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Steroids 74 (2009) 512–519

Contents lists available at ScienceDirect

Steroids

journa l homepage: www.e lsev ier .com/ locate /s tero ids

novel spreadsheet method for calculating the free serum concentrations ofestosterone, dihydrotestosterone, estradiol, estrone and cortisol: Withllustrative examples from male and female populations�

orman A. Mazer ∗

ection of Endocrinology, Diabetes and Nutrition, Boston University School of Medicine, Boston, MA, United States

r t i c l e i n f o

rticle history:eceived 18 December 2008eceived in revised form 25 January 2009ccepted 27 January 2009vailable online 7 February 2009

eywords:athematical model

ree hormonepreadsheetestosterone

a b s t r a c t

In humans, testosterone (T), dihydrotestosterone (DHT), estradiol (E2), estrone (E1) and cortisol (C) bindto the serum proteins sex hormone-binding globulin (SHBG), albumin (Alb) and corticosteroid-bindingglobulin (CBG). Equilibrium dialysis is considered to be the “gold standard” for measuring the free con-centrations of these steroids but is technically difficult and not widely available. Based on a mathematicalmodel of the 5-ligand/3-protein binding equilibria, we developed a novel spreadsheet method for cal-culating the free and bioavailable (free + Alb-bound) concentrations of each steroid in terms of the totalsteroid and protein concentrations. The model uses 15 association constants KSHBG-X, KAlb-X, and KCBG-X

(X = T, DHT, E2, E1 and C) that have been estimated from a systematic review of published binding stud-ies. The computation of the free and bioavailable concentrations uses an iterative numerical method thatcan be readily programmed on a spreadsheet. The method is illustrated with six examples corresponding

stradiolortisol

to young men (YM), old men (OM), obese men (Ob M), young women (YM), pregnant women in the 3rdtrimester (Preg T3) and oophorectomized women on oral conjugated equine estrogens (CEE). The result-ing free hormone concentrations for YM and YW fall within the normal references ranges obtained byequilibrium dialysis for all five hormones. The model also accounts for the competitive binding effectsof high estrogen levels on the free T levels in Preg T3. This novel spreadsheet method provides a “user-

imati

friendly” approach for estand women.

. Introduction

Steroid hormones are transported in serum in the unboundfree) and protein-bound states [1–3]. The free concentration of theormone is generally thought to influence its tissue uptake and bio-

ogical activity, although some debate still exists on this point [4–8].quilibrium dialysis is considered to be the “gold standard” for mea-uring free hormone concentrations (or the free hormone fractionsrom which they are derived) but is technically difficult, sensitive toemperature and not widely available [9–12]. Alternatively, math-matical models based on the principles of chemical equilibria, i.e.

he laws of mass action and mass balance, have been successfullysed for many years to describe protein–steroid interactions and toalculate the free concentrations of testosterone and other steroids3,12–16].

� A preliminary version of this work was presented as poster P2-551 at the 88thnnual Meeting of the Endocrine Society, Boston, June 2006.∗ Current address: Department of Biomathematics, F. Hoffmann – La Roche Ltd.,alzgasse 30, 4052 Basel, Switzerland.

E-mail address: t42nam@comcast.net.

039-128X/$ – see front matter © 2009 Elsevier Inc. All rights reserved.oi:10.1016/j.steroids.2009.01.008

ng the free concentrations of circulating sex hormones and cortisol in men

© 2009 Elsevier Inc. All rights reserved.

While the modeling of chemical equilibria can, in theory, beapplied to any number of steroids and steroid-binding proteins [15],and was previously used to model the binding of 21 endogenoussteroids to sex hormone-binding globulin (SHBG), corticosteroid-binding globulin (CBG) and albumin (Alb) [12], the previouscomputational methods for multiple-ligand/multiple-protein mod-els have used computer programs or software applications thatwere not generally available to most clinical researchers [12,14,15].For this reason the calculation of free hormone concentrations hasbeen limited mostly to a one-ligand/two-protein binding model fortestosterone (T) in which the free T concentration is determinedfrom the concentrations of total T, SHBG and Alb (the latter oftentaken to be 4.3 g/dL, if not measured) [16]. Using the “one-ligandmodel” the calculation of free T requires only the solution of aquadratic equation and can be readily performed on a spreadsheetor with other statistical analysis programs. Empirically derivedequations have also been proposed for estimating the concentra-

tions of free or non-SHBG-bound T but lack any physico-chemicalbasis or generalizability [17,18].

