WO2009156194A1

WO2009156194A1 - Computation of drug composition

Info

Publication number: WO2009156194A1
Application number: PCT/EP2009/053739
Authority: WO
Inventors: Aslak Tveito; Glenn T. Lines
Original assignee: Simula Innovation As
Priority date: 2008-06-23
Filing date: 2009-03-30
Publication date: 2009-12-30

Abstract

The invention relates to the computing of a desired composition of a drug. A test drug is defined in terms of a number of components, where each component represents an effect to normal cells and to pathogenic cells. A drug model is accessed, and based on the drug model, the weight of each component of the test drug is determined by minimizing the difference in a predefined physiological property. The minimization is done between the normal cell after the test drug has been applied and the normal cell prior to applying the test drug, and between the pathogenic cell after the test drug has been applied and the normal cell prior to applying the test drug. The desired composition of the test drug is set as the composition defined by the weights of the components. In embodiments, the physiological property is the action potential of cardiac cells, the pathogenic cells are ischemic cells and the desired drug is an anti-arrhythmic drug.

Description

COMPUTATION OF DRUG COMPOSITION

FIELD OF THE INVENTION

The invention relates to the computing of a desired composition of a drug.

BACKGROUND OF THE INVENTION

The heart is an intricate system which functioning is vital for the well being of every person and severe malfunctioning of the heart may be fatal. A large group of cardiac malfunctioning relates to various forms of cardiac arrhythmia.

Various types of treatment for cardiac arrhythmia have been developed. A widely used type of treatment relates to the use of anti-arrhythmic drugs. In the development of new improved drugs, computer models can be used to study the effect of known anti-arrhrythmic drugs. In such models, a specific mathematical model of cardiac cells is typically considered, and the application of a certain drug is analyzed in terms of changed properties of the cell in order to identify candidate drugs which are likely to reduce certain types of arrhythmia. However, despite intense research into identifying drug candidates, about 300.000 people die of cardiac arrhythmia in the form of ventricular fibrillation per year in the US alone.

The inventors of the present invention have appreciated that an alternative approach to finding candidate drug wound be of benefit, and have in consequence devised the present invention.

SUMMARY OF THE INVENTION

It is an object of the present invention to provide an approach which may lead to the identification of improved drugs which may be used to treat cell related malfunctioning, such a malfunctioning of cadiomyocytes. To this end, the inventors of the present invention have had the insight that the composition of a drug may be determined by computing desired properties of the drug.

According to a first aspect of the present invention there is provided, a method for computing a desired composition of a drug, the method comprising : - defining a test drug in terms of a number of components, where each component represents an effect to normal cells and to pathogenic cells;

- accessing a drug model, wherein the drug is modelled in terms of known effects to normal cells and to pathogenic cells, wherein the effect of the test drug to the normal cells and to the pathogenic cells can be determined, and wherein a physiological property of the normal cells and of the pathogenic cells can be determined by the drug model;

- determining the weight of each component of the test drug by minimizing the difference in the physiological property:

between the normal cell after the test drug has been applied and the normal cell prior to applying the test drug, and

between the pathogenic cell after the test drug has been applied and the normal cell prior to applying the test drug

- setting the desired composition of the test drug as the composition defined by the weights of the components.

In the present invention, the cells are divided into two collections, one collection of normal cells and one collection of pathogenic cells; and the drug is represented by a number of components each representing an effect to normal cells and to pathogenic cells. This effect can relate to a physiological property of the cell, such as to the alteration of a given ionic density in an ionic channel. The invention relies on known drug models which are capable of assessing the effect of a drug component to both the normal cells and the pathogenic cells. Such models are known in the art. Examples of specific models are provided in the description of embodiments below. The gist of the present invention is to determine the drug composition such that the mix of the two types of cells is as well-behaved as possible after the drug has been applied. The actual computation is performed by finding drug components which at the same time minimizes the difference in the given physiological property of the drugged normal cells and the non-drugged normal cells in addition to the difference in the physiological property of the drugged pathogenic cells and the non-drugged normal cells. Thus, the computation determines the drug composition which ensures that the behaviour in terms of the physiological property of the pathogenic cell comes as close as possible to the behaviour of the physiological property of the normal cells, while at the same time ensuring that the drugged normal cells are effected as little as possible by the drug.

