Abstract
Mathematical methods exist to determine the fractions of sex hormones bound to albumin, bound to sex hormone binding globulin (SHBG), or unbound, using total hormone concentration and SHBG concentration. We used data from eight prospective studies of postmenopausal women to assess the validity of these estimates for fractions of estradiol (E2) and to investigate the impact of using calculated values in breast cancer relative risk (RR) models. Comparisons were made between measured and calculated concentrations of free and nonSHBGbound E2 in four studies. Relationships between the hormone fractions were investigated and a sensitivity analysis of the calculation performed. Breast cancer RRs were estimated using conditional logistic regression by quintiles of free E2. There is a high correlation (r > 0.91) between calculated and measured values of both free and nonSHBGbound E2. The calculation is highly sensitive to total hormone concentration but is relatively insensitive to SHBG concentration. In studies with both measured and calculated values, the RRs of breast cancer by quintile of free E2 were almost identical for both estimates; using calculated values in all possible studies the RR in the highest compared with the lowest quintile of free E2 was 2.29 (95% confidence interval, 1.65–3.19). The mathematical method used to calculate fractions of E2 is valid, and RR analyses using calculated values produce similar results to those using measured values. This suggests that for epidemiological studies, it is only necessary to measure total E2 concentration and SHBG concentration, with hormone fractions being obtained by calculation, producing savings in cost, time, and serum.
Introduction
A recent collaborative reanalysis using individual data from prospective studies has demonstrated significant relationships between concentrations of endogenous sex hormones and breast cancer risk in postmenopausal women (1) . Sex hormones generally circulate bound to plasma proteins with only a very small fraction circulating unbound (between 1 and 5%). The majority of the protein binding for estradiol (E2) is provided by albumin and sex hormone binding globulin (SHBG). Initially, it was hypothesized that it was only the free (unbound) fractions of sex hormones that were biologically active (2) . Additional work (3 , 4) showed that sex hormones have a much lower binding affinity for albumin than for SHBG and that the fractions of these hormones bound to albumin are readily dissociable, whereas the fractions bound to SHBG are not, suggesting that the sum of free and albuminbound fractions of these hormones might be classed as bioavailable.
Assays are available to measure not only the overall concentration of hormones but also the concentrations or proportions of free and bound fractions. A number of authors (2 , 3 , 5, 6, 7) have proposed simple mathematical methods based on the law of mass action to calculate the various hormone fractions using total hormone concentration, SHBG concentration, and albumin concentration. Previous validation studies (6 , 8) have demonstrated the usefulness of these mathematical methods albeit with small sample sizes. The aims of this article are 2fold. First, we examine the validity of the mathematical calculation method for both free E2 and nonSHBGbound E2 in a large sample of postmenopausal women and investigate the sensitivity of the calculation. Second, we examine the use of calculated versus measured hormone fractions in breast cancer relative risk (RR) models.
Materials and Methods
Data Collection.
Data from nine prospective epidemiological studies investigating the relationship between endogenous hormones and breast cancer risk in postmenopausal women have been brought together by the Endogenous Hormones and Breast Cancer Collaborative Group (1) . Eight of these studies (9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19) contribute to the current analysis, and collaborators were asked to provide data on concentrations of hormones (measured by a number of different assay methods; Table 1⇓ ), reproductive, and anthropometric factors for each woman in their study. Women who were using hormone replacement therapy or other exogenous sex hormones at the time of blood collection were excluded.
After preliminary inspection of the data, it was noted that a small number of hormone concentrations were implausibly high or low, and a general exclusion criterion on the basis of the distance of extreme values from the interquartile range (IQR) was adopted. This type of criterion ensures that only very extreme values are excluded and does not enforce the automatic exclusion of observations. A hormone concentration was considered to be valid if it was in the range (Q_{1} − 1.75 × IQR, Q_{3} + 1.75 × IQR), where Q_{1} is the first quartile and Q_{3} is the third quartile of the studyspecific log hormone concentrations. The multiplier of 1.75 was chosen such that ∼1% of the available data for each hormone were excluded.
Calculation of Hormone Fractions.