Using experimentally determined values for the respec-tive protein–steroid association constants (KSHBG-T = 1 × 109 L/mol;KAlb-T = 3.6 × 104 L/mol) Vermeulen and others have shown that

ids 74 (2009) 512–519 513

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N.A. Mazer / Stero

he calculated free T values from the single-ligand model aren reasonably good agreement with equilibrium dialysis mea-urements made at 37 ◦C on a wide range of samples [11,16]. Inermeulen’s validation study a notable exception was observed

n pregnant women, whose calculated free T concentrations werepproximately 30% lower than the measured concentrations [16].his discrepancy was attributed to the extremely high estro-en concentrations associated with pregnancy (on the order0,000–20,000 pg/mL) that would compete with testosterone forHBG binding sites and effectively raise the actual free T concen-ration above the value obtained from the one-ligand model. Basedn this important finding we were motivated to develop a moreuser-friendly” multi-ligand/multi-protein model that would allowalculation of the free concentrations of T and other steroid hor-ones and could potentially account for the competition between

hem.In the present paper we describe a novel spreadsheet method

or simultaneously calculating the free concentrations of T, dihy-rotestosterone (DHT), estradiol (E2), estrone (E1) and cortisol (C)1

rom their total concentrations and the concentrations of SHBG, Albnd CBG (if measured). The model uses 15 protein–steroid associa-ion constants that have been estimated from a systematic reviewf the steroid-binding literature. The theoretical and computationalspects of the 5-ligand/3-protein model are presented in sufficientetail that the interested reader may replicate the model on his/heromputer. A working copy of the spreadsheet is also provided in theupplementary material to this manuscript. Illustrative simulationsf the free hormone profiles in male and female populations arehen provided and compared with normal reference ranges derivedrom equilibrium dialysis methods [19–21].

We believe that the spreadsheet method will be useful for aumber of reasons. It will provide clinical researchers with a sim-le and easy to use tool for calculating the free concentrations of T,HT, E2, E1 and C under a wide range of experimental conditions.he method can be easily modified to accommodate improvedstimates of the protein–steroid association constants should theyecome available. Lastly we believe that the present model andomputational methodology will be useful to basic researchersnd mathematical modelers interested in further characterizingrotein–steroid binding interactions and the role of free hormoneoncentrations in steroid transport and action.

. Methodology

.1. Chemical equilibria in the model

The model describes the interactions of five hormone ligandsT, DHT, E2, E1 and C) with three binding proteins (SHBG, Alb andBG), as depicted schematically in Fig. 1. Implicit in this represen-ation is the assumption that the ligands compete independentlyor the same binding sites on each protein, and that only the bind-ng affinities differ between them. The 15 association constants areymbolized by KSHBG-X, KAlb-X, and KCBG-X; where X corresponds tohe five ligands T, DHT, E2, E1 and C. CBG and Alb are both thoughto have a single steroid-site per molecule [22]. In the case of SHBG,hich is a homo-dimer, there is a current controversy as to whether

here are one or two equivalent binding sites per dimer [23–26] orwo non-equivalent sites [27]. For simplicity we will assume that

single binding site exists per SHBG dimer. However, given that

HBG concentration assays are typically calibrated by measure-ents of SHBG binding capacity, which determine the total number

f steroid binding sites available, the distinction between one or two

1 Although the letter “F” has historically been used as an abbreviation for cortisol,e use the letter “C” in this work as it is more intuitively recognized.

Fig. 1. Chemical equilibria in the 5-ligand/3-protein hormone-binding model.

binding sites per dimer is essentially moot, as long as the two sitesare equivalent.

