Three Key Concepts in Our Analysis

Part 1. Breast-Cancer: Production-Rate vs. Incidence-Rate

The purpose of this chapter is to draw distinctions between three concepts, because the distinctions will make the method of this study readily understandable.

A. Annual production-rate of breast-cancers by radiation.

B. Annual delivery-rate (incidence-rate) of radiation-induced clinical breast-cancers.

C. The "Law of Equality," which refers to the situation when the annual clinical incidence-rate is equal to the annual production-rate,

despitethe latency period.

Production versus Delivery: Cases on the "Shelf"

The breast-cancers produced in a population, by radiation received during a single year, do

notall occur exactly 3, 10, 20, 30, 40, or 50 years later. Rather, a certain number arecommittedin a single year, and this number is spread outclinicallyover many years, as explained in Chapter 2. We shall use the terms "committed" and "produced" as equivalent. The concept of production simply means that, once the irradiation has occurred, some cells of female breasts have experienced an addition of lesions essential for developing breast-cancer. Some fraction of the females who received such lesions will later develop overt, clinical breast-cancer.We can think of a simple shelf. The irradiation received in a particular year puts "on the shelf" a certain number of future breast-cancers. These are the produced breast-cancers or committed breast-cancers, from this one year of radiation exposure. At some later time, and over a period of many years, those produced breast-cancers come "off the shelf" and are "delivered." By delivered, we mean that they become

clinicalbreast-cancers, or clinicallydetectedbreast-cancers.If readers just keep the distinction in mind between annual

production-rate and annualdelivery-rate, they will already be well advanced into understanding the method of our study.

Relationship of Annual Incidence-Rate to Annual Production-Rate

Now we come to the third concept: Years when the annual incidence-rate is equal to the annual production-rate,

despitethe variable and often long period of latency. We will demonstrate the "Law of Equality" in Part 2.Application of this "law" achieves almost everything! In other words, this concept explains how we will approach the problem of

finding outwhat fraction of recent, current, and future breast-cancer is due to past irradiation. Readers who intend to follow the method on which our findings rest, will need to grasp this law. The "easy readers" can do without it, of course, or can come back to it after a first "flying read" of the book's more narrative parts. However, we encourage everyone to try Part 2 of this chapter, which is really easier than some readers may imagine.Part 2. A First Demonstration of the "Law of Equality"

How can annual incidence-rate become equal to annual production-rate, despite the variable latency period? In a problem such as this, it helps a great deal to start with some simplified or idealized conditions, so that the mind is not diverted by real but momentarily deferrable details, to which we will attend gradually.

In our first demonstration, readers should note three key conditions: The same age for everyone who is irradiated, the same number of such people, year after year, and the same radiation dose, year after year. By people, we mean female people, since our focus in this book is breast-cancer.

So, for the first demonstration, we arbitrarily say that the age is 5 years of age, the number of such female children is constant but not specified, and the radiation dose to the breasts each year is a dose which produces (commits) a total of 100 cases of breast-cancer, excluding any cases which occur sooner than ten years after production.

o -Each box in the grid represents 2.5 cases of clinical breast cancer.

o -Each horizontal row of 40 boxes represents 100 breast cancers -- the numberproducedby a single year of irradiation. So each row represents one year of production. In Fig. A, the group irradiated in 1920 is age-5. In 1921 (next row), it's a new group, age-5, etc.

o -Each vertical column represents the number of radiation-induced breast cancers clinicallydetectedin a single year.

o -Bothshadedcolumns have 40 vertical boxes, as do the columnsbetweenthe two shaded columns. Such columns represent 100 clinicallydetectedbreast-cancers per year, in the years 1970 through 1986. These are the columns which demonstrate "the law of equality" under the conditions specified in the text, Part 2.