Once the desired properties of the test drug have been determined, the person skilled in the art may build a candidate drug using known effects of drug components to mimic the constitution of the test drug as defined by the weights of the components. The present invention does not provide the specific components of a candidate drug, but the desired properties of the drug. The actual realization of the drug itself is a task for the skilled drug designer.

In an embodiment, finding drug components which ensure that the behaviour of the physiological property is as well behaved as possible may advantageously be dealt with in terms of minimizing a distance function measuring the distance between the cells collections under drugged/non-drugged conditions. Minimization over several parameters is a computational difficult task, by formulating the problem as a problem of minimization a distance function a well-known mathematical tool box is available to the skilled person.

In an advantageous embodiment, the physiological property may be the action potential of the cells. The action potential is well-suited for the minimization, since appropriate drug models which describe the effect of a drug component on the action potential is available in the art. Of especially suited drug models are drug models based on cell dynamics. In general other or even all variables describing a physiological property in the drug model may be used for minimization of the system.

The overall method may, with the proper selection of the drug model, be applied to muscle cells in general. However, it may nevertheless be especially suited for searching for candidate drug related to cardiac cells. In a specific embodiment, the drug is an anti-arrhythmic drug. The pathogenic cell may relate to any appropriate pathogenic state of the used muscle cells. Cell dynamics can be modelled in various ways, however with respect to drug properties, a modelling based on ionic current densities and related ionic channel species of the cell is well suited, since there is a profound knowledge in the art related to the influence from drugs on ionic channel conductance. A clear links is thereby established between the effect of the drug components and the desired properties of the test drug. This link is rendered even more direct by modelling the effect of the drug in terms of a scaling of the ionic current densities.

In an advanced embodiment, the time dependence of the effects of the drug may be accounted for. The aspect may readily be incorporated into the method according to the embodiments of the present invention. The minimization task however becomes more complex.

In a second aspect, the invention relates to a computer program product, when in use, or when running, on a computer causes a system to perform the method of the first aspect. The system may be a general computing system, such as a stand-alone computing unit or a cluster of computing units connected in a network.

In a third aspect, the invention relates to a decision support system for computing a desired composition of a drug, the system may be a computing system implemented to carry out the method of the first aspect. The decision support system may be implemented by loading the computer program product of the second aspect into one or more computing units.

In general the various aspects of the invention may be combined and coupled in any way possible within the scope of the invention. These and other aspects, features and/or advantages of the invention will be apparent from and elucidated with reference to the embodiments described hereinafter.

BRIEF DESCRIPTION OF THE DRAWINGS

Embodiments of the invention will be described, by way of example only, with reference to the drawings, in which FIG. 1 illustrates the distance function, D(d), close to the optimized drug vector, by use of the Luo-Rudy I model;

FIG. 2 illustrates the resulting action potentials as a function of time using the Luo-Rudy I model;

FIG. 3 illustrates the resulting action potentials as a function of time using the Hund-Rudy model.

FIG. 4 displays the spatial distribution of the transmembrane potential of the Luo- Rudy I model; and

FIG. 5 displays the spatial distribution of the transmembrane potential of the

Hund-Rudy model.

DESCRIPTION OF EMBODIMENTS

Embodiments of the present invention are described in terms of a mathematical model relating to the computation of an optimal drug related to a collection of normal cardiac cells and a collection of ischemic cells. However, the disclosure of the specific computations merely serves as illustrations of the methodology and should not be construed to limit the scope of the invention only to this type of computation.

The drug model is described in terms of a mono-domain model combined with a system of ordinary differential equations modelling the cell dynamics:

V_t = <5(V,_XX + V_}7) -J(V, S) ,

S_t=F(v,s). where v denotes the transmembrane potential which satisfies a no-flux boundary condition, δ is the spatial diffusion constant, J is the total ionic current density scaled by the cell membrane capacitance, and s carries gating variables and ionic concentrations governed by the nonlinear function F.

In the case of spatially uncoupled cells the dynamics of a normal single cell is governed by:

S'N = F_N(V_N^S_N). where the current J_N is a sum of specific currents: n S_N).

