JP7281633B2 - How to Determine Mean Red Blood Cell Age - Google Patents

Description

The present invention relates to a method for determining mean cell age and a system and program for determining mean cell age.

Although it is important to keep blood glucose levels low in diabetic patients and blood glucose monitoring is essential, plasma glucose levels can easily fluctuate due to diet and other factors. The hemoglobin A1c (HbA1c) value, which indicates the ratio of glycated hemoglobin to total hemoglobin, fluctuates reflecting the average blood glucose level over the past one to several months, and is used as a biomarker of average blood glucose (AG) level.

In order to estimate the average blood glucose level from the hemoglobin A1c level, many studies have been reported assuming a linear relationship. Proposed:

Formula (1) is widely used among physicians by recommendation of the American Diabetes Association (ADA). Also, although the linear relationships proposed so far, including Equation (1) above, are functional and valid in actual clinical use, they contain some mathematical contradictions.

First, in these equations hemoglobin is glycated even in the absence of glucose in plasma (AG = 0 mg/mL). When graphing these equations, they must pass through the origin.
Second, these formulas exceed the theoretical limit of 100% for hemoglobin A1c values. Even when red blood cells are soaked in saturated sugar solution (about 200,000 mg/mL), the hemoglobin A1c value should not exceed 100%.

On the other hand, mean corpuscular age (M _RBC ) is known as a biomarker for diagnosing hemolytic anemia (Non-Patent Document 2), gastrointestinal bleeding (eg, bleeding caused by colon cancer), and the like. For example, the lifespan of red blood cells is normally about 120 days, but when there is bleeding or hemolytic disease, the number of red blood cells produced by the bone marrow increases, which is 6 to 8 times the normal lifespan. , shortened to 100 days or less in hemolytic anemia. When the amount of red blood cells in the blood tends to be insufficient, that is, when the decomposition and metabolism of red blood cells are accelerated for some reason, and red blood cells tend to be insufficient, hematopoietic stem cells in the bone marrow tissue proliferate and differentiate. to produce more red blood cells. Therefore, unless there is severe hematopoietic abnormality, massive bleeding, or hemolysis, the blood red blood cell count does not change significantly. Therefore, it has been difficult to detect abnormalities such as bleeding and hemolysis in the body from the blood red blood cell count.

As a conventional method for measuring erythrocyte lifespan, ⁵¹ Cr, a radioactive isotope, or artificially chemically modified (for example, biotin-labeled) erythrocytes are administered to the human body as injections, and the reduction rate of their blood amount is measured. There are ways to do that, and they are in clinical use. However, these methods have the risk of serious side effects such as internal exposure of the subject and anaphylactic shock due to repeated administration of chemically modified red blood cells that are immunologically foreign, and are limited according to the severity of the disease. It was only used purposefully. Against this background, the development of a non-radioactive and safe erythrocyte lifespan measurement method has been awaited.

Nathan, D. M., Kuenen, J., Borg, R., Zheng, H., Schoenfeld, D., Heine, R. J., 2008. Translating the A1C assay into estimated average glucose values. Diabetes Care 31 (8), 1473-1478 . Usuki, K., 2015, Hemolytic anemia: diagnosis and treatment, Journal of the Japanese Society of Internal Medicine, Vol.104, No.7, pp.1389-1396 Shrestha, R. P., Horowitz, J., Hollot, C. V., Germain, M. J., Widness, J. A., Mock, D. M., Veng-Pedersen, P., Chait, Y., 2016. Models for the red blood cell lifespan. J Pharmacokinet Pharmacodyn 43 (3), 259-274. Beach, K. W., 1979. A theoretical model to predict the behavior of glycosylated hemoglobin levels. J Theor Biol 81 (3), 547-561. Higgins, P. J., Bunn, H. F., 1981. Kinetic analysis of the nonenzymatic glycosylation of hemoglobin. J Biol Chem 256 (10), 5204-5208. Ladyzynski, P., Wojcicki, J. M., Bak, M., Sabalinska, S., Kawiak, J., Foltynski, P., Krzymien, J., Karnafel, W., 2008. Validation of hemoglobin glycation models 22 using glycemia monitoring in vivo and culturing of erythrocytes in vitro. Ann Biomed Eng 36 (7), 1188-1202. Lledo-Garcia, R., Mazer, N. A., Karlsson, M. O., 2013. A semi-mechanistic model of the relationship between average glucose and HbA1c in healthy and diabetic subjects. J Pharmacokinet Pharmacodyn 40 (2), 129-142. Malka, R., Nathan, D. M., Higgins, J. M., 2016. Mechanistic modeling of hemoglobin glycation and red blood cell kinetics enables personalized diabetes monitoring. Sci Transl Med 8 (359), 359ra130.

