JPH06194362A - Metabolism analysis method - Google Patents

Abstract

PURPOSE:To represent temporal variation of the concentration of drug or metabolite thereof in humor including blood accurately by approximating the temporal variation of the concentration of a labeled body, e.g. a drug, using a function shown by a specific formula CONSTITUTION:A drug, a poison, or a substrate is given to an animal and the temporal variation of the concentration of the substance and/or the metabolite thereof in humor, e.g. blood, is measured in order to analyze metabolism in the body of the animal. Temporal variation of concentration Cp in the humor is approximated by a function; Y=Aexp(-k1t)+B(1-exp(-k1t))-(1-exp(-k2t)exp(-k3t), or Y=A(exp(-k1t)+b-(1-exp(-k1t))-(l-exp(-k2t))exp(-k3t)). In the formula, (t) represents the time to be elapsed after dosage, and A, B, b, k1, k2, k3 represent constants. The analysis method can be applied to phleboclysis, endoceliac injection, or oral dispensation, and especially useful for phleboclysis.

Description

Detailed Description of the Invention

[0001]

FIELD OF THE INVENTION The present invention relates to a method for analyzing the kinetics of drugs, substrates, etc.
The present invention relates to a kinetic analysis method for drugs, substrates, etc., which can obtain quantitative information on the metabolic capacity and the metabolic rate as well as the size of the metabolic pool that is the rate-determining step of the metabolic pathway.

[0002]

2. Description of the Related Art Drugs, toxins, or substrates labeled with a suitable radioisotope such as carbon, hydrogen, sulfur, or phosphorus are administered to animals by intravenous injection, intraperitoneal injection, oral administration, etc. By measuring the change in radioactivity concentration over time,
Metabolism in animals can be analyzed. Alternatively,
Metabolism in the body is analyzed by administering a drug labeled with an appropriate stable (non-radioactive) isotope, such as nitrogen, to animals by intravenous injection, etc., and measuring the time-dependent change in labeled concentration in blood. can do.

When a radiolabeled substance is intravenously injected, the time-dependent change in the radioactivity concentration in the blood is often the true number of the time after administration on the horizontal axis of the graph, and the logarithm of the blood radioactivity concentration on the vertical axis. When plotted on an axis, it was analyzed by utilizing the fact that it is approximately represented by a straight line or a curve containing two or more straight line portions.

In the former case, that is, when the temporal change in the radioactivity concentration in the blood is approximated to a straight line on a semilogarithmic graph, the temporal change in the radioactivity concentration in the blood Cp is Cp = Co exp (-λt). (T is the time after administration, Co and λ are constants), and it is estimated that there is no metabolic pool involved in the metabolism of the labeled substance. On the other hand, in the latter case, that is, when the graph consists of two or more straight lines, the blood radioactivity concentration Cp
The change over time of Cp ＝ C ₁ exp (-λ ₁ t) ＋ C ₂ exp (-λ ₂ t) ＋ C ₃ exp (-λ ₃ t) ＋ ・ ・ ・ (1) (C ₁ , C ₂ , C ₃ ; λ ₁ , λ ₂ , λ ₃ etc. are constants), and the existence of some metabolic pool is estimated.

From the corresponding combination of Cp, t values, C ₁ ,
Calculation programs for obtaining optimum values of parameters such as C ₂ and C ₃ and λ ₁ , λ ₂ and λ ₃ are also commercially available.

However, on the graph, a region where t is sufficiently small (for C ₁ , λ ₁ ) and a region where t is sufficiently large (C ₂ ,
λ ₂ etc.) to C ₁ , C ₂ , C ₃ etc. and λ ₁ , λ ₂ , λ ₃
If the constant (parameter) is sufficiently greater than ,, lambda ₁ being asked accurately has lambda _2, etc. etc., or lambda ₂
Is limited to the case where is sufficiently larger than λ ₃ etc. If the difference between λ ₁ and λ ₂ etc. is relatively small, the proportion occupied by the straight line part will be small and the accuracy of C ₁ , C ₂ and λ ₂ etc. obtained from the graph will be poor, and the measured data of the obtained graph The conformity with is extremely poor. In such a case, the accuracy of calculation by an arithmetic program that calculates each parameter without using a graph becomes extremely poor.

