Breast cancer and microbial cancer incidence in female populations around the world: A surprising hyperbolic association

Abstract

Current literature on cancer epidemiology typically discusses etiology of cancer by cancer type. Risks of different cancer types are, however, correlated at population level and may provide etiological clues. We showed previously an unexpected very high positive correlation between breast cancer (BC) and young-adult Hodgkin disease incidence rates. In a population-based case–control study of BC, older ages at the first Epstein–Barr virus exposure, indicated by older ages at onset of infectious mononucleosis, were associated with elevated BC risk. Here we examine BC risk in association with microbial cancer (MC) risk in female populations across the world. MC cancers are cervical, liver and stomach cancers with established causal associations with human papillomaviruses, hepatitis viruses, and helicobacter pylori, respectively. We examined age-adjusted BC and MC incidence rates in 74 female populations around the world with cancer registries. Our analysis suggests that BC and MC rates are inversely associated in a special mathematical form such that the product of BC rate and MC rate is approximately constant across world female populations. A differential equation model with solutions consistent to the observed inverse association was derived. BC and MC rates were modeled as functions of an exposure level to unspecified common factors that influence the 2 rates. In conjunction with previously reported evidence, we submit a hypothesis that BC etiology may have an appreciable link with microbial exposures (and/or immunological responses to them), the lack of which, especially in early life, may elevate BC risk. © 2008 Wiley-Liss, Inc.

The etiology of cancer is usually considered separately by the type of cancer. Doll and Peto, in their seminal publication on causes of cancer in 1981,1 stated: “Gradually, it has come to be realized that agents or habits which greatly increase or decrease the likelihood of 1 particular type of cancer arising (in humans or experimental animals) may have little effect on most other types of cancer, so that the prevention of each type also must be considered separately. This realization reinforces the need to consider cancers of different organs as largely independent diseases, just as we have to consider separately different infectious diseases such as syphilis, smallpox, and tuberculosis.” Consistent with Doll and Peto, the current literature on cancer causes and epidemiology typically discusses etiology of cancer by cancer type.

In this article, we consider 2 different types of cancer jointly as a way of providing new insights into etiology. If 2 types of cancer show a certain pattern of risks, where the causes are largely known for 1 type but unknown for the other, discovery of the relational pattern between the 2 should provide new etiologic clues.

Previously, we showed a very high positive correlation between breast-cancer incidence rates and young-adult Hodgkin disease incidence rates in female populations around the world and within the United States.2 The strong positive association between the epithelial cancer and the lymphoma had not previously been described and was higher than any other pairs of cancer types that are well known to be correlated at the population level (e.g., colon and breast cancer; cervix and stomach cancer). Here, we show that female breast-cancer incidence rates have a surprising hyperbolic association with microbial-cancer incidence rates at the population level. The implications of this finding are discussed in view of generating novel hypotheses on cancer etiology.

Material and methods

Incidence data

Throughout this article, we refer to stomach, liver, and cervix cancer as microbial cancer, because their etiology involves known infectious agents: Helicobacter pylori for stomach cancer; hepatitis-B virus for liver cancer; and specific human papillomaviruses for cervix cancer.3, 4 To gain insight into the relation between incidence rates of MC and those of BC in females, we used data in the most recent 2 volumes, 7 and 8, of Cancer Incidence in 5 Continents (CI5),5, 6 published by the International Agency for Research on Cancer. The CI5 reports incidence data in almost all functional population-based cancer registries covering areas within Africa, Americas, Asia, Europe, and Oceania. Volumes 7 and 8 provided data for over 180 and 200 populations worldwide, respectively, generally for the 5-year time periods of 1988–1992 and 1993–1997, respectively. Age-specific incidence data are given by tumor site. Rates for cancer in volumes 7 and 8 were based upon the ICD-97 and ICD-108 site classification, respectively, and we extracted the four cancer types of interest using the following ICD codes: breast (ICD-9 174, ICD-10 C50); stomach (ICD-9 151, ICD-10 C16); liver (ICD-9 155, ICD-10 C22); and cervix uteri (ICD-9 180, ICD-10 C53).