The calculation of E2 fractions (free, SHBGbound, and albuminbound) was based on the following equation derived from the law of mass action (2) : where [E2] is total E2 concentration, [E2_{F}] is free E2 concentration, [SH] is SHBG concentration, [A] is albumin concentration, and K_{E}^{A} and K_{E}^{SH} are association constants for the binding of E2 to albumin and E2 to SHBG, respectively. Free E2 is found by numerical solution of 1.1, albuminbound concentration can then be calculated as follows where [E2_{A}] is albuminbound E2 concentration. NonSHBGbound E2 concentration is found by summing the calculated concentrations of free E2 and albuminbound E2.
We assumed the following association constants as used previously (2 , 3) : K_{E}^{A} = 6 × 10^{4}; and K_{E}^{SH} = 0.68 × 10^{9}. A fixed albumin concentration of 4 g/dl (equivalent to 5.97 × 10^{−4} mol/liter) was assumed; from equation 1.2 , it is apparent that calculated nonSHBGbound E2 concentration is therefore simply a multiple of calculated free E2 concentration. It was assumed that there were two binding associations for E2, one between E2 and SHBG and the other between E2 and albumin. We did not allow for competition between E2 and testosterone (or any other sex hormone) for binding sites on SHBG or albumin following previous work (8) , which has shown that such additional complexity has a negligible effect on the calculated values (2) .
Statistical Analysis.
Because neither measured nor calculated values of the hormone fractions were normally distributed, Spearman correlations were used to measure the correlation between measured and calculated values. Linear regression was used to model the relationship between logtransformed measured and calculated fractions of the hormones, and a BlandAltman plot (20) was used to assess the extent of agreement between the two measures. As there were significant betweenlaboratory variations in mean hormone concentrations, we used percentage of total hormone concentration rather than absolute values to allow direct comparisons of hormone fractions between different studies.
Sensitivity to changes in total E2 concentration and SHBG concentration was assessed by replacing either measured values of total E2 or measured values of SHBG by a common fixed concentration in the calculation. The calculated free E2 from each of these changes was then compared with measured free E2 by Spearman correlations.
Conditional logistic regression was used to calculate the RRs of breast cancer (and their 95% confidence intervals) in relation to fifths of free E2 using studyspecific cut points defined by the quintiles of free E2 among the control subjects. Linear trends in RRs were assessed by χ^{2} tests.
All analyses were performed using the data analysis language R (21) .
Results
Validity of Calculation Method.
Four of the nine studies (9 , 10 , 12 , 17, 18, 19) allowed comparisons between measured and calculated free E2 in a total of 1055 women. There is a very high correlation (r = 0.96) between the measured values and the calculated values of free E2 (Fig. 1)⇓ , and the degree of agreement illustrated by the BlandAltman plots is very high and unbiased. Four of the nine studies (10 , 12 , 16 , 17) allowed comparisons between measured and calculated nonSHBGbound E2 in 1277 women. Although the calculated values for nonSHBGbound E2 are still highly correlated (r > 0.91) with the measured values, the absolute values differed systematically in studies using an ammonium sulfate precipitation assay to determine nonSHBGbound E2 concentrations (Fig. 1)⇓ . The median values for measured and calculated percentage of free and percentage of nonSHBGbound E2 by study are given in Table 1⇓ . The calculated values of free and nonSHBGbound E2 are similar across all studies.
Sensitivity of Calculation.
The relationship between total E2 and measured free E2 is surprisingly linear considering that SHBG concentration, as well as total E2 concentration, affects the concentration of free E2. The observed correlation between free and total E2 (r ≥ 0.93 for each study) is highly positive, whereas the correlation between total E2 and SHBG is slightly negative (r ≤ −0.09), with a modest negative correlation between free E2 and SHBG (r ≤ −0.26). As a consequence of these relationships, it is apparent that the effect of SHBG on the determination of free E2 in the calculation will be only modest, with total E2 concentration dominating. The impact of substituting a constant value of SHBG for individual measured values does not alter the Spearman correlation between measured and calculated free E2 (r = 0.96). However, substituting a constant value of total E2 for individual measured values results in the Spearman correlation between measured and calculated free E2 falling to 0.32.
