WO2004016807A1

WO2004016807A1 - Method for determining free thrombin concentration

Info

Publication number: WO2004016807A1
Application number: PCT/SE2003/001273
Authority: WO
Inventors: Göran Hansson
Original assignee: Astrazeneca Ab
Priority date: 2002-08-16
Filing date: 2003-08-13
Publication date: 2004-02-26
Also published as: AU2003248589A1; SE0202454D0

Abstract

A method for determining concentration of free thrombin in a sample as a function of time uses a substrate that reacts with thrombin to form a product. A physical property of the product, such as optical absorbance or fluorescence emission, is measured as a function of time. A time derivative of the property is computed, preferably using a time averaging technique, to determine a rate of reaction from substrate to product. The time derivative is then used to determine the concentration of α2MT as a function of time and the time derivative of the concentration of α2MT used to determine a concentration of free thrombin. The computation minimises the effects of variability in the α2MT -dependent decay parameter, k and the parameter kcat2 expressing a rate of turnover of substrate by α2MT.

Description

METHOD FOR DETERMINING FREE THROMBIN CONCENTRATION

The present invention relates to techniques for the measurement of free thrombin as a function of time in blood, blood plasmas and other types of samples.

Accurate measurement of the generation of free thrombin as a function of time enables determination of the endogenous capacity of a sample to generate thrombin and is useful as a tool to evaluate the effect of different antithrombotics in drug development. The effect on thrombin generation can be assessed before and after administration of drugs to an individual.

The paper "Continuous Registration of Thrombin Generation in Plasma, Its Use for the Determination of the Thrombin Potential" by H C Hemker et al,

Thrombosis & Haemostasis, 1993, Vol. 70, pp. 617 - 624 describes a method in which the time course of thrombin generation in plasma can be obtained from continuous optical density recording of p-nitroaniline (pNA) production. Although the method described recognises that the optical density indicating pNA production includes a first contribution from free thrombin generation and a second contribution from α₂ macroglobulin- thrombin complex (hereinafter "α MT"), the paper describes a simple mathematical treatment that assumes foreknowledge of factors k (the α₂MT- dependent decay constant) and/(the ratio k_cat2 /

of the rate constants for the turnover of substrate by α₂MT and thrombin respectively).

The present inventors have determined that this approach does not accurately account for the contribution of the α₂MT complex in the formation of pNA. The factors k and/may vary from sample to sample, and even from measurement to measurement. Therefore, it is desirable to determine these factors in respect of each measurement rather than implement them as constants.

It is an object of the present invention to provide a method and apparatus providing a more accurate determination of free thrombin concentration in a sample.

It is a further aspect of the invention to provide a method and apparatus for determining free thrombin concentration in a sample in a manner such that the effects of variability in the c^MT-dependent decay parameter, k and the parameter k_ca.₂ expressing a rate of turnover of substrate by α₂MT are minimised.

According to one aspect, the present invention provides a method for determining an amount or concentration of free thrombin in a sample as a function of time, comprising the steps of: providing a substrate that reacts with thrombin to form a product; measuring a physical property of the product as a function of time; computing a time derivative of said physical property to determine a rate of reaction from substrate to product; determining, from said time derivative, the concentration of α₂MT as a function of time; and determining, from a time derivative of said concentration of α₂MT, a concentration of free thrombin.

Preferably the physical property of the product that is measured is an optical property, such as absorbance or emission in the ultra-violet or visible spectrum. In a preferred aspect, the determining steps are effected by iteratively varying values for: (i) a parameter k, expressing a relationship between a rate of change of concentration of α₂MT and a concentration of free thrombin; and (ii) a parameter k_cat2, expressing a rate of turnover of substrate by 0-₂MT, to reduce an optimisation value.

Once parameter values for k and k_cat have been established for which the optimisation value is a minimum, these values are used in the determination of thrombin concentration as a function of time.

According to another aspect, the present invention provides an apparatus for the determination of free thrombin concentration, as a function of time, in a sample, comprising: means for measuring a physical property of the product; means for computing a time derivative of said physical property to determine a rate of reaction from substrate to product; means for determining, from said time derivative, the concentration of α₂MT as a function of time, and means for determining, from a time derivative of said concentration of α₂MT, a concentration of free thrombin.

