Valid Relevance in Medical Practice: The Inadequacy of the Linear Model of Health and DiseaseThe Weisenfeld Lecture

Presumably there is a real world out there. That is, existence, reality, does not require the presence of human beings. Throughout history, some individuals have tried to describe that reality. Some have done it amazingly well—those like Hippocrates, Vesalius and Shakespeare, or Kepler and Copernicus, as the latter two charted the course of the planets. Thinkers have tried to understand those observations—some, like Socrates, Montaigne, and Wittegenstein primarily using words. Others, such as Euclid and Newton chose mathematics, the latter to understand the careful observations made by Kepler and Copernicus on how the planets move. Newton solved the two-body problem, explaining mathematically the orbit of the moon around the earth. Fundamental to that Newtonian conception of existence is linearity. Linearity does not mean a straight line, but rather a continuous line. In an arithmetic system, linearity can be exemplified by a series, such as 1, 2, 3, 4, etc., or by such numbers in a squared series or a cubed series, or by many other convolutions and patterns. Linearity refers to any continuous line that can be validly extended based on known data points. That belief not only was used to explain mathematical problems, it also became generalized as a basic conception of reality. 

Today, in medical practice, we are driven by this same metaphysics—specifically, a belief in a world ruled by linear laws. We believe in, and we try to practice, evidence-based research and care, with conclusions growing out of the belief that valid data points are linear. If those data points are not linear, then they are believed to be (1) invalid, (2) confounded by other factors, or (3) outliers—that is, not able to be understood in any linear system. Outliers in any data set are currently routinely ignored, or more precisely, eliminated. The smoothed data set, one that then conforms to a linear model, can be analyzed using the basic methodologies of the Newtonian concept of reality. 

Good observers have long been concerned that the Newtonian mathematical explanation of existence does not include everything. This awareness resulted in the inclusion in most cosmologies and mythologies of acceptance of the extraordinary or the “supernatural.” There was, of course, the linear Apollo, rising every day and driving his chariot across the sky in an orderly fashion. But there was also the unpredictable Dionysus, inebriated with wine or joy. And there are also concerns in hard science. For example, Newton's system could be used to calculate the motions of the moon around the earth, the so-called two-body problem, though this required many correction factors. But neither Newton, nor anybody else, could or ever will solve, by using linear mathematics, the three-body problem—specifically, how the moon orbits around the earth in the presence of another mass, such as Jupiter or the sun. Solutions required the construction of a “new” conception of reality, a nonlinear conception, to explain the motion of the moon around the earth in the presence of Jupiter. The discovery of “alternative mathematical systems” by Poincaré, Einstein, Heisenberg, Mendelbort, and others ¹ led to the world of quantum physics and fractals, systems that are noncontinuous and nonlinear, but not random. 

This report advances the thesis that our current linear conception of biology and of health and disease is inadequate to explain many aspects of biology, health, and disease. It also has a second and related goal, specifically to suggest that without a dogged and continuing attention to the basic purposes of medical care, those purposes are not likely to be advanced. (More on that later.) 

Currently, and for many years, we have relied on a linear conception of reality for understanding biological phenomenon and for diagnosing and treating patients. However, I hope to show here what most of us know intuitively: It does not work adequately. We constantly need and use “smoothing factors” to eliminate the variability that is found in every system and every study. Probably, the major systematic error—systematic bias—of modern medicine is an unshakable belief in the validity and adequacy of a linear conception of reality. Like those trying to explain the three-body problem using a Newtonian conception of reality, we today force the bell-shaped curve or another linear curve onto our data, even when it does not fit. Here, I will use the field of glaucoma as a case to discuss how a linear interpretation of reality is inadequate. 

