Analysis of Total and Transferrin-Bound Iron from Serum Samples

Abstract:
Results from assays conducted independently of one another can produce findings that are incompatible with underlying physiologic constraints as a result of random errors. The authors investigate one such instance here, relating to assays of total iron and transferrin-bound iron concentrations in sera. A method of adjusting the observed total and transferrin-bound iron concentrations in these settings was outlined. The key of this approach was to satisfy the intrinsic physiologic constraint that the total serum concentration of iron is at least as great as the serum concentration of transferrin-bound iron. The adjustment method proposed by the authors can be readily applied in other settings that have physiologic constraints.

The FDA has issued a draft guidance for in vitro and in vivo studies of intravenously injected iron sucrose (1). One of the recommendations is that total iron and transferrin-bound iron concentrations are to be assessed simultaneously in sera from healthy male and female volunteers, prior to further determinations of area under the curve measures.

These recommendations are eminently reasonable, but practical implementation may not necessarily be straightforward. Assays of total iron and transferrin-bound iron concentrations are typically done independently, but can result in violations of obvious physiologic constraints. Suppose, for example, total iron is assayed in assay A, and transferrin-bound iron in assay B. Total iron should incorporate both transferrin-bound and non-transferrin-bound (e.g., free) iron, hence, one might expect that assay A should lead to a quantitative estimate of total iron no smaller than the amount of transferrin-bound iron calculated from assay B. This is not necessarily the case, however, as described in a study by Goggin et al. (2). These authors developed a spectrophotometric method for measuring total iron and transferrin-bound iron in human serum, and found a strong correlation (r=0.97) between the two measurements in naïve serum samples (n=341). On the other hand, they found a non-negligible proportion of paired values in which assayed transferrin-bound iron concentrations exceeded the assayed total iron concentrations. The purpose of this article is to outline a method of adjusting the observed total and transferrin-bound iron concentrations in such settings, so as to satisfy the intrinsic physiologic constraint that the total concentration of iron is at least as great as the concentration of transferrin-bound iron.

Methods

Let T and B generically refer to the serum concentrations of total iron and transferrin-bound iron, respectively, commonly in units of μg/dL. A serum sample was taken from a random individual; from this serum sample, assay A reports X_T as the concentration of total iron, and assay B reports X_B as the concentration of transferrin-bound iron (both in units of μg/dL). If X_T ≥ X_B, it would be accepted that X_T and X_B are valid estimates of the serum concentrations of total iron and transferrin-bound iron respectively in this individual. But if X_T < X_B, these observed values are at odds with the physiologic constraint that bound iron cannot exceed total iron. In this situation, the authors propose a likelihood approach for estimation of total and transferrin-bound iron concentrations. The approach is standard, but differs in the details, depending on what is known about the operating characteristics of the assays. 

Assay variances known

It is assumed that as random variables, X_T and X_B have normal (Gaussian) distribution with means μ_Τ and μ_Β respectively, and variances σ_T² and σ_B².  For now, it is assumed that σ_T² and σ_B² are known, as might be ascertained by reference to the assay manufacturers. There is the physiologic constraint that μ_Τ ≥  μ_Β , but otherwise these underlying means are unknown. If X_T < X_B is observed, the authors propose maximizing the joint likelihood of (X_T, X_B) over {μ_Τ,  μ_Β}, subject to the inequality constraint μ_Τ ≥  μ_Β;  and  are used, with the values of μ_Τ and μ_Β, respectively, that maximize this likelihood, as the estimates of total iron and transferrin-bound iron concentrations, respectively, in this situation.

In fact, these estimates can be easily derived using the method of Lagrange multipliers. It turns out that

[Eq. 1]

Assay coefficients of variation known

Operating characteristics of assays are not necessarily reported in terms of variances or standard deviations (SDs); a common alternative is to report coefficients of variation (CVs) rather than SDs, especially if assay variability tends to increase linearly with mean levels. The authors outline how CVs can be incorporated into the likelihood formulation.