Embodiments of the present invention will now be described by way of example and with reference to the accompanying drawings in which:

Figure 1 is a schematic diagram of a process for thrombin induced conversion of a substrate S to product P yielding a measurable change in optical properties of product P;

Figure 2 is a flow chart showing the procedural steps for determining an optimised set of values for k and k_cat2 for use in determining a concentration of thrombin as a function of time, [T];; Figure 2a is a flow chart showing alternative procedural steps to those of figure 2; and

Figure 3 is a schematic diagram of apparatus suitable for carrying out the method steps of figure 2.

The invention provides a method for determining concentration of free thrombin in a sample as a function of time. The expression "free thrombin" is used herein to indicate that the thrombin is essentially not bound to another component of the sample and that it is active in converting substrate to product, as defined hereinafter. Thus, free thrombin does not include prothrombin and it does not include α₂MT or thrombin which is bound to other components of plasma such as antithrombin or thrombomodulin.

Free thrombin typically is produced by triggering the conversion of a latent or precursor form of thrombin (such as prothrombin) which is present in the sample into free thrombin. Thus, conveniently, the sample, such as plasma, is triggered to produce thrombin from prothrombin by the addition of calcium ions and/or the addition of tissue factor in the presence of phospholipid or by the addition of other triggering agents. It will therefore be appreciated that the method is suitable for measuring the capacity of a sample (such as plasma) to produce free thrombin.

Thus, the method may additionally comprise the step of triggering the production of free thrombin from a precursor in the sample.

It will appreciated that it is preferred if the triggering of the production of free thrombin occurs when the substrate is already present so that free thrombin produced can act immediately on the substrate. Variations of how the sample, triggering agent and substrate are mixed may readily be determined by the person skilled in the art. With reference to figure 1, there is shown a schematic diagram of a method according to one embodiment of the present invention. The method is based on the ability of thrombin, T, to create p-nitroaniline (pNA), which absorbs light at the wavelength 405 nm, through the reaction

H - β - Ala - Gly - Arg - pNA → H - β - Ala - Gly - Arg + pNA . ( 1 )

For simplicity, the product pNA will hereinafter be referred to as "P" and the substrate H-β-Ala-Gly-Arg-pNA will hereinafter be referred to as "S".

While the specific substrate and product identified above are presently preferred, it will be understood from the teaching of this specification that other substrates and products can be used in accordance with the present invention by determining appropriate parameters k, k_catl, K_M, ε and b as defined hereinafter.

The term "substrate" as used throughout the present specification is therefore intended to include any material that reacts with thrombin to form a product that has a measureable physical property, and preferably an electromagnetic absoφtion or emission characteristic, eg. a product which absorbs light, emits fluorescence, or otherwise modifies an optical property of an excitation beam. Preferred techniques utilise absoφtion in the ultraviolet and/or visible spectrum, but the method also includes use of mass spectroscopy to monitor the reaction and the use of luminescence, typically of the product.

Similarly, the term "product" as used throughout the present specification is therefore intended to include any material that provides such a measurable physical property such as an electromagnetic absoφtion or emission characteristic as a function of concentration, eg. that absorbs light, emits fluorescence, or otherwise provides a measurable optical property as a function of concentration. The measurable physical property or characteristic of the product should be detectable while the product is in the presence of the substrate and not be confused by any similar physical property of the substrate.

The increase in product P, caused by the reaction, results in a change in the optical properties of the product. In the preferred embodiment, the optical property monitored is the absorbance A of product P. The change in absorbance is proportional to the concentration of P, denoted [P]. To be more precise, AA - εl[p] by the Beer-Lambert law, where ε is the extinction coefficient and / is the path length.

In another embodiment, the fluorescence of the product is measured. The fluorescence F is related to the concentration of P through the formula

F = I₀kφ_f (1 - X^d[P]) ^« IoW_f4P] ( )

where ε and 1 are the same as above, k_P is the fraction of the emitted photons that can be measured and is determined by the geometry of the particular fluorimeter used. I₀ is the power of the exciting radiation and φ_f is the quantum efficiency (the fraction of excited molecules that fluoresce). This formula is the same as the Beer-Lambert Law where εl is replaced by dl₀k_pφ_f and AA is replaced by F. For this reason, the method as described herein in relation to absorbance measurements can be used when measuring fluorescence if the in-parameter εl is replaced by εll₀kφ_f .