Even when the accuracy of parameters such as C ₁ and C ₂ and λ ₁ and λ ₂ is relatively good, the values of those parameters are used as the pool volume and blood flow velocity in the estimated metabolic model. It was difficult to connect directly.

The conventional method (1) (the sum of simple exponential functions) is applied when the blood concentration Cp is constantly decreased, and the slope of the decrease is midway. When the value increases with (in this case, an inflection point appears), especially the density
It was difficult to apply Cp when it showed a marked maximum with time.

[0009]

Therefore, it is possible to accurately express the temporal change of the concentration in body fluid such as blood by a function of a relatively simple form, and to estimate the metabolic model and There is an urgent need for a method for analyzing the kinetics of drugs, toxins, substrates, etc., which enables at least semi-quantitative estimation of the size of the metabolic pool of E. coli. Even if the time-dependent change in concentration Cp shows a complicated profile, such as the decrease gradient increases in the middle or shows a remarkable maximum, the time-dependent change in concentration can be accurately expressed with a relatively simple function. Therefore, a method for analyzing the in vivo kinetics of drugs and the like has been earnestly desired.

The object of the present invention is, firstly, to obtain a function of the time-dependent change in the concentration of a drug or the like and a metabolite thereof in a body fluid such as blood in a relatively simple form even if the profile is complicated. It is to realize a dynamic analysis method for drugs and the like that can be accurately expressed by.

Secondly, the object of the present invention is to estimate the in vivo metabolism model of a drug or the like, and at least semi-quantitatively estimate the size of the metabolic pool therein, the pharmacokinetics of the drug or the like. It is to realize the analysis method.

[0012]

In the present invention, in order to achieve the above-mentioned object, the time-dependent change of the concentration Cp of a labeled substance such as a drug in a body fluid is expressed by a function Y = A exp (-k ₁ t) + B {1−exp (-k ₁ t)} {1−exp (-k ₂ t ')} exp (-k ₃ t) (2) is approximately expressed (t is the time after administration, A, B , K ₁ , k ₂ ,
k ₃ represents a constant, t 'is equal to t-d, d is 0 or a positive real number, when t is not greater than d t' is 0).

The true number of the time t after administration is plotted on the horizontal axis of the graph,
When the logarithm of the blood labeled substance concentration Cp is plotted on the vertical axis,
Approximate A from the intercept of the curve at t = 0, k ₁ from the tangent slope of the curve near t = 0, and k ₃ from the tangent slope of the part where the tangent of the curve is minimum. . If k ₃ = k ₂ , k ₃ = 2k ₂ , k ₃ = 3k ₂ , k ₃ = 4k ₂ , 2k ₃ = k ₂ , 3k ₃ =
When k ₂ and 4k ₃ ＝ k _2, etc., L _m = {1−exp (-k ₂ t)} exp (-k ₃ t) is the maximum t _m (the measured value of Cp around that). Exists t
(may be t _m ), {Cp−A exp (-k ₁ t _m )} / {1−exp (-k ₁ t _m )} {1−exp (-k
₂ t _m )} exp (-k ₃ t _m ) is found (A, k ₁ , k ₃ use the above approximations), and A exp (-k ₁ t) + B {1−exp (-k ₁ t)} {1−exp (-k ₂ t)} exp (-
The value of k ₂ (relative relationship with k ₃ ) that gives the value of k ₃ t) closest to Cp (measured value) was found, and using this k ₂ , Y ＝ A exp (-k ₁ t ) + B {1−exp (-k ₁ t)} {1−exp (-k ₂ t)} ex
Let p (-k ₃ t) be a function representing the change over time in the concentration Cp of the labeled substance in blood.