As before, it is assumed that as random variables, X_T and X_B have normal (Gaussian) distribution with means μ_Τ and  μ_Β respectively, and variances σ_T² and σ_B².  However, now σ_T² and σ_B² are unknown, and instead two different parameters are given, which characterize variability in the underlying assays, namely, CV_T and CV_B , the coefficients of variation of the assays for total iron and transferrin-bound iron, respectively. 

Again as before, if X_T < X_B is observed, the authors propose maximizing the joint likelihood of (X_T, X_B) over {μ_Τ,  μ_Β}, subject to the inequality constraint μ_Τ ≥ μ_Β; in writing this likelihood, σ_T is replaced by CV_T*X_T, and  σ_B by CV_B *X_B. With these substitutions in Equation 1, one can arrive at

as the respective estimates of total iron and transferrin-bound iron concentrations. 

For those with a more numerical bent, Matlab code for this maximization procedure is given in the Appendix, which also includes further technical details.

Example

Figure 1 shows a plot of total iron concentrations and transferrin-bound iron concentrations from 180 individuals. The assays from which these concentrations were derived are not equally reliable; reported coefficients of variation are 8% for total iron and 5% for transferrin-bound iron. It is clear from Figure 1 that for a non-negligible proportion of the 180 subjects, the observed transferrin-bound iron concentrations exceed the observed total iron concentrations. After applying the approach described by the authors, the “corrected” concentrations are shown in Figure 2-which is now tied data [total iron concentration = transferrin-bound iron concentration], but with no violations of the intrinsic ordering of total iron and transferrin-bound iron.

Discussion

On an individual level, tests and assays can yield inconsistent results, as with independent assays for serum concentrations of total iron and transferrin-bound iron. Summary statistics can obscure these inconsistencies, for example, by simple averaging of results over large cohorts of individuals. Nevertheless, the individual consistencies remain, and the suggestion in this article is to address these inconsistencies in a logically coherent manner prior to data summarization. The authors illustrate an approach in the context of independent assays for serum concentrations of total iron and transferrin-bound iron, but the approach remains valid in analogous settings with intrinsic structural (physiologic) constraints. For example, ligand binding assays are commonly used to assess concentrations of monoclonal antibody drugs in plasma or serum samples (3). These assays are designed to measure the total or free forms of monoclonal antibody and the target ligand. Total monoclonal antibody is the sum of bound and unbound forms of the monoclonal antibody drug, hence, the value must exceed that of the free form. This constitutes a physiologic constraint on the assay determinations, completely analogous to the scenario outlined in this note.

It is implicitly assumed that the tests or assays are accurate, that is, there is no systematic bias impinging on the test or assay outcomes. In the setting described by the authors, the assays for serum concentrations of total iron and transferrin-bound iron have been appropriately vetted for accuracy, and the inconsistent findings can be attributed to random error. Nevertheless, in other settings, it may be worthwhile to rule out the possibility of systematic bias.

In practice, summary statistics may give no inkling of individually inconsistent results, and corrections to individual data may be inconsequential. Nevertheless, it can be argued that data analyses and submissions ought to be based on rigorously validated data, and the proposal described in this article is made in this spirit.

Appendix

The following outlines the Matlab code for estimation of the concentrations of total and transferrin-bound iron, in the scenarios described earlier. The authors’ methodology invokes the Matlab command fmincon, which minimizes a multivariate function (the objective function) subject to inequality constraints. In the following examples, the objective functions are saved as Matlab M files in the working directory.

