The Measure of Cancer

Part 1. Quantitation

by Ian Magrath

...all exact science is dominated by the idea of approximation.

The Step Pyramid of King Djozer, built around 2650 BCE at Saqqara, Egypt, is the first known monumental stone structure. The pyramid and adjacent temple complex were designed by Imhotep, High Priest of Heliopolis and styled “son of Ptah”, the sun god. Also astrologer, scribe and physician, he is believed to be the author of the Edwin Smith medical papyrus and was identified by the ancient Greeks with Asclepius, their God of Medicine.

Yet despite their beguiling mystery, it was not the nature of the heavenly bodies themselves that stirred the imagination of the first astronomers, for this was the province of ancient lore, but rather the uncanny association of their movements with time and the seasons. Generations of observation eventually led, when men (or more likely, women) learned the art of agriculture, to the emergence of calender making - an empirical science laced with superstition and jealously guarded by the astronomer-priests who developed it. Crops were first cultivated in settled hunter-gatherer communities in the Near East, such as the one at Abu Hureyra in the Euphrates valley that took advantage of the routes of migrating gazelle. By 8000 BCE the settlers of Abu Hureyra had domesticated sheep and goats and were growing pulses and other cereals. By 6000 BCE, farming had spread to the Nile and Indus valleys and to other regions of Asia, including China. Crops were also being cultivated in the Andes and parts of central and North America. But farming soon created a need for mathematics that went beyond the priestly skills of calender making. It led to the growth of walled cities (Jericho, in the Jordan valley, was some nine acres in size by 6500 BCE), the bartering of goods, and systematic warfare. Trade required keeping tallies of goods while geometry (world measurement) was essential for navigation between the ancient entrepots, and for the construction of new cities and their palaces and temples. The pyramids of Egypt - gigantic tombs of the Pharoahs of the old Kingdom constructed almost 5000 years ago - stand as lasting monuments to the mathematical skills of ancient civilizations.

Arithmoi

Mathematics began - it still does - with counting. Notched bones dating from 35000 BCE provide evidence of primitive record-keeping well before the agricultural revolution, and such tally “sticks” afforded some of the earliest known aids, beyond parts of the body, to human memory. Cardinal systems of this kind, in which numbers (without requiring a concept of number) are represented by a series of identical units that can be compared, one-on-one, to a collection of objects or animals, were the forerunners of more sophisticated ordinal systems, in which numbers have a specific sequence or order, each with its own symbol. The almost universal sign for the number one, and perhaps other numbers, such as the “handfuls” 5 (V) and 10 (X), probably evolved from notches on bones or sticks. Calculation, however, a natural step beyond tally keeping and the foundation of science and hence material progress, required more dynamic systems. The hand has always been the calculating (and counting) machine par excellence but pebbles (calculi in Greek), shells or clay tokens have also been extensively used from the earliest times. Counting, i.e, the process of generating arithmetical data, whether for simple record keeping or calculation is, unfortunately, not quite as simple as it might at first appear. The counted units must be “like elements,” the definition of which varies according to the context of the counting. Counting apples versus oranges may present few difficulties, but whether heifers should be counted with cows might depend upon the use to which the data is to be put. The notion of defining objects for counting is inherent in the modern mathematical concept of “sets.” A set can be defined in any way and may contain zero elements. Examples might be the set of all cancers, or of a particular cancer in the world, or the set of sets of different types of cancers in a defined geographic region. Counting, that is, the quantitation of multitude, is the process of enumerating the elements (or units) in a specified set, i.e, the sequential matching (or mapping) of each of its elements with the next number in a hierarchically ordered series of natural whole numbers (positive integers) until all the elements in the set are exhausted.

For much of human history, the counting numbers have been conceived as inseparably linked to the objects enumerated. Indian astronomers, in this tradition, gave individual numbers multiple names related to objects imbued with the sense of the number. The number one, according to Brahmagupta, writing in the seventh century CE about long-standing practices, was called adi, the beginning, or Tanu, the body, as well as many other things, each based on an object or event denoting something unique. Two was associated with Ashvin, the twin gods, netra, the eyes and various other “twosomes” or dualities. The ancient Greek word arithmoi, often translated as number, in fact meant a number of things and did not encompass the modern abstract concept of a pure number. The idea inherent in arithmoi, which was the ancient norm, resulted in a mental barrier that inhibited the evolution of mathematics, but counting things nevertheless remains a fundamental element of a broad range of human endeavors.