Assuming that {1−exp (−k ₁ t _m )} can be approximated to 1, the expression {Cp−A exp (−k ₁ t _m )} / {1−exp (−k ₂ t _m )} exp You may use (-k ₃ t _m ) to find a temporary approximation to B.

In the above calculation, b = B / A
It is easier to find (b) by rewriting (2) as follows. Y ＝ A [exp (-k ₁ t) ＋ b {1−exp (-k ₁ t)} {1−exp (-k ₂ t)} exp (-k ₃ t)] (3) b is 1 It never grows.

When there is a relation of k ₂ = mk ₃ (m is a positive real number), t _{m at which} L = {1−exp (-k ₂ t)} exp (-k ₃ t) becomes maximum is t _m = {Ln (m + 1)} / mk ₃ . The maximum value of L obtained at k ₂ ＝ mk ₃ (t _m ＝ {ln (m + 1)} / mk ₃ is {1-1 / (m + 1)} / {(m + 1) / m} For example, when k ₂ = k ₃ , t _m = ln 2 / k ₃ and the maximum of L is 1 / 4.When k ₂ = 2k ₃ , t _m = 1
It is n3 / (2k _3). The maximum value of L is (2/3) (3) -1/2 or about 1 / 2.6. When m> 1, the maximum value of L is k ₂ =
It is always larger than the maximum value 1/4 when k ₃ .

When there is a relation of k ₃ = nk ₂ , {1-exp (-k ₂
t)} exp (-k ₃ t) is the maximum tm is t _m = {ln (n + 1) −ln (n)} / k ₂ = {ln (n + 1) −ln (n)} / (k ₃ / n). The maximum value of L obtained at t _m = {ln (n + 1) −ln (n)} / (k ₃ / n) is [1− {n / (n + 1)}] {n / (n + 1)} n. For example, k ₃ = 4k ₂
Then the maximum of L is (1/5) (4/5) 4, or about 0.08.
Then, the maximum value of L when n> 1 is always smaller than the maximum value 1/4 when k ₂ = k ₃ .

Utilizing such a relationship, for example, first,
Assuming that k ₂ = k ₃ , {Cp / A −exp (-k ₁ t _m )} / [{1−exp (-k ₁ t _m )} / 4] = 4 {Cp / A−exp (- k ₁ ln2 / k ₃ )} / {1−exp (-k ₁ ln2 / k ₃ )} is a temporary value of b, and the function (3), that is, Y ＝ AY ₀ ＝ A [exp (-k ₁ t) ＋ b {1−exp (-k ₁ t)} {1−exp (-k ₂
t)} exp (-k ₃ t)], and assuming that k ₂ = 2k ₃ , {Cp−A exp (-k ₁ t _m )} /2.6 {1−exp (-k ₁ t _m )} t _m ＝
LN3 / a (2k ₃₎ as the value of the temporary b, obtains the value of the function (3), or the difference between the measured value of Cp is reduced, or increases, or not. If the difference from the measured value | Y−Cp | decreases as a whole,
By further increasing the value of m = k ₂ / k ₃ to 3 or 4, or higher, find the value of m = k ₂ / k ₃ where | Y−Cp | becomes the smallest.

When k ₂ = 2k ₃ , if the difference | Y −Cp | from the measured value increases when k ₂ = k ₃ , then k ₃ = nk ₂
(n = 2, 3, 4, etc.), find the optimal n by the same procedure as above.

The difference between the value of the function (3) and the measured value of Cp (expressed as the sum of standard deviations) is calculated for the value of b obtained by using this optimum value of m or n, and b and k A value of ₂ ,
If necessary, modify the values of k ₁ and k ₃ to obtain the optimal function (2)
Or decide (3).