Similarly, the dynamics of a single ischemic cardiac cell is given by:

S¹^Fi(V₁-S₁). where

Ji(VuS,) - YJ u( ^'Vx -Si)-

The drug is defined to be a vector d containing n non-negative real components. It is assumed that the dynamics of a normal cell after the drug has been applied, can be modelled by replacing J_Nrl with dp_Nrl, i.e.:

V_{N d} — -J_N d(Vn,d_' ^SNd_'d)_' s_N' _d =F_N(^_Λ,._d,s_Nd,d), where π

J NM ( ^VN d ^{■ s}N.d) — 2_^ djJ_Ni ( VN. d , ⁵N d ) ^■ 1-1

No drug has been applied if all components of d=l. After the drug has been applied, the dynamics of a single ischemic cell is given by:

^V _I'_Δ ^~ -Ji_,d(Vid.Si_d,d), S_I'M ^F_N(v_Ld,s_Ld,d), where

J_lJVld-.Sld)

The task is to compute the drug vector d such that, after the drug has been applied, a selected physiological property of a normal cell and an ischemic cell both are as close as possible to the original normal cell, which is known to be well behaved. In an embodiment, the physiological property is selected as the action potential. Below specific known cell models of action potential are used. More specifically, the model published by Luo and Rudy in "A model of the ventricular cardiac action potential: depolarisation, repolarisation, and their interactions", Ore. Res.68 (1991) 1501 (hereafter the Luo-Rudy I model) and the model published by Hund and Rudy in "Rate dependence and regulation of action potential and calcium transient in a canine cardiac ventricular cell model", Ore. Res. 110 (2004) 3168 (hereafter the Hund-Rudy model). Both models are equipped with original parameters.

Thus the test drug is defined in terms of a number of components, where each component represents an effect of the drug to the ionic density of the ionic channels of the normal cells and the pathogenic (ischemic) cells.

The drug vector of the test drug is computed by defining a distance between two action potentials. The distance is defined such that an optimal drug results in action potentials that inherit the good physiological properties of the normal cardiac cell. The following distance function may be used :

(v_{l d k}(t) - v_{N k}(t)f dt

+ ∑ / (J_{N d k}(t) -J_{N k}(t))²dt + ⁽J_{l d k}(t) -J_{N k}(t))² dt \₇ t^t ^[J° J° J where k runs over m different simulations, and the integrals are defined over the associated action potential. Λ/,fc(t) is shorthand for J_N/k(v_N/k(t); s_N/k{t)); similarly for -?/v,d,fc(t) and J_I/d/k(t). In evaluations of the function D(d) we have used m = 6 stimulations with the pacing intervals 450, 400, ..., 250 ms for the Luo-Rudy I model and the intervals 300, 250, ..., 100 for the Hund-Rudy model. The distance function D{d) is minimized by using the Nelder-Mead method implemented in Matlab. The Nelder-Mead method is published in "A simplex method for function minimization" Comput. J. 7 (1965) 308. The search is divided into two steps; first 100 random initial guesses is picked for the c/-vector and the Nelder-Mead algorithm is run until the changes in the c/-vector, measured in the max-norm, are less than 1/10. The 10 best choices are picked as measured in D(d) and the Nelder-Mead algorithm is re-run until convergence according to the default stopping criterion used in Matlab. Of these 10 alternatives, the c/-vector with the lowest D(c/)-value is picked.

Ischemia is modelled by increasing the extracellular potassium concentration [K⁺]_o from 5.4 mM in normal cells to 10 mM in ischemic cells. The following parameters were set: the spatial domain Ω = 100x 100 mm², the mesh consisted of 401x401 computational nodes, the time step was 0.1 ms, and the diffusion constant was δ = 1 mm²/s. All single cell computations was performed in Matlab, and the 2D computations where performed on a Linux cluster. In order to maximize the chance of initiating a spiral, the second stimulus was applied when approximately half the tissue had recovered. The second stimulus is normally referred to as S2 in the art. The time to recovery varies under the different conditions, and the timing of the second stimulus was thus adjusted accordingly.

The results of the minimization of D{d) is disclosed in the following in connection with FIGS. 1 to 5.

