Part 1. BreastCancer: ProductionRate vs. IncidenceRate
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 productionrate of breastcancers by radiation.
B. Annual deliveryrate (incidencerate) of radiationinduced clinical breastcancers.
C. The "Law of Equality," which refers to the situation when the annual clinical incidencerate is equal to the annual productionrate, despite the latency period.
Production versus Delivery: Cases on the "Shelf"
The breastcancers produced in a population, by radiation received during a single year, do not all occur exactly 3, 10, 20, 30, 40, or 50 years later. Rather, a certain number are committed in a single year, and this number is spread out clinically over 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 breastcancer. Some fraction of the females who received such lesions will later develop overt, clinical breastcancer.
We can think of a simple shelf. The irradiation received in a particular year puts "on the shelf" a certain number of future breastcancers. These are the produced breastcancers or committed breastcancers, from this one year of radiation exposure. At some later time, and over a period of many years, those produced breastcancers come "off the shelf" and are "delivered." By delivered, we mean that they become clinical breastcancers, or clinically detected breastcancers.
If readers just keep the distinction in mind between annual productionrate and annual deliveryrate, they will already be well advanced into understanding the method of our study.
Relationship of Annual IncidenceRate to Annual ProductionRate
Now we come to the third concept: Years when the annual incidencerate is equal to the annual productionrate, despite the 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 out what fraction of recent, current, and future breastcancer 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 incidencerate become equal to annual productionrate, 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 breastcancer.
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 breastcancer, excluding any cases which occur sooner than ten years after production.
The Various Rows of FigureA
The first demonstration corresponds with FigureA. Let us focus on the bottom row of boxes. Each box represents 2.5 cases of clinical breastcancer. The bottom row of 40 boxes represents 100 clinical breastcancers produced by irradiation during 1920. We will pretend that there was no breastirradiation before 1920, and that 1920 initiates the annual production of 100 radiationinduced breastcancers.
How do we distribute the cases produced during 1920? We know that the latency period is variable. For simplicity, FigureA arbitrarily shows 2.5 clinical cases observed per year, after an initial latency period of 10 years, (1921 through 1930), and this detectionrate 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 FigureA shows that, in 1921, the same thing occurs: The same number of 5yearolds (who are not the same children as those who were age 5 during 1920) receives the same dose (which puts another 100 breastcancers "on the shelf"). And FigureA, with its 56 rows, shows this productionrate continuing every year through 1975  an arbitrary "cutoff" date. Because of the initial latency period of ten years, the first delivery from 1975production occurs in 1986, as shown by the top row.
The Various Columns of FigureA
Each vertical column depicts the number of clinical breastcancers 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 incidencerate of radiationinduced breastcancer reaches 100 cases  which is equal to the annual productionrate  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 FigureA 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 incidencerate of radiationinduced breastcancer is equal to to annual productionrate of radiationinduced breastcancer, and this annual incidencerate will endure indefinitely, if we maintain the annual productionrate. And this occurs despite the variable latency period for the cases produced in a single year.
FigureA also shows what happens if the annual productionrate is not maintained. We arbitrarily made 1975 the last year of irradiation (with delivery beginning in 1986, after an initial latency). Although the annual productionrate goes suddenly to zero in 1976 and thereafter, the annual incidencerate of radiationinduced cases falls gradually (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 FigureA, 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 minimum latency 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 surely not exceed 50 years in women irradiated at age 55, because of the natural lifespan of humans.
Part 3. The Law's Validity under RealWorld 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 realworld conditions.
For example, the law applies no matter what the deliverypattern may be for the cancers committed during a single year of production. There was nothing "magical" about the pattern illustrated by FigureA. 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 mixed ages, when the cases committed by a single year's radiation exposure are delivered "from the shelf" at a nonuniform rate per year. This situation is a very close approximation to reality, and will constitute our final demonstration.
FigureB: 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 FigureB. Nonetheless, we make radical changes from FigureA 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).

Each box in the grid represents 5 cases of clinical
breast cancer.
 Each horizontal row of 20 boxes represents 100 breast cancers  the number produced by a single year of irradiation. So each row represents one year of production.  Each vertical column represents the number of radiationinduced breast cancers clinically detected in a single year.  Both shaded columns and all columns between the two shaded columns, have 20 vertical boxes. Such columns represent 100 clinically detected breastcancers 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 FigureB, each box represents 5 cases of clinical breastcancer (not 2.5 cases). The bottom row of 20 boxes represents 100 clinical breastcancers 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. FigureB, with its 56 rows, shows this productionrate continuing every year through 1975  an arbitrary "cutoff" date. Because of the 30year initial latency period, the first delivery from 1975production occurs in the year 2006, as shown by the top row.
Each vertical column depicts the number of clinical breastcancers 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 incidencerate of radiationinduced breastcancer reaches 100 cases  which is equal to the annual productionrate  and the equality (100 cases) is shown to continue annually through the year 2006, because the annual productionrate was maintained through 1975. Thus, despite a radical change in initial latency period and delivery rate, the "Law of Equality" is validated by FigureB.
FigureC: The Third Demonstration
FigureC deals with the serious problem of cancers delivered "from the shelf" very quickly (in less than ten years). Again, we pretend that breastirradiation 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.

