Prediction of unfavorable outcome of acute decompensation of diabetes mellitus

Abstract

The aim of the study

Using the method of spectral-probability analysis, to evaluate the possibility of predicting an unfavorable outcome of acute decompensation of diabetes mellitus in patients hospitalized in the intensive care unit using a mathematical model. In relation to clinical practice, the implementation of the proposed algorithm for mathematical processing of a set of test data provides the physician with an additional significant criterion for assessing the probability of a tendency to develop type 1 diabetes in healthy children being examined whose brothers or sisters suffer from this disease.

Materials and methods

A retrospective analysis of 103 medical records of patients hospitalized in the intensive care unit for acute decompensation of diabetes mellitus was conducted.

Results

With regard to the set of analyses of patients with acute decompensation of diabetes mellitus, carried out at the time of admission to hospital, a group of mathematical criteria has been defined that makes it possible to identify patients with a high risk of an unfavorable course of the disease.

Introduction

Acute decompensation of diabetes mellitus (DM) is a serious critical, life-threatening condition caused by a significant increase in blood glucose and associated metabolic disorders [1,2,3]. Diabetic ketoacidosis (DKA) and hyperglycemic hyperosmolar state (HGS) are characterized by varying degrees of insulin deficiency, excessive production of counter-insular hormones, and dehydration [4, 5]. Evaluation of the prognosis of patients with this pathology is important for choosing the tactics of management and observation [6, 7].

It should be noted that the conducted study is aimed at identifying fundamental patterns of properties of the ensemble of laboratory parameters considered as a holistic ensemble of random variables. The identified patterns can be considered as a basis for creating a fairly simple method for mathematical modeling of the disease. A feature of the proposed technique is the ability to continuously improve the accuracy of the results obtained as additional groups of both patients and healthy siblings are included in the study. Predicting the development of T1DM at an early preclinical stage and identifying children at high risk is important not only from the point of view of timely administration of insulin therapy to prevent acute life-threatening metabolic disorders during disease manifestation, but also the possibility of early therapeutic immunomodulatory intervention to prevent or delay the development of T1DM. Generally accepted methods for predicting T1DM at a preclinical stage have not been developed to date. This work is aimed at studying the possibility of mathematical modeling of T1DM prediction based on the assessment of laboratory and genetic parameters. This article further analyzes the results of a cross-sectional cohort study that yielded criteria for assessing the likelihood of healthy siblings developing the disease. These results allow us to define a method for identifying the risk of developing type 1 diabetes.

Materials and methods

A retrospective analysis of 103 medical records of patients hospitalized in the intensive care unit for acute decompensation of diabetes mellitus in the period 2022–2023 was conducted. Of these, 42 men (41%) and 61 women (59%) aged 18 to 88 years. The average age of men was 47 + 17 years, women 58 + 19 years. 37 patients had type 1 diabetes mellitus (36%), 58 patients type 2 diabetes mellitus (56%) and 8 patients pancreatogenic diabetes mellitus (8%). At the prehospital stage, 49 patients (48%) received insulin, 19 patients (18%) oral hypoglycemic agents, 25 patients (24%) did not receive hypoglycemic therapy and in 10 patients the treatment is unknown (10%). For the number of ${N}_{G}=83$ patients with favorable and ${\text{N}}_{R}=20$ patients with unfavorable course of the disease, the data of the total number of ${N}_{a}=20$ laboratory parameters were analyzed, including plasma glucose on admission, ketonuria parameters, acid–base balance data (pH, lactate, osmolality, sodium, potassium), glomerular filtration rate, blood leukocytes. Therapeutic measures, such as the initial dose of intravenous infusion of short-acting insulin and potassium chloride, the volume of rehydration in the first 4 h of admission were evaluated. The results obtained are based on a study of a set of 24 genetic and 8 laboratory parameters of patients’ blood.

To solve the problem, we will apply a solution similar to that indicated in works [8,9,10] and based on the assumption that any considered set of indicators of the human body’s performance $x\left( j\right)$, where $j=1, 2, \dots 20$, can be represented in the form

$$x\left( { j} \right) = x_{1} \left( { j} \right) + x_{2} \left( { j} \right)$$

(1)

In this case, ${x}_{1}\left( j\right)$ is background noise, which is a consequence of the action of a complex of phenomena and processes associated with the normal (not associated with serious pathologies) physiological functioning of the patient's health. At the same time, the term ${x}_{2}\left(j\right)$ describes the deviation of a set of laboratory indicators or parameters, which is a consequence of the launch of a pathological process. The ensemble of these parameters is considered as a sequence of complexly interconnected random variables.

In this case, in addition to the assumption used in [8,9,10] about the weak dependence of the two sequences of random variables (RV), the hypothesis about significantly different properties of the statistics of the RV set ${x}_{2}\left(j\right)$ for patients with subsequent recovery or, conversely, a fatal outcome of the disease is allowed for consideration. In other words, we believe that for the corresponding patients, with a high probability, there is a difference in the properties of the experimental probability density corresponding to the set of results of the initial medical tests at the time of admission to the hospital. Further, this assumption is verified using the example of a favorable and unfavorable course of acute decompensation of diabetes in patients admitted to the hospital due to acute decompensation of diabetes mellitus.

In an effort to clarify the calculations made, we present Fig. 1 below. In Fig. 1a, the wide "rough" curve $\rho_{1} \left( w \right)$ corresponds to the probability density of the complex of values ​​of the RV ${x}_{1}\left(j\right)$. According to the relation (Eq. 3) written below, the dependence $\rho_{1} \left( w \right)$ is discrete, but in Fig. 1, for greater clarity, it is schematically presented as a continuous function. The small-scale roughnesses of the curve $\rho_{1} \left( w \right)$ shown in Fig. 1a are associated with a deviation from some unknown "ideal" distribution and are explained by the limited number of measurements taken. The effective period of such small-scale roughnesses of the function $\rho_{1} \left( w \right)$ is equal to the discretization interval specified when dividing the range of values ​​of the results of the medical tests into separate intervals. This interval can be chosen to be very small, if necessary. In Fig. 1a also schematically demonstrates the "narrow" curve $\rho_{2} \left( w \right)$ which corresponds to the probability density for the complex of RV ${x}_{2}={x}_{2}(j)$, associated with the action of pathology. Just like $\rho_{1} \left( w \right)$ for greater clarity, the discrete dependence $\rho_{2} \left( w \right)$ is schematically depicted as a continuous curve.