2.2. Estimation of protein–steroid association constants

Based on a systematic review of in vitro protein–steroid bind-ing studies conducted at 37 ◦C, we tabulated the experimentallydetermined values for KSHBG-X, KAlb-X, and KCBG-X; where X cor-responds to T, DHT, E2, E1 and C (Tables 1–3, respectively). Foreach binding protein, we found six or more studies characteriz-ing its interaction with some or all of the five ligands of interest[12,13,28–36]. In the case of SHBG and Alb, the association constantfor T measured in each study was used to normalize the associa-tion constants of the other ligands reported (Tables 1 and 2). In thecase of CBG the association constant for C was used to normalizethe other association constants (Table 3). Based on the tabulatedstudies the mean value for KSHBG-T was 0.998 × 109 L/mol, with arange from 0.35 to 1.66 × 109 L/mol; the mean value for KAlb-T was3.57 × 104 L/mol, with a range from 2.53 to 4.06 × 104 L/mol; and themean value for KCBG-C was 5.23 × 107 L/mol, with a range from 2.56to 9.40 × 107 L/mol. Although the data are sparse in some cases (e.g.for DHT, E1 and C), the summary statistics of the reported and nor-malized values generally show smaller CV values for the normalizedvalues (ratios) compared to the un-normalized values. This sug-gests that the relative binding affinities are more consistent acrossstudies than the absolute values. As the mean values obtained forKSHBG-T and KAlb-T were virtually the same as the values used byVermeulen [16], we have used them as the values in our model. ForX = DHT, E2, E1 and C the model values used for KSHBG-X and KAlb-Xwere taken as the products of the values for T with the mean of thenormalized values for these ligands (last row in Tables 1 and 2). Inthe case of CBG, the mean value for KCBG-C was used as the modelvalue and the products of this value with the mean of the normal-ized values were used for the other ligands (X = T, DHT, E2 and E1)(last row in Table 3). It may be further noted that the inter-subjectvariation of KSHBG-T, KSHBG-E2 and KCBG-C reported from individuallaboratories [3,10,23,34] has a smaller CV (5–35%) than the vari-ation of the mean values reported from the different laboratories(Tables 1 and 3). This suggests a relatively small degree of biologicalvariation in the binding constants themselves.

2.3. Equations in the model

Based on the laws of chemical equilibria (mass action) and

mass balance, the 5-ligand/3-protein model can be expressed interms of eight equations in a particularly compact and symmetricform (Fig. 2). Eq. (1A–E) express the free concentrations of eachof the five ligands (X = T, DHT, E2, E1 and C) in terms of their totalconcentrations (XTotal), the concentrations of unoccupied protein

514 N.A. Mazer / Steroids 74 (2009) 512–519

Table 1Association constants between SHBG and steroids at 37 ◦C (L/mol); normalized by testosterone.

References T DHT E2 E1 C

L/mol L/mol Normalized L/mol Normalized L/mol Normalized L/mol Normalized

Vermeulen [28] 0.80 × 109

Anderson [29] 1.66 × 109 0.64 × 109 0.39Rosner and Smith [23] 0.35 × 109 0.99 × 109 2.83 0.22 × 109 0.63Dunn et al. [12] 1.60 × 109 5.50 × 109 3.44 0.68 × 109 0.43 0.15 × 109 0.094 0.0016 × 109 0.001Moll et al. [30] 0.98 × 109 0.50 × 109 0.51Södergård et al. [13] 0.60 × 109 1.07 × 109 1.79 0.31 × 109 0.53

n 6 3 3 5 5 1 1 1 1Mean 0.99 × 109 2.52 × 109 2.69 0.47 × 109 0.50 0.15 × 109 0.094 0.0016 × 109 0.001S.D. 0.53 × 109 2.58 × 109 0.83 0.20 × 109 0.09CV (%) 53.4 102.4 31.0 42.6 19.1

Model value 1.00 × 109 2.69 × 109 0.50 × 109 0.094 × 109 0.0010 × 109

Table 2Association constants between albumin and steroids at 37 ◦C (L/mol); normalized by testosterone.

References T DHT E2 E1 C

L/mol L/mol Normalized L/mol Normalized L/mol Normalized L/mol Normalized

Clark and Bird [31] 2.53 × 104 4.24 × 104 1.68Anderson [29] 3.60 × 104 5.55 × 104 1.54Vermeulen [28] 3.60 × 104

Dunn et al. [12] 4.00 × 104 8.00 × 104 2.00 6.00 × 104 1.50 4.00 × 104 1.00 0.30 × 104 0.075Moll et al. [30] 3.60 × 104 3.50 × 104 0.97Södergård et al. [13] 4.06 × 104 6.61 × 104 1.63 4.21 × 104 1.04

n 6 3 3 4 4 1 1 1 1Mean 3.57 × 104 6.28 × 104 1.768 4.82 × 104 1.263 4.00 × 104 1.000 0.30 × 104 0.075S.D. 0.55 × 104 1.90 × 104 0.202 1.16 × 104 0.300C

M 104

bcestssottttc

TA

R

V

DMPKG

nMSC

M

V (%) 15.4 30.3 11.4 24.1

odel value 3.60 × 104 6.36 × 104 4.55 ×

inding sites (SHBG, Alb and CBG) and the corresponding asso-iation constants with the particular ligand. Likewise Eq. (2A–C)xpress the three concentrations of the unoccupied protein bindingites (SHBG, Alb and CBG) in terms of the total protein concentra-ions (assumed to represent the maximal concentration of bindingites), the free concentrations of each ligand (X) and the corre-ponding association constants with each ligand. The appearancef the molecular weights of Alb and CBG in these equations reflect

he use of weight/vol concentrations for these proteins, in contrasto SHBG which is typically expressed in molar units correspondingo the ligand binding capacity. For a given set of the total concen-rations of ligands, proteins and the 15 protein–ligand associationonstants, these eight equations can be solved simultaneously to

able 3ssociation constants between CBG and steroids at 37 ◦C (L/mol); normalized by cortisol.