In order to resolve some mathematical inconsistencies in the above linear relationship, the present inventors sought a more accurate relationship between the average blood glucose level and the hemoglobin A1c level, and found a new and accurate We succeeded in obtaining a nonlinear relational expression (Equation (21) described later). Furthermore, based on this relational expression, the present inventors have succeeded in deriving a relational expression or an approximation thereof that can easily and accurately determine the average corpuscular age from the average blood sugar level and the hemoglobin A1c value.

Therefore, an object of the present invention is to propose a relational expression or an approximation thereof that can determine the average corpuscular age from the average blood glucose level and the hemoglobin A1c value of a subject, and a method for determining the average corpuscular age using these expressions. and to provide a system and program for determining mean red blood cell age.

The measurement principle of the present invention is a method of measuring erythrocyte lifespan by utilizing the non-enzymatic continuous saccharification of erythrocytes newly generated in the body by blood glucose throughout the 120 days of erythrocyte lifespan.
The longer the new erythrocytes are exposed to blood glucose and the higher the blood glucose concentration, the more glycation progresses, and the Schiff bases formed in the binding of glucose and erythrocyte proteins are chemically transformed over time. change to a stable bond. As a result, saccharification accumulates irreversibly. Therefore, the longer the erythrocyte lifespan, the more saccharification progresses.
Since the mechanism of red blood cell glycation is a non-enzymatic reaction, it is not affected by qualitative and quantitative changes in other enzymes and proteins, and a stable glycation reaction occurs continuously in the body. It can also be reproduced in vitro. The present invention provides an erythrocyte lifespan measurement technique that utilizes such in vivo homeostatic, non-enzymatic saccharification reaction.
As mentioned above, homeostasis is maintained so that the amount of red blood cells in human blood is always kept constant. Therefore, when there is continuous bleeding or hemolysis in the body, the bone marrow tissue produces red blood cells to try to restore the normal value. In this case, red blood cell supply and demand (metabolism) rates increase and red blood cell life span is shorter than normal. Therefore, these erythrocytes are exposed to glucose in the blood for a short period of time, that is, the erythrocytes end their lives with low saccharification.
The technique of the present invention is a method of calculating the erythrocyte life span from the exposure time of erythrocytes to blood glucose by utilizing the aforementioned non-enzymatic saccharification mechanism of erythrocytes which physiologically exists in the human body.

［式中、Ｍ_ＲＢＣは平均赤血球年齢（day）であり、ＨｂＡ１ｃはヘモグロビンＡ１ｃ値であり、ｋ_ｇは糖化率（dL/mg/day）であり、ＡＧは平均血糖値（mg/dL）である］
からなる群から選択される式に代入することにより、平均赤血球年齢（Ｍ_ＲＢＣ）を決定する、［１］の方法；
［３］（１）平均血糖（ＡＧ）値とヘモグロビンＡ１ｃ（ＨｂＡ１ｃ）値を入力する手段；
（２）入力された前記平均血糖値とヘモグロビンＡ１ｃ値に基づいて、平均赤血球年齢を算出する演算手段；
（３）算出された前記平均赤血球年齢を出力する手段；
を含む、平均赤血球年齢を決定するシステム；
［４］前記演算手段が、下記式（３０）、（３４）、又は（３５）：

［式中、Ｍ_ＲＢＣは平均赤血球年齢（day）であり、ＨｂＡ１ｃはヘモグロビンＡ１ｃ値であり、ｋ_ｇは糖化率（dL/mg/day）であり、ＡＧは平均血糖値（mg/dL）である］
の少なくともいずれか１つに基づいて、平均赤血球年齢を算出する、［３］のシステム；
［５］コンピュータを下記の手段（１）、手段（２）、手段（３）として機能させるための、平均赤血球年齢（Ｍ_ＲＢＣ）を決定するプログラム：
（１）平均血糖（ＡＧ）値とヘモグロビンＡ１ｃ（ＨｂＡ１ｃ）値を入力する手段；
（２）入力された前記平均血糖値とヘモグロビンＡ１ｃ値に基づいて、平均赤血球年齢（Ｍ_ＲＢＣ）を算出する演算手段；
（３）算出された前記平均赤血球年齢を出力する手段；
［６］前記演算手段が、下記式（３０）、（３４）、又は（３５）：