In the increasing region of the function Y ₀ , if the value of │Y-Cp│ does not become small even if the value of k ₂ is modified,
By substituting for t a variable t'which is equal to t-d when t is greater than a positive constant d and is 0 when t is not greater than d, and choosing an appropriate value for d The accuracy of can be improved.

The above calculation can be easily performed by using an electronic calculator. The electronic computer, for example, has significant digits 1 to 9.999 for A, decimal digits 0.01 to 1000,
Significant digits 1 to 9.9 or 9. for b, k ₁ , k ₂ , k ₃ .
If the numerical signals up to 99 and decimal digits from 0.001 to 10 can be selectively output to the arithmetic unit, the purpose is sufficiently achieved.

The metabolic analysis method of the present invention can be applied to various administration methods such as intravenous injection, intraperitoneal injection, and oral administration, but is particularly useful when administered by intravenous injection. When the radiolabel is administered by intravenous injection and the time change of the radioactivity concentration in blood is measured, the function used in the present invention agrees very well with the measured value of the time change of the radioactivity concentration in blood. . The same applies when a compound labeled with a stable isotope is used instead of the radiolabel. It is also effective when tracing at least a part of the compound itself and its metabolites by using a microanalysis method (liquid chromatography, gas chromatography, etc.) without using a labeled substance.

The metabolic analysis method of the present invention is extremely useful in clarifying its pharmacokinetics in the development of new drugs.

[0025]

EXAMPLES The following examples are given to further illustrate the present invention. [Example 1] 0.05 cc of a solution containing 2.3 μmol of inulin labeled at the methoxy position with 34 Cq / mmol of ¹⁴ C was injected into the portal vein of a Wistar male rat weighing 220 g for 15 seconds, 75 seconds, 3 seconds.
Minutes, 5 minutes, 10 minutes, 20 minutes, 30 minutes, 40 minutes, 50 minutes, and 60 minutes later, about 100 microliters of blood was collected from the hepatic vein, plasma was obtained by a conventional method, and its radioactivity was counted by liquid scintillation counting. It was measured by the method. The measurement results are shown in Table 1.

[0026]

A graph was prepared by plotting the true number of the time t after administration on the horizontal axis and the logarithm of the relative radioactivity intensity Cp on the vertical axis,
A straight line passing through the nearest point corresponding to t = 15 seconds, 75 seconds, and 3 minutes was drawn, and the slope and the intercept of the vertical axis (t = 0) were obtained.
The slope was -0.6 and the intercept was 1530. Further, when the gradient of the straight line passing through the nearest points of t = 20, 30, 40, 50, 60 minutes was calculated on the above graph, it was -0.05.

Then, suppose that 1530 is used as A and 0.6 is used as k _1.
Is selected as k ₃ , 0.05, and then the value of k ₂ is determined. In order to select m in k ₂ = mk ₃ , first estimate t _m that gives the maximum value of L (L = {1−exp (-k ₂ t)} e
xp (-k ₃ t)). When k ₃ = 0.05, t _m is between m and
Have the relationship shown in.

[0029]

20, 10 and 2 are selected as m corresponding to the values 3.04, 4.80 and 11.0 of t _{m which} are close to 3 minutes, 5 minutes and 10 minutes where the measured values are, and the value of L at t _m is set to these values. I calculated. Table 3 shows the calculation process of the value of L. L has the relationship of L = {1−exp (-mk ₃ t _m )} exp (-k ₃ t _m ) between m, k ₃ and t _m , but 1−exp (-mk ₃ t _m ) = 1-exp {-ln (m + 1)} = 1-1 / (m + 1) holds, so L is L = {1-1 / (m + 1)} exp (-k ₃ t _m ). Therefore, for m = 20, 10, 2
The values are shown in the right column of.