Each box in the grid represents 5 cases of clinical
breast cancer.
 Each horizontal row of 4 boxes represents 20 shortlatency breast cancers  the number produced by a single year of irradiation. So each row represents one year of production.  Each vertical column represents the number of radiationinduced shortlatency breast cancers clinically detected in a single year.  Both shaded columns, and all the nonshaded columns between the two shaded columns, have 4 vertical boxes. Such columns represent 20 clinically detected shortlatency breastcancers 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 FigureC, 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 productionrate 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 FigureC is comparable to Figures A and B.
During 1927, the situation is reached where the annual clinical incidencerate of radiationinduced breastcancer is equal to the annual productionrate of radiationinduced breastcancer: 20 cases per year. And this equality continues year after year, as long as the annual productionrate is maintained. So, the "Law of Equality" is validated again. The "shortlatency" cases obey the principle with no deviation.
FigureD: The Key Demonstration
In FigureD, 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 realworld situation: A nonuniform rate of delivery. However, we want to emphasize that the deliverypattern chosen for FigureD is only illustrative of countless possible deliverypatterns.

Each box in the grid represents 5 cases of clinical
breast cancer.
 The total number of breastcancers produced by one year of irradiation = 100 cases. The annual production is delivered as follows: An initial 9year latency period, followed by four successive years of 15 clinical breastcancers 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.  Each year of production has its own letter, so that readers can distinguish one year's commitment from the next year's commitment. FigureD depicts 20 successive years of production (letters A through T).  Each vertical column represents the number of radiationinduced breast cancers clinically detected during a single year. Example: The column for Year14 shows (5 cases from Year1 irradiation, indicated by one "A") + (15 cases from Year2 irradiation) + (15 cases from Year3 irradiation) + (15 cases from Year4 irradiation) + (15 cases from Year5 irradiation). Sum = 61 clinical cases.  During Year21, the annual clinical incidencerate 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 productionrate of 100 cases put "onto the shelf"  every year. For delivery of each year's production, we use a nineyear initial latency period, with delivery beginning during the tenth year and distributed as follows:
Four successive years of 15 clinical breastcancers per year (= 60 cases delivered), followed by eight successive years of 5 clinical cases per year (= 40 more cases delivered).
In FigureD, each letter represents 5 cases of clinical breastcancer. 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 productionyears. The vertical columns have the same meaning as in the preceding figures: They depict the total incidence of radiationinduced breastcancers delivered during a single year.
Examination of the strangelooking result shows that, during the 21st year, the annual clinical incidencerate of radiationinduced breastcancer reaches 100 cases  which is equal to the annual productionrate  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 productionrate. The last productionyear represented by "T" boxes is Year19. Delivery of "T" cases begins in Year29 (after the latency). In Year30, there are no "U" cases for delivery because none were produced. If we had not stopped the steady, annual productionrate, the constant annual clinicalrate would have continued at 100 cases per year indefinitely.
So, FigureD proves that the result becomes "neat," despite some interesting irregularities which occur in the annual incidencerate before it 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 incidencerate of radiationinduced clinical breastcancers. The constancy endures as long as a constant annual productionrate endures among a population of constant size. It follows that any combination of constant incidencerates will itself become a constant total rate, for all the possible "exercises" combined.
This concept has great importance for handling a population of mixed ages, which we must do in our analysis.
Suppose that 09 yearold children differ from the 1019 yearolds in radiationsensitivity, or in the deliverytimes for radiationinduced cancer. And suppose these agegroups differ in sensitivity and latency periods from women in the agegroup 2029 years. We could go on describing numerous differences until every agegroup is considered.
Despite this diversity, if breastirradiation 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 incidencerate of radiationinduced breastcancer in this population becomes equal to the combined annual productionrate of such breastcancer, despite the mixed ages of the population. And this combined annual incidencerate will endure indefinitely too, until the annual productionrate is altered.
This means: If we can figure out the annual productionrate from radiation for a specific period of years, then we will know the annual incidencerate from radiation which will occur decades later  and thus we learn what fraction of the total breastcancer problem in those later decades was caused by ionizing radiation.
"The simplest questions are the hardest to answer."  Northrup Frye
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