eferences C T DHT

L/mol L/mol Normalized L/mol

ermeulen (C) [28],Vermeulen et al. (T)[32]

3.00 × 107 0.15 × 107 0.050

unn et al. [12] 7.60 × 107 0.53 × 107 0.070ickelson et al. [33]a,b 4.00 × 107 0.24 × 107 0.060 0.083 × 107

ugeat et al. [34] 4.80 × 107

losterman et al. [35] 9.40 × 107

ayrard et al. [36] 2.56 × 107

6 3 3 1ean 5.23 × 107 0.307 × 107 0.060 0.083 × 107

.D. 2.71 × 107 0.199 × 107 0010V (%) 51.9 64.8 16.5

odel value 5.23 × 107 0.313 × 109 0.057 × 107

a Values for T and E2 are estimated from data obtained at 4 ◦C.b Value for E1 is estimated from value for E2 based on the ratio of cortisone to cortisol

23.7

3.60 × 104 0.27 × 104

give the free concentrations of ligand and unoccupied concentra-tions of protein binding sites that would be attained at equilibrium.Interestingly it is not necessary to combine these equations intoa single polynomial equation, nor is it possible to obtain a closedform expression for the solution of the free ligand concentrations(X’s) in order to solve the model. In fact a surprisingly simple com-putational approach allows one to solve the two sets of equationsiteratively and accurately as shown in the next section.

2.4. Computational strategy

Fig. 3 illustrates the computational strategy for solving Eqs.(1A–E) and (2A–C) in an iterative fashion that can be programmed

E2 E1

Normalized L/mol Normalized L/mol Normalized

0.011 0.00044 × 107 0.00011 0.00004 × 107 0.000009

1 1 1 1 10.011 0.00044 × 107 0.00011 0.00004 × 107 0.000009

0.00057 × 107 0.00005 × 107

binding affinities (Dunn et al. [12], Mickelson et al. [33]).

N.A. Mazer / Steroids 74 (2009) 512–519 515

FaaE

ocucToTptswt2wetttE(vpAa(

Fht

Fig. 4. Convergence plot of the incremental percent changes (absolute values) of the

ig. 2. Mathematical equations derived from the laws of mass balance and massction in the 5-ligand/3-protein hormone-binding model. The unsubscripted vari-ble X corresponds to the free concentration of the particular ligand (X = T, DHT, E2,1 or C).

n a Microsoft® Excel© spreadsheet. We begin by taking the totaloncentration of each protein as the first estimate (i = 1) of thenoccupied binding site concentrations and then use Eq. (1A–E) toompute the first estimates of the free ligand concentrations (i = 1).hese values are then substituted into Eq. (2A–C) to obtain the sec-nd estimates for the unoccupied binding site concentrations (i = 2).he resulting values are then substituted into Eq. (1A–E) to com-ute the second estimates of the free ligand concentrations andhe cycle repeats until the concentrations of unoccupied bindingites and free ligands have converged to fixed values. In practicee compute the absolute value of the percentage change in each of

he parameters (from iteration i to i + 1) and have found that with0 iterations stable values are attained that vary by less 0.0001%ith any further iterations. A typical plot of the convergence of the

ight parameters of the model is shown in Fig. 4. To accommodatehe different units used for measuring total ligand and total pro-ein concentrations all values are converted to molar units usinghe known molecular weights of T (288.4), DHT (290.4), E2(272.4),1(270.4), C (362.5) and the proteins Alb (69,000; [16]) and CBG52,000; [22]). The calculated free concentrations are then con-

erted back to the conventional units. Missing data, e.g. ligand orrotein concentrations are entered as zero (with the exception oflb which is taken to be 4.3 g/dL). The spreadsheet treats zero valuess 10−70 to prevent divisions by zero. In the limiting case of 1-ligandtestosterone) and 2-binding proteins (SHBG and Alb) the present

ig. 3. The iterative computational strategy used to solve 5-ligand/3-proteinormone-binding model (see text for explanation). The estimation of the concen-rations of unoccupied binding-sites and free hormone stops after 20 iterations.

concentrations of free hormones and unoccupied binding sites versus computationaliteration. The horizontal line (1 × 10−4) corresponds to an incremental change of 1part per million. By 20 iterations the incremental percent changes in all parametershave fallen below this threshold.