Fig. 1

Schematic view of the dependencies ${\uprho }_{\text{1,2}}\left(w\right)$ (a), as well as a schematic view of the probability density $\uprho \left(w\right)$ in the presence of an independent random term ${x}_{2}\left(t\right)$ in (Eq. 1) (b)

As is known, the probability density $\rho \left( w \right)$ of the sum of two statistically independent random variables ${x}_{1}$ and ${x}_{2}$ with probability distributions $\rho_{1} \left( w \right)$ and $\rho_{2} \left( w \right)$ can be written using an integral convolution of the form

$$\uprho \left(\text{w}\right)=\underset{-\infty }{\overset{\infty }{\int }}{\uprho }_{1}\left(w-w{\prime}\right){\uprho }_{2}\left(w{\prime}\right)dw{\prime}$$

(2)

(Since natural processes are considered continuous, we consider relation (Eq. 2) to be valid, including in the case of a weak dependence of the RV ${x}_{1}(j)$ and ${x}_{2}(j)$.) We write the distributions $\rho_{1,2} \left( w \right)$ in (Eq. 2) using sums of the form

$${\uprho }_{1,2}\left(w\right)=\sum_{m=0}^{N-1}{P}_{\text{1,2}}\left(m\right)\updelta \left(w-m\right).$$

(3)

In (Eq. 3), the factor $\updelta \left(Z\right)$ is the Dirac delta function, and the coefficients $\rho_{1,2} \left( w \right)$ are the probabilities of the values of random variables ${x}_{\text{1,2}}(j)$ falling into intervals of the form

$${x\left( j\right)}_{min}+mh\le x\left( j\right)<{x\left( j\right)}_{min}+\left(m+1\right)h,$$

(4)

$$N =\left[\frac{{x\left( j\right)}_{max}-{x\left( j\right)}_{min}}{h}\right], 0\le m\le N-1.$$

In (Eq. 4), $h$ is the size of the discrete interval specified above, ${x\left( j\right)}_{max}$ и ${x\left( j\right)}_{min}$ are, respectively, the maximum and minimum values of the data set $x\left( j\right)$, $j=1, 2, \dots 20$, corresponding to one patient, and the square brackets denote the integer part of the number.

Figure 1b schematically shows (also in the form of a continuous curve) the “total” probability density $\rho \left( w \right)$ specified in (Eq. 2), corresponding to the presence in (Eq. 1) of an independent term ${x}_{2}(j)$, for the width $2{\upsigma }_{2}$ of the probability density of which (in this case ${\upsigma }_{2}$ is the standard of fluctuations of the random variable ${x}_{2}={x}_{2}( j)$) the condition is introduced

$$2{\upsigma }_{2}\ge \left(2\div 3\right)h.$$

(5)

According to (Eq. 3), if (Eq. 5) is true, the small-scale “roughness” of the ${\uprho }_{1}\left(w\right)$ dependence, the effective period of which, as indicated above, is h, will be averaged. The consequence of this process will be a decrease in the characteristic amplitude of such random small fluctuations. Consequently, the “total” dependence $\uprho \left(\text{w}\right)$, as schematically demonstrated in Fig. 1b, with a high probability may correspond to a much “smoother” curve than the one presented by ${\uprho }_{1}\left(w\right)$ in Fig. 1a. This difference between the studied dependences can be recorded by mathematical means, such as, for example, the Fourier transform. In this case, the degree of this “smoothness”, determined by a decrease in the amplitude of small-scale disturbances, with a high probability will differ for patients with an upcoming favorable or, conversely, unfavorable course of the disease. Such a property can be used as a diagnostic feature. Below, this hypothesis will be verified using data from analyses of a group of patients with diabetes mellitus with favorable and unfavorable disease progression during their hospital stay.

In order to enhance the mathematical effect under consideration, we will further apply a nonlinear transformation of the form

$$F\left[ {x\left( j \right)} \right] = {\text{sin}}\left[ {W(\left| {x\left( { j} \right)} \right| + \Delta )^{\alpha } } \right]$$

(6)

which depends on the parameters $W$ and $\alpha$, to the directly obtained results $x\left(j\right)$ of medical tests. Here the term $\Delta ={10}^{-7}$ is introduced in order to avoid division by zero in (Eq. 6) at $x\left(j\right)=0$ in the case of negative powers of $\alpha$. For all $|x\left( j\right)|\ne 0$ the value of $\Delta$ must satisfy the condition $\frac{\Delta }{{Min\left\{ {\left| {x\left( { j} \right)} \right|^{\alpha } } \right\}}} \ll h$. The statistical meaning of the transition to the study of the statistics of the ensemble of random variables $F\left[x\left(j\right)\right]$ instead of the original set $x\left(j\right)$ of test results is as follows: the probability density function ${\uprho }_{F}\left(\text{w}\right)$ corresponding to $F\left[x\left(j\right)\right]$ has an integrable singularity. In particular, in the simplest case $W=1$, $\alpha \, = \,1$ replacing the absolute value $|x\left( j\right)|$ on $x\left(j\right)$, discarding the term $\Delta$ and the validity of the condition -$-\uppi /2\le x\le\uppi /2$ we obtain

$${\uprho }_{F}\left(\text{w}\right)={\left[1-{w}^{2}\right]}^{-1/2}{\uprho }_{1}\left\{\text{arcsin}(w)\right\} .$$

(7)