Glaucoma is unquestionably related to the level of pressure within the eye—that is, the so-called intraocular pressure. Armaly ² (and others) showed an apparently linear relationship between the development of visual field loss and the level of intraocular pressure (Fig. 1). Many years before, careful observers had noted that eyes that felt hard often demonstrated patterns of damage that fitted into the disease that was called “chronic glaucoma.” Scientists then developed methods to measure eye pressure more easily, though, interestingly, not more validly, than using fingers. These instruments were used to survey vast numbers of people, producing data on intraocular pressure that showed that pressure was distributed in a bell-shaped curve, skewed slightly to the right, that is to the side with higher eye pressures (Fig. 2). The validity of both curves, that is, the bell-shaped curve of distribution of intraocular pressure in a large population and the curve showing that with increasing pressure there was an increasing likelihood of developing visual field loss, seemed indisputably valid. The next obvious step was to combine them, resulting in the belief that pressure above a particular level was going to be associated with the development of visual field loss due to glaucoma. 

Figure 1.

 

Relationship between the likelihood of a visual field defect being present and the level of intraocular pressure. Modified from Graham P. Epidemiology of chronic glaucoma. In: Heilmann K, Richardson KT, eds. Glaucoma: Conceptions of a Disease, Pathogenesis, Diagnosis, Therapy. Stuttgart, Germany: Georg Thieme; 1978.

Figure 2.

 

Distribution of intraocular pressures in a population aged 40–73. Seventy-five years living in Ferndale, Wales, showing an excess of high intraocular pressures. A standard distribution is shown in the shaded bars. From Graham P. Epidemiology of chronic glaucoma. In: Heilmann K, Richardson KT, eds. Glaucoma: Conceptions of a Disease, Pathogenesis, Diagnosis, Therapy. Stuttgart, Germany: Georg Thieme; 1978.

Glaucoma is a terrifying condition. It is the single greatest cause of irreversible blindness in the world. ³ Blindness is feared universally to the same degree as cancer and other devastating diseases. Melissa and Gary Brown (Brown et al. ⁴ ) have shown that people equate blindness with the disability caused by a severe stroke. 

Treatment for glaucoma is designed at lowering the pressure within the eye, because it has been definitively demonstrated that such pressure lowering can preserve vision. But in whom does the pressure need to be lowered, and how much, and for how long? 

In 1960 when I was a resident, those questions were easily answered. The average intraocular pressure was determined by excellent population studies to be 15 mm Hg, with a standard deviation of 3 mm Hg in most populations. The upper limit of normal, then, was 21 mm Hg. According to standard statistical therapy, based on linearity, individuals with intraocular pressures greater than two standard deviations above the mean are assumed to be sufficiently abnormal that there is a 95% chance that they will have glaucoma. Consequently, individuals with pressures above 21 mm Hg were said to have glaucoma and were advised to have treatment because they “had glaucoma.” 

For those familiar with current concepts of glaucoma, it may be surprising to consider that diagnosing glaucoma was that “simple.” However, in 1960, it was. Specifically, those with pressures over 21 mm Hg had glaucoma. Because “glaucoma” causes blindness (I will discuss this fallacy later), those with IOP over 21 mm Hg were, as I have already mentioned, advised to have treatment. Because “glaucoma causes blindness,” the fact that the treatments designed to lower intraocular pressure sometimes caused retinal detachment, blurring of vision, or, occasionally, death, was acknowledged, but considered “collateral damage”—that is, the cost of preventing blindness. The fact that these treatments appeared to keep around 95% of individuals from developing visual field loss was considered proof that (1) glaucoma was intraocular pressure over 21 mm Hg and (2) that treatments that were being used worked. 

Some careful observers, however, noted that, on the one hand, not all those with elevated intraocular pressure developed glaucoma, and, on the other hand, some of those who had glaucoma had pressures in the “normal range.” Understanding this nonlinearity was difficult. New terms were developed such as, “pseudoglaucoma,” or, amazingly, “normal-pressure glaucoma” to describe those with pressures in the average range who developed glaucomatous visual field loss. At the other end of the spectrum, the term “ocular hypertension” was developed to characterize those with statistically elevated pressures but no visual field loss. 