［式中、Ｍ_ＲＢＣは平均赤血球年齢（day）であり、ＨｂＡ１ｃはヘモグロビンＡ１ｃ値であり、ｋ_ｇは糖化率（dL/mg/day）であり、ＡＧは平均血糖値（mg/dL）である］
の少なくともいずれか１つに基づいて、平均赤血球年齢を算出する、［５］のプログラム；
により解決することができる。 The object is, according to the present invention,
[1] A method for determining the average corpuscular age based on the average blood glucose (AG) and hemoglobin A1c (HbA1c) values of a subject;
[2] The average blood glucose (AG) and hemoglobin A1c (HbA1c) values of the subject were calculated using the following formulas (30), (34), and (35):

[In the formula, _MRBC is the average corpuscular age (day), HbA1c is the hemoglobin A1c value, _kg is the glycation rate (dL/mg/day), and AG is the average blood glucose level (mg/dL). be]
The method of [1], wherein the mean corpuscular age (M _RBC ) is determined by substituting into a formula selected from the group consisting of;
[3] (1) Means for inputting average blood glucose (AG) and hemoglobin A1c (HbA1c) values;
(2) calculating means for calculating average corpuscular age based on the inputted average blood glucose level and hemoglobin A1c value;
(3) means for outputting the calculated average corpuscular age;
a system for determining mean red blood cell age;
[4] The computing means is represented by the following formula (30), (34), or (35):

[In the formula, _MRBC is the average corpuscular age (day), HbA1c is the hemoglobin A1c value, _kg is the glycation rate (dL/mg/day), and AG is the average blood glucose level (mg/dL). be]
The system of [3], wherein the mean corpuscular age is calculated based on at least one of
[5] A program for determining mean corpuscular age (M _RBC ) for causing a computer to function as means (1), means (2), and means (3) below:
(1) means for inputting average blood glucose (AG) and hemoglobin A1c (HbA1c) values;
(2) calculation means for calculating the average corpuscular age (M _RBC ) based on the input average blood glucose level and hemoglobin A1c value;
(3) means for outputting the calculated average corpuscular age;
[6] The computing means is represented by the following formula (30), (34), or (35):

[In the formula, _MRBC is the average corpuscular age (day), HbA1c is the hemoglobin A1c value, _kg is the glycation rate (dL/mg/day), and AG is the average blood glucose level (mg/dL). be]
The program of [5], which calculates mean corpuscular age based on at least one of
can be solved by

According to the present invention, the average corpuscular age can be determined more easily and accurately than in the past by measuring the blood glucose level and collecting blood without relying on conventional isotope administration or administration of chemically modified erythrocyte injections. .

In addition, blood vessels become fragile in diabetic patients, and intra-tissue bleeding may occur in severe cases. For such bleeding, there are comparatively inconvenient examination methods such as sensitive detection of red blood cells in stool for gastrointestinal bleeding. It wasn't discovered until after his condition became serious.
The method of the present invention can detect the life span of erythrocytes, i.e., the presence of bleeding and hemolysis, using only examination methods normally applied to diabetic patients, such as measuring HbA1c in blood and measuring blood glucose concentration in diabetic patients. can. In other words, it is a new technique that can determine the erythrocyte life span only from the data of examination methods performed on ordinary diabetic patients. The technique of the present invention with such specifications is the first method that can calculate erythrocyte life span without any additional, invasive procedure to the patient.
On the other hand, at the site of bone marrow transplantation, the red blood cell supply rate from the transplanted bone marrow can be monitored by using the method of the present invention. Since the method of the present invention does not involve the risk of exposure to radiation or the risk of repeated drug administration, it has become possible to continuously measure erythrocyte life span.
Since the technique of the present invention is a method configured according to such specifications, it does not require special medical facilities or specialized technicians associated with radiation treatment, and it is possible to easily measure erythrocyte life span in clinical settings. ing.