Counting Cancers

Counting the number of cases of a specific type of cancer in a given region or population provides data that can be used to determine the resources required for cancer control in the region. For purposes of comparison with other populations, however, it is necessary to calculate an incidence rate, i.e., the number of new cases occurring for each subset of a specific number of individuals in the population in a defined period of time (most often, the number of cases per 100,000 per year). Differences in incidence rates reflect differences in exposure to environmental risk factors for cancer, tempered, of course, by genetic predisposition. Thus, incidence rates provide a necessary first step towards understanding factors which cause or predispose to cancer, and an assessment, if measured over time, of the success of preventive measures. Their determination is a primary function of cancer registries. Mortality rates, in conjunction with incidence rates, provide information on the success of treatment at a public health (i.e., population) level.

Accurate counting of cancer sets or subsets requires accurate definition of the counted elements - in this case, with rare exceptions, a pathological diagnosis. The quality of the diagnosis, however, will vary with the skill and resources available to the pathologist. Many pathologists, particularly those in developing countries, have little or no access to modern diagnostic techniques such as the detection of protein expression patterns or, for some cancers, tumor-specific molecular abnormalities. Cancer subtypes, however defined, may have a different etiology (cause), so that unless cases are diagnosed with a high degree of precision, the strength of the association between risk factors and specific cancers will have a built-in and variable degree of imprecision. Accurate diagnosis is only one element in the calculation of accurate incidence rates. There must also be an active process of case ascertainment that includes all institutions, regardless of location, in which cancers from the population in question may be diagnosed (cases remaining undiagnosed are effectively beyond reach). Cancer registries in single institutions provide information that is largely of value only to the institution itself, although cancer frequencies (i.e., the relative proportions of different cancers) in comprehensive regional centers may approximate population-based data. Accurate incidence rates, of course, also require accurate estimates of the size of the relevant population - whether an entire regional population, or a defined subset based on age, sex, ethnicity, religion or any other desired criterion or set of criteria. There remains a paucity of high-quality incidence data in the world, particularly in poorer countries, which rarely have population-based registries.

Less Developed Countries (2000) Figure 1. Another kind of pyramid! This one shows the numbers of individuals in each age group, males and females, in the less developed countries. Data from Globocan 2000, IARC.

For accurate comparison of incidence rates between entire populations rather than between narrow age subsets, it is necessary to correct for differences in the age structure between populations (i.e., the proportion of individuals in different age-groups), since cancer incidence varies markedly with age. Children (age 0-14 years) comprise some 30-50% of the populations of developing countries, but generally no more than 15% of populations in high-income countries (Figures 1 and 2) so that the set of cancers that occur particularly or exclusively in children account for a higher proportion of all cancers in developing countries even though their incidence rates may be similar to those in affluent countries. Conversely, the fraction of individuals above 65, who have the greatest risk of developing cancer, is much higher in most high-income nations than in developing countries. Thus, while crude (i.e., uncorrected) incidence rates provide an accurate measure of the actual cancer burden of a population, regardless of its age structure, age-standardized rates, whereby the crude rates are standardized, e.g., to the age structure of the world population, give a better perspective on incidence rate differences that relate to variable exposure to carcinogens. In spite of the many potential sources of error in counting cancer cases and people, the range of differences in cancer incidence in defined populations is generally sufficiently large, compared to the size of the inevitable errors, that cancer registries do, in fact, provide a great deal of valuable information (Table 1), particularly when collected over many years such that time-trends become apparent.

More Developed Countries (2000) Figure 2. Population pyramid for more developed countries.Data from Globocan 2000,IARC.

Geometria

A second major element of mathematics is measurement, i.e., the quantitation of magnitudes, namely mass and dimension. Measurement, like counting, entails comparison, but with a series of units of defined magnitude, such as length (e.g., centimeters, kilometers) or duration (e.g., seconds, hours) rather than the series of counting numbers. The ancient Greeks viewed the numerical expression of measurement, which requires the concept of fractions, as being entirely different from the process of counting. Nevertheless, Pythagoras and the members (called mathematikoi) of his secretive philosophical and religious school founded around 518 BCE in Croton, southern Italy (then, Magna Grecia), observed that certain numbers of pebbles could be arranged in specific geometric shapes such as triangles, e.g., 3, 6, 10, or squares, e.g., 4, 9, 16. Pythagoras had lived in both Egypt and Babylon, and doubtless imbibed much of his mathematical knowledge from these more ancient civilizations - including, perhaps, the famous theorem named after him, which was known to the Babylonians and other early agriculturalists at least a thousand years before, and which to this day is a central element in the calculation of space-time coordinates. He believed that the universe is governed by rational numbers, i.e., numbers that can be expressed as a ratio of two other numbers (including all natural numbers and fractions). Consequently, he attempted to suppress the disconcerting fact that the length of the hypotenuse of a right-angled triangle with sides of one unit is a number (2) that cannot be expressed as a ratio of whole numbers, i.e., is alogon (irrational, and also unspeakable). Irrational numbers invoke the concept of infinity - anathema to the ancient Greeks. They fit into the infinite number of gaps between the rational numbers arranged along an imaginary “number line” and when expressed as decimal fractions create an infinite expansion of seemingly randomly arranged digits.