model produces values that are virtually identical to the quadraticsolutions of the Vermeulen model [16], which are also provided inthe spreadsheet. This implies that the inclusion of saturable bindingfor Alb, as indicated in Eq. (2B), has a negligible effect on the calcu-lation, as assumed in the Vermeulen model [16]. In addition to thefree hormone concentrations the spreadsheet also computes the socalled bioavailable concentrations of T, DHT, E2 and E1 which arethe sum of the free concentrations plus the Alb-bound concentra-tions. For each ligand X, the latter are the product of KAlb-X, Alb andX. A working copy of the Microsoft® Excel© spreadsheet which per-forms these calculations is provided in the supplementary materialto this manuscript.

3. Results

3.1. Illustrative examples in male and female populations

Table 4 provides six illustrations (three from men and threefrom women) in which the multi-ligand model is used to calcu-late the free and bioavailable concentrations of T, DHT, E2, E1 andC based on the total hormone concentrations and the concentra-tions of SHBG, albumin and CBG from representative literaturedata. The three illustrations in men correspond to healthy youngmen (YM), elderly men over 65 years of age (OM) and morbidlyobese men (Ob M). The three illustrations in women correspondto healthy young women in the follicular phase of the menstrualcycle (YW), pregnant women in the third trimester (Preg T3), andoophorectomized women receiving oral conjugated equine estro-gens (CEE). Also shown for comparison are the free T concentrationscalculated from the 1-ligand/2-protein Vermeulen model [16]. Theconcentrations of total hormone and binding proteins used in theseillustrations are derived from one or more studies carried out inthe respective populations. The values correspond to mean data asreported in those studies or are derived by averaging means frommultiple studies [37–50].

Looking across all six illustrations the calculated free T concen-trations from the multi-ligand model are observed to be within 1.4%of the Vermeulen model [16] with the exception of the Preg T3 case,where it is 31.5% higher. As discussed below this disparity reflectsthe competition effect between E2, E1 and T for the SHBG bind-ing sites. Interestingly, despite the markedly different hormone and

protein levels seen in Preg T3 compared to YW, the free and bioavail-able T concentrations calculated by our model are quite similar. Themuch smaller calculated free T in oophorectomized women on oralCEE (compared to YW) is attributable to the marked increase inSHBG as well as the absence of ovarian androgen secretion.

516 N.A. Mazer / Steroids 74 (2009) 512–519

Table 4Illustrative examples of calculated free and bioavailable hormone concentrations in male and female populations computed from the concentrations of total hormone andbinding proteins using the multi-ligand model and the Vermeulen model [12] (YM = young men, OM = old men, Ob M = obese men, YW = young women, Preg T3 = pregnantwomen in 3rd trimester, CEE = oophorectomized women receiving oral conjugated equine estrogens). Input data are derived from the publications listed below each column.

YM OM Ob M YW Preg T3 CEE

Input dataT total (ng/dL) 600 350 300 34 120 21DHT total (ng/dL) 50 35 20.3 12.4 18 7.6E2 total (pg/mL) 22 19 48 130 18000 36E1 total (pg/mL) 30 28 57.6 95 16000 148Cortisol total (�g/dL) 13 11 13 13 30 13Alb total (g/dL) 4.5 4.3 4.5 4.5 3.7 4.3SHBG total (nmol/L) 30 55 18.1 60 335 210CBG total (mg/dL) 4 3.4 2.15 4.5 11.7 6.5

Calculated resultsa

cFT (pg/mL) 131.50 49.85 77.83 4.04 4.49 0.90cFDHT (pg/mL) 5.22 2.14 2.57 0.63 0.26 0.13cFE2 (pg/mL) 0.53 0.37 1.28 2.19 123.85 0.27cFE1(pg/mL) 1.14 1.01 2.24 3.18 371.93 3.45cFC (ng/mL) 4.96 4.80 12.31 4.19 3.82 2.57cBT (ng/dL) 321.88 116.81 190.50 9.90 9.12 2.11cBDHT (ng/dL) 22.18 8.72 10.90 2.66 0.92 0.52cBE2 (pg/mL) 16.30 10.74 39.35 67.17 3142.56 8.00cBE1 (pg/mL) 27.89 23.68 54.75 77.76 7551.09 80.91

Vermeulen model [16]cFT (pg/mL) 133.36 49.76 77.42 4.06 3.42 0.90% differenceb −1.4 0.2 0.5 −0.5 31.5 −0.3

ReferencesEsoterix [37] Kenny et al. [39] Isidori et al. [41] Shifren et al. [45] Zhang et al. [47] Shifren et al. [45]Kudielka et al. [38] Vermeulen et al. [40] Blanchette et al. [42] Fernandez-Real et al. [46] Soldin et al. [48] Pripp et al. [50]

Kudielka et al. [38] Zumoff et al. [43] Esoterix [37] Mazer [49]Purnell et al. [44]

a centra

mcTSst

aaaElt

tboa

3d

e[mFfmof

Calculated free concentrations are denoted cFX and calculated bioavailable conb Defined as 100(multi-ligand model − Vermeulen model)/Vermeulen model.