Our analysis included only those populations for which data are available in both volumes 7 and 8 to enable us to calculate incidence rates for a 10-year time interval across the periods reported in the 2 volumes, 1988–1997 for most populations. Further, to obtain statistically stable estimates of incidence rates, we considered only those populations that have a female population size greater than or equal to 500,000. New Mexico and Uruguay, Montevideo were excluded from the analysis because of data problems described in the appendix to this article. With these inclusion/exclusion criteria, 74 female populations remained and were analyzed: 2 from Africa; 22 from Americas; 19 from Asia; 25 from Europe; and 6 from Oceania.

Statistical analysis

For each of these 74 populations, we calculated the age-standardized breast, stomach, liver, and cervix cancer incidence rates for the time interval that combines the reported time periods of the 2 volumes, using the standard world population given in the CI5 as the reference population for the age distribution.5, 6 For each population, we added the stomach, liver, and cervix cancer rates and we refer to the sum of the 3 rates, as the age-standardized MC incidence rate.

To assess the functional relationship between age-standardized BC rates and MC rates, we fitted 3 models on the 74 pairs of incidence rates: (i) a linear model y = a + bx with 2 parameters, slope b and intercept a; (ii) a hyperbolic model xy = C with 1 parameter C; and (iii) a natural-cubic-spline model y = β₀ + β₁x + β₂(x − k₁)³₊ + β₃(x − k₂)₊³ + β₄(x − k₃)³₊ + β₅(x − k₄)₊³ with 4 knots k₁, k₂, k₃, k₄ placed at 4 quantiles of the BC rates9, 10 with 4 β parameters. The model (iii) with a natural cubic spline has versatility in approximating a wide variety of nonlinear functions. This model was considered here to give (approximately) the best smooth functional fit to the data. The other 2 models, the linear (i) and hyperbolic (ii) models, were considered in order to assess whether a simpler functional form suffices as a description of the relationship between the age-standardized BC rates and MC rates. Specifically, if the scatter plot of the (BC, MC) rates lies approximately on a straight line, the fit of the linear model (i) would be close to the approximately-best functional fit given by the model (iii). If the scatter plot of the (BC, MC) rates lies approximately on a rectangular hyperbola, on the other hand, the fit of the hyperbolic model (ii) would be close to the approximately-best functional fit given by the model (iii).

All 3 models were fitted based on the same principle: minimizing the sum of the squared Euclidian distances from 74 points of our data to the model. The Euclidian distance between a point (p,q) and a curve y = f(x) is the shortest distance from the point to some point on the curve, i.e. . We chose this two-dimensional least-square (Euclidian distance) criterion as opposed to the standard least-square criterion on the y-axis direction because the 2 variables, the BC rates and the MC rates, have exchangeable roles and are not in a predictor–response relation as in a standard regression analysis. We did not assign any weight to the Euclidian distance of each point based on, for example, the size of the population to which the point corresponds. Under a stochastic model for cancer incidence, rates for small populations have larger variances than rates for large populations and, therefore, weighting according to the size of each population would be appropriate. In the CI5 case, however, stochastic variation is only 1 part of the large variation in rates (e.g., the quality of cancer registration is another); as noted, all populations studied comprised at least 500,000 women. To make statistical inferences on the fits of the 3 models, we used a bootstrap procedure with 5000 iterations.11

In addition, under a set of assumptions on the 2 types of cancer incidence rates at the population level, we derived a differential-equation model, the solutions to which are consistent with the hyperbolic model. The reason for considering the differential-equation model was to derive BC and MC rates as functions of the level of exposures to unspecified factors. By solving the differential-equation system with a set of initial value conditions derived from the data on the 74 populations, theoretical BC and MC rates were obtained as functions of the putative exposure level. These theoretical BC and MC rates were plotted, together with the observed rates, against an estimated value of the putative exposure level for each of the 74 populations; the exposure level was estimated by minimizing the sum of squared relative errors between the theoretical and observed rates of BC and MC. These estimates of the exposure levels are tabulated with the observed BC and MC rates (see Table II).

To further test the likelihood that any relationship found might be due to chance or some trivial property of the datasets or their relationships, we undertook some additional analyses: first, we replaced the dataset on BC with comparable data on colon cancer; second, we replaced the summed rates of MC (cervix, stomach, and liver) by the sum of colon, lung, and pancreas cancer rates; and third, we examined the relationship between BC and each of the 3 microbial cancers separately.

