EP1549326A1

EP1549326A1 - Dry oral phlebotonic and vasculoprotective formulation for the treatment of venous insufficiency, capillary fragility and haemorrhoids, in the pharmaceutical form of a chewable tablet containing diosmin

Info

Publication number: EP1549326A1
Application number: EP03780209A
Authority: EP
Inventors: Catherine Meignant; Corinne Vieillard-Baron
Original assignee: Laboratoires Innothera SAS
Current assignee: Laboratoires Innothera SAS
Priority date: 2002-10-11
Filing date: 2003-10-09
Publication date: 2005-07-06
Also published as: FR2845597B1; FR2845597A1; WO2004032942A1; AU2003288308A1

Abstract

The invention relates to a dry oral phlebtonic and vasculoprotective formulation for the treatment of venous insufficiency, capillary fragility and haemorrhoids, which takes the pharmaceutical form of a chewable tablet containing diosmin. The inventive formulation comprises a mixture of at least one active ingredient containing granular or pure diosmin and a combination of excipients. Said formulation takes the pharmaceutical form of a tablet which can be bitten, chewed or sucked. The aforementioned excipients can comprise at least two polyols, such as xylitol and sorbitol, as well as a compression agent, such as dicalcium phosphate.

Description

Dry oral formulation for phlebotonic and vasculoprotective purposes for the treatment of venous insufficiency, capillary fragility and hemorrhoidal crisis, in pharmaceutical form of chewable tablet containing diosmin

The invention relates to a new dosage form ("pharmaceutical form") for a tablet containing diosmin. Diosmin is a naturally occurring bioflavonoid found, for example, in the peel of citrus fruits. It has already been proposed to use flavonoids other than diosmin in various indications, for example a water-soluble flavonoid combined with a metallic compound, to combat conditions of the ENT sphere, as described in WO- A-02/09699 (Immupharm APS). Diosmin, on the other hand, is specifically known for its veinotonic properties.

Diosmin used in the pharmaceutical industry is a hemisynthesis product in the form of a hygroscopic powder, practically insoluble in water but soluble in DMSO, obtained from the natural product hesperidin. a series of extraction treatment, processing and ^{purification.} We can for example refer to FR-A-2 760 075 and FR-A-2 760 016 (Innokem) which describe two industrial manufacturing processes for diosmin by oxidation with iodine. The semisynthetic resulting product is a pharmaceutical active ingredient described in "Diosmin" monograph of the European Pharmacopoeia, 4 ^th edition.

Due to its origin, this active ingredient is a mixture of several substances, including pure diosmin and related substances such as hesperidin, linearin, diosmetin. The water content of the product obtained is of the order of 1 to 5%, typically of the order of 3%. The content of pure diosmin is at least 90% and at most 101% compared to the anhydrous substance. The term "native diosmin" will be used subsequently to designate this active principle as described in the European Pharmacopoeia, in the form of the hemisynthesis product available industrially from the various producers of this substance, and by "titrated diosmin" pure diosmin included in the mixture of substances forming the native diosmin. Diosmin is known for its venotonic properties by vasoconstrictor action and increase in venous tone, decrease in the volume of venous stasis, lowering of average venous pressure, increase in systolic and diastolic blood pressure in post-operative orthostatic hypotension . It also has vasculoprotective properties by increasing the resistance of vessels and decreasing their permeability.

Its main therapeutic indications are venous insufficiency (in particular the improvement of symptoms related to venolymphatic insufficiency such as heavy legs, pain, impatience of the primo-decubitus), capillary fragility and the treatment of related functional signs. to hemorrhoidal crisis.

Advances in the study of the pharmacological properties of diosmin have led in recent years to an increase in daily dosages.

We have thus gone from a daily dosage of 2 x 150 mg twenty years ago to a dosage of 600 mg in a single dose, up to 2 x 600 mg to 3 x 600 mg in certain indications like treating hemorrhoidal crisis.

These dosages lead to a large mass and therefore to a large volume of active principle, all the more so since this active principle is not a pure substance, for a form dosed for example with 600 mg of titrated diosmin, it is it is necessary to provide, for example, around 660 mg of active ingredient native diosmin.

