 , 0 _ 1n 2000 2500 t/sec 2500 3000 3000 t/sec x 104 5_ , O1E a. a. 2 n' D1 -EI'' 2000 2500 3000 r.E,_ 2000 t/sec 2500 t/sec Figure 6-71: Patient 6, Segment 6: Original Signals 145 3000 8 TV! I.! 6 _ O_ ° 60 Ans . 2000oo 2500 3000 2500 t/sec 3000 t/sec In-I UUU III I-- I 95 )> 500 nI __ 2500 /sec R2 oo 3000 2o00 2500 3000 t/sec n x 104 Iv 00 01 5 I _ n1 (l 2ooo00 rx 1. -1. 2500 t/sec 3000 2%00 2500 3000 t/sec 104 - 1.6 . _ aL a- CE 1.4 1 nr _ 2000 uJ 2500 1 _ '2'00 3000 t/sec 2500 t/sec Figure 6-72: Patient 6, Segment 6: Filtered Signals (L = 61) L=61 mult(HR, SV, CO) (Correct) 146 3000 n OU a6 50 m co At A 2500 2boo 200 3000 t/sec 10 fnnn I UUU I M 95 > 500 uE vll I 2500 N2 oo 3000 2000oo 10 ,x 2500 3000 2500 3000 tsec t/sec 4 3000 t/sec I fV 2500 4 IJ 0 5 0 1 II 2'000 2500 _ 2o00 3000 t/sec 1.2 a. a. t/sec x 104 4 - ' 1.4 w 1 W. 0.8 - 2000 LU I 2500 t/sec _ '200 3000 2500 t/sec Figure 6-73: Patient 6, Segment 6: Filtered Signals (L = 121) L=121 mult(HR,ABPM, RPP) (Correct) mult(HR, SV, CO) (Correct) 147 3000 8 IL 50 m 6n O_ o6 2 dn 2500 20oo 3000 t/sec 4 3000 2500 3000 I---n I UUU nf IIB .vv CT I 95 > 500 n1 200oo 7 2500 t/sec 1\ _ .. 2500 3000 2o00 t/sec t/sec 4 x 10 11 00 AII f. 0 5 I 1 rI 2oo 2500 3000 200oo t/sec .2 aa- x 104 I w 1.6 1.4 0 .I n a I 2000 2500 3000 t/sec 1 Ir 2500 3000 t/sec v4 - 20(30 2500 t/sec Figure 6-74: Patient 6, Segment 6: Filtered Signals (L = 241) L=241 mult(HR,ABPM, RPP) (Correct) 148 3000 60 a co 50 2 6- _ An 200ooo - o - A I Jl z 2500 t/sec 3000 2000 1000 95 > 500 nn 10 o 3000 2500 t/sec 3000 2000 2500 3000 t/sec 5 I n _ I 2000 x 10o -1 · 3000 · i 1.4 1 _ _1_ n U.o0 2500 t/sec 1 .z cc | I | i-L 2500 t/sec x 104 n 2000 aa- [ rI __ 2000 3000 t/sec 100 YI 2500 w 2000 - 0 s 2500 1 F _ 200 3000 t/sec 2500 t/sec Figure 6-75: Patient 6, Segment 6: Filtered Signals (L = 361) L=361 M+(SV, CO) Both rose after blood infusion. deriv(SV, CO) (Spurious) rmult(HR, SV, CO) (Correct) 149 3000 n, Du r Tc "7 - a 50 6 0 n4F - 2500 20 00 2600 3000 t/sec 14I*nfr I VUU I 95 > 500 on I! )00 0 2500 I 5 _- In _ lU 3000 ! 2'o00 t/sec 1 2500 n 20ooo 3000 'oo 2500 3000 t/sec X 104 I- . ·I I- 1.4 .U 1 na 3000 Vsec t/sec aa: 2500 4 200 1. 11J_ 3000 t/sec 100 ,x10 2500 uJ w 0 2500 t/sec 14 e l- 2000 2000 3000 2500 t/sec Figure 6-76: Patient 6, Segment 6: Filtered Signals (L = 481) L=481 M+(SV, CO) Both rose following blood infusion. deriv(SV, CO) (Spurious) mult(HR,ABPM, RPP) (Correct) mult(HR, ABPM, VC) (Spurious) 150 3000 -- %1- UV 50 0 r,r An Y000 2500 t/sec 3000 2o00 1 uuu IUU 2500 t/sec 3000 2500 3000 l- > 500 95 on - l A ! $ 2-000 2500 3000 2'000 Vsec X 10 Vsec 4 IV 0 Ai A n 2'000 2500 3000 2'000 t/sec 1.2 x104 3000 1.6 a1 w 0.800 2000 2500 t/sec 2500 1.4 1.2 3000 t/sec t/sec Figure 6-77: Patient 6, Segment 6: Filtered Signals (L = 601) L=601 M-(HR, AT) HR decreased following blood infusion. Lagging effect of vasoconstriction caused AT to increase. M- (HR, SV) HR decreased and SV increased following blood infusion. M+(SV, AT) deriv(ABPM, RPP) (Spurious) deriv(ABPM, VC) (Spurious) deriv(SV, CO) (Spurious) mult(HR, ABPM, RPP) (Correct) mult(HR, ABPM, VC) (Spurious) mult(HR, CVPM, VC) (Spurious) mult(HR, SV, CO) (Correct) 151 Chapter 7 Discussion The goal of this thesis is to develop a system for learning qualitative models from physiological signals. In Chapter 1 we mentioned two potential applications for such a system. First, the system could be a useful tool for the knowledge acquisition task of inducing useful models from vast amounts of data. Second, the system could be incorporated into an intelligent patient monitoring system to perform adaptive model construction for diagnosis in a dynamic environment. In this chapter, we evaluate the performance of the system in knowledge acquisition and identify sources of error. We then examine its applicability in diagnostic patient monitoring. 7.1 Validity of Models Learned The results in Chapter 6 show that reasonable qualitative models can be learned from raw clinical data. Model constraints in our "gold standard" model constructed in Section 4.2 and other useful constraints showed up repeatedly in the models learned from our clinical data. These include: * constraints valid in general such as mult(HR, SV, CO) and mult(HR, ABPM, RPP). * constraints valid in specific patient conditions, possibly representing compensatory mechanisms, such as M- (HR, CO) and M- (CO, AT) in hypovolemia. * constraints valid under the effect of certain drugs. For example, M+ (SV, CO) showed up in patients on beta-blockers because of their steady heart rate, and M+ (ABPM, AT) showed up in patients with an increased dosage of GTN causing vasodilation. 