To get a more detailed picture of the situation it is necessary to take into account that thrombin, whenever present, will react with ₂-macroglobulin (0-₂M) to form α₂ macroglobulin-thrombin complex (α₂MT) which also has the ability to cause the reaction from S to P.

The reaction rate for the reaction from S to P is given by the Michaelis- Menten equation as

d[P} _ = a_λ[τ]+ a₂[a₂MT] (3) dt

where

The Michaelis constants for T and 0.₂MT are assumed to be the same, K_MX = K_M2 = K_M . The coefficients ai and a₂ are functions of time since [S] will change during the reaction. The formula for the reaction may be given as S -» bP + R where R is the remaining part of the substrate after the product (pNA) has been cleaved and released. In the preferred embodiment, R = H-β-Ala-Gly-Arg. Thus, b moles of product are created from each mole of substrate, and the substrate concentration will change as

where [S]₀ is the initial substrate concentration. We also know that the reaction rate for the reaction from T to C-₂MT is proportional to the concentration of T,

if the concentration of α₂M is large. Therefore, according to a preferred embodiment, sufficient quantity of α₂M is provided such that all the generated thrombin can be transformed to α₂MT, and the measurements made, in enough time for the transformation to finish.

By combining these two equations we get:

with the solution¹

a k (note that -2- = -ss≤- is constant).

«ι ^kc n

Since we start measuring before any α₂MT or T has been created, we know that:

giving us:

ke - f e ^a- _i- J(tVt' . (10) The absorbance measurements over time provide a set of absorbance values A_f , i = l,...,N . In a preferred embodiment, the absorbance or fluorescence measurements are taken every two seconds.

To get the thrombin concentration it is necessary to obtain the derivative:

Since each of the measured values will generally contain a random measurement error it is desirable to use some kind of averaging to filter noise. The raw signals from the measuring instrument are often irregular with random noise. The quality of result obtained according to the present invention is greatly improved if a time averaged value is used in respect of each sample value. In the preferred embodiment, a moving average window technique is used. The derivative is calculated at each point (time t'_k) by finding the slope of a regression line created from n equally spaced points within a window (ie. window size equal to n), centred on the point t'_k for which the derivative is being calculated. Typical values for n would be in the range 20 - 40. In a preferred embodiment this corresponds to a moving average window size of approximately 40 - 80 seconds.

The equation used for calculating the slope of the regression line is:

slope (12)

f'(x) + af(x) = g(x) ---> f(x) =

which, in the preferred embodiment, is implemented in the Microsoft Excel 2000 function "slope". Since average values have been taken over a large number of points, it is necessary to calculate average times as well to make sure that every averaged point gets its correct placement in the middle between the first and last point used for averaging.

We calculate values for —[ —p] at the times dt

I +n t_k' = — Yf, , for 0 < / < N - n + l , (13)

using the formula:

through the Excel function "slope" as mentioned above. Since t_Q' for large values of n will be much bigger than the time step in the original measurement series, leaving a big gap before the first point, we fill out with values at equally spaced times before t₀' and from the last time t_N' __n+l to the

time t_N , using before t' and after t __Λ+1 , giving us a

new series of values at the times t₀" to t_N" , without any gap in the

beginning or the end. For simplicity we will revert to calling the time values

The next step is to calculate the concentration of α₂MT as:

In the preferred embodiment, this is performed by approximating the integral by a sum:

This can be rewritten as:

giving a faster way to perform the actual calculations. The coefficient a^t) is calculated as:

combining the formulas for aj and [S](t) as given above.