It is easy to see that for $|\text{w}|\to 1$, small and fine-scale roughnesses of the probability density, which will take place both for $\rho_{1} \left( w \right)$ and for $\rho_{1} \left\{ {{\text{arcsin}}\left( w \right)} \right\}$, will increase significantly in amplitude as a result of multiplication by the factor ${\left[1-{w}^{2}\right]}^{-1/2}$ tending to infinity on the right-hand side of (Eq. 7). This effect is highly likely to lead to an increase in the differences between the $\rho_{F} \left( {\text{w}} \right)$ dependencies in the absence (see the wide “rough” curve $\rho_{1} \left( {\text{w}} \right)$ in Fig. 1a) and in the presence (“smooth” dependence $\rho \left( {\text{w}} \right)$ in Fig. 1b) of the random term ${x}_{2}\left(j\right)$ on the right-hand side of (Eq. 1). (As follows from the analysis of (Eq. 7), the use of a nonlinear transformation of the form (Eq. 6) causes a kind of “magnifying glass” effect, which makes it possible to study the differences in the properties of individual sections of the $\rho \left( {\text{w}} \right)$ dependence.) The parameters $W$ and $\alpha$ introduced in (Eq. 6) determine the length of the interval of pronounced tendency to infinity for the integrable dependence $\rho_{F} \left( {\text{w}} \right)$

Next, by analogy with [1,2,3], the set of analyses for each patient is compared with a statistical functional of the form

$${\text{L}}\left( {\text{p}} \right){\text{ = L}}\left( {{\text{p,h,W,}}\alpha } \right){\text{ = ~}}\left| {{\text{Arg}}\,\left( {\mathcal{A}} {\sum\limits_{{m = 0}}^{{N - 1}} {P_{m} \exp \left( {ipm} \right)} } \right)\,} \right|$$

(8)

In (Eq. 8), the real multiplier $\mathcal{A}$ is used to move to a more convenient range of values for the $L\left( {p,h,W,\alpha } \right)$ dependence (in this case, it turned out to be convenient to choose the value $\mathcal{A}={N}_{b}$+1, where ${N}_{b}=103$ is the total number of patients considered in this article). Further, by analogy with (Eq. 4), in relation (Eq. 8), any multiplier ${P}_{m}$ under the sum sign is the probability of the value of the function $F\left[x\left(j\right)\right]$ falling into interval number $m$ of the range of values, defined as follows:

$$F{\left[x\left(j\right)\right]}_{min}+mh\le F\left[x\left(j\right)\right]<F{\left[x\left(j\right)\right]}_{min}+\left(m+1\right)h,$$

(9)

$$0\le m\le N-1, N=2\left[{h}^{-1}\right]$$

where the square brackets in the expression for the total number of intervals $N$ denote the integer part of the number. (In (Eq. 9) it is taken into account that for the function (Eq. 6) the difference$F{\left[x\left(j\right)\right]}_{max}-F{\left[x\left(j\right)\right]}_{min}\le 2$, from which we obtain the relation for$N$.) Finally, in (Eq. 8) the argument $p$ is a Fourier variable, the factor$i=\sqrt{-1}$, and the function $\text{Arg}(\text{Z})$ is the argument of a complex quantity on the plane of the complex variable (for example, $\text{Arg}\left(1\right)=0,\text{Arg}\left(i\right)=\frac{\uppi }{2}$, $\text{Arg}\left(-1\right)=\uppi$ и $\text{Arg}\left(-i\right)=\frac{3\uppi }{2}$).

Below we demonstrate the possibility that the application of statistical functional of type (Eq. 8) with high probability allows for effective differentiation of variants of high probability of the upcoming favorable course of the disease or high risk of its unfavorable development for new patients. This type of diagnostic differentiation is implemented based on the comparison of two ensembles of dependencies (Eq. 8), each of which corresponds to its own patient with an already known fact of recovery or death. The criterion obtained as a result of the comparison makes it possible to draw a conclusion about the high probability of recovery or, conversely, death for newly admitted patients. Such information allows immediately after admission to the hospital to identify patients with a particularly high risk, which makes it possible to more carefully monitor laboratory parameters and timely correct therapy to increase the chances of a favorable course of the disease.

Results and discussions

Each of the figures considered below shows the dependences $L\left( {p,h,W,\alpha } \right)$ as a function of the Fourier variable ${\mathfrak{p}}$ for different values of the parameters $h$, $W$, and $\alpha$. The initial set of test results for the corresponding patient, to which this statistical functional corresponds, includes the results of the above analyses. For each of the 103 patients considered below, hospitalized with a diagnosis of decompensated diabetes mellitus, all the indicated information corresponds to the moment of admission to the hospital. These patients are randomized into two groups depending on the outcome of the course of acute decompensation of diabetes. One of the groups included ${N}_{G}=83$ patients with a favorable course of the disease; in all Figs. 2, 3, 4, they correspond to the curves $L\left( {p,h,W,{ }\alpha } \right)$ or circles (Fig. 3) in green. The other group includes ${N}_{G}=20$ patients with a fatal outcome. These patients are compared with curves $L\left( {p,h,W,{ }\alpha } \right)$ or circles (Fig. 3) in red.

Fig. 2

a The ensemble of ${N}_{R}=20$ red curves corresponds to the dependencies $L\left( {p,h,W,{ }\alpha } \right)$ at $h=0.15, W=1$ and $\alpha = - 1.495$ for patients with an unfavorable course of the disease. Blue dotted curves I and II are the upper and lower boundaries of the curvilinear channel (10). b Part of a inside the area highlighted with a yellow background. c Red and blue dotted curves are the same as in a, b. Ensembles of ${N}_{G}=83$ green and ${N}_{R}=20$ red lines are the $L\left( {p,h,W,{ }\alpha } \right)$ dependencies (constructed at $h=0.15, W=1$ and $\alpha = - 1.495$) for recovered patients and, respectively, for patients with a fatal outcome. Figure 2d. The blue dotted boundaries of the channel (Eq. 10) are the same as in a, b, c. The yellow background highlights one of the regions of exit from the boundaries of the channel (Eq. 10) for one of the green curves $L\left( {p,h,W,{ }\alpha } \right)$ shown in c. d The region highlighted with a yellow background in d. The differences $S_{2} \left( {p_{l} } \right) - L_{M} \left( {p_{l} , h,W,{ }\alpha } \right)$ are symbolically depicted by red vertical segments

Fig. 3

Logarithmic representation for 83 average values ​​${\mathcal{L}}_{G, M}$ and 20 similar averages ${\mathcal{L}}_{R, M}$ (green and red sequences of circles, respectively)