152 7.1.1 Model Variation Across Time As discussed in Chapter 3, GENMODEL learns a qualitative model by creating an initial search space of all possible QSIM constraints, and successively pruning inconsistent con- straints upon each given system state. Therefore if the system changes within the modeling period (in our case 16.7 minutes) resulting in a different underlying model, neither the old model nor the new model may be obtained. Constraints in the old model are pruned be- cause they are inconsistent with the states after the system change. Constraints in the new model are pruned before the system change because they are inconsistent with the previous system. This may explain cases when we obtain very few or no model constraints. For example, if a patient is previously stable with an increasing relationship between the heart rate (HR) and the cardiac output (CO): M+(HR, CO) but develops hypovolemia in the middle of a modeling process, resulting in a decreasing cardiac output and a compensatory mechanism involvingan increasing heart rate, the new valid constraint is: M-(HR, CO) But this will not appear in the final model because at the onset of hypovolemia, this constraint has already been pruned by GENMODEL according to states corresponding to the previously stable condition. Furthermore, the previously valid constraint M+(HR, CO) will be pruned because it is now inconsistent with the system states corresponding to hypovolemia. 7.1.2 Model Variation Across Different Levels of Temporal Abstraction The models learned in Chapter 6 varied across different levels of temporal abstraction, represented by the filter length L. For example, constraints which involve the skin-to-core temperature gradient AT representing the level of vasoconstriction in the body, generally appeared only under large values of L, i.e. in coarser time scales. This means the response of AT generally lags behind the responses of other parameters. This may be due to the considerable heat capacity of the body causing a delay in the measurable effects of 153 vasoconstriction. In general, we observe that fewer model constraints were learned with decreasing L or finer time scales. This can be due to the following reasons: * As discussed in Section 5.2.4, smaller values of L and therefore smaller values of a correspond to larger cutoff frequencies in the lowpass Gaussian filters, and larger band- widths in the bandpass filtering operation equivalent to the cascade of the Gaussian filter with the differentiator. This reduces the amount of noise rejection achieved, and results in noise sensitivity problems in detecting zero crossing points and therefore less accurate segmentation. This additional amount of noise may have caused correct constraints to be pruned, resulting in fewer or even no constraints left in the final model. * Smaller values of L correspond to faster processes which may have more dynamic models. As discussed in Section 7.1.1, a system change within a modeling period can cause constraints belonging to both the previous and the current model to be pruned, resulting in a smaller model or even one with no constraints. 7.1.3 Model Variation Across Different Levels of Fault Tolerance We observe that in general the size of the model learned increases with increasing levels of fault tolerance. A fault tolerance level of q7means that GENMODEL allows for inconsistent states up to a fraction a constraint. of the total number of states in the system behavior before pruning Therefore with larger 7, fewer constraints will be pruned and the resulting model will contain more constraints. An indication of r7being set too high is that conflicting constraints start to appear. For example, in Patient 5 with L = 61 (Figure 6-30), both M+(CVPM, AT) and M-(CVPM, AT) appear in the model learned. This is because both CVPM and AT are relatively steady and contain only few inc and dec segments which distinguish between the M+ and M154 constraints. Within a high level of tolerance, the distinction is obscured. 