With respect to DHT, the calculated free concentrations in theale populations are approximately 1/25 of the respective free T

oncentrations. This is lower than the 1/10 ratio of total DHT to totallevels, and reflects the 2–3-fold greater binding affinity of DHT toHBG and Alb compared to T (Table 1). Analogous relationships areeen in the free and total DHT concentrations in YW and CEE relativeo the corresponding T levels.

The calculated free E2 and E1 concentrations vary markedlycross the six examples, reflecting the aromatization of T in mennd the changes in SHBG associated with aging, obesity, pregnancynd oral CEE administration. It is interesting to note that the free2 levels in all three male populations actually exceed the free E2evels in oophorectomized women treated with oral CEE, althoughhe calculated free E1 levels are higher in the latter group.

In contrast to the sex hormones, the calculated free C concen-rations show less variation with gender and aging, but appear toe relatively higher in obese men and relatively lower in women onral CEE. The latter cases reflect the reduction in CBG with obesitynd elevation with oral estrogen.

.2. Comparison to normal reference ranges from equilibriumialysis

The normal reference ranges of T, DHT, E2, E1 and C derived fromquilibrium dialysis measurements in healthy men and women19–21] are compared with the corresponding calculations for

ales (YM, OM and Ob M) and females (YW, Preg T3 and CEE) in

ig. 5A and B, respectively. The calculated values for YM and YWall within or on the boundary of the normal ranges for all five hor-

ones. In contrast the calculated free T for OM is at the lower limitf the normal range; while the calculated values for free E2 and E1or Ob M are at or above the upper limits of the normal ranges for

tions are denoted cBX, where X corresponds to T, DHT, E2, E1 and C.

men. For Preg T3 the calculated free E2 and E1 values are about 40-and 100-fold higher than the corresponding normal ranges; whilefor oophorectomized women on CEE, the calculated values of freeT, DHT, E2 and C are below the corresponding normal ranges forwomen.

3.3. Influence of C, CBG on calculated free T concentrations

In view of the high molar concentrations of C and CBG, weexplored the possible influence of these variables on the calculatedfree T concentrations for the six illustrative examples in men andwomen. Table 5 shows the effect of eliminating C and CBG individu-ally and together on the calculated free T concentrations. Comparedto the full model, eliminating C results in a small decrease (−1.07 to−3.23%) in the free T concentrations; while eliminating CBG resultsin a modest increase (4.01 to 26.49%) in the free T concentrations.The elimination of C and CBG together tends to cancel the twoeffects and results in a small (0.3 to 3.2%) increase in free T con-centrations. Thus neglecting both C and CBG in the full model has arelatively minor influence on the calculated free T concentrations.Interestingly, the effect of eliminating T and SHBG individually ortogether has virtually no effect (<1%) on the calculated free C con-centrations (data not shown).

3.4. Influence of equine estrogens on calculated free hormoneconcentrations

A number of conjugated and unconjugated estrogenic com-pounds have been measured in serum following oral administrationof CEE [51]. The conjugated species, being polar compounds, wouldnot be expected to bind to serum proteins; whereas the unconju-gated species, which total 330 pg/mL at a CEE dose of 1.25 mg, would

N.A. Mazer / Steroids 74 (2009) 512–519 517

Table 5Influence of C and CBG on calculated free T concentrations (pg/mL) (YM = young men, OM = old men, Ob M = obese men, YW = young women, Preg T3 = pregnant women in3rd trimester, CEE = oophorectomized women receiving oral conjugated equine estrogens).