Fig. 4

An ensemble of 40 solid green and 20 red curves $L\left( {p,h,W,{ }\alpha } \right)$, as well as the boundaries of the channel (Eq. 10) (blue dotted curves I and II) at $h=0.175, W=5$ and $\alpha = - 1.495$ for the local fragment $2.10\le p\le 2.24$ of the abscissa interval corresponding to row 2 of Table 1

Figure 2 shows the dependencies $L\left( p \right) = L\left( {p,h,W,{ }\alpha } \right)$ for $h=0.15, W=1$ and $\alpha = - 1.495$. Figure 2a shows 20 red curves $L\left( {\mathfrak{p}} \right)$, each of which corresponds to a patient with an unfavorable outcome. Here, the upper and lower blue dotted curves I and II correspond to the dependencies $S_{1} \left( {\mathfrak{p}} \right)$ and, respectively, $S_{2} \left( {\mathfrak{p}} \right)$ of the form

$$S_{1} \left( p \right) = Max \left\{ {L\left( {p,h,W,{ }\alpha } \right)} \right\} + \varepsilon \left( p \right)$$

(10)

$$S_{2} \left( p \right) = T\left( p \right)H\left[ {T\left( p \right)} \right], T\left( p \right) = Min \left\{ {L\left( {p,h,W,{ }\alpha } \right)} \right\} - \varepsilon \left( p \right)$$

where $H\left[X\right]=1$ for $X>0$ and $\left[X\right]=0$ for $X\le 0$ is the unit Heaviside function of the variable $X$. The factor $H\left[ {T\left( {\mathfrak{p}} \right)} \right]$ is necessary to ensure the condition $S_{2} \left( {\mathfrak{p}} \right) \ge 0$.

In (Eq. 10), the functions $S_{1,2} \left( {\mathfrak{p}} \right)$ and $\varepsilon \left( {\mathfrak{p}} \right)$ are constructed based on the Chebyshev inequality and for each p are calculated based on the values of the ensemble ${N}_{R}=20$ red curves for fatal patients. In more detail, in (Eq. 10), the maximum and, accordingly, the minimum are taken for a given value of the argument p over the entire set of “red” dependencies $L\left( {p,h,W,{ }\alpha } \right)$, and the term $\varepsilon \left( {\mathfrak{p}} \right)$ (we will call it the critical deviation) is written as

$$\varepsilon \left( p \right) = \sqrt {\frac{{\sigma^{2} \left( p \right)}}{{P_{crit} }}} + {\upnu }\left( p \right)$$

(11)

In (Eq. 11), the denominator ${P}_{crit}$ is the probability that the curve $L\left( {\mathfrak{p}} \right)$, corresponding to a new patient being examined whose health condition at the time of admission to the hospital corresponds to a high risk of a fatal outcome, goes beyond the boundaries of the curvilinear channel defined by the curves $S_{1} \left( {\mathfrak{p}} \right)$ and $S_{2} \left( {\mathfrak{p}} \right)$. These curves in Figs. 2 and 4 are designated by the Roman numerals I and II, respectively. In all cases, we further assume that ${P}_{crit}=0.05$. Thus, with regard to any of Figs. 2 and 4, the value of ${P}_{crit}$ can be considered as a lower estimate for the probability of “missing the target”, that is, a false conclusion about the low danger of an unfavorable course of the disease for a patient with a high risk of a given outcome of the disease. In addition, in (Eq. 11), the numerator ${\upsigma }^{2}(p)$ of the fraction under the square root sign and the term

$${\text{v}}\left( p \right) = \frac{{Max\left\{ {L\left( {p,h,W,{ }\alpha } \right)} \right\} - Min \left\{ {L\left( {p,h,W,{ }\alpha } \right)} \right\}}}{{N_{R} - 1}}$$

(12)

there is a dispersion and, accordingly, an average value for an ensemble of ${N}_{R}-1=19$ vertical distances

$$l_{m} \left( p \right) = L^{{\left( {m + 1} \right)}} \left( {p,h,W,{ }\alpha } \right) - L^{\left( m \right)} \left( {p,h,W,{ }\alpha } \right)$$

(13)

between the nearest vertical (i.e. considered for the same argument $p$) red curves. In other words, in (Eq. 13) the ensemble of values $L^{\left( m \right)} \left( {p,h,W,{ }\alpha } \right), m = 1, 2, \ldots N_{R}$, is a sequence ${N}_{R}$ of values of the functions $L\left( {p,h,W,{ }\alpha } \right)$, corresponding to different patients and sorted in ascending order for the same value of $p$. As an example, in a more detailed Fig. 2b, which corresponds to the interval $1.178\le p\le 1.194$ (see the area highlighted in yellow in Fig. 2a), for the selected value of the Fourier variable ${\mathfrak{p}} = const$ , the intervals $l_{1} = L^{\left( 2 \right)} \left( {p,h,W,{ }\alpha } \right) - L^{\left( 1 \right)} \left( {p,h,W,{ }\alpha } \right), l_{15} = L^{{\left( {16} \right)}} \left( {p,h,W,{ }\alpha } \right) - L^{{\left( {15} \right)}} \left( {p,h,W,{ }\alpha } \right)$ and $l_{19} = L^{{\left( {20} \right)}} \left( {p,h,W,{ }\alpha } \right) - L^{{\left( {19} \right)}} \left( {p,h,W,{ }\alpha } \right)$, which are part of the general ensemble ${l}_{1}, {l}_{2}, \dots {l}_{19}$ and correspond to the given value of the variable $p$, are marked with vertical blue segments. In addition, for the same value of ${\mathfrak{p}}$, points $a$ and $b$ are marked, corresponding to the values of $Max\left\{ {L\left( {p,h,W,{ }\alpha } \right)} \right\}$ and, respectively, $Min\left\{ {L\left( {p,h,W,{ }\alpha } \right)} \right\}$ in (Eq. 10) and (Eq. 12). Obviously, for different $p$, the same dependence $L\left( {p,h,W,{ }\alpha } \right)$ may correspond to values ​​of $L^{\left( m \right)} \left( {p,h,W,{ }\alpha } \right)$ with different indices m. Figure 2c shows together the complete set ${N}_{G}=83$ green curves, which correspond to the dependences $L\left(p\right)$ for recovered patients, as well as the ensemble of all ${N}_{R}=20$ red lines, already considered in Fig. 2a,b, corresponding to cases of fatal outcome. For Fig. 2c, curves Ι and ΙΙ are the same as in Fig. 2a, b.