7.1.4 Sources of Error False Positives In the models learned, we observe that spurious constraints sometimes appeared in the resulting model. For example, in Patient 1 (L = 601), we obtained the spurious constraint: inv.deriv(ABPM,RPP) This can be due to several possible reasons: * The waveforms are relatively smooth with few critical points. This results in a system behavior with few states, corresponding to few examples for learning. Now it is relatively probable that these examples are consistent with the incorrect constraint. For instance, if whenever ABPM decreases, RPP is positive, then the above incorrect invderiv constraint will be learned. * The level of fault tolerance is set too high resulting in incorrect constraints not being pruned. False Negatives We observe that even constraints that are generally valid in all conditions, such as mult(HR, SV, CO) did not appear in every model learned. There are several possible reasons for this: * Since few states are available in the data segment resulting in few examples for learning, if these examples are corrupted by noise, the correct constraint will be pruned. * Values corrupted by noise are recorded as corresponding values by the system. This problem is explained further in Section 8.1.1. * The level of fault tolerance is set too low resulting in correct constraints being pruned. 155 Landmark Values Temporal abstraction refers to how close two times have to be before we label them as the same distinguished time point. Similarly we have to decide how close two function values have to be before we label them as the same landmark value. If the tolerance is set too low, we may amplify trends of relatively steady signals. This is the case in the heart rate signals of Patients 4 and 5 (Figures 6-23 and 6-30) which are relatively steady due to the effect of beta-blockers. The fluctuations within 2-3 beats per minute are amplified into a series of inc (increasing) and dec (decreasing) segments. The whole segment might well have been labeled as std (steady) if we had set the tolerance appropriately. 7.2 Applicability in Diagnostic Patient Monitoring From the models shown in Chapter 6, we see that constraints of models learned do track changes in patient condition over time. For example, the following changes were tracked: Compensatory mechanisms during shock e.g. the constraints M-(HR, CO) and M- (CO, AT) learned when the patient experienced hypovolemia. Effects of drugs e.g. the constraint M+ (SV, CO) tracked the effect of beta-blockers because of the patient's relatively constant heart rate, and the constraint M+ (ABPM, AT) tracked the effect of an increased dosage of GTN causing vasodilation. 7.2.1 A Learning-Based Approach to Diagnostic Patient Monitoring Since model constraints learned track patient condition over time, we might be able to build a diagnostic patient monitoring system based on our learning system. The patient monitoring system continually learns models from patient data and detects changes in the models learned. Diagnoses are made based on these changes. This learning-based approach to diagnostic patient monitoring is summarized in Figure 7-1. The traditional history-based approach to diagnostic patient monitoring goes in the opposite direction. It generates histories based on different models. These histories are matched with the patient data. Diagnoses are based on models corresponding to histories that best match the patient data. This approach can be achieved by a hypothesize-testrefine cycle as shown in Figure 7-1 [6]. 156 Learning-based History-based Approach: Approach: Diagnosis Refine Figure 7-1: Two approaches to diagnostic patient monitoring. In the learning-based approach, models are continually learned from the patient data. In the history-based approach, a hypothesize-test-refine cycle is used to generate models that best match the patient data. In each approach, diagnoses are made based on the current model. The learning-based approach may be more efficient since the hypothesis model is generated directly from the patient data and there is no need for a hypothesize-test-refine cycle. 7.2.2 Generating Diagnoses Based on Models Learned We need to be careful in making diagnoses based on models learned. The same model constraint can correspond to different diagnoses in different contexts. For example, * The constraint M + (HR, CO) can correspond to a healthy person performing physical exercise, when the rise in heart rate increases the cardiac output. It can also correspond to a hypovolemic condition when the compensatory effects of increased heart rate, increased myocardial contractility and vasoconstriction successfully increase the cardiac output, but the patient is still in hypovolemia. * The constraint M- (HR, CO) can correspond to a person in hypovolemic shock when the increasing heart rate is incapable of compensating for the decreasing cardiac output. The same constraint can also be obtained when blood infusion is performed in response to the hypovolemia. In this case, the cardiac output will increase because of 157 the blood infusion, and the heart rate will decrease to normal. Therefore different patient conditions may result in the same model constraint. However, this may not be a problem for applying the learning approach to diagnostic patient monitoring for the following reasons: * Most changes in patient condition will involve changes in many parameters and in multiple constraints. * If only few parameters are monitored, a semi-quantitative representation may be used. This may allow more fine grained distinctions between patient conditions. 