It is well-known that distinct biological indices (analytes) have distinct variability. We try to use some mathematical algorithms to pick out a set of blood parameters which give an opportunity to retrieve the initial volume of the blood spotted, and use it to calculate exact concentrations of analyts interesting to a physician. For our analysis we used the database of biochemical blood parameters obtained in Russian Scientific Center of Roentgen-Radiology during 1995-2000, which includes more than 30000 of patients.
Determination of different biological factors on the base of dried blood spot technology has a great practical importance in investigation of back lands populations, in epidemiological studies or in special people contingents monitoring, see [1], [2]. This technology presumes that blood sampling is performed by patient himself, the sample is spotted on a dry, as a rule, porous surface (filter paper, cellulose acetate membrane, etc.), and the posterior transportation to a laboratory, for example, by post in a standard or a special envelope.
Modern diagnostic equipment gives an opportunity to investigate many characteristics of dried blood spot, such as metabolites (see [3], [4], [5], [6], and [7]), hormones (see [8], [9], [10]), glycated hemoglobin (see [11]), and even some immune system parameters (see [12]). The possibility of DNA and RNA investigations in such samples is of great importance, since it gives, for example, an opportunity for mass investigations of socially important infections such as AIDS, hepatitis, etc.
In 1992, in USA a laboratory standard for dried blood spot testing (DBS) was elaborated, see [13]. In 2001, in Russia, Application Instruction on Alcor Bio Ltd Reagent Kit for immuno-enzymatic determination of thyrotropic hormone in dried blood spot of newborn was approved.
Under the dried blood spot technology using, the problem of the sample spotted volume determination remains one of the main practical questions. The existing versions of the technology may assume spotting of a known blood volume by means of some dosing device and a posterior elution, or using of special filtering device for the plasma separation under dried spot preparation, see [14].
But there are no any universal method calculating the volume of the blood spot, which does not use a dosing device. The solution of this problem gives an opportunity to increase essentially the accuracy of the results and to simplify the blood sampling procedure. There is a series of articles, where the calculations are based on the concentration of some electrolyte and on a correction of the plasma volume by the hematocrit value, see [16]. In the present work we try to elaborate a universal technology for the spotted volume calculations.
The solution of this problem is also important for such branches of Medicine as Catastrophe Medicine and Forensic Laboratory, where non-standard situations are typical, and, for example, there is no opportunity to sample patient’s blood, and so it is necessary to use the remains of the patient’s blood on some other objects, instead of the standard blood sample. Such a situation can appear under investigations of the victims to traffic accidents, or other catastrophes.
It is well-known that distinct biological indices (analytes) have distinct variability, see [15]. We try to use some mathematical algorithms to pick out a set of blood parameters which give an opportunity to retrieve the initial volume of the blood spotted, and use it to calculate exact concentrations of analyts interesting to a physician. For our analysis we used the database of biochemical blood parameters obtained in Russian Scientific Center of Roentgen-Radiology during 1995-2000, which includes more than 30000 of patients.
Let us describe the mathematical model of the problem. Let x i , 1 ≤ i ≤ m, stand for the value of the result of the laboratory analysis on the ith molecular compound content. The value x i obtained as a result of the blood sample analysis depends on two following parameters at least: the patient p which is selected from some collection ¶, and the volume λ of the blood sample under the analysis. Thus, the value x i is a function of two parameters:
The problem is to find a function f (y 1 , y 2 , . . . , y m ) whose value f x 1 (p, λ), x 2 (p, λ), . . . , x m (p, λ) is close to λ from the statistical point of view.
Notice that due to the uniform distribution of the molecules under consideration in the blood, a k-multiple extension of the volume must lead to the same increasing of all the indices x i . In other words, we have x i (p, k λ) = k x i (p, λ). Therefore, if f approximates the blood volume, then the following relation must be valid:
This notice is a natural motivation to look for the function f in the class of positively homogeneous functions of degree 1, i.e., we assume that the equality f (k y 1 , k y 2 , . . . , k y m ) = k f (y 1 , y 2 , . . . , y m ) holds for each positive k. Such functions are uniquely defined by their values at the unit sphere S m-1 defined by the equation
. By g we denote the restriction of the function f onto this unit sphere.
Polynomials form the simplest but rich class of functions. Let us look for g among the functions which are the restrictions of the polynomials onto S m-1 . Our statistical experiments show that it is enough to consider the polynomials of degree two vanishing at the origin. In other words, we put ρ = y 2 1 + • • • + y 2 m , and look for g in the form
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