This can cause a number of practical difficulties, as this medication is administered orally. One solution consists in developing a drinkable liquid form, typically from a powder or granules for oral suspension packaged in sachets, as described by EP-A-0 711 560 (Adir et Cie) for effervescent granules, FR-A-2 661 830 (J. Corbière) for a soluble or dispersible powder or FR-A-2 661 610 (Rhône-Poulenc Santé) for a lyophilized form to disperse or dissolve. This drinkable liquid form, if it makes it possible to condition significant quantities of active principle, is not always well accepted by the patients taking into account the manipulations involving tearing the sachet, pouring and suspending the powder or the granules. For this reason, the most common pharmaceutical forms of the drugs containing diosmin are the dry forms, the active ingredient being mixed with various excipients and prepared in the form of a naked tablet (tablet stricto sensu), film or coated. The diosmin tablets that are currently found on the market are therefore tablets having a fairly large mass and volume: for example, in the case of the specialty Diovenor® 600 mg sold by Innothéra Laboratories, the unit mass of the tablet is approximately 925 mg per 600 mg of titrated diosmin. These tablets may be difficult to swallow for some patients, especially the elderly. However, the treatment of venous insufficiency is often a prolonged treatment, typically of several weeks at least, and, if the daily taking of the drug causes an inconvenience, this treatment risks being badly observed by the patient - reaction which one encountered in particular for bulky swallow tablets which are difficult to swallow. In addition, it is in practice impossible to produce in swallowing tablets unit doses greater than 600 mg of titrated diosmin, as this would lead to tablets which are extremely difficult to administer. It would also be desirable to be able to have higher unit doses, for example titrated to 1200 mg of diosmin, but it has hitherto been impossible to conceive of a corresponding dry form which is acceptable by the majority of patients.

A first line of research consisted in trying to reduce the total mass of the tablet by reducing the mass of its excipients (while keeping the same dose of active ingredient), for example by trying to reduce the total mass to a value of around 750 mg for a tablet dosed with 600 mg of titrated diosmin.

This research revealed a certain number of technological difficulties - resulting from the very small proportion of excipients -, with in particular phenomena of cleavage of the tablet after compression, as well as difficulties during the filming stage. So it was not possible to cut in this way to a satisfactory product, the shape obtained proving to be too fragile.

The basic idea of the invention consists, on the contrary, in increasing the mass of the excipients to remedy this fragility, with a particular choice of these excipients making it possible to achieve a chewable form.

Here and in the continuation, one will hear by “chewable tablet” (pharmaceutical form known in the English terminology under the term of chewable tablet) as well a tablet “chewable” in the strict sense, as a tablet “to suck "or" to chew ", terms which designate here equivalent pharmaceutical forms.

The chewable forms are in the form of relatively bulky tablets, typically from 2000 to 3000 mg of unit mass, which makes it easier to choose excipients which can give the tablet the desired mechanical properties. In addition, the chewable forms are generally very well accepted by patients, since the choice of excipients makes it possible to give them a pleasant flavor and to mask an unpleasant consistency or taste of the active principle (diosmin is indeed present in the mouth a slightly earthy consistency, which should be masked). Thus, in a different field, FR-A-2 774 291 (Innothera Laboratory) describes a chewable tablet whose particular composition makes it possible to effectively mask the unpleasant taste of an iron salt present in small amounts to improve it. patient acceptance. In general, the object of the invention is to propose a new pharmaceutical form of tablets containing diosmin as active ingredient, the combination of excipients of which meets both technical and patient acceptability constraints. In other words, the object of the invention is to provide a tablet:

- which can be implemented on an industrial scale and at no additional cost with the usual manufacturing techniques,

- which has sufficient final mechanical cohesion and stability, which is not likely to break during packaging and handling by the patient,

- which is however not too hard to be chewed easily, - and which has a pleasant flavor and consistency in the mouth. More specifically, in its most general aspect, the invention provides, as a new product, a dry oral formulation for phlebotonic and vasculoprotective purposes for the treatment of venous insufficiency, capillary fragility and hemorrhoidal crisis , this formulation being in pharmaceutical form of tablet comprising, as a mixture, at least one active principle containing diosmin and a combination of excipients, characterized in that said pharmaceutical form is a form of chewable, suckable or chewable tablet . The hardness of the tablet is advantageously between 150 and 250 N, preferably between 150 and 200 N.