Studies by Armaly ⁵ in the United States, Bankes et al. ⁶ in England, Linnér and Strömberg ⁷ in Scandinavia, and Schappert-Kimmijsen ⁸ and others demonstrated that the percentage of individuals whose pressures were above 21 mm Hg and who developed visual field loss within 5 years, was actually around 5%, not 100%. It was not surprising, then, that around 95% of those being treated for “glaucoma” were “successfully treated,” in the sense that they did not have progressive glaucomatous visual field loss; 95% of those being treated for glaucoma would not develop visual field loss, because they did not have glaucoma—that is, even without treatment, they would not have developed field loss. Next, Hollows and Graham ⁹ showed that around one third of those with glaucoma had pressures below 21 mm Hg—that is, in the “normal” range. 

These observations were deeply threatening to ophthalmologists. They were threatening because discarding a long-standing belief is upsetting. Ophthalmologists in the 1960s all “knew” that glaucoma was a disease of elevated pressure within the eye. In fact, most of us still “know” that, judging by how we act. Even today the success of a surgical procedure for glaucoma is usually defined as a procedure that lowers intraocular pressure below 21 mm Hg. A recent survey by Cantor (personal communication, 2005) showed that approximately two thirds of ophthalmologists currently use intraocular pressure as the basis for diagnosing glaucoma or for changing treatment. There was a second, even more threatening aspect to the new findings about the relationship between intraocular pressure and development of glaucomatous visual field loss. In actuality, these findings were not new, but they had been discredited because they did not fit into a linear system of health and disease, of reality. The second reason, was, of course, that these findings challenged the fundamental conception of existence. If it is certain that there is a direct relationship between intraocular pressure and “getting glaucoma,” then there must be no threshold defined by statistics to separate “glaucoma” from “no glaucoma” on the basis of the pressure within the eye. If there is no threshold, then, the rules of linear cause and effect are invalid. That is, this new way of thinking meant that the existing conception of reality was invalid. If one were to accept a nonlinear way of relating variables to each other, understanding what was happening in glaucoma became much more complex. However, the change did not mean to suggest that intraocular pressure is unrelated to the development of glaucoma. Rather, it was that the relationship between pressure and development of glaucoma is not linear. 

The Advanced Glaucoma Intervention Study (AGIS), ¹⁰ Collaborative Initial Glaucoma Treatment Study, ¹¹ Early Manifest Glaucoma Treatment Study, ¹² European Glaucoma Treatment Study, ¹³ and Glaucoma Laser Trial ¹⁴ all showed that it is possible to prevent or slow down the development of continuing visual field loss in individuals with glaucoma by lowering their intraocular pressure. The graph in the publication of the results of the Advanced Glaucoma Intervention Study (AGIS) ¹⁰ has become an icon, probably familiar to most ophthalmologists around the world (Fig. 3). It seems that lowering the intraocular pressure to a particular level prevents the development of glaucoma in all individuals. Recall that between 1960 and 2000, concepts of glaucoma were changing markedly. The belief that 21 mm Hg was a magic number allowing the diagnosis of glaucoma was no longer considered valid, and there was confusion over whether intraocular pressure played a role at all. The AGIS graph, then, provided, for many, reassurance that there is a linear relationship between intraocular pressure and visual field loss. It was just that the level of pressure that was believed to be damaging was lower than had been thought. In clinical practice, then, the “acceptable” or “target” pressure was lowered from 21 to 12 mm Hg. Those viewing the AGIS graph were likely to conclude that all those whose intraocular pressures were consistently below 12 mm Hg remained stable; in contrast, those with pressures in the range of 20 mm Hg were more likely to develop visual field loss. 

Figure 3.

 

The relationship between level of intraocular pressure and the development of visual field loss in four groups divided according to their level of intraocular pressure. AGIS Investigators: Advanced Glaucoma Intervention Study (AGIS): 4. Comparison of treatment outcomes within race: 7 yr results. Ophthalmology. 1998;105:1146–1164.