Associations between Hormone Fractions and Breast Cancer Risk.
In the four studies where it was possible to obtain both a calculated and measured value of free E2, estimates of the RR of breast cancer by quintiles of measured and calculated free E2 were nearly identical across all of the quintiles (results not shown).
Table 2⇓ shows the RR of breast cancer by quintile of calculated free E2 in all eight studies with a total of 632 cases and 1544 controls. The RRs are in close agreement with those obtained previously (1) for measured free E2 where the RR in the top quintile in the four studies where a measurement was made was 2.58 (95% confidence interval, 1.76–3.78).
Discussion
This study has confirmed the validity of using a simple mathematical formula for determining the unbound and bound fractions of E2 in a large sample of postmenopausal women, with the calculated values comparing well with those determined by a range of measurement techniques (22, 23, 24) . It has been shown that values calculated using this simple method correlate closely with measured values of free and nonSHBGbound E2 (r > 0.91 for all comparisons) using a range of assay methods. The calculated values for free and nonSHBGbound E2 are relatively insensitive to variations in SHBG concentration but highly sensitive to variations in total E2 concentration. This complements previous work (8) , which showed that using a random value of SHBG uncorrelated with an individual’s true level, Spearman correlations between calculated and measured free E2 remained very high. It is possible that this calculation method is equally valid for use in premenopausal women; however, with premenopausal women having much higher concentrations of E2, further work is needed to verify this.
Studies using the ammonium sulfate precipitation (2 , 25) method to determine nonSHBGbound E2 gave concentrations that were lower than the calculated concentrations, whereas studies using other methods gave broadly similar values (Table 1)⇓ . In a subset of 112 of the controls in the Nurses’ Health Study (12) , two assays were used to determine percentage of nonSHBGbound E2—the ammonium sulfate precipitation method and the Sepharose method (26) ; the Spearman correlation between the two sets of results was 0.91 (12) . Clearly, the method chosen affects the absolute concentrations of nonSHBGbound E2, but our analyses confirm that the ranking of individuals is virtually unaffected (Fig. 1)⇓ .
The impact of using calculated hormone fractions in RR calculations was also investigated. The estimated RRsh of breast cancer using measured and calculated values of free E2 are very similar and inferences using either set of values would be the same. We did not evaluate the effect of using calculated values of nonSHBG bound E2 because, under the assumption of a constant albumin concentration, calculated values of nonSHBGbound hormone are directly proportional to calculated values of free E2, and therefore, RR calculations would produce identical results. However, it has been demonstrated that using measured albumin has virtually no effect on the calculation of free testosterone using the mass action method (7) , and therefore, it is hypothesized that RR estimates for free and nonSHBGbound fractions of E2 are likely to be very similar.
The analysis presented here, and previous validation studies, suggest that it is not necessary to carry out assays for free or nonSHBGbound fractions of E2 in epidemiological studies. From a study design perspective, this implies that only two assays are required in epidemiological studies—one for the total E2 concentration and one for SHBG concentration. An assay with high precision for the determination of total E2 concentration is essential; the precision of the assay for SHBG is less critical, although in practice, many excellent kit methods to measure SHBG concentration are available. These conclusions provide investigators with two main benefits; in general, studies will be more cost and time effective as the assays will be simpler and cheaper to perform, and perhaps more importantly, they will require less serum per study subject.
Members and Affiliations of the Endogenous Hormones and Breast Cancer Collaborative Group
Affiliation of the Secretariat.
T. J. Key, P. N. Appleby, G. K. Reeves, and A. W. Roddam, Cancer Research UK Epidemiology Unit, University of Oxford.
Collaborating Studies
Columbia, United States.
J. F. Dorgan, Fox Chase Cancer Center (Philadelphia, PA); C. Longcope, Departments of Obstetrics and Gynecology and Medicine, University of Massachusetts Medical School; F. Z. Stanczyk, Department of Obstetrics and Gynecology, University of Southern California School of Medicine; H. E. Stephenson, Jr., Department of Surgery, University of Missouri Health Sciences Center; R. T. Falk, Division of Cancer Epidemiology and Genetics, National Cancer Institute; R. Miller, Cancer Screening Services, Ellis Fischel Cancer Center (Columbia, MO); and A. Schatzkin, Division of Cancer Epidemiology and Genetics, National Cancer Institute.