Figure - A "Law of Equality": Demonstration 1

The Various Rows of Figure-A

The first demonstration corresponds with Figure-A. Let us focus on the bottom row of boxes. Each box represents 2.5 cases of clinical breast-cancer. The bottom row of 40 boxes represents 100 clinical breast-cancers

producedby irradiation during 1920. We will pretend that there was no breast-irradiation before 1920, and that 1920 initiates the annual production of 100 radiation-induced breast-cancers.How do we distribute the cases

producedduring 1920? We know that the latency period is variable. For simplicity, Figure-A arbitrarily shows 2.5clinicalcases observed per year, after an initial latency period of 10 years, (1921 through 1930), and this detection-rate of 2.5 cases per year goes on for 40 years. Thus, the bottom row is depicting 40 different latency periods. The irradiation during 1920 has put 100 cases "on the shelf," and every one of those cases labeled "produced" ultimately becomes one which is delivered, labeled "clinically detected." The delivery, from the 1920 production, begins during 1931 and is completed during the year 1970 --- a total of 40 years.The next row of Figure-A shows that, in 1921, the same thing occurs: The same number of 5-year-olds (who are not the same children as those who were age 5 during 1920) receives the same dose (which puts another 100 breast-cancers "on the shelf"). And Figure-A, with its 56 rows, shows this production-rate continuing every year through 1975 --- an arbitrary "cut-off" date. Because of the initial latency period of ten years, the

firstdelivery from 1975-production occurs in 1986, as shown by the top row.

The Various Columns of Figure-A

Each vertical column depicts the number of

clinicalbreast-cancers detected (delivered) in a specific year. Each box represents 2.5 clinical cases, so the number detected during 1931 = 2.5 cases, but the number detected during 1940 has risen to 25 cases. In 1933, age 18 delivers the bottom box; age 16 delivers the top box.The key point is that during 1970, the

incidence-rate of radiation-induced breast-cancer reaches 100 cases --- which is equal to the annualproduction-rate --- and the equality (100 cases) is shown to recur in 1971, 1972, 1973, and in every year through 1986. The columns for these years of equality are the shaded columns and those between the two shaded columns.

Statement of the "Law of Equality"

So Figure-A has demonstrated the "Law of Equality": If the same level of irradiation is maintained year after year, and if the number of irradiated females is the same every year, we finally reach the situation where the annual clinical incidence-rate of radiation-induced breast-cancer is equal to to annual production-rate of radiation-induced breast-cancer, and this annual incidence-rate will endure indefinitely, if we maintain the annual production-rate. And this occurs

despitethe variable latency period for the cases produced in a single year.Figure-A also shows what happens if the annual production-rate is

notmaintained. We arbitrarily made 1975 the last year of irradiation (with delivery beginning in 1986, after an initial latency). Although the annual production-rate goes suddenly tozeroin 1976 and thereafter, the annual incidence-rate of radiation-induced cases fallsgradually(that is, the number of boxes per vertical column declines, until there is just one box occurring in the vertical column for the year 2025).

Duration of the Radiation Effect

In Figure-A, the total duration of the radiation effect is 50 years --- 10 years of an initial latency period plus 40 years of "delivery." This initial latency period and total duration were chosen to be only illustrative. We are quite confident that, in reality, there is no

minimumlatency period, as noted in Chapter 2. And in reality, the duration of the radiation effect may exceed 50 years in people who are irradiated as children. By contrast, the duration of effect will surelynotexceed 50 years in women irradiated at age 55, because of the natural lifespan of humans.Part 3. The Law's Validity under Real-World Conditions

Because the "Law of Equality" is central to the method of our study and therefore central to our findings, we intend to prove, below, that the law also applies to real-world conditions.

For example, the law applies no matter what the delivery-pattern may be for the cancers committed during a single year of production. There was nothing "magical" about the pattern illustrated by Figure-A. The patterns in Figures B and C will each be very different from Figure A, in initial latency periods and in speed. Most importantly, we will demonstrate that the law applies to a population of

mixedages, when the cases committed by a single year's radiation exposure are delivered "from the shelf" at anon-uniformrate per year. This situation is a very close approximation to reality, and will constitute our final demonstration.