A second part of this present report deals with relevance. The AGIS graph is a powerful exhibit in that discussion. Those whose pressures were, on average, higher, clearly were demonstrated to get worse more rapidly than those whose pressures were, on average, lower. The question, however, is whether that finding has any relevance for the diagnosis or treatment of patients with glaucoma. The graph suggests a great deal of visual loss, because the units on the y-axis end at 6, suggesting that 4 units is as much as can be lost. In actuality, the denominator is 20 units. Thus, those patients who “fared worse” lost approximately 3 of 20 units of vision over an 8-year period. It is likely that the average number of years of life remaining in the patients in the AGIS study was approximately 15. (I am aware that I am lapsing into thinking along linear lines.) If this supposition is correct, then the average amount of visual loss in the “worst group” of patients in AGIS should be expected to be around 7 units before the time of death. Is that amount of visual loss going to interfere with the ability to perform the activities of daily living? It seems unlikely, but the answer cannot be given with any authority, because that issue was not studied as part of AGIS; in fact, it was not even addressed. (More on that later.) 

But, back to the question of validity. There are no confidence limits on the AGIS graph (Fig. 3). In actuality, not all those whose intraocular pressure was in the range of 12 mm Hg remained stable. Fourteen percent of those in the 12-mm Hg group had 4 units of visual field loss in the same 8 years. This 4-unit loss is greater than the mean visual field loss of 3 units for those whose pressures were 12 mm Hg. Furthermore, about half of those whose pressures were 20 mm Hg did not show visual field loss. This graph, then, establishes the important fact that lowering intraocular pressure is an effective way to prevent visual field loss in some individuals, but it does not establish a level of pressure that will be safe for all individuals or even that lowering intraocular pressure is effective in all individuals. 

Figure 4 shows an analysis of the AGIS data performed by Caprioli and colleagues. ¹⁵ The graph shows their conclusions about what characteristics predicted who was going to develop visual field loss. The rate at which the visual field was deteriorating and the age of the patient were both strongly indicative of who was going to have field loss. Note that they were equally significant from a statistical point of view (P < 0.001). Also note that the level of intraocular pressure was not a statistically significant indicator of who would develop visual field loss (P < 0.1). 

Figure 4.

 

Statistical significance of various characteristics in patients with glaucoma, related to the likelihood of their developing visual field loss. The slope of the visual field refers to the rapidity with which the visual field is worsening. AGIS VFS, the amount of visual field loss according to the Advanced Glaucoma Intervention Study system of staging visual fields; IOP mean, mean intraocular pressure; IOP fluctuation, the fluctuation of intraocular pressure. Eye (right) refers to comparison of the right eye versus the left eye. Histogram based on data from Nouri-Mahdavi K, et al. IOVS 2005;46:ARVO E-Abstract 3729.

Figure 5 shows the same information, but analyzed by me in a different way. The individual data points of those who got worse or who did not get worse were tabulated. The data were then divided into two columns for each characteristic: those individuals who got worse (the dark bars) and those who did not get worse (the light bars). There is a marked difference between those with rapidly developing visual field loss (toward the top of the y-axis) and those with more slowly developing visual field loss (toward the bottom of the y-axis). The mean of those who got worse (the dark bars) is clearly higher than the mean of those who did not. However, there is also a point, a threshold, above which everybody got worse. Phrased differently, when the rate of deterioration was above a particular threshold, all individuals with that rate developed progressive visual field loss. This was not a matter of “perhaps they would,” or “perhaps they would not.” All individuals above that threshold had progressive visual field loss. 

Figure 5.