FIG. 1A is the erythrocyte probability density function p(t) and FIG. 1B is the normalized erythrocyte life span distribution. Figure 2A shows a three-compartment model of hemoglobin glycation and Figure 2B shows a simplified two-compartment model. FIG. 3A is a graph visualizing the relationship between the average blood sugar and HbA1c shown in Equation (21), and FIG. 3B is a partially enlarged graph thereof. FIG. 4A is a graph showing an approximation of the relationship between mean blood glucose and HbA1c together with the full underlying relationship (21), and FIG. 4B is a partially enlarged graph. FIG. 5 is a graph showing the relationship between mean corpuscular age (M _RBC ) and iA1c/(1000−2/3×iA1c).

In the method for determining the mean corpuscular age of the present invention (hereinafter sometimes referred to as the method of the present invention), the mean blood glucose (AG) value and the hemoglobin A1c (HbA1c) value obtained from the subject are used to determine the mean corpuscular age (M _RBC ) can be determined. As described later, the present invention provides a plurality of specific approximation formulas that can determine the average corpuscular age, but the essence of the present invention is not limited to these individual approximation formulas. It is based on a novel technical idea that the mean corpuscular age can be determined from two test values, the hemoglobin A1c value and the hemoglobin A1c value.

The average blood sugar level and hemoglobin A1c level can be obtained by a conventional method. In particular, the average blood sugar level can be measured over time over a long period of time using a continuous blood sugar measuring device that can be attached to the body. Note that HbA1c is the ratio of glycated hemoglobin to total hemoglobin, but in each formula in the present specification, the value when 100% is 1 (for example, 0.055 if 5.5%) use.

［式中、ＨｂＡ１ｃはヘモグロビンＡ１ｃ値であり、ＡＧは平均血糖値（mg/dL）であり、ｋ_ｇは糖化率（dL/mg/day）であり、α、βはガンマ分布のパラメータである］
を元にして導き出したものである。 The relational expression or approximate expression thereof between the average blood sugar level, the hemoglobin A1c value, and the average corpuscular age used in the method of the present invention, or the system of the present invention or the program of the present invention, which will be described later (hereinafter, the relational expression or the approximate expression thereof in the present invention ), and in particular, the approximate expression is the relational expression (21) between the average blood glucose level and the hemoglobin A1c level, as described in detail in the examples below:

[In the formula, HbA1c is the hemoglobin A1c value, AG is the average blood sugar level (mg/dL), _kg is the saccharification rate (dL/mg/day), and α and β are parameters of the gamma distribution. ]
It was derived based on

［式中、α、βはガンマ分布のパラメータであり、Γはオイラーのガンマ関数を意味する］
に基づくものである。 Here, the above formula (21) is the probability of erythrocyte death according to the gamma distribution, which the present inventor has determined to be the most preferable among the three types of distributions proposed by Shrestha et al., 2016 (Non-Patent Document 3). Density function p(t):

[Where α, β are the parameters of the gamma distribution and Γ means Euler's gamma function]
It is based on

［式中、Ｍ_ＲＢＣは平均赤血球年齢（day）であり、ＨｂＡ１ｃはヘモグロビンＡ１ｃ値であり、ｋ_ｇは糖化率（dL/mg/day）であり、ＡＧは平均血糖値（mg/dL）である］
を挙げることができる。 The relational expression or its approximation in the present invention is not limited to the following, but for example,

[In the formula, _MRBC is the average corpuscular age (day), HbA1c is the hemoglobin A1c value, _kg is the glycation rate (dL/mg/day), and AG is the average blood glucose level (mg/dL). be]
can be mentioned.

Equation (30), which is a hyperbolic approximation of Equation (21), Equation (34), which is an approximation of Equation (21), and Equation (35), which is a linear approximation of Equation (21), are all used to determine mean cell age. (30 ), Equation (34) is preferred, and Equation (30) is more preferred as it is the closest approximation.

k _g (saccharification rate, unit: dL/mg/day) in the above formula is reported to be 6 to 10×10 ⁻⁶ dL/mg/day as an estimated value, as shown in Table 1 below. For example, it can be appropriately selected within this range. It is also possible to measure mean corpuscular age, hemoglobin A1c value and mean blood glucose level in a population and determine _kg based thereon. Furthermore, as a utilization aspect of the present invention, it is possible to determine whether the average corpuscular age including _kg is relatively high or low with respect _to the normal value. You can get useful information without

The system for determining the average corpuscular age of the present invention (hereinafter sometimes referred to as the system of the present invention) is not limited to the above-described relational expression or approximation thereof in the present invention, but for example,
(1) means for inputting average blood glucose (AG) and hemoglobin A1c (HbA1c) values (hereinafter referred to as input means);
(2 ₎ computing means (hereinafter referred to as computing means called);
(3) means for outputting the calculated average corpuscular age (hereinafter referred to as computing means);
can include

The input means in the system of the present invention is not particularly limited as long as the average blood sugar level and hemoglobin A1c value can be input so that the calculation means can use it. For example, keyboard, touch panel, optical reader, voice recognition equipment and the like.