Results

Figure 1 shows the scatter plot of the BC and MC rates of the 74 female populations. Contrary to our initial expectation, the model with the smallest sum of squared Euclidian distances to the 74 points was the hyperbolic model with a sum of 3173.8, followed by the cubic spline model with 3447.1. The linear model had a substantially poorer fit with 5330.9. The parameter estimates of the 3 models are given in Table I. The plot of the hyperbola is depicted in Figure 1.

Figure 1. The scatter plot of the sum of cervix, liver and stomach cancer rates versus breast cancer rate for 74 world female populations. Rates (per 100,000 person-years) are age-standardized to the world population. The fitted hyperbola that minimizes the sum of squared Euclidian distance from the 74 points is drawn.

Table I. The Fit of the Three Models: Linear, Hyperbolic, and Natural Cubic Spline, to Breast Cancer and Microbial Cancer Rates in the 74 World Female Populations

Model	Estimated parameter values	Sum of squared Euclidian distances	Difference of the squared Euclidian distances from hyperbola	95% Confidence interval of the difference from hyperbola	p-value vs. hyperbola
Hyperbolic, xy =C	C= 1106.55	3173.8	–	–	–
Linear, y =a +bx	a = 53.83, b = −0.50	5330.9	2157.1	(194.9, 4568.3)	0.026
Natural cubic spline, y = β0 + β1x + β2 (x − k1)+^3 + β3(x − k2)+^3 + β4 (x − k3)+^3 + β5 (x − k4)+^3 Knots—k1 = 19.99; 5% ile, k2 = 41.52; 35% ile, k3 = 74.66; 65% ile, k4 =94.12; 95% ile	β0 = 1.00 × 10^2	3447.1	273.3	(−1609.1, 2805.0)	0.72
	β1 = −2.01 × 10^0
	β2 = 7.03 × 10^−4
	β3 = −1.31 × 10^−3

Bootstrap-based 95% confidence intervals for the differences in the sum of the squared Euclidian distances between models are shown in Table I, taking the best-fitted hyperbolic model as the reference. The fit of the hyperbola is statistically significantly better than the fit of the linear model with a p-value of 0.026, but the fits of the hyperbola and the cubic spline are comparable.

One possible explanation for the observed hyperbolic relation between BC rate and MC rate can be given by a differential equation model described below.

Let f be the level of exposure to unspecified risk factors for BC. Denote by r_B(f) and r_M(f) the rates of developing BC and MC, respectively, in a female population exposed to factors characterized by f. Assume that each individual of the population is in 1 of 3 states: BC; MC; or healthy.

The rates r_B(f) and r_M(f) are by definition the ratios of BC cases K_B(f) and of MC cases K_M(f), respectively, to the total person-years at risk T. We consider 2 populations with exposure levels of f and f + Δf, where Δf is a small positive increment. If the BC risk-factor increases level from f to f + Δf, then we observe an increase in the number of BC cases and a decrease in the number of MC cases. It is reasonable to assume that: (i) the increase K_B(f + Δf) − K_B(F) in BC cases is due only to the individuals that are healthy when exposed to f and develop BC when they are exposed to f + Δ f; (ii) the increase in BC cases is proportional to the number of healthy individuals living at exposure f, the BC rate r_B(f) caused by the exposure f, and the exposure increment Δf; (iii) the decrease K_M(f) − K_M(f + Δf) in MC cases is due only to the individuals that develop MC when exposed to f but would be healthy when exposed to f + Δf; (iv) the decrease in MC cases is proportional to the number of individuals that are healthy when exposed to f + Δf, the MC rate r_M(f + Δf) caused by the exposure f + Δf, and the exposure increment Δf. These assumptions are summarized by the equations below

• 

These equations can be modified by dividing both sides by TΔf

• 

When the increment, Δf, converges towards 0 we obtain 2 differential equations that describe the changes of BC rate and MC rate with respect to f.