158 Chapter 8 Conclusion and Future Work This thesis describes work in learning qualitative models from clinical data corrupted with noise and artifacts. The learning system consists of three parts: Front-end Processing removes artifacts and smooths the signals according to the level of temporal abstraction desired. Segmenter segments the signals at critical points producing a qualitative behavior of the system. GENMODEL learns a qualitative model from the qualitative behavior. We tested the system on clinical data obtained from six patients during cardiac bypass surgery. Useful model constraints were obtained, representing the following categories of information: General Physiological Knowledge e.g. mult(HR, SV, CO), mult(HR, ABPM, RPP). Compensatory Mechanisms During Shock e.g. M-(HR, CO) and M-(CO, AT) representing the mechanisms of tachycardia and vasoconstriction to compensate for a hypovolemic shock. Effects of Drugs e.g. M+(SV, CO) representing the effect of beta-blockers producing a steady heart rate, and M+(ABPM, AT) representing the effect of vasodilators such as GTN causing vasodilation which decreases both the arterial blood pressure and the skin-to-core temperature gradient. 159 The learning system might be used in a diagnostic patient monitoring system to learn models from patient data. Based on these models, diagnoses of the patient condition can be made. This provides a learning-based approach to diagnostic patient monitoring, which is different from the traditional history-based approach. The learning-based approach may be more efficient because it avoids the hypothesize-test-refine cycle. 8.1 Future Work Promising directions for future work include: Front-end Processing * Investigate wavelet transforms for multiresolution signal analysis in achieving different levels of temporal abstraction in the resulting qualitative behavior [35]. GENMODEL * Investigate more robust ways of dealing with data corrupted by artifacts, e.g. examine the applicability of results in PAC learning from data corrupted with attribute noise. * Investigate methods of using corresponding value information in noisy data. (See Section 8.1.1.) Patient Monitoring * Investigate more precisely the relationship between levels of temporal abstraction and delays in physiological responses. For example, the system can be tested in the domain of pharmacokinetics to relate the level of temporal abstraction at which appropriate constraints are learned to the onset of action of different drugs. * Investigate the context information necessary for accurate diagnoses based on learned models. 8.1.1 Using Corresponding Values in Noisy Learning Data In Section 3.4.4 we mentioned that since we are learning from noisy signals, we have to allow for errors in the qualitative state history produced from the signals by the front-end 160 processor and the segmenter. We accommodated this noise in the qualitative behavior by introducing fault tolerance into GENMODEL. Specifically, we set a noise level r7 to a fraction of the total number of qualitative states, such that a constraint is not pruned until it has failed for this many states. The noisy learning data introduce another difficulty in GENMODEL - that of locating corresponding values. When two or more functions reach landmark values in the same qualitative state, these landmark values are noted to be corresponding values. However, if the learning data is noisy, we do not know which states are valid and which are not. If we continue to label all landmark values which coexist in the same state as corresponding values, we will come up with a number of incorrect corresponding value sets. These incorrect sets will consistently cause constraints that check for corresponding values (i.e. M + , M-, add, mult) to fail and ultimately be pruned, even if they are valid constraints. This is in fact the approach currently taken in our learning system, and can be regarded as a source of error. One simple solution to this problem is to give up corresponding value checking and simply rely on direction-of-change consistency for checking monotonic constraints. Without the magnitude information provided by the corresponding