Further, as will be described later, in a device in which the system of the present invention is incorporated in a continuous blood sugar measuring device, the continuous blood sugar device itself can function as an input means, and the average blood sugar level output from the continuous blood sugar device can be Inputs can be made available to the computing means. In this case, the hemoglobin A1c value can be separately input using various input means (for example, a keyboard, etc.) exemplified above.

The computing means in the system of the present invention, from the average blood sugar level and hemoglobin A1c input from the input means, is not limited to the following, for example, average Calculation means capable of calculating erythrocyte age can be exemplified, for example, a CPU of a computer.

The output means in the system of the present invention is not particularly limited as long as it can output the average corpuscular age calculated by the calculation means so that the system user can recognize it or can be used in another device. For example, a display, a printer, a speaker, etc. can be mentioned. Further, when the average corpuscular age calculated by the computing means is used in another device (for example, a comprehensive diagnosis system), the transmission means to the device can be used as the output means.

The system of the present invention includes the above-described input means, computing means, and output means. Furthermore, a configuration (eg, a cloud system) that connects via a communication line (eg, the Internet) can also be used.

An example of a configuration incorporated in another device is a device in which the system of the present invention is incorporated in a continuous blood glucose measurement device. The apparatus can calculate the mean corpuscular age simply by inputting the hemoglobin A1c value.

Examples of the mode of connection via a communication line include a mode in which the input means and the output means are installed at locations accessible to system users, and the computing means is installed at an arbitrary location.

The program for determining the average corpuscular age of the present invention (hereinafter sometimes referred to as the program of the present invention) is a program for causing a computer to function as the above-described input means, arithmetic means and output means. As for the "input means", "computing means", and "output means" in the program of the present invention, the description in the system of the present invention can be applied as it is.
A computer-readable recording medium recording the program of the present invention is also included in the present invention.

The process of each formula derived by the present inventor in the present invention will be specifically described below, but these are not intended to limit the scope of the present invention.

［式中、α、βはガンマ分布のパラメータであり、Γはオイラーのガンマ関数を意味する］

<<1. Erythrocyte Lifespan》
The probability density function p(t) of erythrocyte death is the following gamma distribution, which the present inventor has determined to be the most preferable among the three types of distributions proposed by Shrestha et al., 2016 (Non-Patent Document 3). can be represented as:

[Where α, β are the parameters of the gamma distribution and Γ means Euler's gamma function]

［式中、R(t)は赤血球の誕生からt日後の赤血球数であり、R_０（/day）は赤血球産生速度である］
p(t)及びR(t)を視覚化したグラフを図１に示す。 The number of red blood cells decreases with p(t).

[wherein R(t) is the red blood cell count t days after the birth of the red blood cells, and R ₀ (/day) is the red blood cell production rate]
A graph visualizing p(t) and R(t) is shown in FIG.

In FIG. 1, FIG. 1A is the erythrocyte probability density function p(t), and FIG. 1B is the normalized erythrocyte lifespan distribution, both of which are examples. Since we assumed a constant erythropoiesis rate R ₀ , the lifespan distribution will be the same as the current age distribution of erythrocytes. Solid line, α = 7.83, β = 12.72, life expectancy = 99.59 days, M _RBC = 56.16 days, dotted line α = 46.38, β = 2.77, life expectancy = 128.47 days, M _RBC = 65.62 days.

前記式（５）から導かれた以下の式を使用した。

Red blood cell count (RBC) is calculated as follows:

The following equation derived from equation (5) above was used.

従って、M_ＲＢＣは

となる。 Mean red blood cell age (M _RBC ) can be calculated as follows.

Therefore, the M _RBC is

becomes.