• (1)

Any solution of this system has the property that the product, r_B(f)r_M(f), is constant, as we can see from the computation below

• 

Following the steps shown in Appendix B, the solution of the differential system [1] is given by:

• 

where C = r_B(0)r_M (0) and

• 

We investigated how well the solution of the system [1] fits our data from 74 pairs of cancer incidence rates. For our analysis, we choose the solution of system [1] that has the initial condition: r_B(0) = 9.99 per 100,000 person-years, the lowest BC rate in the dataset and r_M(0) = C/r_B(0) with C = 1106.55 per 10¹⁰ squared person-year, the estimated value obtained previously (Table I). For a population P, with the observed BC rate r_B(P) and MC rate r_M(P), we estimated the corresponding exposure level f_P by taking the value of f that minimizes the sum of squared relative errors . We estimated the exposure level f for each of the 74 populations considered (Table II). Figure 2 shows the graphs of the solutions r_B(f) and r_M(f) of the system [1], together with the observed BC and MC rates versus the estimated levels of exposure. The observed BC rates and MC rates are in reasonable agreement with the solution of the differential system [1].

Figure 2. The plot of the solutions of the differential system [1] with initial values r_B(0) = 9.99 and r_M(0) = 110.77 per 100,000 person-years [the blue line is the breast cancer (BC) rate and the red line is the microbial cancer (MC) rate]. The observed age-standardized BC rates (blue crosses) and MC rates (red crosses) are shown according to the estimated exposure level of the BC risk factors.

Table II. Observed Age-Standardized Breast Cancer (BC) and Microbial Cancer (MC) Rates (Per 100,000 Person-Years) and Estimated Level of Exposures f for 74 World Female Populations

Population	Age-standardized BC rate	Age-standardized MC rate	Estimated level of exposures
Zimbabwe, Harare: African	20	81.63	0.32
Thailand, Khon Kaen	9.99	57.24	0.62
Japan, Hiroshima	34.97	59.4	0.72
Costa Rica	31.24	56.22	0.75
China, Qidong County	10.62	47.27	0.76
Uganda, Kyadondo County	20.35	49.98	0.78
Japan, Yamagata Prefecture	25.52	49.87	0.82
Ecuador, Quito	26.53	49.76	0.84
Japan, Nagasaki Prefecture	28.45	47.7	0.89
Thailand, Chiang Mai	15.46	39.6	0.91
Japan, Osaka Prefecture	26.11	45.19	0.91
India, Chennai	23.67	41.7	0.95
Brazil, Goiania	46.41	51.93	1.05
Japan, Miyagi Prefecture	32.13	41.31	1.05
India, Bangalore	21.04	31.03	1.08
China, Shanghai	26.75	31.59	1.14
Viet Nam, Hanoi	19.98	23.7	1.16
Belarus	31.34	32.86	1.18
China, Tianjin	24.4	22.83	1.22
Poland, Kielce	29.3	26.39	1.24
China, Hong Kong	35.1	32.05	1.25
India, Mumbai	28.53	23.98	1.26
Poland, Lower Silesia	35.72	31.77	1.26
Estonia	38.97	32.2	1.31
Singapore: Chinese	42.36	33.9	1.35
Latvia	36.54	24.9	1.36
Slovakia	40.49	29.49	1.37
Kuwait	32.77	11.91	1.44
Croatia	42.43	25.63	1.45
Yugoslavia, Vojvodina	46.54	29.83	1.48
Philippines, Manila	52.69	37.23	1.52
Czech Republic	49.11	28.07	1.55
Slovenia	48.7	25.34	1.56
Poland, Warsaw city	49.09	24.18	1.58
Spain, Asturias	47.47	18.72	1.6
Spain, Murcia	47.35	16.09	1.61
USA, Puerto Rico	49.83	19.95	1.62
USA, California, Los Angeles: Hispanic White	59.78	28.76	1.74
Norway	58.79	19.32	1.77
Germany, Saarland	66.33	21.56	1.88
Italy, Florence	69.93	24.03	1.92
Finland	68.5	14.2	1.93
Switzerland, Zurich	69.7	13.6	1.95
UK, Scotland	74.2	19.93	1.99
Italy, Venetian Region	74.41	19.4	2
Australia, New South Wales	74.22	13.99	2.01
Australia, Victoria	74.29	14.36	2.01
Sweden	74.83	14.78	2.02
Canada, Quebec	74.56	12.82	2.02
Australia, South	74.78	12.48	2.02
New Zealand	76.3	17.57	2.03
Denmark	77.36	20	2.03
USA, Utah	75.85	10.76	2.04
Australia, Western	77.22	15.63	2.04
Canada, Saskatchewan	77.15	12.12	2.05
Canada, Alberta	78.48	12.79	2.07
Canada, Ontario	78.62	13.25	2.07
USA, Michigan, Detroit: Black	81.05	19.77	2.08
USA, Hawaii	81.01	17.43	2.09
USA, California, Los Angeles: Black	81.57	18.88	2.09
France, Bas-Rhin	81.15	15.99	2.09
Canada, Manitoba	81.74	13.29	2.1
Canada, British Columbia	81.53	11.69	2.1
The Netherlands	83.05	13.11	2.12
USA, Iowa	84.57	10.81	2.14
France, Isere	85.4	14.73	2.14
USA, Louisiana, New Orleans	86.59	15.15	2.16
Israel: Jews born in Israel	89.35	11.22	2.19
USA, Michigan, Detroit: White	89.76	12.02	2.2
USA, Georgia, Atlanta: White	93.96	10.67	2.24
USA, Washington, Seattle	94.43	11.56	2.25
USA, Connecticut: White	95.51	11.49	2.26
USA, California, Los Angeles: Non-Hispanic White	103.4	11.66	2.34
USA, California, San Francisco: Non-Hispanic White	106.44	10.11	2.37