For the Luo-Rudy I model, the following weights of the components of the desired composition of the test drug are found : Gfi = Gf_Na = 0.906; Gf₂ = cf_κ = 1.323; Gf₃ = Gfκi = 0.294; Gf₄ = cf_Kp = 1.811; Cl₅ = d_b = 0.046; Gf₆ = Gfs, = 0.655.

FIG. 1 illustrates the function D{d) close to the optimal drug given the above components

to d&. In the top left panel, the function D is shown as it varies as a function of d_Na, with all other components of the drug kept at their optimal value as given above. The other panels show similar plots for the other components of the drug vector. These plots show that the drug defined by the above components indeed is a local minimum of the function D(d). It cannot, however, be guaranteed that this provides a global minimum; i.e. even better drug vectors may exist. This is a normal situation found in non-linear optimization.

For the Hund-Rudy model, maximum conductance of the /_Na, J-Kr, J-Ki, J-NaK, J-NaCa and /caL currents are optimized. The procedure described above provides the following optimal drug vector components: Gfi =d_Na = 0.5873; Gf₂ = d_Kr = 1.7142; d₃ = 0.0073; d_Λ = d_NaK = 1.5766; Gf₅ =d_NaCa = 0.0; Gf₆ = dcai = 1.1783. FIG. 2 illustrates the resulting action potentials as a function of time using the Luo-Rudy I model with standard parameters, and similarly in FIG. 3 using the Hund-Rudy model.

FIG. 2A shows the action potential of a normal cell 20 and the action potential of an ischemic cell 21. The well-known elevated resting state and early repolarization is displayed. FIG. 2B shows the action potential of a normal cell 20 without any medication (that of FIG. 2A), and the action potential of an ischemic cell 22 when the drug has been applied. FIG. 2C again shows the action potential of a normal cell 20 but here compared to the action potential of a normal cell 23 after the drug been applied. FIG. 2A illustrates that the action potentials of a normal cell and an ischemic cell are quite different. This difference is believed to be arrhythmogenic. FIG. 2B shows that the drug changes the ischemic cell such that the action potential is closer to the action potential of a normal cell. In FIG. 2C it is seen that the drug does not change the action potential of a normal cell very much. The purpose of the drug is to alter the normal cells as little as possible whereas the ischemic cells should be changed such that their behaviour is as close as possible to the normal undrugged cells. The computed drug using the Luo-Rudy I model is observed to have this effect on the action potentials.

FIG. 3 shows similar plots using the Hund-Rudy model exposed to the drug found using this model. As is seen in FIG. 3A, the healthy normal cells 30 and the ischemic cells 31 behave quite differently. As is seen in FIG. 3B the drug changes the action potential of the ischemic cell 32 in a favourable way, whereas the normal cells exposed to the drug 33 is quite similar to the undrugged normal cells 30 as is illustrated in FIG. 3C.

To further illustrate the disclosed method, FIGS. 4 and 5 display the spatial distribution of the transmembrane potential of the Luo-Rudy I model (FIG. 4) and the Hund-Rudy model (FIG. 5). Each column (Cl to C4) shows the spatial distribution at various time instants, whereas each row (Rl to R4) shows a given collection of cells. The computation is initiated by an S1-S2 stimulus adjusted in order to create computational fibrillation in the ischemic case. Sl and S2 stimuli are known in the art. Row Rl of FIG. 4 illustrates the transmembrane potential of a collection of normal cells at the four time instants Cl to C4 being : 100 ms, 200 ms, 500 ms, 1000 ms. The row R2 illustrates a simulation of a collection of ischemic cells. It is observed that the ischemic parameters lead to computational fibrillation (C3 and C4). The row R3 shows a simulation of normal cells after application of the drug. It is observed that this solution is well behaved and similar to the normal case where no drug has been applied (Rl). In the row R4 the solution for the case where the drug has been applied to the ischemic cells is shown. The computational fibrillation as observed in row R2 where no drug was applied is now replaced with a more stable solution.

FIG. 5 is similar to FIG. 4 except that the Hund-Rudy model is used. Again, in Rl all cells are normal; R2 illustrates the ischemic cells (exhibiting re-entry). Row R3 illustrates the normal cells exposed to the drug obtained by the Hund-Rudy model, and in R4 ischemic cells are exposed to the drug. FIG. 5 illustrates the four time instants Cl to C4 being : 150 ms, 200 ms, 250 ms and 1000 ms. Again, it is observed that the drug mends the arrhythmogenic behaviour of the ischemic cells. In fact, for both models it has not been possible to provoke computational fibrillation or even reentry in the case of drugged ischemic cells; a series of numerical experiments has been performed for this purpose.