 

Individual data points from the analysis made by Nouri-Mahdavi K et al. in Figure 4 (IOVS 2005;46:ARVO E-Abstract 3729) analyzing those patients whose visual fields got worse and those whose fields did not get worse in four categories, specifically related to the rapidity of worsening visual field, the age of the patient, the amount of visual field loss, and the mean intraocular pressure. The dark bars represent those who developed visual field loss, and the light bars those who did not. The figures at the top of each bar are the mean values. The units on the y-axis represent the ranges, adjusted so as to have a similar denominator. Zero represents the lowest value and 100 the highest.

Recall that age was considered to be as powerful a statistical indicator of who will get worse as was rate of change (P < 0.001). However, age did not distinguish with certainty which individuals will get worse. The oldest individuals in the population did not get worse (Fig. 5). Thus, when considering a particular person, rate of change of visual field can provide definitive information about whether he or she will get worse, but age cannot. 

These characteristics about intraocular pressure, rate of change, and age can be shown to apply to other biological characteristics, such as the nature of the optic disc in patients with glaucoma, an area we will now examine. Specifically, we will show that there is a definite relationship, but that it is not linear. 

Glaucoma is currently defined by many as “a characteristic optic neuropathy.” Therefore, abnormality of the optic disc is central to determining whether glaucoma is present and to managing the condition. Just as it is possible to measure intraocular pressure and develop a mean and standard deviation, so is it possible to characterize the size of the cup of the optic nerve, the cup/disc ratio, and analyze that measurement statistically. This graph (Fig. 6, also taken from Caprioli et al. ¹⁶ ) shows the well-known phenomenon that bigger cups are commoner in patients with glaucoma. The difference between the average cup size of a person who does not have glaucoma and one who does is significant and is of clinical importance. In the Collaborative Initial Glaucoma Treatment Study (CIGTS), the presence of a cup/disc ratio of a certain size (0.6 where 0.0 is the healthiest and 1.0 is the sickest) was considered sufficiently valid evidence of abnormality that the patient could be said to have glaucoma and be enrolled as a glaucoma patient, even when visual field loss was not present. Consider the data points in the individuals who comprised the group of normal subjects, glaucoma suspects, and glaucoma patients in Figure 6. The largest cup/disc ratio that was found in a patient who was normal was 0.8. Thus, a large number of normal individuals had cups that were markedly larger than two standard deviations above normal. Note also that there were individuals who had cup/disc ratios of 0.6 or 0.7, significantly above the average for normal, but did not have glaucoma. Note also that the patient who had the smallest cup/disc ratio actually had glaucoma. That is, just as with intraocular pressure, values that were statistically abnormal, but not in the “always abnormal” range (here above 0.8) were not valid markers indicating that the individual had glaucoma. An individual could have a cup/disc ratio at the mean of normal and still be abnormal. Conversely, an individual could have a cup/disc ratio well above two standard deviations—that is, well above “normal limits”—and yet not have glaucoma. When values were not in the “always abnormal” range, they were essentially useless in distinguishing between individual patients with glaucoma and those without glaucoma. This is a very threatening thought, because it says that a value in the normal range cannot be considered proof of the absence of abnormality. 

Figure 6.

 

The mean and standard deviations of the cup/disc ratio for normal subjects, glaucoma subjects and those with glaucoma (left). Distribution of cup/disc ratios for the same groups (right). Modified from Caprioli J, Miller JM. Videographic measurements of optic nerve topography in glaucoma. Invest Ophthalmol Vis Sci. 1988;29(8): 1294–1298.

In the second part of this report, I wish to consider the relevance of any particular finding, such as an intraocular pressure of 15 or of 50 mm Hg, a small or large cup/disc ratio, or the presence or absence of visual field loss. Relevance refers to closeness to the issue being considered. In the area of glaucoma, there are two relevant issues for patients: (1) Does, or will, the glaucoma or its treatment interfere with how the patient wants to function, and (2) does, or will, the glaucoma or its treatment cause a decrease in the quality of the patient's life? Intraocular pressure, optic disc, or visual fields are not of concern to patients until they have been educated, or rather miseducated, to believe that there are linear relationships between those variables and what they can do and what they cannot do, or how well they feel or how sick they feel. It is worth considering these issues carefully. 