When we examined the relationship between MC and colon cancer as a sensitivity analysis of our findings on BC and MC cancer, we found that the relationship does not fit well with the hyperbola or the differential equations consistent with the hyperbola (see Supplementary Material, Figs. 12). In addition, when we examined the relationship between BC and the sum of colon, lung, and pancreas (rather than MC), we found that replacing MC with some arbitrary 3 cancers results in a very poor fit (see Supplementary Material, Figures 3, 4). Using each of the 3 MCs separately (instead of all 3 together) with BC cancer resulted in fairly good, but poorer than the sum of the 3 MC cancer types, fits (see Supplementary Material, Figures 5–10).

Discussion

Our analysis of the CI5 data showed that the product of BC rate and MC rate is approximately constant across world female populations. Note, again, that this is not just any inverse relationship: it is a hyperbolic association. Two epidemiologic implications of this specific form of association, derived mathematically, are as follows. Denote by r_B(P) and by r_M(P), the risks of developing BC and MC, respectively, in a female population P. Consider 2 female populations P and Q; note that P and Q are the names of the populations and not the levels of the factors to which the populations are exposed. The first implication of the hyperbolic relationship is that r_B(P)/r_B(Q) =r_M(Q)/r_M(P), or that the relative risk of BC in Population P versus Population Q is identical with the relative risk of MC in Population Qversus Population P. This holds for any 2 female populations. The second implication is that

• 

or that the population attributable fraction12–14 of BC in Population P is identical to that of MC in Population Q. That is, the proportion of BC risk attributable to the effects of risk factors, environmental and/or genetic, in Population P is equal to the proportion of MC risk attributable to the effects of risk factors in Population Q. This, also, holds for any 2 female populations.

The hyperbolic association of BC and MC rates at the population-level, characterized by the very specific forms of relative risk and attributable fraction relationships described above, naturally leads us to consider the level of factors (“exposures” in epidemiology) f the increase of which elevates BC risk, but lowers MC risk, in the manner that is consistent with the special mathematical form of association. Given that the known causes of MC are viruses and bacteria, it may be inferred that 1/ f is an indicator of the risk of specific (or general) infectious diseases in a population. The fact that BC rates are positively associated with f in the manner depicted in Figure 2 is consistent with some hypotheses proposed previously for BC. For example, Richardson15 proposed that BC risk is associated with late exposure to common viruses and Yasui et al.2 provided supporting evidence for the hypothesis that “delayed” exposure to Epstein–Barr virus is a risk factor for BC. Although our results do not specify what f is, we provide estimates of f for the 74 populations in Table II, which might be useful in further identifying the as-yet-unspecified risk factor(s).