Table 3 m 1 / (m + 1) 1-1 / (m + 1) k ₃ t _m exp (-k ₃ t _m ) L 20 0.0476 0.952 0.152 0.854 0.818 10 0.0909 0.909 0.240 0.787 0.715 2 0.333 0.667 0.549 0.578 0.386

On the other hand, Cp / A -exp (-k for t _m triplicate
The value of ₁ t _m ) was calculated using the measured value of Cp. The calculation process and results are shown in Table 4.

Table 4 mt _m exp (-k ₁ t _m ) t Cp / A Cp / A −exp (-k ₁ t _m ) 20 3.04 0.142 3 0.258 0.116 10 4.80 0.050 5 0.182 0.132 2 11.0 0.0025 10 0.101 0.0985

For m = 20, 10, 2, the ratio {Cp / A-exp
The value of (-k ₁ t _m )} / L is m = 20 t = 3 0.116 / 0.818 = 0.141 m = 10 t = 5 0.132 / 0.715 = 0.185 m = 2 t = 10 0.0985 / 0.386 = 0.255 It was The value of this ratio can be used as a temporary value for b.

Therefore, as the values of k ₂ and b, _{three combinations} of k ₂ = 20k ₃ = 1.0 b = 0.14 k ₂ = 10k ₃ = 0.5 b = 0.18 k ₂ = 2k ₃ = 0.1 b = 0.25 are selected. , Y was calculated (as described above, A was 1530, k ₁ was 0.6, and k ₃ was 0.0.
5 were used respectively). As a result, it was the combination of k ₂ = 10k ₃ = 0.5 b = 0.18 that had the smallest difference from the actually measured value. However, this combination gives Y that is about 10% larger than the measured value of Cp at t ≥ 20, so if the value of b is reduced to 0.16, the error between the value of Y and the measured value of Cp at t ≥ 20 becomes extremely small. Moreover, the error at t <20 was sufficiently small.

After all, a function that approximately represents the temporal change of Cp
As Y Y = 1530 [exp (-0.6t) ＋0.16 {1−exp (-0.6t)} {1−exp (-
0.5t)} exp (-0.05t)] was obtained. Table 5 compares the Y value and the measured Cp value at each measurement time. Further, it is shown as a graph in FIG. In FIG. 1, the abscissa represents the true number of the time t 1 after administration, and the ordinate represents the logarithm of the relative radioactivity intensity Cp.

[0037]

The excretion rate of inulin the initial concentration of the labeled inulin value 1530 in plasma of A, the value 0.6 of k ₁ is the rate of migration to the tissues such as the liver of the inulin, the value 0.05 of k ₃ is urine It is considered that k ₂ represents the resupply rate of inulin from tissues such as liver to blood, and b represents the magnitude of its contribution.

[Embodiment 2] The above calculation may be performed by using a personal computer, and the calculation can be performed in a short time without any trouble. In advance on the personal computer
Significant digits 1 to 9.999 for A in function (3) above
Up to 10 decimal digits from 0.1 to 1000, b, k ₁ , k ₂ , k ₃ significant digits from 1 to 9.99 and decimal digits from 0.001 to 10 can be selectively output to the arithmetic unit. Keep it. The same result as above can be obtained by inputting the measurement result of radioactivity into a personal computer and performing calculation according to a preset program.

[Third Embodiment] In the first embodiment, exp (-k ₂ t _m ) is negligible as compared with 1 when t is 20 minutes or more. Hence t
The average value of the ratios of Cp / A and (exp-k ₃ t) was calculated for the measured Cp values for ≧ 20, and 0.16 was set as the temporary value of b. Similar to the first embodiment, the value of L is calculated for m = 20, 10, 2, and {C
The ratio of p / A-exp (-k ₁ t _m )} / L is m = 20 t = 3 0.116 / 0.818 = 0.141 m = 10 t = 5 0.132 / 0.715 = 0.185. Although there is no measurement point between 3 minutes and 5 minutes, it is estimated by interpolation that m = 16 is optimal, so if k ₂ is 0.8,
Table 6 shows the relationship between the Y value and the measured Cp value for each measurement time up to 10 minutes.