The degree to which the green lines exit the boundaries of the channel (Eq. 10) will be characterized by a sum of the form

$$G_{M} \left( {h,W,{ }\alpha } \right) = \,\sum\limits_{l = 1}^{{N_{p} }} {{\text{S}}_{M,l} }$$

(14)

Where

$${\text{S}}_{M,l} = \left[ {L_{M} \left( {p_{l} , h,W,\alpha } \right) - S_{1} \left( {p_{l} } \right)} \right]H\left[ {L_{M} \left( {p_{l} , h,W,\alpha } \right) - S_{1} \left( {p_{l} } \right)} \right] + \left[ {S_{2} \left( {p_{l} } \right) - L_{M} \left( {p_{l} , h,W,\alpha } \right)} \right]H\left[ {S_{2} \left( {p_{l} } \right) - L_{M} \left( {p_{l} , h,W,\alpha } \right)} \right]$$

The subscript $M$ here and below is the number assigned to the corresponding patient. In this case, the dependence $L\left( {p,h,W,{ }\alpha } \right)$ corresponding to the patient number $M$ will also be denoted as $L_{M} \left( {p,h,W,{ }\alpha } \right)$, if necessary. For patients with a favorable development of the disease, $M$ takes the values $M=1, 2, \dots 83$, and for patients with a negative outcome, $M=1, 2, \dots 20$. In addition, in (Eq. 14), the factor $H\left(X\right)$ is the unit Heaviside function ($H\left(X\right)=1$ for X > 0 and $X>0$ for $X\le 0$), the horizontal coordinate ${p}_{l}=l\Delta p$, the interval $\Delta p=\frac{\uppi }{{N}_{p}}$, where ${N}_{p}=500$ is taken. It is easy to understand that the values ${\mathcal{S}}_{M,j}$ are always non-negative and are greater than zero only if the curve $L_{M} \left( {p,h,W,{ }\alpha } \right)$ exits the boundaries of the channel (Eq. 10).

Figure 2d shows channel (Eq. 10) and depicts one of the green curves $L\left( {\mathfrak{p}} \right)$ from those shown in Fig. 2c, and Fig. 2d examines in more detail one of the regions where this curve exits the limits of channel (Eq. 10); this region is highlighted with a yellow background in Fig. 2d. In Fig. 2d, with regard to the specified curve $L_{M} \left( {p, h,W,{ }\alpha } \right)$, differences of the form $S_{2} \left( {p_{l} } \right) - L_{M} \left( {p_{l} , h,W,{ }\alpha } \right)$ are symbolically depicted by red vertical segments. In this case, for the region highlighted with a yellow background, the first term $\left[ {L_{M} \left( {p_{l} , h,W,{ }\alpha } \right) - S_{1} \left( {p_{l} } \right)} \right]H\left[ {L_{M} \left( {p_{l} , h,W,{ }\alpha } \right) - S_{1} \left( p \right)} \right]$ in sum (Eq. 14) is not taken into account. Indeed, within the specified region, the curve $L_{M} \left( {p, h,W,{ }\alpha } \right)$ is located below the lower boundary $S_{2} \left( {\mathfrak{p}} \right)$ of the channel (Eq. 10), and therefore, it obviously passes below its upper boundary $S_{1} \left( {\mathfrak{p}} \right)$. Therefore, the difference $L_{M} \left( {p_{l} , h,W,{ }\alpha } \right) - S_{1} \left( {p_{l} } \right) < 0$, and, therefore, the factor $H\left[ {L_{M} \left( {p_{l} , h,W,{ }\alpha } \right) - S_{1} \left( {p_{l} } \right)} \right] = 0$.

In Fig. 3 for ${N}_{G}=83$ patients with numbers $M=1, 2, \dots 83$ (green sequence of circles), which correspond to recovered patients,

$$h = 0.15,W = 1,\alpha = \alpha_{j} = - 1.479 + 0.02\left( { j - 1} \right), j = 1, 2, \ldots {\mathcal{N}} = 15$$

(15)

where we assume $\mathcal{N}=15$, the average values ​​of the type are given in a logarithmic scale

$${\text{L}}_{G, M} = \frac{1}{{\text{N}}} \mathop \sum \limits_{j = 1}^{{\text{N}}} G_{M} \left( {h,W,{ }\alpha_{j} } \right)$$

(16)

(In other words, in (Eq. 16) each term $G_{M} \left( {h,W,{ }\alpha_{j} } \right)$ corresponds to the same quantities $h$ and $W$, but to different values of the power $\alpha = \alpha_{j} = - 1.479, - 1.477, - 1.475 \ldots - 1.451$, see (Eq. 6).)

In Fig. 3, the vertical axis corresponds to the values of the function ${\text{log}}_{10}({\mathcal{L}}_{G, M}+0.01)$. Here, the term 0.01 under the decimal logarithm sign is introduced in order to avoid the logarithmic function going to $-\infty$ when ${\mathcal{L}}_{G, M}=0$. We emphasize that the case ${\mathcal{L}}_{G, M}=0$ means the complete absence of an exit of the curve $L_{M} \left( {p, h,W,{ }\alpha } \right)$, corresponding to patient number $M$ with a favorable course of the disease, from the boundaries of all $\mathcal{N}=15$ considered channels of the form (Eq. 10). (When ${\mathcal{L}}_{G, M}=0$, the value ${\text{log}}_{10}({\mathcal{L}}_{G, M}+0.01)=-2$ corresponds to the green circles at the bottom of Fig. 3.) In the same figure, also on a logarithmic scale, the average values of the form are similarly presented

$${\text{L}}_{R, M} = \frac{1}{{\text{N}}} \mathop \sum \limits_{j = 1}^{{\text{N}}} R_{M} \left( {h,W,{ }\alpha_{j} } \right)$$

(17)

(red sequence of circles; the values of $h,W,{ }\alpha_{j}$ are also given according to (Eq. 15)) for ${N}_{R}=20$ patients with a fatal outcome. In (Eq. 17), the values of $R_{M} \left( {h,W,{ }\alpha_{j} } \right)$, $M=1, 2, \dots 20$, correspond to such patients and are calculated similarly to ${G}_{M}$, see the remark below. In the case of ${\mathcal{L}}_{R, M}=0$, the value of ${\text{log}}_{10}({\mathcal{L}}_{R, M}+0.01)=-2$ corresponds to the red circles in the lower part of Fig. 3.

