By Gilbert R. Hillman
Those who were in high school after the mid-1960s were probably exposed to the “New Math,” which includes the basics of set theory. Treating objects as belonging, or not belonging, to a set provides a clear way to express and visualize logical ideas. For example, “A is a member of set B, set B is contained in set C, so A is a member of set C” is an easy deduction to understand and to visualize with the usual “Venn diagram” of overlapping circles.
But in 1965, Lotfi Zadeh, a computer scientist at the University of California, Berkeley, introduced a new idea: Suppose an individual item didn’t have to either wholly belong or not belong to a set, but could partially belong? He called a set that can have partial memberships a “fuzzy set,” and he developed the mathematics that lets us define how much an item belongs to a set and what happens when fuzzy sets are combined. Now instead of saying”“A belongs to C,” we can talk about how much it belongs to C. The process of making decisions using fuzzy sets is called “fuzzy logic.” The term doesn’t mean “logic in which we don’t know quite what we are doing” (though it is used jokingly that way sometimes). Rather, it means “logic using numbers and rules that have a known amount of uncertainty built into them.”
Zadeh’s mathematical idea is easy to apply to the biomedical world because so many concepts are not exactly defined; we use this kind of thinking all the time. Usually we use familiar words as the names of fuzzy sets. Suppose we want to predict the chance that someone will be diabetic. We might say, “Overweight people are at risk of diabetes.” Now we have mentioned two fuzzy sets (“overweight people” and “[people with] diabetes”) and a fuzzy logic relationship (“at risk of”).
“Overweight” is fuzzy because it is a matter of degree, but we can describe quite accurately what we mean by that, in terms of a person’s body mass index (BMI, defined as the person’s weight in kilograms divided by height in meters squared). We make a table, or a graph, showing what BMI value corresponds to what degree of overweight. It’s not a straight-line relationship, because anyone with BMI 25 or less isn’t overweight at all (their membership in the “overweight” set is zero), and anyone with a BMI of 40 or more is definitely overweight (their membership is 1). For BMI values in between, there would be memberships having fractional values between 0 and 1. This table or graph, showing how membership in a fuzzy set is determined by some piece of information, is called a “membership function.” “Having diabetes” also can have degrees of severity, so that is fuzzy too, and the relationship “at risk of” can take into account the fact that the more overweight you are, the more severe the diabetes may be.
By using meaningful words to name the fuzzy sets, the process is easy to understand and can be built up intuitively. What about body-builders who don’t have increased diabetes risk but have a large BMI due to big muscles with little fat? We might create a fuzzy set, “bodybuilder”; membership might be based on numbers of pushups or percent body fat. Then we make the rule that the degree of “overweight” is determined by the combination of BMI and not being a bodybuilder. The way to calculate “and not” is part of fuzzy-set mathematics, and is simply arithmetic. But the sets that we choose to make, and the rules for combining them, contain our special knowledge of the relationship between weight and diabetes. This little calculation is on its way to becoming an expert system for predicting risk of diabetes. It is easy to add as many more rules and conditions as we can think of to make the system as accurate as possible. How about a “family history of diabetes” fuzzy set? One distant relative = low membership; both parents = high membership. The combination of “overweight”and (which is a fuzzy set operation) “family history” is probably a more reliable predictor of diabetes than either one alone.
Anatomy is a fine example of a property that is fundamentally fuzzy. Everyone has more or less the same parts, but their size, shape, and position vary among individuals, and some parts are more variable than others. The changes that happen with disease also are variable. If you wanted to make a computer system that helped with diagnosis, or that measured the extent of a disease or the effect of a drug, the computer system would need to use information about normal and abnormal variability in whatever is being measured, and it is easy to accomplish this using fuzzy sets.
My collaborator Thomas A. Kent (now of Baylor College of Medicine in Houston) and I have been interested in building computer systems that can make measurements of disease- or injury-related changes in the brain when seen in MRI. MRI images have a scale of brightness from black to white. A radiologist is trained to recognize, for example, brightness where the image should be dark, and to interpret that as a particular kind of damage. Probably no computer system can outdo a radiologist at seeing that there is something there that shouldn’t be there. But for some purposes, for example a trial of a new treatment, it may be important to determine not only whether it’s there, but the size or severity of the abnormality, so that the effect of treatment can be measured.
"Anatomy is a fine example of a property that is fundamentally fuzzy."
Humans are very good at recognizing by eye what they are looking at, but computers are better at counting and measuring. Fuzzy logic is very helpful in guiding the computer to find the right thing to measure. For example, suppose we want to measure the extent of subcortical (near the center of the brain) strokes, as opposed to strokes in the cortex of the same brain. The stroke-affected areas look brighter in MRI than the rest of the brain. We make a “bright” fuzzy set, defined by comparing each point with the average intensity of the brain in each image, and a “near the cortex” fuzzy set, defined by the distance from the surface of the brain. Then we could make a “subcortical stroke” set, defined as “bright” and not “near the cortex.” This logic captures some of the neuroradiologist’s understanding of how you know which thing to measure. Once the right bright blob is found in the MRI image, the computer measures its size and brightness, and because “subcortical stroke” is a fuzzy set, it can handle structures that have varying brightness and blurred edges.
All these calculations might be accomplished with conventional statistical methods if we preferred to do so. But fuzzy logic is an intuitive way to solve measurement problems, using computer programs that are easy to construct and to understand, and that run quickly in routine use.