YM OM Ob M YW Preg T3 CEE

Full model 131.50 49.85 77.83 4.04 4.49 0.90-C 127.68 48.81 75.32 3.96 4.41 0.89% differencea −2.91 −2.08 −3.23 −1.98 −1.79 −1.07-CBG 141.47 53.77 80.95 4.45 5.68 1.00% differencea 7.58 7.87 4.01 9.97 26.49 11.76- 78% 0

bewtE3jotlc

Ftd[g(lCmh

C, -CBG 135.72 50.40differencea 3.20 1.11

a Defined as 100(reduced model − full model)/full model.

e expected to exhibit protein binding. To simulate the potentialffect of the unconjugated species on free hormone concentrationse assumed that their protein-binding affinities would be similar

o E1 or E2. Using the CEE data set (Table 4), the concentrations of1 and E2 were individually increased from their initial values to30 pg/mL in order to represent the total concentration of uncon-ugated equine estrogens. In each case the calculated free fractions

f T, DHT, E2, E1 and C increased by less than 0.4%. These simula-ions suggest that the additional estrogenic compounds in CEE areikely to have little influence on the estimation of free hormoneoncentrations.

ig. 5. Comparison of the calculated free hormone concentrations for the illustra-ive examples in men and women with the respective normal reference rangesetermined by equilibrium dialysis by Quest Laboratories for T [19,20], DHT19–21], E2 [19,20], E1 [19] and C [20]. Panel (A) corresponds to the three maleroups (YM = young men, OM = older men, Ob M = morbidly obese men). PanelB) corresponds to the three female groups (YW = young women in the follicu-ar phase of the menstrual cycle, Preg T3 = pregnant women in the 3rd trimester,EE = oophorectomized women receiving oral CEE). The horizontal dashed line seg-ents represent the upper and lower limits of the normal reference ranges for each

ormone.

.06 4.10 4.53 0.90

.30 1.25 0.86 0.60

4. Discussion

4.1. Comparison to other models

We have developed a novel spreadsheet method for simultane-ously calculating the free and bioavailable serum concentrationsof T, DHT, E2, E1 and C from the total concentrations of thesehormones and their binding proteins SHBG, Alb and CBG. Whileother authors have mathematically modeled steroid/protein bind-ing equilibria in the past, we believe that our 5-ligand/3-proteinmodel is much simpler to implement than the multi-ligand mod-els described by Dunn et al. [12] and Feldman et al. [15] as wellas the 2-ligand (T, E2)/2-protein (SHBG, Alb) model described bySödergård et al. [13] and recently by de Ronde et al. [52]. The spread-sheet method will enable investigators to go beyond the calculationof free T levels using the 1-ligand/2-protein Vermeulen model [16]and calculate the free and bioavailable concentrations of other clin-ically relevant androgenic and estrogenic steroids. In this regardthe measurement (or calculation) of free DHT and free E2 levelshas not been routinely available in most research or clinical ref-erence laboratories and has limited the assessment of hormonestatus in men and women and the effects of exogenous hormonetherapy.

With the exception of pregnant women, the differences betweenthe multi-ligand model and the Vermeulen model [16] are generallysmall for free T concentrations (Table 3), implying that the competi-tion between steroids for the protein binding sites is usually of littlequantitative importance. However in pregnancy the extremely highconcentrations of E2 (18,000 pg/mL) and E1 (16,000 pg/mL) are ableto effectively compete with T for SHBG binding sites and elevate thefree T concentration predicted by our model by 31.5% comparedto the Vermuelen model. This difference is quite consistent withVermeulen’s own observation of the disparity between measuredfree T levels in pregnant women and the calculated values from thesingle-ligand model [16]. Our observation that the calculated freeand bioavailable T levels in late pregnancy are comparable to YWis also consistent with experimental data from pregnant women[53].

In contrast to the competition effects of high E2 and E1 levels inpregnancy, we have shown explicitly that the inclusion of C and CBGin the 5-ligand/3-protein model has little impact on the calculatedfree T levels, despite the fact that the C concentrations are one totwo orders of magnitude higher (in molar units) than testosteroneconcentrations in men and women (Table 5). In this case the muchweaker binding affinity of C for SHBG and T for CBG (Tables 1 and 3)mitigates the competition effects. While Vermeulen suggested longago that the binding of T to CBG would have a negligible effect onfree T levels [28,32], the possibility of cortisol displacing T from CBG

or SHBG binding sites has not been widely examined. It is reassur-ing that both effects tend to cancel each other, as seen in Table 5.Likewise the simulated effects of unconjugated equine estrogenson the free concentrations of T, E2 and E1 are also shown to benegligible.