Guernsey, United Kingdom.
D. S. Allen, I. S. Fentiman, T. J. Key, D. Y. Wang, Cancer Research UK; M. Dowsett, Academic Department of Biochemistry, Royal Marsden Hospital; and H. V. Thomas, Department of Psychological Medicine, University of Wales College of Medicine.
Nurses’ Health Study, United States.
S. E. Hankinson for the Nurses’ Health Study Research Group, Channing Laboratory, Department of Medicine, Brigham and Women’s Hospital and Harvard Medical School; and Department of Epidemiology, Harvard School of Public Health.
New York University Women’s Health Study, New York, United States.
P. Toniolo and A. Akhmedkhanov, Department of Obstetrics and Gynecology, New York University School of Medicine; and K. Koenig, R. E. Shore, and A. ZeleniuchJacquotte, Nelson Institute of Environmental Medicine, New York University School of Medicine.
Study of Hormones and Diet in the Etiology of Breast Tumors, Italy.
F. Berrino, Division of Epidemiology, Istituto Nazionale per lo Studio e la Cura dei Tumori; P. Muti, Department of Social and Preventive Medicine, University at Buffalo, State University of New York; Istituto Nazionale per lo Studio e la Cura dei Tumor; and A. Micheli, V. Krogh, S. Sieri, V. Pala, E. Venturelli, G. Secreto, Istituto Nazionale per lo Studio e la Cura dei Tumori.
Rancho Bernardo, United States.
E. BarrettConnor, G. A. Laughlin, Department of Family and Preventive Medicine, University of California at San Diego.
Radiation Effects Research Foundation, Japan.
M. Kabuto, Environmental Risk Research Division, National Institute for Environmental Studies, Ibaraki; S. Akiba, Department of Public Health, Faculty of Medicine, Kagoshima University; R. G. Stevens, Department of Community Medicine, University of Connecticut Health Center; K. Neriishi, Department of Clinical Studies, Radiation Effects Research Foundation; and C. E. Land, Radiation Epidemiology Branch, National Cancer Institute.
Study of Osteoporotic Fractures, United States.
J. A. Cauley and L. H. Kuller, Department of Epidemiology, University of Pittsburgh; and S. R. Cummings, Departments of Medicine and Epidemiology and Biostatistics, University of California, San Francisco; and the Study of Osteoporotic Fractures Research Group.
Washington County, United States.
K. J. Helzlsouer, A. J. Alberg, T. L. Bush, and G. W. Comstock, Department of Epidemiology, The Johns Hopkins University School of Hygiene and Public Health; G. B. Gordon, Oncology Center and Department of Pharmacology and Molecular Sciences, The Johns Hopkins University School of Medicine; S. R. Miller, Department of Health Policy and Management, The Johns Hopkins University School of Hygiene and Public Health; and C. Longcope, Department of Obstetrics and Gynecology and Medicine, University of Massachusetts Medical School.
Acknowledgments
We thank the women who participated, the research staff, the collaborating laboratories, and the funding agencies in each of the studies.
Footnotes

Grant support: The central pooling and analysis of these data were supported by Cancer Research UK.

The costs of publication of this article were defrayed in part by the payment of page charges. This article must therefore be hereby marked advertisement in accordance with 18 U.S.C. Section 1734 solely to indicate this fact.

Note: See Appendix section for members and affiliations of the Endogenous Hormones and Breast Cancer Collaborative Group.

Requests for reprints: Dr. Andrew Roddam, Endogenous Hormones and Breast Cancer Collaborative Group, Cancer Research UK Epidemiology Unit, University of Oxford, Gibson Building, Radcliffe Infirmary, Oxford OX2 6HE, United Kingdom. Phone: 441865302221; Fax: 441865310545; Email: andrew.roddam{at}cancer.org.uk
 Received April 25, 2003.
 Revision received August 6, 2003.
 Accepted August 7, 2003.