Geometria, as rigorously expounded by Euclid of Alexandria in the Elements, dating to 300 BCE, dealt with points, lines and circles. Such mental constructs fitted nicely into Plato’s concept of “ideal forms” that he believed lay behind all earthly objects, such that ancient geometry, like arithmetic, was a tool that dealt with the tangible - the world of the senses. As with counting, the most immediately available units of measure were parts of the human body - e.g., a hand or foot-length, the distance from outstretched fingers to the elbow (a cubit) or to the tip of the nose (a yard). Whilst some of these measurements are still used today, the obvious disadvantage of variability in limb size has been eliminated by “standardization” of the chosen unit of length. At first, this was achieved by simply using a particular individual’s appendage, e.g., the King’s foot, but later, a reference length was used, against which all measuring devices were standardized. Whilst clearly superior, reference lengths do not provide absolute precision. Metal bars, for example, change length according to temperature, and the dimensions and mass of objects would appear to differ when the measuring point and the measured event are in a state of relative motion - a difference that is irrelevant to everyday life, but which becomes important when elementary particles are accelerated to close to the speed of light, as occurs, for example, in radiation therapy. Standards also differ in different countries. After the French Revolution, Talleyrand, while President of the French National Assembly in 1791, stated the principle, subsequently enacted into law, that units of measure must be defined against an agreed upon standard and moreover, that for any standard to be used internationally, it should not be “arbitrary” or contain “anything specific to any people on the globe.” Two years later, the meter - defined as a ten millionth part of a quarter of the Earth’s meridian - was introduced as the standard unit of length. Sufficiently non-partisan, the Earth’s meridian suffered from the problems of accurate measurement and accessibility. In 1799, therefore, a standard meter bar was placed in the archives of the new French Republic, along with a mass of 1000 cubic centimeters of water at a temperature of 4 degrees centigrade (the standard definition of a kilogram). These standards have been subsequently improved upon several times. A meter, for example, was most recent.ly defined in the XVIIth General Conference on Weights and Measures (1983) - as the distance traveled by light in space in 1/299,792,458 of a second, a second having been defined at an earlier conference as the duration of 9,192,631,770 cycles of microwave light absorbed or emitted by the hyperfine transition of cesium-133 atoms in their ground state undisturbed by external fields - in essence, a measurement based on the wavelength of a highly coherent microwave beam.

Measuring Masses

In the context of cancer, measurement is a means of assessing the burden of disease in an individual. This does not require a high degree of accuracy. For many years, comparison of the size of tumor masses with commonplace objects, such as a nutmeg or an orange, was sufficient, since, apart from the possibility of surgical removal in some cases, there was generally little, until the 20th century, that could be done to alter the natural history of the disease. Even today, ultra-precise measurements of the size of an individual tumor are of limited value. More important is the likelihood that the tumor will cause compression, erosion or invasion of adjacent organs or tissues, (which depends upon location and biological properties as well as size), and the ability to define the margins of the tumor, such that local therapy (surgery or radiation therapy) is maximally effective. Reduction in size is, however, the primary measure of the effectiveness of treatment, and is central to the assessment of the efficacy of new drugs. Precise measurement of tumor size is difficult, since tumors tend not to develop as geometric shapes susceptible to the accurate calculation of volume. They are often irregular and sometimes ill-defined, because of the invasion of surrounding tissues. Tumor cells may also float in “serous” fluid in body compartments, such as the pleural or peritoneal cavities. In glandular tissue such as the breast and pancreas, they tend to fill the gland ducts early in their evolution. They may eventually break through the basement membrane surrounding the ducts, thereby gaining entrance to the rest of the organ and potentially spreading to other parts of the body. Similarly, leukemias - neoplasms of blood cells and their precursors - diffusely involve the bone marrow and circulate in the bloodstream, making direct assessment of total tumor burden difficult. Here, treatment response is based on the ratio between malignant cells and normal cells (other than red cells) in a bone marrow sample, once circulating cells are eliminated. In all cases, tumor masses contain some - often a great deal - of normal tissue, as well as some dead tumor cells, such that even if precise measurements of a mass were possible, the proportion of contained tumor would remain, at best, an estimate.