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.2. Determination of association constants

An essential aspect of the present work was the compilationf the 15 steroid–protein association constants used in the model.e observed greater consistency between studies for the normal-

zed binding affinities of the steroids to SHBG and Alb (relativeo T) and to CBG (relative to C), in comparison to the abso-ute affinities (see CV values in Tables 1–3). In view of the verylose correspondence of the mean literature values we obtainedor KSHBG-T and KAlb-T (Tables 1 and 2) to Vermeulen’s values of× 109 L/mol and 3.6 × 104 L/mol, respectively [16], we used the

atter values to anchor all of the SHBG and Alb association con-tants (Tables 1 and 2). In the case of CBG, where the binding dataor the sex hormones is scant, we used the mean literature valuef KCBG-C (5.23 × 107 L/mol) to anchor the other CBG associationonstants (Table 3). It should be appreciated that most of the bind-ng studies in the literature were performed 25–35 years ago usingnalytical methods that have generally been surpassed today. Theecent SHBG-binding studies by Cherkasov [54] were performed at◦C and were therefore not included in our compilation of dataeasured at 37 ◦C. In the event that new research leads to more

ccurate and/or accepted values for these constants the spread-heet method can easily be updated. Similarly in the event thatHBG is found to have two non-equivalent or cooperative bindingites as suggested by a recent study [27], the model can, in prin-iple, be adapted to reflect the more complex behavior; althoughHBG-binding studies will be required for all ligands of interest.s noted previously the current model should provide an accu-ate representation of two identical SHBG binding sites, as longs the SHBG concentration used in the model corresponds to theotal binding capacity, i.e. twice the concentration of SHBG homo-imers. Lastly, it should be appreciated that using “mean values” forhe binding constants to calculate free hormone concentrations ofndividual subjects implicitly assumes that the biological variationf these constants is relatively small. Based on the available datahis assumption appears justified.

.3. Validation

Validation of a new methodology is an ongoing process. Thelose correspondence observed between the calculated free hor-one concentrations for healthy young men and young womenith the normal ranges for these hormones determined by an

stablished reference laboratory using equilibrium dialysis [19–21]epresents an important first step (Fig. 5A and B). Given the dis-arate sources of data from which the YM and YW values werealculated and the large gender differences for the free sex hor-one concentrations, the observed consistency with the normal

anges is quite satisfying. The ability to quantitatively account forhe competitive effects of E2 and E1 on T binding in the pregnanttate provides additional validation for the present model.

A limited comparison of individual free T concentrations mea-ured by dialysis and calculated using an earlier version of theresent model was previously reported in a crossover study of7 post-menopausal women treated with oral conjugated equinestrogens vs. transdermal estradiol therapy by Shifren et al. [55].n that study the calculated free T concentrations were approxi-

ately 14% lower than the dialysis values and Lin’s Concordanceorrelation coefficient was 0.901 across all treatments. Whilehese findings are reassuring, a more systematic comparison ofndividual samples analyzed by dialysis and computed with the

-ligand/3-protein spreadsheet model will be needed for rigor-us validation of this approach, particularly for DHT, E2, E1 and. Systematic discrepancies, if observed, could result from struc-ural limitations of the model, inaccurate parameter values, andr the non-standardization of hormone and protein assays used to

(2009) 512–519

determine the association constants and analyze the clinical sam-ples. As noted by Vermeulen and others [16,11] it is particularlyessential that the assays used to measure SHBG concentrations inclinical samples become standardized and that they perform com-parably to the assays that were (or will be) used to determine theassociation constants between SHBG and the various steroids. Stan-dardization of the CBG assay is likewise important. In conductingfuture validation studies it is also important to recognize that thedialysis method, as well, is sensitive to temperature, dilution effects,the stability and purity of radiolabeled compounds, and the properfunctioning of the dialysis membranes and cells.

Once validation and/or optimization of the model have beenachieved, it will be useful to generate reference ranges for eachof the calculated free and bioavailable hormone concentrationsin healthy men and women as a function of age. The effects ofmenopause and pregnancy in women should also be carefully stud-ied. Such reference ranges will provide a necessary framework forthe analysis and treatment of hormone deficiency states associatedwith aging and other conditions.

4.4. Conclusions

A new spreadsheet method for simultaneously calculating thefree and bioavailable serum concentrations of T, DHT, E2, E1 andC has been presented and illustrated with examples from maleand female populations. We believe that the new method will beuseful to clinical investigators wanting to estimate these free hor-mone concentrations in their studies as well as to basic researchersand mathematical modelers interested in further characterizingprotein–steroid binding interactions and the role of free hormoneconcentrations in steroid transport and action. Although the newmethod is consistent with a body of free hormone data obtainedby equilibrium dialysis, further validation of this method is stillneeded.

Acknowledgements

The author acknowledges the support of Dr. Shalender Bhasinand the funding provided by his NIH Grants: 1RO1AG31206,5UO1AG14369 and 1P30AG031679.

Appendix A. Supplementary data

Supplementary data associated with this article can be found, inthe online version, at doi:10.1016/j.steroids.2009.01.008.

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