<<2. Hemoglobin Saccharification》
Figure 2 shows two models of hemoglobin glycation. Figure 2A is a three-compartment model of hemoglobin glycation and Figure 2B is a simplified two-compartment model. HbA _ald is the aldimine complex (intermediate) and k ₁ , k ₂ , k ₃ , k _g are rate constants.

［式中、[G]はグルコース濃度（定常状態ではAGと同じ）であり、HbA_ａｌｄはアルジミン複合体（中間体）濃度である］

dHbA_ａｌｄ/dtは定常状態において無視することができ、Hb(t)+HbA_ａｌｄ+HbA1cは定数であるので、次式が成立する。

従って、ヘモグロビンは、インタクトのヘモグロビンに比例して糖化され、前記モデルは、２コンパートメントモデルに簡略化することができる（図２Ｂ）。

［式中、k_ｇは糖化率（dL/mg/day）である］
式（１６）は、総糖化率がk_１（HbA_ａｌｄに変換する力）とk_３/(k_２+k_３)（HbA1cに変換する保持画分）との積であることを示している。過去に報告されたk_ｇの推定値（非特許文献４～８）を表１に示す。 Hemoglobin glycation is assumed to follow a three-compartment model (Fig. 2A).

[wherein [G] is the glucose concentration (same as AG at steady state) and HbA _ald is the aldimine complex (intermediate) concentration]

Since dHbA _ald /dt can be ignored in steady state and Hb(t)+HbA _ald +HbA1c is a constant, the following holds.

Thus, hemoglobin is glycated in proportion to intact hemoglobin and the model can be simplified to a two-compartment model (Fig. 2B).

[Wherein, k _g is the saccharification rate (dL/mg/day)]
Equation (16) indicates that the total saccharification rate is the product of k ₁ (power to convert to HbA _ald ) and k ₃ /(k ₂ +k ₃ ) (retained fraction to convert to HbA1c). . Table 1 shows estimated values of k _g reported in the past (Non-Patent Documents 4-8).

［式中、Hb(t)は、t日齢の赤血球における非糖化ヘモグロビンの値であり、Hb_０は、赤血球誕生時のインタクトヘモグロビンの値である］
従って、

は、t日齢の赤血球における糖化ヘモグロビンの割合を示す。

[Where Hb(t) is the value of non-glycated hemoglobin in t-day-old red blood cells, and Hb ₀ is the value of intact hemoglobin at birth of the red blood cells]
Therefore,

indicates the percentage of glycosylated hemoglobin in t-day-old erythrocytes.

従って、HbA1cは

となる。 The collected blood contains red blood cells of various ages. Here we assume that the rate of hemoglobin synthesis (R ₀ Hb ₀ ) does not change.

Therefore, HbA1c is

becomes.

FIG. 3 shows a graph visualizing the relationship between average blood sugar (AG) and HbA1c expressed by Equation (21).
Fig. 3A shows AG and HbA1c under conditions of α = 40, β = 4, M _RBC = 82 (thick gray solid line) and α = 6, β = 12, M _RBC = 42 (thin black solid line). indicates a complete relationship with k _g was set to 8×10 ⁻⁶ .
FIG. 3B expands the relationships in the physiological range and shows them with their tangent lines at AG=200 (dotted lines).

式（２１）、式（２２）における関係を視覚化したグラフを図３Ｂに示す。
AG_０= 0における接線は、テーラー展開を用いることにより得られる。

<<3. Origin of intercept - tangent line
The tangent of equation (21) around AG ₀ is (x, y denote AG, HbA1c at the tangent, respectively), indicating the positive intercept:

FIG. 3B shows a graph that visualizes the relationships in Equations (21) and (22).
The tangent at AG ₀ = 0 is obtained by using the Taylor expansion.

<< 4. Approximation of derivation relation
Equation (21) gives an accurate relationship between AG and HbA1c. However, this equation is too complicated to obtain the inverse function needed to convert HbA1c to AG. We also want to reduce the two parameters α, β to one parameter M _RBC .

この式から、M_ＲＢＣを簡単に求めることができる〔式（３５）〕。 To begin with, equation (23) gives the following linear approximation (Fig. 4).

From this equation, M _RBC can be easily obtained [equation (35)].

式（２５）を式（２１）に代入すると、

α>> 1であるので、

この式は、二次元（２Ｄ）近似を与える（図４）。 As a better approximation, the Taylor expansion gives:

Substituting equation (25) into equation (21) yields

Since α >> 1,

This formula gives a two-dimensional (2D) approximation (Fig. 4).