[0041]

From this result, the function y at 15 and 75 seconds
Value increases the error of + to Cp Found, as k ₂
0.8 was too large, and rather 0.6 was judged appropriate.

Example 4 Sprague Dawley rats were administered 20 mg / kg body weight of inaperisone (muscle relaxant) from the portal vein, serum was obtained from the collected venous blood, and separated by gas chromatography and mass spectrometry. The concentrations were as shown in Table 7.

[0044]

When a function approximating the concentration Cp of inaperisone was determined according to the present invention, the following function was obtained. Y = 4.7 [exp (-0.115t) +0.6 {1−exp (-0.115t)} {1−exp
(-0.02t)} exp (-0.028t)] The relationship between the Y value and the measured Cp value is shown in Table 8.

[0046]

The function Y = 5.0 [exp (-0.115t) +0.6 {1−exp (-0.115t)} {1−exp
(-0.018t ')} exp (-0.028t)] t' = t−5, the value of Y becomes as shown in Table 9, 30 minutes or
The accuracy of the 45-minute region approximation is improved.

[0048]

[0049]

EFFECTS OF THE INVENTION According to the method for analyzing the kinetics of drugs, toxic substances, substrates, etc. of the present invention, the temporal change in the concentration of the labeled substance in the body fluid such as blood can be accurately expressed with a relatively simple function. be able to. In particular, a change having a complicated profile, such as showing a maximum or an inflection point, is difficult to be expressed by the conventional compartment model, but according to the present invention, a comparison is performed without performing a complicated calculation such as convolution. Can be expressed accurately with a simple function.

Further, according to the present invention, the size of the metabolic pool in the in vivo estimated metabolic model of drugs, poisons, substrates, etc. can be estimated at least semi-quantitatively. Not only that, but the relative relationship between the size of the metabolic pool and the body fluid flow rate or tissue uptake rate can be estimated,
If there is information about one, it can be estimated about the other.

[Brief description of drawings]

FIG. 1 is a graph showing a function showing a temporal change in radioactivity intensity and an actually measured value.

Claims (11)