The concept of “exposures f” needs further discussion. Certainly, r_B(f)r_M(f) is highly unlikely to represent the level of a single factor. Further, it does not need to be a set of factors that work in the same directions (i.e., increasing or decreasing BC or MC risk). For example, a factor X1 may increases BC risk while having no effect on MC risk, and a factor X2 may decrease MC risk while having no effect on BC risk. For the BC and MC rates to exhibit the hyperbolic association, however, the X1 and X2 must follow a very specific form of correlated changes.

One biologic hypothesis is sketched in Figure 3, where consideration is given to the difference between early and late exposure to specific (or general) microbial agents. With lower hygiene and greater crowding, there are higher rates of early childhood exposure to specific agents, resulting in microbial tolerance, and chronic infection/inflammation, eventually leading to oxidative DNA damage and subsequent MCs. With improved hygiene and reduced crowding, the exposure to agents is generally later in childhood/adolescence. In this setting, specific immune responses may be mounted to the agents, leading, in some instances, to a differentsource of inflammatory responses and DNA damage in different organs.

Figure 3. Possible biological mechanisms that can increase either microbial cancer risk or breast cancer risk.

APPENDIX A

Two registries, New Mexico and Uruguay, Montevideo stood out as having major errors in their reported cancer rates, and consequently we excluded them from our analysis. New Mexico reports cancer incidence for the entire population of the state, and also separately for its Hispanic white, non-Hispanic white and American Indian populations. When we added the person-years reported in volume 8 for the Hispanic white, non-Hispanic white and American Indian female populations of the state, we found that the total person-years are half of the person-years reported for the entire female population of the state. This is in sharp contradistinction to the ethnic breakdown of New Mexico population displayed in the hard-copy of volume 8, which shows that the majority of New Mexico population is formed by the 3 ethnic groups. For this reason, we believe that the age-standardized cancer rates reported for women of New Mexico are about half the true values. The age-standardized BC rate for women of Uruguay, Montevideo registry is the highest reported in volume 8, higher than the BC rate of populations of United States. Because women in the United States are known to have 1 of the highest BC rates and because volume 8 warns that, for the Uruguay registry, a high proportion of BC cases are diagnosed on clinical imaging only, it appears likely that BC cases are markedly over-reported and therefore the rates are erroneous.

APPENDIX B

The system [1] can be solved analytically. The fact that the product r_B(f)r_M(f) is constant is crucial for the solution, because it allows us to simplify and integrate the differential equations. In the literature on equations of classical mechanics, a constant function like r_B(f)r_M(f) that involves the variables of the system is called an “integral of motion” for the system.

Denote by C the constant value of r_B(f)r_M(f). The constant C depends only on the initial conditions of the system and its value is the product r_B(0)r_M(0) of the 2 cancer rates if the factor f is 0. The rates r_B(0) and r_M(0) are positive numbers smaller than 1. Also the sum r_B(0) + r_M(0) is smaller than 1 because each individual is classified as either MC, BC, or healthy. From these inequalities, it follows that C = r_B(0) r_M(0) = r_B(0)(1 − r_B (0)) <1/4.

We use this information to write r_M(f) = C/r_B(f), and to find a differential equation involving only r_B(f), i.e. dr_B/df = (1 − C/ r_B)r_B. This equation is separable and we can integrate it by separating all the terms that contain r_B on 1 side and the terms that contain only f on the other side.16 This is a standard method to solve separable equations. We integrate both sides, writing the left hand side in the appropriate form for integration

• 

Then,

• (2)

where K is a constant that depends only on the initial conditions of the system r_B (0), r_M(0). It was crucial to integrating the equation that C is less than 1/4.

To be able to solve for r_B(f) in equation [2], we need to know the sign of the fraction under the absolute value sign. This is established based on the properties of the initial conditions r_B(0), r_M(0). The fraction evaluated at f = 0 is positive and this follows from the inequality 0 < r_B(0) (1 − r_B(0) − r_M(0)). Because of continuity, when f increases we do not expect a change of sign of the fraction. This says that, at least for small values of f, the fraction remains positive.We conclude by writing the solution of the differential system [1]

• 

where and

•