が成立するので、式（２７）は以下のように再編成できる：

この式は、双曲線近似を与える（図４）。 When x is infinitely close to 0,

holds, so equation (27) can be rearranged as follows:

This formula gives a hyperbolic approximation (Fig. 4).

この式を使って、M_ＲＢＣを求めることができる：

Now we can easily derive the following inverse function:

Using this formula, we can find the M _RBCs :

An approximation of the relationship between average blood glucose (AG) and HbA1c is shown in FIG. 4, together with the original full relationship (equation (21) under conditions of α=40, β=4). In FIG. 4, "hyperbolic" is the hyperbolic approximation of equation (28), "2D" is the two-dimensional approximation of equation (27), and "linear" is the linear approximation of equation (24).

xが限りなく０に近づくとき、

が成立するので、

を代入すると、

従って、M_ＲＢＣは

となる。 Substituting equation (25) into equation (21), the following equation can be derived from second-order approximation.

When x is infinitely close to 0,

holds, so

by substituting

Therefore, the M _RBC is

becomes.

<< 5. Consideration》
The inventors have succeeded in proposing a novel and accurate non-linear relationship between AG and HbA1c at steady state, Equation (21), where the intercept in the linear relationship between AG and HbA1c is derived was presented (Fig. 3). This is the first equation showing the relationship between AG and HbA1c based on the exact erythrocyte lifespan distribution (Shrestha et al., 2016). The linear relationship proposed earlier is an approximation of the non-linear relationship.

Our equation (21) shows that AG increases and accelerates with increasing HbA1c. Equation (29), a hyperbolic approximation, indicates that the variation of AG values between 13%-18% of HbA1c is 1.12 times greater than the variation of AG values between 5%-10% of HbA1c. This relationship deserves consideration for physicians who deal with diabetic patients in the actual clinical setting.

Approximation is a kind of technique. There are many functions that are close to a certain function within a certain range. We need to determine/select the appropriate formula. The inventors have succeeded in approximating their own proposed equation (21) to the hyperbolic function (28). This approximate hyperbolic function was very close to the original function (21) (Fig. 4). In addition, we were also able to obtain a plurality of approximation formulas.

<<6. Verification of Approximation Formulas》
(1) Determination of glycation rate ( _kg ) For 31 hemolytic anemia patients and 76 non-anemic subjects, erythrocyte creatinine (EC) values and hemoglobin A1c values (according to International Federation of Clinical Chemistry (IFCC) units; hereinafter referred to as iA1c) ) were measured, and the mean corpuscular age (M _RBC ) and iA1c/(1000−2/3×iA1c) were calculated from these measurements.

As for the hemoglobin A1c value, the clinically used National Glycohemoglobin Standardization Program (NSGP) unit (HbA1c _NGSP ) and the iA1c can be converted by the following conversion formula.

Also, M _RBC was calculated by the following formula proposed by the present inventors.

Furthermore, iA1c/(1000−2/3×iA1c) is based on the following formula derived from the approximation formula (30).

FIG. 5 shows the relationship between the calculated mean corpuscular age (M _RBC ) and iA1c/(1000−2/3×iA1c). In FIG. 5, Δ indicates hemolytic anemia patients, and ◯ indicates non-anemic patients. A linear relationship was observed, as shown in FIG.

Subsequently, saccharification rate (k _g ) was calculated by two methods (slope method and direct method). Table 2 shows the saccharification rate ( _kg ) calculated by each method for the total population (ie, hemolytic anemia patients and non-anemic subjects) and the non-anemic population.

なお、平均血糖（AG）値として、過去の複数の報告から、糖尿病患者を除いた正常値である100 mg/dLを使用した。 In the slope method, the slope of the regression line passing through the origin (horizontal axis: M _RBC , vertical axis: iA1c/(1000−2/3×iA1c)) of the equation (39) was determined by a least-squares model.
In the direct method, it was directly calculated by the following formula, which is a modified formula (39).

As the average blood glucose (AG) value, 100 mg/dL, which is the normal value excluding diabetic patients, was used from multiple past reports.