It is obvious that a person with no vision is less likely to function normally than a person who has no visual loss. The assumption is that there is a linear relationship between vision and function. (I here define function as the ability to perform the activities of daily living.) 

Figure 7 illustrates the data points from a study in which individuals with various stages of glaucoma were tested to determine how they could perform the activities of daily living. The test, The Assessment of Disability Related to Vision (ADREV), has been well studied, is reproducible, and is internally valid. ¹⁷ The wide scatter on this scattergram could be the result of inaccurate data points with regard to ADREV or inaccurate data points with regard to the visual field. These explanations seems unlikely, however, because the degree of accuracy of the determinations of ADREV and of visual fields has been studied and does not show the variability that would be necessary to explain data points with such a wide scatter. A more likely explanation is that the relationship between function and field is in fact not a linear one, but rather a “chaotic” one. Some individuals with poor visual field function well, and some with excellent field function poorly. The nature of the relationship of two variables associated with each other chaotically is not random, but also it is not linear. It is not possible, as it is with linear relationships, to determine the reliability of future data points based on existing ones. One can only determine (and this can be done reliably) the range of future data points that might occur. “Chaotic” relationships are not random; they are just not linear. There is a definite limitation to the position of electrons as they circle the nucleus of an atom. However, it is not possible to determine validly exactly in what position a particular electron will be at any particular moment. One only can discuss the ranges in which they will be found. Quantum mechanics and fractals obey laws, just not Newtonian laws. 

Figure 7.

 

Comparison between the ability to perform the activities of daily living and binocular visual field in patients with various stages of glaucoma. The ability to perform the activities of daily living was assessed using the Assessment for Disability Related to Vision, ¹⁷ in which 0 indicated an inability to perform any of the nine tests and 63 an ability to perform all nine tests perfectly. The Esterman Binocular Visual Field is an estimate of binocular visual field in which 0 represents no detectible field and 100 a full field.

It seems highly likely that the relationship between function and vision is a chaotic one. This association should be studied using different approaches than have been used in the past. One reason that this particular type of study has not been performed is that the type of analysis has not been possible until recently, because it is only in the fairly recent past that performance-based measures have been used to study function. ^{17 –21}  

Figure 8 shows the relationship between the ability to perform the activities of daily living and the quality of life. Again, there is a relationship, as one would expect. However, in this scattergram the points are distributed widely. ¹⁷ Interestingly, Mills et al. ²² found a remarkably similar type of scattergram when comparing visual field with quality of life. 

Figure 8.

 

Comparison of the ability to perform the activities of daily living with quality of life as estimated by the NEI-VFQ-25 survey. The units for the Assessment for Disability Related to Vision are explained in Figure 7. The higher the score on the NEI-VFQ-25, the better the quality of life.

Time does not permit a more penetrating analysis of the inadequacy of using only a linear conception of reality as a basis for medical research and practice. In conclusion, I have briefly suggested that the current method used to conceptualize reality and as the basis for medical and biological research and practice is not sufficient to understand biological phenomena such as health and disease. Clearly, there is great power in standard statistical theory based on linearity. However, the problem arises when nonlinear data are forced into a linear system. Consideration of what is “always abnormal” can be powerful, but is rarely made. For example, intraocular pressure above 34 mm Hg appears always to be associated with visual loss. ²³ Knowing that, it is reasonable to say that a person with intraocular pressure greater than 34 mm Hg will develop glaucomatous damage. The problem, however, is that it is not possible to say with certainty whether a person who is not in the “always abnormal” range does or does not have glaucoma, even when the intraocular pressure is five standard deviations above the mean. People with eye pressures of 30 mm Hg who do not have glaucoma are not outliers. They are representatives of a biological nature in which intraocular pressure does not cause optic nerve damage. Basing diagnosis and treatment solely on a linear system leads to (1) overtreatment of those “outside normal limits,” and (2) undertreatment of those “within normal limits,” and failure to appreciate the seriousness of those “always abnormal.” 