[Claims] 1. Analysis of metabolism in an animal body by administering a specific substance to an animal and measuring the time-dependent change in the concentration of the substance and its metabolite or any one of them in body fluid such as blood. In the method, the time change of the concentration Cp in the body fluid is calculated by a function Y = A exp (-k ₁ t) + B {1−exp (-k ₁ t)} {1−exp (-k ₂ t)}. ex
p (-k ₃ t) or Y ＝ A [exp (-k ₁ t) ＋ b {1−exp (-k ₁ t)} {1−exp (-k ₂ t)} ex
p (-k ₃ t)] (t is the time after administration, A, B, B, k ₁ , k ₂ , k ₃ are constants)
A metabolic analysis method characterized by being represented approximately by. 2. The metabolic analysis method according to claim 1, wherein the specific substance is a drug, a toxic substance or a substrate. 3. The specific substance is a drug, toxin, or substrate labeled with a radioactive or stable isotope,
The metabolic analysis method according to claim 2, wherein the metabolism in the animal body is analyzed by measuring the time-dependent change in the concentration of the labeled substance in the body fluid. 4. The metabolic analysis method according to claim 3, wherein the specific substance is a drug, a toxic substance, or a substrate labeled with a radioisotope. 5. The metabolic analysis method according to claim 1, wherein the specific substance is administered to an animal by intravenous injection. 6. The metabolic analysis method according to claim 1, wherein the body fluid is blood. 7. The true number of the time t after administration is plotted on the horizontal axis of the graph, and the logarithm of the measured value Cp of the concentration of the specific substance and its metabolite or one of the metabolites in the body fluid is plotted on the vertical axis. , Approximate value of k ₁ is obtained from the gradient of the tangent line at or near t = 0 of the curve on this graph, and approximate value of A is obtained from the intercept of the vertical axis, and t is larger than the area where the approximate value of k ₁ is obtained. In the area,
From the tangent slope where the slope of the curve is approximately constant
obtains an approximate value of k _3, if _{k 3 = mk 2 (m =} 1,2,3,4, ····) or _{nk 3 = k 2 (n =} 1,2,3,4, ···・), For which k ₂ corresponds to at least one of m or n, {1−exp (-k ₂ t)} exp (-k ₃ t) is a maximum t, or it has a measured value. Close t
t _m, and at t _m {Cp−A exp (-k ₁ t _m )} / {1−exp (-k ₁ t _m )} {1−exp (-k
₂ t _m )} exp (-k ₃ t _m ), and use it as the temporary value of B, or {Cp / A −exp (-k ₁ t _m )} / {1−exp (-k ₁ t _m )} {1−exp (-k ₂
t _m )} exp (-k ₃ t _m ), and the temporary value of b is obtained, and the temporary value of B or b, the approximate value of A, the approximate value of k ₁ , and k ₃ The function Y ＝ A exp (-k ₁ t) + B {1−exp (-k ₁ t)} {1−exp (-k ₂ t)} ex
p (-k ₃ t) or Y ＝ A [exp (-k ₁ t) ＋ b {1−exp (-k ₁ t)} {1−exp (-k ₂ t)} ex
The value of p (-k ₃ t)] is to select k ₂ to provide the value closest to Cp, modify the value of B or b as necessary, a difference of Y and Cp over the entire region of t is Using the approximate value of A, k ₁ , k ₃ , the value of B or b, and the value of k ₂ , or a value obtained by modifying them so that the difference between Y and Cp becomes small. , Function Y ＝ A exp (-k ₁ t) ＋ B {1−exp (-k ₁ t)} {1−exp (-k ₂ t)} ex
p (-k ₃ t) or Y ＝ A [exp (-k ₁ t) ＋ b {1−exp (-k ₁ t)} {1−exp (-k ₂ t)} ex
p (-k ₃ t)] is a function representing the temporal change of the concentration Cp in the body fluid,
The metabolic analysis method according to claim 1. 8. The time change of the concentration Cp is expressed by a function Y = A [exp (-k ₁ t) + b {1−exp (-k ₁ t)} {1−exp (-k ₂ t)} ex
The method of metabolic analysis according to claim 7, which is approximately represented by [p (−k ₃ t)]. 9. The time change of the concentration Cp is approximately represented by the function Y, the true number of the time t after administration is on the horizontal axis of the graph, and the logarithm of the measured value Cp of the concentration is on the vertical axis. , And the approximate value of k ₁ is calculated from the tangent slope of the curve on this graph at or near t = 0, and the approximate value of A is calculated from the intercept, and t is larger than the area where the approximate value of k ₁ is calculated. In the area,
From the tangent slope where the slope of the curve is approximately constant