The robustness in the sense that small changes in the parameters should not induce large changes in the results has been tested by performing a series of numerical experiments using the Luo-Rudy I model. These experiments indicate that small changes in the optimal drug vector, does not change the main feature of the drug; computational fibrillation does not seem to arise using any drug close to the optimal value.

The numerical experiments as illustrated in FIGS. 1 to 5 show that the method in accordance with embodiments of the present invention is capable of computing properties of a prospective drug that changes the physiological property, in the form of the action potential, in a favourable manner. The results as disclosed here are provided under simplified conditions. However, other more complicated cell models can be implemented in the general method of the various embodiments of the present invention. Moreover, the distance function D{d) used here may be redefined for instance in order to try to prevent oscillations in other variables. In principle, it is possible to invoke a distance function including the spatial distribution of the variables, but this would lead to very challenging minimization problems. Since many different types of cardiomyocytes will be affected by the same antiarrhythmic drug, it is possible to extend the distance function to cover many different cell types.

A basic assumption of the disclosure is that the drug only affects the maximum conductance of the channels, i.e. it is assumed, for instance, that the gating dynamics are unaffected by the drug. There are, however, cases where the drug is use-dependent, i.e. only attaches when the channel is open. For such a drug the degrees of freedom in the distance function could be the binding and unbinding rates of the drug.

The invention can be implemented in any suitable form including hardware, software, firmware or any combination of these. The invention or some features of the invention can be implemented as computer software running on one or more data processors and/or digital signal processors. The elements and components of an embodiment of the invention may be physically, functionally and logically implemented in any suitable way. Indeed, the functionality may be implemented in a single unit, in a plurality of units or as part of other functional units. As such, the invention may be implemented in a single unit, or may be physically and functionally distributed between different units and processors.

Although the present invention has been described in connection with the specified embodiments, it is not intended to be limited to the specific form set forth herein. Rather, the scope of the present invention is limited only by the accompanying claims. In the claims, the term "comprising" does not exclude the presence of other elements or steps. Additionally, although individual features may be included in different claims, these may possibly be advantageously combined, and the inclusion in different claims does not imply that a combination of features is not feasible and/or advantageous. In addition, singular references do not exclude a plurality. Thus, references to "a", "an", "first", "second" etc. do not preclude a plurality. Furthermore, reference signs in the claims shall not be construed as limiting the scope.

Claims

1. A method for computing a desired composition of a drug, the method comprising :

- defining a test drug in terms of a number of components, where each component represents an effect to normal cells and to pathogenic cells;

2. The method according to claim 1, wherein the physiological property is the action potential.

3. The method according to any of the preceding claims, wherein the drug model is based on cell dynamics.

4. The method according to any of the preceding claims, wherein the normal cells and the pathogenic cells are cardiac cells.

5. The method according to any of the preceding claims, wherein the cell dynamics is modelled by use of a set of ionic current densities, where each component in the set of ionic current densities is related to an ionic channel species of the cell.

6. The method according to any of the preceding claims, wherein the drug is defined as a vector containing real non-negative components, where each component represents the effect to normal cells and to pathogenic cells.

7. The method according to any of the claims 1-5, wherein the effect of the drug to the normal cells and pathogenic cells is represented as scaling of the ionic current densities.

8. The method according to any of the preceding claims, wherein the minimization is performed in terms of minimizing a distance function measuring the distance between the normal cell after the test drug has been applied and the normal cell prior to applying the test drug, and between the pathogenic cell after the test drug has been applied and the normal cell prior to applying the test drug.

9. The method according to any of the preceding claims, wherein the effects of the drug is time dependent.

10. The method according to claim 4, wherein the pathogenic cells are ischemic cardiac cells.

11. The method according to claim 1, wherein the drug is an anti-arrhythmic drug.

12. A computer program product, when in use on a computer, to cause a system to perform the method of any of the preceding claims.

13. Decision support system for computing desired composition of a drug, the system being based on the method of any of the claims 1-11.