Global variables are used to pass parameter values to the objective functions. The global variables are denoted: 

tval

observed value of total iron concentration

bval

observed value of transferrin-bound iron concentration

sig2t

variance of total iron assay (if known)

sig2b

variance of transferrin-bound iron assay (if known)

cvt

coefficient of variation of total iron assay (if known)

cvb

coefficient of variation of transferrin-bound assay (if known) 

It is assumed that assay reproducibility is expressed either in terms of variances (or standard deviations) of replicate measurements, or with coefficients of variation.
The likelihood approach outlined previously is predicated on the assumption of normality for replicate assay results. Let

represent the probability density function of a normally distributed random variable with mean μ and variance σ². Then, the joint likelihood L of (tval, bval) is the product of the two underlying normal densities

and the goal is to maximize this joint likelihood over {μ_T, μ_B}, subject to the simple linear constraint μ_Τ ≥ μ_Β. This maximization is equivalent to the minimization of the negative log likelihood - log L, for which fmincon is invoked.  

The objective functions specified in the Matlab M files below are proportional to the negative log likelihoods (but with extraneous constants omitted), and maximization of the likelihoods is equivalent to minimization of the objective functions.

objfunc1.m
function z = objfun1(x) 
% Objective function to be minimized if sig2t and sig2b are known
%
global tval bval sig2t sig2b
z = (x(1) - tval)^2/sig2t + (x(2) - bval)^2/sig2b;
return
end

objfunc2.m
function z = objfunc2(x)
% Objective function to be minimized if cvt and cvb are known 
%
global tval bval cvt cvb
z =(x(1) - tval)^2/(cvt*tval)^2 + (x(2) - bval)^2/(cvb*bval)^2;
return
end

For completeness, the objective functions are provided if sig2t and cvb, or cvt and sig2b are given.
objfunc3.m
function z = objfunc3(x)
% Objective function to be minimized if sig2t and cvb are known 
%
global tval bval sig2t cvb
z =(x(1) - tval)^2/sig2t +(x(2) - bval)^2/(cvb*bval)^2;
return
end

objfunc4.m
function z = objfunc4(x)
% Objective function to be minimized if cvt and sig2b and are known 
%
global tval bval cvt sig2b
z =(x(1) - tval)^2/(cvt*tval)^2 +(x(2) - bval)^2/sig2b;
return
end

Here is a prototypical Matlab program for the maximization procedure:

main.m
global tval bval sig2t sig2b cvt cvb

% Specify the observed tval and bval values, and the measures of variability % of the two assays, here

%  The linear constraint μ_Τ ≥  μ_Β is rendered in Matlab as A*x ≤ b, % where x denotes the vector (μ_Τ; μ_Β)

A =[-1   1]; b=[0];
%  The vector x0 contains starting values for the minimization procedure 
% through fmincon. We have found the following starting values to work
% well.

x0 = .5*(tval+bval)*[1; 1];

Notes:

1. One can invoke fmincon regardless of the ordering of tval and bval. If in fact tval ≥ bval, the linear constraint is not active, and fmincon will return x = (tval; bval).
2. If bval ≥ tval, the linear constraint is active, and fmincon will return the value specified in text equation (1) for x(1) and x(2).
3. If minimization is a one-off operation, then anonymous functions might be an appropriate approach for the minimization problem in Matlab. Separate function files are used, with parameters passed into the functions via global variables, for clarity.

References

1. FDA, Draft Guidance on Iron Sucrose (Rockville, MD, November 2013) www.fda.gov/downloads/drugs/guidancecomplianceregulatoryinformation/guidances/ucm297630.pdf.
2. M.M. Goggin et al., Bioanalysis 3 (6) 1837-1846 (2011).
3. J.W. Lee et al., AAPS Journal 13 (1) 99-110 (2001).

Submitted: Feb. 24, 2017
Accepted: Mar. 6, 2017

Article Details

Pharmaceutical Technology
Vol. 41, No. 7
Pages: 100–103

Citation

When referring to this article, please cite it as J. A. Koziol and M. A. Grossman, “Analysis of Total and Transferrin-Bound Iron from Serum Samples,” Pharmaceutical Technology 41 (7) 2017.