Michael A. Grossman is president of Strategic Advantage Inc., Wellesley, MA 02481, StrategicAdvantage@comcast.net.

Michael A. Grossman

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The FDA has issued a draft guidance for 

 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.

	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

 as the concentration of total iron, and assay B reports X

 as the concentration of transferrin-bound iron (both in units of μg/dL). If X

 are valid estimates of the serum concentrations of total iron and transferrin-bound iron respectively in this individual. But if X

, 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. 

	It is assumed that as random variables, X

 have normal (Gaussian) distribution with means μ

 are known, as might be ascertained by reference to the assay manufacturers. There is the physiologic constraint that μ

 , but otherwise these underlying means are unknown. If X

 is observed, the authors propose maximizing the joint likelihood of (X

}, subject to the inequality constraint μ

;  and  are used, with the values of μ

, 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

	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

 are unknown, and instead two different parameters are given, which characterize variability in the underlying assays, namely, CV

 , the coefficients of variation of the assays for total iron and transferrin-bound iron, respectively. 

	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.

	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 

 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 

-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.

	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.

	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: 

observed value of total iron concentration

observed value of transferrin-bound iron concentration

variance of transferrin-bound iron assay (if known)

coefficient of variation of total iron assay (if known)

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 σ

. 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

}, subject to the simple linear constraint μ

. 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

 is rendered in Matlab as A*x ≤ b, % where x denotes the vector (μ

)
	
	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];

	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.

 (Rockville, MD, November 2013) www.fda.gov/downloads/drugs/guidancecomplianceregulatoryinformation/guidances/ucm297630.pdf.
	2. M.M. Goggin et al., 

3 (6) 1837-1846 (2011).
	3. J.W. Lee et al., 

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

	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,” 

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