Now the thrombin concentration can be calculated using the equation:

1 d[a₂MT]

[T] = (20) k dt This is calculated in the same way as — — but without averaging, using dt n=2, giving us the simplified formula:

[^ W rW i (21) -t i.-l

[S], K_m and k_catl must be known for the algorithm to work, / and k_cat2 can be determined, or optimal values for them found, by using two constraints that arise from the physical situation. These are that the concentration of thrombin eventually will become zero when all thrombin has reacted with α₂M (the reaction should be given enough time for this to happen - eg. approximately 20 minutes) and secondly that the sum of the calculated values for the thrombin and α₂MT concentrations should equal:

at all times. In practice the optimisation is accomplished by varying k and k_cat2 to minimise an optimisation value, OV =i? SE(l + lOOOOfr]) , where [j] is the average thrombin concentration during a previous predetermined period of time (eg. the last two minutes) and RMSΕ is a measurement of the difference (root mean square error) between the calculated values of:

as defined above. The left hand expression is the sum of the calculated time traces of T and α₂MT. The right hand expression is originated from the derivation of the absorbance trace. If optimisation has been performed well, these two expressions should be close in value. To be more precise:

The factor 10000 in the optimisation value is preferred since

tends to be much smaller than RMSE. It is desirable to select this factor such that | j and RMSE have the same or similar level of influence on the optimisation The preferred value of 10000 may be determined by trial and error on typical experimental data. Ideally, it reflects the difference in range of RMSE and T, and that T is more important to optimise than RMSE. The T value will normally become very close to zero, whereas RMSE will not.

The optimisation may be done using Excel' s solver function, which uses a quasi-Newton method with forward approximation of the derivatives.

After the optimisation is done, an accurate assessment of the thrombin concentration can be achieved for all tj using the respective values of k and k_cal2 for which the optimisation value - eg. RMSE(1 + 1000θ [τ]) - is minimised, in the equations 18 and 21 to determine [T];.

With reference to figure 2, there is shown a flow chart indicating the steps for carrying out the iterative determination of k and k^ in order to determine [α₂MT]_{ and [T]_;.

In step 20, initial values for k and k_cat2 are set. Any initial values for k and k_caα could in principle be used. However, a good set of initial values is such that the values are already close to the optimal values. This will minimise the time for optimisation. In a typical example, initial k = 0.2 and initial k_ca-₂, = 3 * k_cati. Since the optimisation usually converges quickly, no upper limit for the values k and k_cat2 need normally be set.

In step 21, the value of [α₂MT]j for each averaged time sample is computed using equation 18, deploying the initial value of k and k_cat2. In step 22, the value of [T]j for each averaged time sample is computed using equation 21, deploying the initial value of k and k_cat2.

Using the determined values of [ct₂MT]j and [T]_i5 in step 23 the value of an optimisation expression RMSE(1 + 1000θ [-Tj) is calculated for the initialised values of k and k^ over all i. The resulting optimisation value OV is stored in step 24.

At this point, the optimisation value is compared (step 25) with previously generated optimisation values to determine whether convergence has been achieved and an optimum value established. Clearly, this will not be the case for the first pass, and the process continues to step 26 where the new optimisation value is added to the existing series of previously established optimisation values. The values for k and k_cat2 are incremented (step 27) and the procedure repeated.

This sequence repeats until decision step 25 indicates convergence has been achieved and a minimum value of OV has been achieved. At this point, the process branches to step 40 to retrieve the values of k and k_cat2 for which OV is minimised.

These values are then used to determine [α₂MT]j for all i (step 41) and [T]j for all i, and generate a time profile of thrombin concentration (step 42). In figure 2a, an alternative (less efficient) scheme is shown, in which a range of possible values of k and k_cat2 are iteratively checked. Steps 120 to 124 correspond to steps 20 to 24 of figure 2. In this scheme, however, iterative increment of k (step 125) up to a final value of k (step 126), is performed for each of a set of incremental values of k_cat2 (step 127) and the iteration process repeated for each new value of k_cat2 up to a final value of k_cat2 (step 128).

With reference to figure 3, an appropriate apparatus 30, suitable for carrying out the method of the invention, is shown.

A suitable light source 31, having emission at 405 nm wavelength, is projected onto a sample chamber 32 containing the sample to be measured, and in particular, the product molecules P. The absoφtion at the 405 nm band is monitored by detector 33, using techniques well known in the art.

