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Appendix. How the standard Gail Model method was modified and implemented in the halls.md breast cancer risk calculator.

Here are the statistical details, the technical stuff, describing the inner workings of the halls.md Breast Cancer Risk Calculator. My summary is: I’ve made some modifications to the published Gail Model method, but my calculator works very well and it can do a few things that other calculators can’t.

Firstly, the standard Gail model

Table 1 shows the relative risk values for Gail model¹ in column X, as published in reference 1. These are also the same relative risks described for NSABP model 2 in reference 5. However, when I tried to implement a NSABP model2 calculator using column X figures, the results didn’t match the output of the NCI’s Breast Cancer Risk Assessment Tool. But when I modified the relative risk values to column Z, then my NSABP model2 calculator results closely emulated the results from the NCI‘s Breast Cancer Risk Assessment Tool.

Pretend I’m a statistician. Yes, (ahem), I see. Column X is the ‘published’ relative risk factors of the Gail Model, and Column Z is the slight modifications made by Dr. Halls, to get the results to match the NCI’s results.

The X and Z numbers are very close to each other anyway.

I suspect that my column Z’s relative risk values have incorporated the mathematical effects of h*I(t) and F(t) into them (and also into the absolute risk curves that I use), whereas those terms are handled by separate mathematical operations in the NCI’s Breast Cancer Risk Assessment Tool.

h*l(t) and F(t) were obfuscated in the original publication, perhaps deliberately omitted to prevent independent verification.

Back when I wrote this in year 2000, I was being polite. Now I can say more directly, Column X data from the original article, was incompletely published, and the column Z values, which I figured out, are what is required.

Note that B. Number of Biopsies column Z uses different values for lifetime risk calculation versus 5-year risk calculation, only for ages under 50, 1 or more biopsies. Although this seems peculiar, using these numbers helps my calculator emulate the NCI’s Tool’s output.

Table 1.	X	Z
A. Age at menarche in years
14 or older	1.00	1.000
12-13	1.10	1.098
11 or younger	1.21	1.188

B. Number of Biopsies
Age under 50 years		life	5-yr
0 biopsies	1.00	1.000	1.000
1 biopsy	1.70	1.278	1.683
2 or more	2.88	1.647	2.750
Age 50 years or older
0 biopsies	1.00	1.000
1 biopsy	1.27	1.237
2 or more	1.62	1.539

C. Number of first degree relatives with breast cancer
Age at first live birth < 20 years
0 relatives	1.00	1.000
1 relative	2.61	2.560
2 or more	6.80	6.168
Age at first live birth 20-24 years
0 relatives	1.24	1.240
1 relative	2.68	2.640
2 or more	5.78	5.318
Age at first live birth 25-29 years
0 relatives	1.55	1.550
1 relative	2.76	2.705
2 or more	4.91	4.591
Age at first live birth 30 + years
0 relatives	1.93	1.930
1 relative	2.83	2.779
2 or more	4.17	3.953

D. Biopsy with atypical hyperplasia?
Not applicable or unknown	1.00	1.00
No atypical hyperplasia	0.93	0.93
Yes, atypical hyperplasia	1.82	1.82

Wow, What are these numbers again?

Relative risks. 1.0 is the magic number meaning ‘the normal amount of risk’. 1.1 would mean “multiply 1.1 * the normal risk, and it becomes 10% higher.

So, relative risk of 1.1 means 10% higher risk than normal.

Next, are shown a series of additional relative risk modifiers. These came from different published studies, for example, about mammographic density, or about taking tamoxifen, or drinking alcohol, etc. If the studies were legitimate and big and provided good reliable relative risk values published, I encorporated them into the calculator.

Mammographic Density

E. Mammographic Density
Not Specified	1.000
0% fatty	0.405
1% to 24%	0.636
25% to 49%	1.000
50% to 74%	1.121
75% to 100%	1.761

The densest breasts have 1.761 relative risk compared to normal and the fatty breasts have the lowest number 0.405. Kindof makes sense.

Tamoxifen

Most people aren’t on this particular medicine, but for those who are, it reduces risk.

F. Taking Tamoxifen?
no	1.00
yes, age under 50 yrs	0.56
yes, age 50 to 60 yrs	0.49
yes, age over 60 yrs	0.45

LCIS – a biopsy that showed Lobular carinoma in situ

Another thing most people don’t have, a history of a suspicious biopsy with LCIS.

H. LCIS on biopsy?
Not applicable	1.00
biopsy at age < 40	8.70
biopsy age 40-44	6.80
biopsy age 45-49	5.60
biopsy age 50-54	4.50
biopsy age >54	3.10

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Alcohol use

Is drinking a lot of alcohol bad? Yes. But suprisingly, not at low amounts.

G. Alcohol use

Not specified 1.00

none 1.00

< 1.5 grams/d 1.07

< 5 grams/d 0.99

< 15 grams/d 1.06

<30 grams/d 1.16

<60 grams/d 1.41

60 or more 1.31

Birth Control Pills

I. Used Oral Contraceptives?
			Current user
Never used		1.00	Age at first use	< 20 yr	1.59
				20-24 yr	1.17
				25-29 yr	1.16
				>29 yr	1.25
Last use 1-4 years ago			Last use 10-14 years ago
Age at first use	< 20 yr	1.49	Age at first use	< 20 yr	1.13
	20-24 yr	1.15		20-24 yr	0.93
	25-29 yr	1.09		25-29 yr	1.06
	>29 yr	1.11		>29 yr	0.95
Last use 5-9 years ago			Last use 15 years or longer
Age at first use	< 20 yr	1.07	Age at first use	< 20 yr	1.14
	20-24 yr	1.09		20-24 yr	1.01
	25-29 yr	1.01		25-29 yr	1.01
	>29 yr	1.18		>29 yr	0.99

Here’s a simple formula. Risk A times Risk B times Risk C, etc, equals the total combined risk.