Approximate value of k ₃ is obtained, and exp (-k ₁ t) is selected from the region where it is negligible compared to 1.
For t {Cp−A exp (-k ₁ t)} / exp (-k ₃ t) or Cp / exp (-k ₃ t
t) to obtain a temporary value for B, or {Cp / A−exp (-k ₁ t)} / exp (-k ₃ t) or Cp / A exp
Calculate the value of (-k ₃ t) and use it as a temporary value of b, and temporarily assume that k ₂ = mk ₃ (m = 1,2,3, ...) Or k ₃ = nk ₂
When (n = 1,2,3, ...), for k ₂ corresponding to at least one of m or n, {1−exp (-k ₂ t)} exp (-k ₃ t) At or near t, where k is a maximum, k ₂ is tentatively selected so that the measured value Cp of the concentration gives a maximum or an inflection point, and the provisional value of B or b, the approximate value of A, and k _Using the approximate value of _1, the approximate value of k ₃ , and the temporary value of k ₂ , the function Y = A exp (-k ₁ t) + B {1−exp (-k ₁ t)} {1−exp (-k ₂ t)} ex
p (-k ₃ t) or Y ＝ A [exp (-k ₁ t) ＋ b {1−exp (-k ₁ t)} {1−exp (-k ₂ t)} ex
p (-k ₃ t)], modify k ₂ so that the value of Y gives the value closest to Cp, and modify the value of B or b as needed to find the total region of t as the difference between the Y and Cp decreases over, the k ₂ values, the a, k _1, the approximate value of k _3, and the value of B or b, or they as the difference between the Y and Cp is reduced, 7. The function Y 1 is defined as a function representing the temporal change of the concentration Cp in the body fluid by using the value corrected to.
The metabolic analysis method according to any one of 1. 10. The time change of the concentration Cp is approximately represented by the function Y, the true axis of the time t after administration is on the horizontal axis of the graph, and the logarithm of the measured value Cp of the concentration is on the vertical axis. , And the approximate value of k ₁ is calculated from the tangent slope of the curve on this graph at or near t = 0, and the approximate value of A is calculated from the intercept, and t is larger than the area where the approximate value of k ₁ is calculated. In the area,
From the tangent slope where the slope of the curve is approximately constant
obtains an approximate value of k _3, if _{k 3 = mk 2 (m =} 1,2,3,4, ····) or _{nk 3 = k 2 (n =} 1,2,3,4, ···・), The t that gives the maximum of {1−exp (-k ₂ t)} exp (-k ₃ t) for k ₂ corresponding to at least one of m or n is the measured value of the concentration. Temporarily select k ₂ so as to correspond to the maximum of Cp or the inflection point or its vicinity, and for the temporary k ₂ , {1−exp (-k ₂ t)} exp (-k ₃ t) is the maximum. And t close to that having a measured value is defined as t _m, and at t _m , {Cp−A exp (-k ₁ t _m )} / {1−exp (-k ₂ t _m )} exp (-k ₃ t _m ), and use it as a temporary value for B, or {Cp / A −exp (-k ₁ t _m )} / {1−exp (-k ₂ t _m )} exp (-k ₃ t _m ) to obtain a temporary value of b, and using the approximate value of A, the approximate value of k ₁ , the approximate value of k ₃ , and the temporary value of B or b, the function Y = A exp (-k ₁ t) + B {1−exp (-k ₁ t)} {1−exp (-k ₂ t)} ex
p (-k ₃ t) or Y ＝ A [exp (-k ₁ t) ＋ b {1−exp (-k ₁ t)} {1−exp (-k ₂ t)} ex
p (-k ₃ t)], modify k ₂ so that the value of Y gives the value closest to Cp, and modify the value of B or b as needed to find the total region of t as the difference between y and Cp decreases over, the k ₂ values, the a, k _1, the approximate value of k _3, and the value of B or b, or they as the difference between the Y and Cp is reduced, 7. The method for metabolic analysis according to claim 1, wherein the function Y 1 is a function representing a temporal change in the concentration Cp of the labeled substance in blood, using the value corrected to. 11. A metabolism of an animal body is analyzed by administering a specific substance to an animal and measuring the time-dependent change in the concentration of the substance and its metabolite or one of them in body fluid such as blood. In the method, the time change of the concentration Cp in the body fluid is calculated by a function Y = A exp (-k ₁ t) + B {1−exp (-k ₁ t)} {1−exp (-k ₂ t ') } e
xp (-k ₃ t) or Y ＝ A [exp (-k ₁ t) ＋ b {1−exp
(-k ₁ t)} {1−exp (-k ₂ t ')} exp (-k ₃ t)] (t is the time after administration, A, B, B, k ₁ , k ₂ , k ₃ are Is a constant, t'is equal to t-d, d is 0 or a positive real number, but t'is 0 when t is not greater than d). Method.