Outliers are today what lepers were 200 years ago: not to be understood, certainly not to be emulated, but rather to be isolated. Both in research and in clinical care, outliers are mysteries that are either ignored or eliminated. However, a study of outliers is more likely to provide an understanding of what leads to health and disease than is a study of the average. For example, what is it that allows some people to have eye pressures of 30 mm Hg and not develop optic nerve damage or to have intraocular pressures of 15 mm Hg and develop damage? What explains why some individuals with 20 dB of visual field loss function well and others with visual field loss of 5 dB function poorly? Linear models provide powerful ways of understanding reality, but they are inadequate. Outliers do not fit into a linear system, but may be understood by using other conceptions of reality, such as those in chaos theory that follow rules, but rules that are different from those of linearity. 

Finally, I have also suggested that we are most likely to achieve our objectives if we concentrate on achieving those objectives, rather than the steps along the way or surrogates with objectives. Our objective, as those interested in health and disease, is to understand health and disease in their totality, not just their individual parts, so that we can restore, preserve, or enhance health and relieve disease. Function and feeling are both integral and essential parts of health. Both function and feeling must be studied with tests of actual function and feeling, not by assuming that surrogate factors or subitems can explain function or feeling adequately. 

A metaphor regarding a valuable work of art may provide some insight into the importance of nonlinearity, and the “individual” in science and medicine, as well as in art. The value of a work of art depends on how powerfully, how validly and relevantly, it conveys a unique original image or idea. Every person is a unique original. Our task and privilege as those working in the field of health is not to standardize individuals by homogenizing them into an indistinguishable average. Trying to make people “normal” will not make them healthier. Rather, we need to create environments that allow unique individuals to function as fully as they wish and to feel as well as they can. Incorporating the concepts of nonlinearity may lead investigators to a better understanding of what constitutes health and disease and may make clinical care more valid and more relevant to its primary purpose—specifically, assisting individuals to function and feel as well as they can. 

Those interested in further reading regarding chaos theory may consult:

•  
  Goldberger AL. Nonlinear dynamics, fractals, cardiac physiology, and sudden death. In: Rensing L, An Der Heiden U, Mackey M, eds. Temporal Disorder in Human Oscillatory Systems. New York: Springer-Verlag; 1987.
•  
  Goldberger AL, Bhargava V, West BJ Mandell AJ. On a mechanism of cardiac electrical stability: the fractal hypothesis. Biophys J. 1985;48:525–528.
•  
  Holden AV, ed. Chaos. Manchester, UK: Manchester University Press; 1986.
•  
  Lorenz EN. Deterministic non-periodic flow. J Atmospher Sci. 1963;20:130–141.
•  
  Mandelbrot BB. The Fractal Geometry of Nature. San Francisco: WH Freeman; 1982.
•  
  Mandell AJ. From molecular biological simplification to more realistic central nervous system dynamics: an opinion. In: Cavenar JO, ed. Psychiatry: Psychobiological Foundations of Clinical Psychiatry. New York: Lippincott, 1985:361–365.
•  
  Swinney HL. Observations of order and chaos in nonlinear systems. Physica D. 1983;7:3–15.
•  
  Taleb NN. The Black Swan. The Impact of the Highly Improbable. New York: Random House; 2007.
•  
  Von Neumann J. Recent theories of turbulence (1949). In: Taub AH, ed. Collected Works. Oxford, UK: Pergamon Press; 1963:437.
•  
  Voss R. Random fractal forgeries: from mountains to music. In: Nash S, ed. Science and Uncertainty. London: IBM UK; 1985.

Footnotes

 Disclosure: G.L. Spaeth, None

The author thanks Kenneth Richardson for his many years of thoughtful analysis of what is valid and relevant, especially for his knowledge of chaos theory.