Figure-B: The Second Demonstration

For ease of comparing Figures A and B, we want to keep the total cancers committed per year of radiation exposure at 100 cases, in Figure-B. Nonetheless, we make radical changes from Figure-A regarding the initial latency period (now 30 years instead of 10 years), and regarding the speed of delivery (now 5 clinical cases delivered per year instead of 2.5 cases per year).

o -Each box in the grid represents 5 cases of clinical breast cancer.

o -Each horizontal row of 20 boxes represents 100 breast cancers -- the numberproducedby a single year of irradiation. So each row represents one year of production.

o -Each vertical column represents the number of radiation-induced breast cancers clinicallydetectedin a single year.

o -Bothshadedcolumns and all columnsbetweenthe two shaded columns, have 20 vertical boxes. Such columns represent 100 clinicallydetectedbreast-cancers per year, in the years 1970 through 2006. These are the columns which demonstrate "the law of equality" under the conditions specified in the text, Part 3.

Figure - B "Law of Equality": Demonstration 2

In Figure-B, each box represents 5 cases of clinical breast-cancer (not 2.5 cases). The bottom row of 20 boxes represents 100 clinical breast-cancers produced by irradiation during 1920. The first cases are delivered during 1951, and the last ones during 1970. The next row represents 100 clinical cases produced by irradiation during 1921. Figure-B, with its 56 rows, shows this production-rate continuing every year through 1975 --- an arbitrary "cut-off" date. Because of the 30-year initial latency period, the

firstdelivery from 1975-production occurs in the year 2006, as shown by the top row.Each vertical column depicts the number of clinical breast-cancers delivered (detected) in a specific year. The number detected during 1951 = 5 cases, and the number detected during 1960 has risen to 50 cases. During 1970, the incidence-rate of radiation-induced breast-cancer reaches 100 cases --- which is equal to the annual production-rate --- and the equality (100 cases) is shown to continue annually through the year 2006, because the annual production-rate was maintained through 1975. Thus, despite a radical change in initial latency period and delivery rate, the "Law of Equality" is validated by Figure-B.

Figure-C: The Third Demonstration

Figure-C deals with the serious problem of cancers delivered "from the shelf" very quickly (in

lessthan ten years). Again, we pretend that breast-irradiation first occurs in 1920. The total production of cases which will occur within ten years = 20 cases (not 100 cases). We choose a lower number just to indicate that such cases are outnumbered by cases with longer latency periods.

o -Each box in the grid represents 5 cases of clinical breast cancer.

o -Each horizontal row of 4 boxes represents 20 short-latency breast cancers -- the numberproducedby a single year of irradiation. So each row represents one year of production.

o -Each vertical column represents the number of radiation-induced short-latency breast cancers clinicallydetectedin a single year.

o -Bothshadedcolumns, and all the non-shaded columnsbetweenthe two shaded columns, have 4 vertical boxes. Such columns represent 20 clinicallydetectedshort-latency breast-cancers per year, in the years 1927 through 1979. These are the columns which demonstrate "the law of equality" under the conditions specified in the text, Part 3.

Figure - C "Law of Equality": Demonstration 3

In our Figure-C, delivery of clinical cases begins after an initial latency period of 3 years, and occurs at the rate of 5 cases per year, so the annual production-rate of 20 cases is delivered over only four years. For cases committed during 1920, delivery begins during 1924 and finishes during 1927. For cases produced during 1921, delivery begins during 1925 and finishes during 1928. The structure of Figure-C is comparable to Figures A and B.

During 1927, the situation is reached where the annual clinical incidence-rate of radiation-induced breast-cancer is equal to the annual production-rate of radiation-induced breast-cancer: 20 cases per year. And this equality continues year after year, as long as the annual production-rate is maintained. So, the "Law of Equality" is validated again. The "short-latency" cases obey the principle with no deviation.