It should be noted that, as indicated above, there is a significant difference in the method for calculating the values of ${R}_{M}$ from ${G}_{M}$. It consists in the fact that when determining ${R}_{M}$ for any red curve with the number $M$, channel (Eq. 10) is constructed taking into account all the other ${N}_{R}-1=19$ red lines, but discarding the contribution from this particular dependence $L_{M} \left( {p, h,W,{ }\alpha_{j} } \right)$. The indicated exclusion from consideration of this dependence allows us to check how strongly the corresponding red curve “gravitates” to the boundaries of channel (Eq. 10), constructed without taking it into account. In other words, in the Cartesian space of five coordinates $\left\{ {h,W,{ }\alpha , p,L} \right\}$, only such areas are considered for which the effect of a significantly greater “attraction” of the red curves corresponding to fatal patients to the area of channels of type (Eq. 10) takes place with a significantly smaller manifestation of such a concentration for the green lines corresponding to recovered patients.

In Fig. 3, the blue horizontal dotted lines 1 and 2 correspond, outside the logarithmic scale, to the values ${T}_{1}=935$ and, accordingly, ${T}_{2}=1473$. As follows from the analysis of Fig. 3, only for one of the 20 patients with an unfavorable outcome is the value ${{\mathcal{L}}_{R, M}>T}_{2}$. For all the others, ${\mathcal{L}}_{R, M}<{T}_{1}$. While for 43 patients with a favorable course of the disease, the value ${\mathcal{L}}_{G, M}$ is located in the region above the dotted line 2. Consequently, for more than half of the recovered patients (which means the probability of such an event ${\mathcal{P}}_{G}>50\%$), the corresponding green curves satisfy the relation

$${\mathcal{L}}_{G, M}>{T}_{2 },$$

(18)

whereas for patients with a fatal outcome the same condition

$${\mathcal{L}}_{R, M}>{T}_{2}$$

(19)

turns out to be true only for one patient out of 20, that is, with a probability of ${\mathcal{P}}_{R}=5\%$. Thus, from the ratio

$${\mathcal{P}}_{G}\gg {\mathcal{P}}_{R}$$

(20)

a fundamentally important conclusion follows about the difference in the probabilistic properties of the ensembles of green and red dependencies. This difference is manifested in the radically higher probability of a significant (compared to the red curves) exit of the green lines beyond the boundaries of the curvilinear channel (Eq. 10).

As a result, we come to the following conclusion. Let the set of analyses of a new patient being examined, carried out upon his admission to the hospital, correspond to an averaging of the type

$$\tilde{L} = \,\frac{1}{N}\,\sum\nolimits_{j = 1}^{N} {G_{M} \,\left( {h,W,\alpha_{j} } \right)}$$

(21)

for which condition (Eq. 18) is true when replacing ${\mathcal{L}}_{G, M}$ with $\widetilde{\mathcal{L}}$ ̃. The symbol "${G}_{M}$" in (Eq. 21) means that for this patient the calculation is carried out in exactly the same way as for patients of group (G). In this case, as before, the values $h=0.15, W=1$ are taken, and the local sequence $\alpha_{j} ,\,j = 1, 2, \ldots {\text{N}}$ corresponds to (Eq. 15). And also let a similar (EQ.18) relation be fulfilled for this patient

$$\widetilde{\mathcal{L}}>{T}_{2 },$$

(22)

We will call the truth condition (Eq. 22) the event ${(A}_{1}$). Then, based on Bayes' theorem, we obtain that for a given patient, the conditional probability ${P}\left(G|{A}_{1}\right)$ that, when the specified event occurs, he belongs to the group (G) of patients with a future favorable course of the disease, satisfies the relation

$$P\left(G|{A}_{1}\right)\ge \frac{P\left(G\right)P\left({A}_{1}|G\right)\cdot 100\%}{P\left(G\right)P\left({A}_{1}|G\right)+P\left(R\right)P\left({A}_{1}|R\right)}=$$

(23)

$$=\frac{P\left(G\right){\mathcal{P}}_{G}\cdot 100\%}{P\left(G\right){\mathcal{P}}_{G}+P\left(R\right){\mathcal{P}}_{R}}=\frac{\frac{83}{83+20}\cdot 0.50\cdot 100\%}{\frac{83}{83+20}\cdot 0.50+\frac{20}{83+20}\cdot 0.05}\approx 97.6\%.$$

In this case, the probability of a “false alarm” is $\left(G|{A}_{1}\right)\le 2.4\%$.

Here it is taken into account that in the case under consideration the probability of belonging to group (G) is $P\left(G\right)=\frac{{N}_{G}}{{N}_{G}+{N}_{R}}=\frac{83}{83+20}$, and for the group (R) of patients with a fatal outcome such probability is $P\left(R\right)=\frac{{N}_{R}}{{N}_{R}+{N}_{G}}=\frac{20}{83+20}$. In this case, the conditional probabilities of the occurrence of event ${(A}_{1}$) for groups (G) and (R) are $P\left({A}_{1}|G\right)={\mathcal{P}}_{G}>0.50$ and, accordingly, $P\left({A}_{1}|R\right)={\mathcal{P}}_{R}=0.05$, see comments to (Eq. 18) and (Eq. 19). Taking into account (Eq. 23), for the probability $P\left(R|{A}_{1}\right)$ of belonging of such a patient to group (R) of patients, which means a high risk of a fatal outcome, we obtain the estimate

$$P\left(R|{A}_{1}\right)<100\%-97.6\%= 2.4\%.$$

(24)

Now let us consider the case when the opposite condition to (Eq. 22) holds for a new patient being examined.

$$\widetilde{\mathcal{L}}<{T}_{2} .$$

(25)

According to Fig. 3, the same relation, i.e. the inverse to (Eq. 18) condition ${\mathcal{L}}_{G, M}\le {T}_{2}$, is satisfied by the test data of 40 of the 83 patients with a favorable course of the disease. (Relation (Eq. 25) means a relatively small exit of the corresponding green curve $L\left( {p,h,W,{ }\alpha } \right)$ from the boundaries of the considered set of curvilinear channels (Eq. 10), constructed according to the test data of patients with a fatal outcome.) Therefore, to determine the probability of belonging of a new patient under examination to group (G) or to group (R), together with the data of 20 patients with a fatal outcome, we will consider only the test results for the specified 40 recovered patients who meet condition (Eq. 25).