Less Developed Countries More Developed Countries Cases Crude rate ASR Cases Crude rate ASR Oral cavity 72687 3 3.7 24466 4 2.4 Nasopharynx 16436 0.7 0.8 2387 0.4 0.3 Other Pharynx 16062 0.7 0.8 6005 1 0.6 Oesophagus 117092 4.9 6.2 16253 2.7 1.3 Stomach 192850 8.1 10 125029 20.5 11 Colon/Rectum 154064 6.4 7.9 291897 47.8 25.4 Liver 132298 5.5 6.8 33680 5.5 2.9 Pancreas 39449 1.6 2.1 61230 10 5.1 Larynx 12390 0.5 0.6 6845 1.1 0.7 Lung 161719 6.8 8.4 175392 28.7 15.6 Melanoma of skin 13904 0.6 0.7 53511 8.8 6.1 Breast 471063 19.7 23.1 579285 94.9 63.2 Cervix uteri 379153 15.8 18.7 91451 15 11.3 Corpus uteri 75336 3.1 3.9 113618 18.6 11.3 Ovary etc. 101060 4.2 4.9 91307 15 9.9 Bladder 27895 1.2 1.4 48129 7.9 4.1 Kidney etc. 22882 1 1.1 47936 7.9 4.6 Brain, nervous system 43076 1.8 2 32538 5.3 4.1 Thyroid 53710 2.2 2.5 35635 5.8 4.4 Non-Hodgkin lymphoma 54659 2.3 2.6 66148 10.8 6.6 Hodgkin's disease 11796 0.5 0.5 12142 2 1.8 Multiple myeloma 11754 0.5 0.6 22705 3.7 1.9 Leukaemia 65366 2.7 3 47388 7.8 5.4 All sites but skin 2561666 106.9 127.9 2175974 356.6 218.3 Table 1. Cases, crude and age-standardized (to the world population) incidence rates (ASR) per 100,000 per annum (in 2000) of major cancers in females in less and more developed countries. Note that some subsets,e.g.,Leukemia, include many different types of cancer. Data from Globocan 2000, IARC.

Patients with most forms of cancer are assigned to a clinical stage, i.e., to one of several hierarchical categories that are designed to indicate progressively more advanced disease. The stage of a tumor is a function of both its physical size and the biological properties of the tumor cells that determine the degree of spread in the body. Although size and biological characteristics are, to a degree, related (in part because the greater the number of tumor cells, the more likely are additional molecular changes to arise), the relationship is not precise. Moreover, tumor volume itself depends upon the biological properties of the tumor cells, including their proliferative rate and death rate, as well as the duration of time that has passed from the onset of a cancer to its diagnosis. Tumor cells disseminate to regional lymph nodes or to distant parts of the body by penetrating lymphatics and blood vessels, but only those capable of surviving and growing in one or more tissues or organs that would be hostile environments to the normal counterpart cells are capable of giving rise to new tumor cell colonies at distant sites. Such colonies are referred to as metastases - literally, tumor that “stands” or “stops” in a different place. Their constituent cells are able to resist the signals that induce apoptosis (programmed cell death) in displaced cells - a mechanism that normally ensures the integrity of organs and tissues. The ability to avoid apoptosis also renders tumor cells relatively resistant to chemother.apy and frequently to radiation therapy. Consequently, the presence of metastases is nearly always associated with a poor treatment outcome, regardless of the tumor burden when treatment begins. Clinical staging, therefore, provides a guide to treatment, since more advanced stages require more intensive therapy and/ or a different blend of local and systemic treatment. In developing countries, patients tend to have higher stage disease at the time of diagnosis than in affluent countries, at least in part because of delay in diagnosis. This must be taken into consideration in determining resources required for treatment and in comparing treatment outcome with that achieved in in more affluent countries. The accuracy of stage assignment, however, is a function of the availability and use of various imaging techniques which must also be taken into account when comparisons are made. Apparent improvements in the survival of patients with localized disease, for example, can result when new techniques that improve the detection of disseminated disease are introduced, eliminating a fraction of patients previously included in this category.

Counting cancers is the foundation of epidemiology and public health. Measuring cancer is necessary for optimal treatment and clinical research.

While clinical stage is usually one of the most important determinants of outcome (particularly in tumors where systemic therapy is ineffective) additional predictive value may be provided by histological features (often also categorized into several grades), including the degree of invasion of adjacent tissues at a microscopic level and by the tumor's molecular profile. The concentration of various tumor markers in the bloodstream, such as hormones, proteins or molecular abnormalities present in circulating tumor cells or DNA, can also permit the prediction of outcome with a particular treatment, provide a more precise measure of treatment response, or indicate imminent relapse. In normal individuals specific markers may be associated with an increased risk of the development of cancer. Finally, many factors other than those associated with the tumor itself can influence both the therapeutic outcome and the toxicity of treatment. These include age, performance status, coexistent chronic infections and malnutrition, inherited variations in the metabolism of chemotherapeutic drugs and the ability of tumor cells and normal tissues to repair radiation or drug-induced damage.