It will be understood that the choice of light source 31 to have an emission spectrum at 405 nm is selected for the product P = pNA. For other substrates that generate different products having different absoφtion spectra, the emission spectrum of light source 31 is selected accordingly.

The absoφtion values, as a function of time, are fed to processor 34 and used to determine d[P]/dt, according to equation 11. Processor 34 carries out the operations described above, and outputs a profile 37 of thrombin concentration as a function of time.

It will be understood that the apparatus is similar in the event that fluorescence measurements are being taken. However, in this instance, the emission spectrum of light source 31 is preferably selected to be non- overlapping with the fluorescence spectrum of product P. For ease of reference, the methods described herein use the following symbology.

[S] the concentration of substrate, S

[P] the concentration of product, P

[T] the concentration of thrombin, T

A absorbance, by P

F fluorescence, by P k_P fraction of emitted photons that can be measured

Io power of exciting radiation φ_f quantum efficiency ε/ the proportionality constant between absorbance (or fluorescence) and concentration of the product (ε = extinction coefficient; / = path length)

K_M the Michaelis constant of the substrate (= K_Mι = K_M2 for T and α₂MT) k_cati the rate of turnover of substrate by thrombin k_cat2 the rate of turnover of substrate by α₂MT k the rate of reaction T - α₂MT as function of T (constant for large α₂M)

T + S *""*-" > T + P

T + a₂M - →a₂MT a₂MT + S ^KM'^kc"'² >a₂MT + P

&ι the reaction rate from S - P by T (equations 3, 4) a₂ the reaction rate from S -» P by α₂MT (equations 3, 4) b the stoichiometric coefficient for P in the reaction formula, S → bP + R n the number of points used in the linear regression when estimating

RMSE (root mean square error) a measurement of the difference between a measured value of the optical property of the product and a calculated combined value related to the concentrations of free thrombin and α₂MT (equation 24)

The effectiveness of a putative anticoagulant can be determined using the method of the invention. Thus, the invention provides a method of assessing the effectiveness of putative anticoagulant the method comprising:

(1) providing a sample, from an individual, containing a thrombin precursor;

(2) determining the concentration of free thrombin in the sample as a function of time using the method and/or apparatus described above; and

(3) assessing whether the rate of free thrombin formation determined in (2) is indicative of anticoagulant activity.

It is noted that determining [r] and [a₂MT] from equation (3) could be performed in three different ways.

Alternative 1 : Equation (3) is rewritten as a function of [r] .

Solving the equation above gives [T] which is further used in calculating [ ₂MT].

Alternative 2: Equation (3) is rewritten as a function of [α₂ -T]. dt k dt ² L ² J

Solving the equation above gives [a₂MT] which is further used in calculating [T] .

Alternative 3: Equation (3) is rewritten as a function of F . d[p] dF

— ^ = a. + a,A: dt dt

Solving the equation above gives F. The function F is further used in

calculating τ] = — — and [a₂MT = kF] independently from each other. dt

This could be performed in any order.

The preferred technique uses alternative 2 as the simplest and most efficient solution.

Other embodiments are intentionally within the scope of the appended claims.

Claims

1. A method for determining a concentration of free thrombin in a sample as a function of time, comprising the steps of: providing a substrate that reacts with thrombin to form a product; measuring a physical property of the product as a function of time; computing a time derivative of said physical property to determine a rate of reaction from substrate to product; determining, from said time derivative, the concentration of α₂MT as a function of time, and determining, from a time derivative of said concentration of ₂MT, a concentration of free thrombin.

2. The method of claim 1 further including the steps of iteratively varying values for: (i) a parameter k, expressing a relationship between a rate of change of concentration of α₂MT and a concentration of free thrombin; and (ii) a parameter k^, expressing a rate of turnover of substrate by α₂MT, to reduce an optimisation value.

3. The method of claim 2 in which the optimisation value incoφorates a value, RMSE, expressing a difference between a measured value of the physical property of the product and a calculated combined value related to the concentrations of free thrombin and α₂MT.

4. The method of claim 3 in which the optimisation value further includes a factor related to the calculated average free thrombin concentration during a preceding period of time.