Table 1 shows the values of $h, W$ and $\alpha$, as well as the boundaries of six intervals ${\mathfrak{p}}_{1} \le {\mathfrak{p}} \le {\mathfrak{p}}_{2}$, within which the properties of the ensemble of green dependencies $L\left( {p,h,W,{ }\alpha } \right)$ are considered, reduced to 40 curves corresponding to the specified patients for whom (Eq. 25) is true. In this case, a condition is introduced that for these intervals there is not a single red curve that would go beyond the boundaries of the corresponding channel (Eq. 10), constructed without taking into account the corresponding dependence $L\left( {p,h,W,{ }\alpha } \right)$, that is, according to the ensemble of the remaining 19 such dependencies for patients with an unfavorable outcome. This fact means that for patients with an unfavorable course of the disease, there is a significantly greater “gravitation” of the curves $L\left( {p,h,W,{ }\alpha } \right)$ to the region inside the specified channel than for recovered patients.

Table 1 Values of $h,W,\alpha$ and the boundaries of the x-axis intervals ${\mathfrak{p}}_{1} \le {\mathfrak{p}} \le {\mathfrak{p}}_{2}$

In the specified table, the value of ${\widetilde{N}}_{G}$ is the number of green curves $L\left( {\mathfrak{p}} \right)$ (the full set of which in this case corresponds to those 40 patients under consideration with a favorable course of the disease, for whom (25) is true) that go beyond the corresponding channel (Eq. 10) on the specified interval of the $p$-axis. In addition, the penultimate column of Table 1 shows the values ${P}_{G}{\prime}$ of the probability that, given the exit of the dependence $L\left( {p,h,W,{ }\alpha } \right)$, corresponding to a newly admitted patient to the hospital, beyond the corresponding channel (Eq. 10), this patient belongs to the "green" group of patients with an upcoming favorable course of the disease under consideration. Finally, the last column shows the probabilities ${P}_{R}{\prime}=1-{P}_{G}{\prime}$ of the reverse event, consisting in the exit of the dependence $L\left( {p,h,W,{ }\alpha } \right)$, corresponding to a newly admitted patient with a high risk of an unfavorable outcome, beyond the corresponding channel (Eq. 10). Note that the requirement introduced above, that for all rows of Table 1 not a single red curve corresponding to patients with a fatal outcome should go beyond the boundaries of the channel (Eq. 10), constructed without taking into account the dependence ${P}_{R}{\prime}=1-{P}_{G}{\prime}$ corresponding to this curve – such a condition illustrates the fundamentally different probabilistic properties of the two ensembles of data curves under consideration.

The total number of green curves under consideration that at least once go beyond the boundaries of channel (Eq. 10) during a complete enumeration of all six sets of values $h, W$, $\alpha$, ${\mathfrak{p}}_{1}$ and ${\mathfrak{p}}_{2}$ specified in Table 1 is 33 out of their total number, which is 40. This means that 7 such dependencies pass inside all the channels under consideration (Eq. 10) for any specified sets of values $h,W,{ }\alpha$, ${\mathfrak{p}}_{1}$ and ${\mathfrak{p}}_{2}$ for all rows of Table 1. And in this case, the same is true for all 20 red dependencies. Consequently, given that (Eq. 24) is satisfied for 19 red lines, the probability ${P}_{spec}$, which determines the degree of specificity of this technique in the case under consideration, is estimated as

$${P}_{spec}=\frac{19}{19+7}\cdot 100\%=73\% .$$

(26)

The value of ${P}_{spec}$ determines the reliability of the belonging of a new patient under examination, for whom condition (Eq. 24) is true and at the same time the dependence $L\left( {p,h,W,{ }\alpha } \right)$ does not go beyond the boundaries of all six channels (Eq. 10) considered in Table 1, to the group (R) of high-risk patients, and not to the group (G) of patients with an upcoming favorable course of the disease. In particular, condition (Eq. 26) means that, on average, out of every four patients whose test data satisfy the condition of not going beyond all six channels corresponding to Table 1 with the simultaneous fulfillment of condition (Eq. 25), three such patients actually belong to the high-risk group, and with respect to the fourth, there is a “false alarm”.

As an example, Fig. 4 shows the set of 40 green and 20 red curves under consideration, as well as the boundaries of the channel (Eq. 10) for the values of $h,W,{ }\alpha$ corresponding to the second row of Table 1, for a part of the section $2.12\le p\le 2.24$ of the abscissa axis corresponding to this row. From the analysis of this figure it follows that there is a fairly high, about 30%, probability of the green curves corresponding to patients with an upcoming favorable course of the disease exiting the boundaries of this channel, constructed based on the set of 20 dependencies $L\left( {p,h,W,{ }\alpha } \right)$, which correspond to patients with an unfavorable outcome. It is important to note that in this case, as for all other rows of Table 1, not a single red curve exits the boundaries of the channel (Eq. 10), constructed without taking into account the dependence $L\left( {p,h,W,{ }\alpha } \right)$ corresponding to this curve.

It should be noted that various machine learning (ML) methods are currently used to create predictive models for type 2 diabetes, including SVM, DT, LR, RF, ANN, Bayesian classifier, etc. At the same time, along with obvious advantages, such an approach has the following disadvantages.

• (a) Any machine learning model is tied to a given set of initial parameters and requires new training when they are replaced. At the same time, the methodology proposed in the article is easily adapted to changes in the specified set. In this case, only the data in Table 1 changes while maintaining the general form of the functional $L\left(p\right)$, see (Eq. 8).