values, a certain level of performance can still be achieved, as in attempts of Bratko et al [4] and Varsek [38] in learning the U-tube model. However, omitting corresponding value checking results in much weaker forms of qualitative constraints, which no longer correspond to Kuipers' original definitions in [20]. For example, without corresponding values, the monotonic constraints M + and .A1- no longer require a one-to-one mapping between values of their function arguments, as they do before (described in Section 2.1). Now if we have M + (a, b), if a increases steeply to the same value a(1l) in two different states, b can increase steeply to a large value b(l) in one state, and increase only slightly to a small value b(2) in the other state, as long as it increases at all. Magnitudes no longer need to correspond. (Figure 8-1) Using landmark intervals to represent corresponding value information in noisy learning data In the previous section we mentioned that exact corresponding values may be difficult to obtain from noisy learning data because we do not know which states are valid and which are not. Here we propose using landmark value intervals together with landmark values to allow 161 a(1 ne I b(1 b(2 t(O) t(1l) t(2) t(3) t(4) Time Figure 8-1: The two functions a and b shown obey direction-of-change consistency for the qualitative constraint M+(a, b), but not magnitude consistency. for an error tolerance in the corresponding values obtained. We show that this is analogous to using a boundingenvelopeto represent a monotonic function in semi-quantitative methods [22, 16]. Error tolerance in corresponding values can be achieved by allowing a point to cor- respond to a region which represents an error band within which the true corresponding landmark value may exist. A landmark value a(ti) corresponds to a closed landmark value interval [b(t2 ), b(t3 )] if and only if there exists a corresponding pair between the landmark value a(ti) and a value within the closed interval [b(t2 ), b(t 3)]. Furthermore, we assume that the noise in the learning data is of a reasonable magnitude, such that a value can be mislabelled by a range of at most one landmark value. Therefore, a landmark value interval can extend by at most two landmark values (one landmark value greater than and one less than the true value). Searching for corresponding sets is similar to before. We proceed down the qualitative state history, looking for landmark values reached in the same state. But now, instead of simply recording each set of values, we check first to see if one of the corresponding values already exists within a corresponding set. If this is the case, and if the other new corresponding value is adjacent to the existing corresponding value in the landmark list, then we extend the existing value into an interval to include the new value. However, if the 162 new corresponding value differs from the existing corresponding value by more than one landmark value, we can conclude that a one-to-one mapping does not exist between the two functions. Therefore, we can prune all monotonic constraints (M+ and M-) between the two functions, and form a new set of corresponding values for constraint checking with add and mult, since add and mult do not require a one-to-one mapping between values of its argument functions. For example, if we have corresponding values (pl, qi), and we come across the state QS(p,t3) = <p, std > QS(q,t3) = < q2, std > Then if q and q2 are adjacent to each other in the landmark list, then we have the new corresponding pair (P1, [ql, q2]) assuming q < q2. However, if ql and q2 are not adjacent to each other in the landmark list, then we can prune the followingconstraints M+ (p, q) M- (p, q) and form the new corresponding value set (pl, q2, .. ) for constraint checking with add and mult. New constraint rules can be added to accommodate intervals in corresponding pairs. For example, a rule similar to Kuipers' Proposition B.1 in [20] can be added as follows: Suppose M + (f, g), with corresponding values (p, [ql, q2]), and QS(f, t1 , t2 ) = < (p,p'), dec >, QS(g, t1 , t 2 ) = < (q, q'), dec >, where q E [ql, q2] Then one of the following two possibilities must be true at t 2: (1) f(t 2 ) =p, g(t2) E [ql, q2]; (2) f(t2) > p, g(t2) > q1. 163 b(t) estimated function upper envelope - boundary / lower envelope , boundary b(l). b(O) ----a(O) a(t) Corresponding value pair: (a(O),[b(O),b(1)]) Figure 8-2: Incorporating intervals into corresponding value sets is analogous to adding a bounding envelope to monotonic functions. 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