As shown in Table 2, the glycation rate (k _g ) of the four types was approximately 7×10 ⁻⁶ , but according to FIG. 5, data were not stable in patients with severe hemolytic anemia, so Of the saccharification rates shown in 2, the values obtained by the whole population/direct method may not be accurate. Therefore, 7.0×10 ⁻⁶ was determined from the three types of saccharification rate ((6.94 to 6.99)×10 ⁻⁶ ) excluding the values from the total population/direct method.

(2) Comparison between the mean corpuscular age determined from the approximate formula of the present invention and the mean corpuscular age measured using ⁵¹ Cr Based on three cases reported in the past, from the approximate formula (30) of the present invention Table 3 summarizes the determined mean corpuscular age (M _RBC ) and the mean corpuscular age measured using ⁵¹ Cr.

Herranz (Herranz, L., Grande, C., Janez, M., Pallardo, F., 1999. Red blood cell autoantibodies with a shortened erythrocyte life span as a cause of lack of relation between glycosylated hemoglobin and mean blood glucose levels in a woman with type 1 diabetes. Diabetes Care 22 (12), 2085-2086.) and Ishii In one case of type 2 diabetes that showed a discrepancy between the HbA1c value and the HbA1c value, Diabetes, _Vol . The average value was also obtained.
Herranz and Ishii determined mean blood glucose (AG) values by calculating the mean from data obtained by self-monitoring of blood glucose (SMBG).
Hiratani (Kazuyuki Hiratani, Toshiki Tozuka, Shinichiro Suemori, Hideho Wada, Masafumi Koga, 2016, A case of type 2 diabetes with stomatocytosis with falsely low HbA1c due to subclinical hemolysis, 59(10):719-723), mean blood glucose (AG) values were determined by continuous glucose measurement (CGM).
Erythrocyte lifespan measurements using ⁵¹ Cr show that the normal value for M _RBC is about 60 days, and the normal range for erythrocyte lifespan (half-life) measured with ⁵¹ Cr is 28-30 days according to Herranz and 28-30 days according to Ishii. According to Hiratani, it is 30± ₅ days, and according to Hiratani, it is 26 to 40 days.

The M _RBC calculated from iA1c [the average value in column (e) of Table 3] was 36.95±5.93, and the M _RBC calculated from ⁵¹ Cr [the average value in column (g) of Table 3] was 41.29±2.22. rice field. The results of the paired t-test were t, -0.9278; df, 2; p (bilateral), 0.4514.

The mean corpuscular age that can be calculated according to the present invention can be used as a biomarker in the diagnosis of, for example, hemolytic anemia and gastrointestinal bleeding (eg, bleeding caused by colon cancer).

Claims (3)

The subject's average blood glucose (AG) and hemoglobin A1c (HbA1c) values were calculated using the following formulas (30) and (34 ):

[In the formula, _MRBC is the average corpuscular age (day), HbA1c is the hemoglobin A1c value, _kg is the glycation rate (dL/mg/day), and AG is the average blood glucose level (mg/dL). be]
A method of determining mean corpuscular age (M _RBC ) based on a subject's mean blood glucose (AG) and hemoglobin A1c (HbA1c) values by substituting into a formula selected from the group consisting of: (1) means for inputting average blood glucose (AG) and hemoglobin A1c (HbA1c) values;
(2) calculating means for calculating average corpuscular age based on the inputted average blood glucose level and hemoglobin A1c value;
(3) means for outputting the calculated average corpuscular age;
A system for determining mean cell age, comprising:
The computing means is represented by the following formula (30) or (34):

[In the formula, _MRBC is the average corpuscular age (day), HbA1c is the hemoglobin A1c value, _kg is the glycation rate (dL/mg/day), and AG is the average blood glucose level (mg/dL). be]
The system, wherein the mean cell age is calculated based on at least one of: A computer with the following means (1), means (2), and means (3):
(1) means for inputting average blood glucose (AG) and hemoglobin A1c (HbA1c) values;
(2) calculation means for calculating the average corpuscular age (M _RBC ) based on the input average blood glucose level and hemoglobin A1c value ;
(3) means for outputting the calculated average corpuscular age;
A program for determining mean corpuscular age (M _RBC ) for functioning as
The computing means is represented by the following formula (30) or (34 ):

[In the formula, _MRBC is the average corpuscular age (day), HbA1c is the hemoglobin A1c value, _kg is the glycation rate (dL/mg/day), and AG is the average blood glucose level (mg/dL). be]
The program for calculating mean corpuscular age based on at least one of

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