• (b) When using ML, a large cohort of patients to be examined is required, on the order of 1000 or more people. Given the possible variability of non-genetic markers of the disease in relation to different national population groups, this circumstance means that it is desirable to conduct appropriate examinations for each new geographic region. At the same time, as follows from the results of this article, to assess the probability of the disease in siblings, it is sufficient to examine about a hundred sick children and a commensurate number of their healthy brothers or sisters.

• (c) Taking into account point (b), the problem of overtraining of artificial neural networks (ANNs) is especially important. This means that a situation arises when the model remembers the properties of the training sample data too well, while losing the ability to generalize and work effectively on new, previously unfamiliar examples. In other words, in this case, the model “adapts” to the training data, but cannot correctly process slightly different in their properties (e.g., significantly more diverse) data of the validation and/or test sample. In particular, the overfitting effect occurs when training is too long, the number of training examples is insufficient, or the structure of the neural network is too complicated.

The key problem here is the fact that the logic of the ANN operation is almost impossible to directly verify, since it is usually too complex and uses a large number of "internal" variables.

At the same time, the logic of the algorithm proposed in this article is very simple and consists of a chain of no more than five to seven consecutive (without branching) steps. This dramatically reduces the number of missed overfitting situations, since, in relation to the testing sample, its consequence will be a complete or almost complete absence of new red curves falling into previously established areas, similar to zones in Fig. 4.

Thus, the approach proposed in this article can be effectively applied in a situation of a relatively small number of examined patients, and can be used as an independent and simple way to verify the correctness of the artificial neural network.

Finally, we note that the overall sensitivity of the approach used, that is, the probability ${P}_{sens}$ of taking into account all patients with an upcoming acute course of the disease, is defined as the ratio of the number of 19 such patients for whom (Eq. 19) is not satisfied, to their total number, equal to 20:

$${P}_{sens}=\frac{19}{20}\cdot 100\%=95\%$$

(27)

Let us emphasize that relation (Eq. 27) is true in both cases under consideration, that is, both when (Eq. 22) is true and when the inverse condition (Eq. 25) is true for the average value of $\widetilde{\mathcal{L}}$, calculated according to (Eq. 21) for the set of test results of the new patient being examined.

Conclusions

Summarizing the obtained results, we come to the following conclusions.

• 1. The basis of the proposed methodology is the transition from taking into account several individual markers (usually about 5 or 6), which are considered particularly informative, to the study of the probabilistic properties of a set of local random sequences consisting of the values ​​of the results of the analyses performed. These properties depend significantly on the statistical relationships between such results. The possible number of such relationships, ranked by the degree of influence on the result of each individual analysis, is proportional to the number of permutations of the entire ensemble of the specified values. Therefore, even for a very limited, about 10 or slightly more, set of such analyses, they can be compared with many millions of considered dependencies of the type $L\left( {p,h,W,{ }\alpha } \right)$; these dependencies change significantly with variations in the values ​​of h, W and α. These variations make it possible to weaken or, conversely, strengthen the influence of certain specified relationships between the results of analyses on the value of the functional $L\left( {p,h,W,{ }\alpha } \right)$.

• As a result, when analyzing such a large possible ensemble of dependencies $L\left( {p,h,W,{ }\alpha } \right)$, it is quite easy to determine the area of ​​features that are characteristic of the entire group of patients with an upcoming unfavorable course of the disease and are completely uncharacteristic of the entire group of recovered patients.

• 2. For a new patient being examined, if the ratio (Eq. 18) is valid, the probability of a fatal outcome from the disease in question is very small and is estimated according to (Eq. 24).

• 3. If the condition (Eq. 25), which is the opposite of (Eq. 18), is true for the specified patient, and the dependence $L\left( {p,h,W,{ }\alpha } \right)$ corresponding to it does not go beyond the boundaries of any of the six channels (Eq. 10) corresponding to the sets of values $h,W,{ }\alpha$ , ${\mathfrak{p}}_{1}$ and ${\mathfrak{p}}_{2}$ given in all rows of Table 1, then the probability of an unfavorable course of the disease is high and is estimated according to (Eq. 27) with specificity (Eq. 26).

• 4. If condition (Eq. 25) is met for the specified patient, and at the same time the curve $L\left( {p,h,W,{ }\alpha } \right)$ exits the boundaries of the channel (Eq. 10) for at least two rows of Table 1, then the probability of a fatal outcome is assessed as low.

• 5. If condition (Eq. 25) is satisfied for the specified patient, and such an outcome occurs only once, then the probabilities ${P}_{G}{\prime}$ and ${P}_{R}{\prime}$, given in the last two columns of Table 1, despite the close-to-absolute values of the probability ${P}_{G}{\prime}$ (which turns out to be much greater than ${P}_{R}{\prime}$), should still be perceived with caution. This is due to the relatively small number of patients with a fatal outcome of the disease taken into account. In this case, one should limit oneself to the conclusion that "there is more data in favor of the hypothesis of a favorable outcome."

• 6. Of fundamental importance is the fact that for each of the patients with a fatal outcome, the probability of the corresponding curve $L\left( {p,h,W,{ }\alpha } \right)$ exiting the limits of channel (Eq. 10) is determined in exactly the same way as for a newly admitted patient with a high risk of an unfavorable course of the disease. This follows from the condition that the boundaries of this channel are always constructed without taking into account the dependence $L\left( {p,h,W,{ }\alpha } \right)$, constructed based on the results of analyses of one of the patients with an unfavorable course of the disease. Such a condition can be considered as an element of verification of the proposed methodology. It means that for a newly admitted patient with a high risk of an unfavorable outcome, the probability of exiting both the region below the dotted line 2 in Fig. 3 and the boundaries of any of the channels (Eq. 10) corresponding to all six lines of Table 1 is obviously small and is estimated as a value of the order of $\sim {N}_{R}^{-1}$, which does not exceed 5%.

• 7. We also note that with the help of the approach described above it becomes possible to rank patients according to the possible prognosis of the course of the disease, using data from a standard set of analyses.