Figure-D: The Key Demonstration

In Figure-D, we treat a far more complex type of delivery of cases "from the shelf." We call this "the key demonstration" of the law, because it so nearly approximates the real-world situation: A non-uniform rate of delivery. However, we want to emphasize that the delivery-pattern chosen for Figure-D is only illustrative of countless possible delivery-patterns.

o -Each box in the grid represents 5 cases of clinical breast cancer.

o -The total number of breast-cancersproducedby one year of irradiation = 100 cases. The annual production is delivered as follows: An initial 9-year latency period, followed by four successive years of 15 clinical breast-cancers per year (= 60 cases delivered), followed by eight successive years of 5 clinical cases per year (= 40 more cases delivered). Thus twelve different latency periods are depicted.

o -Each year of production has its own letter, so that readers can distinguish one year's commitment from the next year's commitment. Figure-D depicts 20 successive years of production (letters A through T).

o -Each vertical column represents the number of radiation-induced breast cancers clinicallydetectedduring a single year. Example: The column for Year-14 shows (5 cases from Year-1 irradiation, indicated by one "A") + (15 cases from Year-2 irradiation) + (15 cases from Year-3 irradiation) + (15 cases from Year-4 irradiation) + (15 cases from Year-5 irradiation). Sum = 61 clinical cases.

o -During Year-21, the annual clinical incidence-rate reaches 100 cases --- which is equal to the annual production rate --- and this equality (100 cases) lasts for nine years. These nine columns demonstrate the "law of equality" under the conditions specified in the text, Part 3.

Figure - D "Law of Equality": Demonstration 4

We return to the annual production-rate of 100 cases put "onto the shelf" --- every year. For delivery of each year's production, we use a nine-year initial latency period, with delivery beginning during the tenth year and distributed as follows:

Four successive years of 15 clinical breast-cancers per year (= 60 cases delivered), followed by eight successive years of 5 clinical cases per year (= 40 more cases delivered).

In Figure-D, each letter represents 5 cases of clinical breast-cancer. Each year of production has its own letter, so that readers can distinguish one year's commitment from the next year's commitment. For example, we can pretend that all the "A" boxes were produced during 1920, all the "B" boxes during 1921, etc. We use 20 different letters, representing 20 different production-years. The vertical columns have the same meaning as in the preceding figures: They depict the total incidence of radiation-induced breast-cancers delivered during a single year.

Examination of the strange-looking result shows that, during the 21st year, the annual clinical incidence-rate of radiation-induced breast-cancer reaches 100 cases --- which is equal to the annual production-rate --- and this equality (100 cases) continues year after year, for a total of nine years. During the years of constancy, each vertical column has a stack of 20 letters.

During the 30th year, the columns begin to lose height (letters) only because we arbitrarily stopped the annual production-rate. The last production-year represented by "T" boxes is Year-19. Delivery of "T" cases begins in Year-29 (after the latency). In Year-30, there are no "U" cases for delivery because none were produced. If we had

notstopped the steady, annual production-rate, the constant annual clinical-rate would have continued at 100 cases per yearindefinitely.So, Figure-D proves that the result becomes "neat," despite some interesting irregularities which occur in the annual incidence-rate

beforeit becomes constant. The "Law of Equality" is validated again.Part 4. The Final Significance of These Proofs

We have shown in Part 3 that various possible "exercises" each lead to a stable, constant, annual incidence-rate of radiation-induced clinical breast-cancers. The constancy endures as long as a constant annual production-rate endures among a population of constant size. It follows that any

combinationof constant incidence-rates will itself become a constant total rate, for all the possible "exercises" combined.This concept has great importance for handling a population of

mixedages, which we must do in our analysis.Suppose that 0-9 year-old children differ from the 10-19 year-olds in radiation-sensitivity, or in the delivery-times for radiation-induced cancer. And suppose these age-groups differ in sensitivity and latency periods from women in the age-group 20-29 years. We could go on describing numerous differences until every age-group is considered.

Despite this diversity, if breast-irradiation is initiated at a certain level into a female population of mixed ages and of constant size, and if irradiation is maintained at that level indefinitely, there necessarily will come a time --- several decades after the initiation --- when the combined annual incidence-rate of radiation-induced breast-cancer in this population becomes

equalto the combined annual production-rate of such breast-cancer, despite the mixed ages of the population. And this combined annual incidence-rate will endure indefinitely too, until the annual production-rate is altered.This means: If we can figure out the annual production-rate from radiation for a specific period of years, then we will know the annual incidence-rate from radiation which will occur decades later --- and thus we learn what fraction of the total breast-cancer problem in those later decades was caused by ionizing radiation.

"The simplest questions are the hardest to answer."

Northrup Fryeo -

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