5. The method of claim 3 or claim 4 in which the value RMSE

comprises: _SE

where [P] is the concentration of product at time t, Ά_X is the reaction rate from substrate to product by thrombin, k_cati is the rate of turnover of substrate by thrombin, and k_cat is the rate of turnover of substrate by α₂MT.

6. The method of claim 4 or claim 5 in which the optimisation value is determined according to an expression such that and R have approximately the same level of influence on the optimisation.

7. The method of claim 6 in which the optimisation value is derived from the expression RMSE(1 + 1000θ[- ]).

8. The method of claim 1 in which the steps of determining the concentration of α₂MT and determining the concentration of free thrombin comprises determining said concentrations at each of a plurality of discrete time values.

9. The method of claim 8 in which the step of determining the concentration of α₂MT includes the step of assuming the initial condition for t = 0 of [α₂MT] = 0.

10. The method of claim 1 further including the step of predetermining a rate of turnover of substrate from thrombin, k_catl.

11. The method of claim 1 further including the step of predetermining the Michaelis constant k_M for the substrate S, thrombin T and α₂MT.

12. The method of claim 1 in which the step of determining a concentration of free thrombin includes the step of minimising the effects of variability in an α₂MT-dependent decay parameter, k and a parameter ^ expressing a rate of turnover of substrate by α₂MT.

13. The method of claim 1 in which the step of determining a concentration of free thrombin includes the step of determining an optimal value for an α₂MT-dependent decay parameter, k and a parameter k_cat2 expressing a rate of turnover of substrate by α₂MT using constraints that the concentration of free thrombin for high values of t tends to zero, and that the sum of calculated values for the thrombin and α₂MT concentrations satisfies the equation:

at all times, where [P] is the concentration of product, a_! is the reaction rate from substrate to product by thrombin, k_<._atι is the rate of turnover of substrate by thrombin, and k^α is the rate of turnover of substrate by α₂MT.

14. The method of claim 1 in which the step of measuring a physical property comprises the steps of making a plurality of discrete measurements of said property at a plurality of corresponding sample times; and wherein the step of computing a time derivative includes the steps of computing a time averaged value in respect of each sample time, each time averaged value being based on a number of said discrete measurements corresponding to said sample time and a previous and/or subsequent sample time.

15. The method of claim 14 in which the time averaged value for each sample time is calculated on a moving window average basis.

16. The method of claim 15 in which the moving window is centred on the sample time for computation of the time averaged value for each sample time.

17. The method of any preceding claim in which the physical property of the product that is measured is an optical property.

18. The method of claim 17 wherein optical property is one of absoφtion or emission of electromagnetic radiation in the visible or ultraviolet spectrum.

19. A method for determining a concentration of free thrombin in a sample as a function of time, comprising the steps of: providing a substrate that reacts with thrombin to form a product; measuring a physical property of the product as a function of time; computing a time derivative of said physical property to determine a rate of reaction from substrate to product; determining, from said time derivative, a concentration of thrombin as a function of time, determining, from a time derivative of said concentration of thrombin, a concentration of 0-₂MT; and deriving corrected values of free thrombin concentration as a function of time based on said determined values of α₂MT concentration.

20. The method of claim 1 or claim 19 further including the step of triggering the production of free thrombin from a precursor in the sample.

21. Apparatus for determining a concentration of free thrombin in a sample as a function of time, comprising: means for measuring a physical property of the product; means for computing a time derivative of said physical property to determine a rate of reaction from substrate to product; means for determining, from said time derivative, the concentration of α₂MT as a function of time, and means for determining, from a time derivative of said concentration of 0-₂MT, a concentration of free thrombin.

22. A method for determining a concentration of free thrombin in a sample substantially as described herein with reference to the accompanying drawings.

23. Apparatus substantially as described herein with reference to the accompanying drawings.

24. A method of assessing the effectiveness of putative anticoagulant, the method comprising the steps of: providing a sample, from an individual, containing a thrombin precursor; determining the concentration of free thrombin in the sample as a function of time using the method according to any one of claims 1 to 18 and/or the apparatus of claim 19; and assessing whether the rate of free thrombin formation determined is indicative of anticoagulant activity.