Is that legitimate?

It’s straight out of probability textbooks. To combine two probabilities, you multiply them together, and so on.

Summary Relative Risk is calculated using this formula:
Summary Relative Risk = A x B x C x D x E x F x G x H x I

For example, a 30 year old woman with menarche at age 12, one benign biopsy, no relatives with breast cancer, no atypical hyperplasia, mammographic density unknown, not taking tamoxifen, not drinking alcohol, no LCIS, no oral contraceptives:
Summary Relative Risk = 1.10 x 1.70 x 1.00 x 0.93 x 1.00 x 1.00 x 1.00 x 1.00 x 1.00 = 1.7391

After calculating summary relative risk, then the absolute risk is determined using polynomial equations. For example, the equation: Y = -0.0102*X² + 1.619*X + 0.1418 describes the black part of the curve shown below. For the Gail model1 calculator, there are separate curves for 10-year, 20-year and 30-year absolute risk, for ages 20, 30, 40 and 50. For the NSABP model2 calculator, there are separate curves for 5-year and lifetime absolute risk, for ages 20, 30, 40, 50, 60 and 70.

Is this stuff about risk curves, legitimate?

Yes. it came straight out of the publications. It’s how you convert a Summary relative risk, into an absolute risk. You look it up from a chart with a curved relationship between the X and Y axes.

breast cancer relative risk curve This example graph shows the absolute 10-year risk curve for a 30 year old (for Gail model¹) The black part of the curve comes from reference 1, and the gray part of the curve is my own estimate of the curve extended for higher relative risks, which might occur if many high risk modifiers were selected. (The vast majority of women will have Summary Relative Risk values within the black part of the curve). From this curve, if Summary Relative Risk was 20, then the Absolute Risk of developing breast cancer within 10 years, for a woman age 30, would be 28.4%.

The X-axis value of 20, and a Y value of 28.4%, are where the black curve is at.

For ages that fall in-between the decade values of 20,30,40,50,60, and 70, a simple linear equation scales the absolute risks. For example, a 35 year old women would be assigned an absolute risk value half-way between age 30 and 40.

After determining absolute risks using the curves available, linear equations estimate any missing values. For Gail model¹, the 5-year and lifetime risk are estimated by linear interpolation from the 10-year, 20-year and 30-year risks. For NSABP model², the 10-year, 20-year and 30-year risks are estimated by linear interpolation from the 5-year and lifetime risks.

Back to the actual Breast Cancer Risk Calculator.

References

Benichou J, Gail MH, Mulvihill JJ, Graphs to estimate an individualized risk of breast cancer. J Clin Oncol 1996; 14:103-110.
Gail MH, Brinton LA, Byar DP, Corle DK, Green SB, Chairer C, Mulvihill JJ, Projecting individualized probabilities of developing breast cancer for white females who are being examined annually. J Natl Cancer Inst 1989; 81:1879-1886.
Spiegelman D, Colditz GA, Hunter D, et al.: Validation of the Gail et al. model for predicting individual breast cancer risk. J Natl Cancer Inst 1994; 86:600-607.
Bondy ML, Lustbader ED, Halabi S, et al.: Validation of a breast cancer risk assessment model in women with a positive family history. J Natl Cancer Inst 1994; 86: 620-625.
Costantino JP, Gail MH, Pee D, Anderson S, Redmond CK, Benichou J, Wieand S, Validation studies for models projecting the risk of invasive and total breast cancer incidence. J Natl Cancer Inst 1999; 91:1541-1548.
Byrne C, Schairer C, Wolfe J, Parekh N, Salane M, Brinton LA, Hoover R, Haile R, Mammographic features and breast cancer risk: Effects with time, age and menopause status. J Natl Cancer Inst 1995; 87:1622;1629.
Fisher B, Costantino JP,Wickerham DL at al, Tamoxifen for prevention of breast cancer: Report of the national surgical adjuvant breast and bowel project P-1 study. J Natl Cancer Inst 1998; 90:1371-1388.
Spiegelman D, Colditz GA, Hunter D, Hertzmark E. Validation of the Gail et al. model for predicting individual breast cancer risk. J Natl Cancer Inst 1994; 86:600-607.
Smith-Warner SA, Spiegelman D, S Yaun, et al. Alcohol and breast cancer in women: A pooled analysis of cohort studies. JAMA 1998; 279:535-540.
Bodian CA, Perzin KH, Lattes R, Lobular Neoplasia. Long term risk of breast cancer and relation to other factors. Cancer 1996:78:1024-1034.
Collaborative Group on Hormonal Factors in Breast Cancer. Breast cancer and hormonal contraceptives: collaborative reanalysis of individual data on 53 297 women with breast cancer and 100 239 women without breast cancer from 54 epidemiological studies. Lancet 1996; 347:1713-1327.

These published medical journal articles should be available to the public in most medical libraries at medical schools, major hospitals, cancer treatment centers, etc.