The diosmin used can be a native diosmin, the residual water content of which is between 1 and 5%, preferably between 2 and 4%, very preferably of the order of 3%, and the average diameter, between 20 and 50 .mu.m. The diosmin used can also be a granulated diosmin, the residual water content of which is between 4 and 8%, and the average diameter, between 50 and 150 μm.

In a preferred embodiment, the combination of excipients comprises at least one polyol, and the ratio of the mass of the polyol, or of all of the polyols, to the titrated mass of diosmin is then preferably between 2 and 3. The polyol, or one of the polyols, is advantageously sorbitol.

The combination of excipients can in particular comprise at least two polyols, advantageously xylitol and sorbitol, in a mass ratio of sorbitol to xylitol preferably greater than 1, very preferably between 1, 20 and 1.50. Preferred combinations of excipients include:

- a polyol, a sweetener, a compression agent, a lubricant, an acidifier and a flavor; - a polyol, a dare, a compression agent, a lubricant, an acidifier and a flavor;

- sorbitol, a compression agent, a lubricant, an acidifier and a flavor;

- xylitol, sorbitol, a compression agent, a lubricant, an acidifier and a flavor. The compression agent can be chosen from the group comprising dicalcium phosphate, lactose and their mixtures, preferably dicalcium phosphate.

The lubricant can be chosen from the group comprising talc, stearic acid, magnesium stearate and mixtures thereof.

The combination of excipients can also comprise, when the active principle is a granulated diosmin, an additional granulating agent, in particular chosen from the group comprising hydroxypropylmethylcellulose, polyvinylpyrrolidone, starches, dextrins, oses, polyols and their mixtures.

Alternatively, in the various formulations above with two polyols, one of the sorbitol or xylitol polyols is replaced by mannitol. A preferred formulation comprises the following constituents: native diosmin; xylitol; sorbitol; dicalcium phosphate; lubricant selected from the group comprising talc, stearic acid, magnesium stearate and mixtures thereof; acidifier and aroma.

The preferred quantitative formula is: titrated mass of diosmin 600 mg; xylitol 600 to 800 mg, preferably 700 mg; sorbitol 750 to 950 mg, q.s. one 2400 mg unit mass tablet; dicalcium phosphate 10 to 60 mg, preferably 25 mg; talc 20 to 50 mg, preferably 30 mg; stearic acid 10 to 30 mg, preferably 15 mg; magnesium stearate 30 to 60 mg, preferably 45 mg; flavor 15 to 40 mg, preferably 25 mg; citric acid 3 to 8 mg, preferably 5 mg; or a quantitative homothetic formula.

0

We will now detail each of the constituents of the formulation, namely active principle (diosmin) and various excipients, then describe a preferred, but non-limiting example of implementation of the invention.

Active ingredient (diosmin)

As indicated above, it is possible to use, for the production of the chewable tablet of the invention, native diosmin or else granulated diosmin.

In the first case, for the manufacture of the tablet is used directly, in admixture with the other constituents, diosmin of Pharmacopoeia quality as commercially available, while in the second case the diosmin is not used as such (native) but after having undergone a preliminary stage of wet granulation before its incorporation into the mixture to be compressed.

Native diosmin

Native diosmin (European Pharmacopoeia, 4 ^th edition) is a gray to light yellow, hygroscopic powder, practically insoluble in water but soluble in DMSO. The native diosmin contains at least 90% and at most 101% of titrated diosmin with respect to the anhydrous substance. Its water content, determined by microdosing, is not more than 6%.

For the implementation of the invention, it is possible to use a native diosmin having a water content of between 1 and 5%, preferably between 2 and 4%, very preferably around 3%. Here and in the sequel, the "water content" is determined according to the analytical methods specified in the European Pharmacopoeia, ^4th edition, under 2.5.12 "Semi-micro water." Native diosmin is in the form of a sparse powder, its density being of the order of 0.4 to 0.45. Its particle size is a monomodal distribution characterized by an average diameter of the order of 20 to 50 μm.

Diosmin is naturally compressible, with a compressibility index varying between 20 and 50, a Hausner ratio greater than 1 and a compaction capacity greater than 20. Although compressible, native diosmin lacks cohesion between particles, so it will be necessary to choose excipients (see below) allowing to bring a satisfactory cohesion. Diosmin, which is not soluble in water as indicated, becomes quite pasty on the palate, with a consistency that can be described as earthy or chalky, which should be masked by a choice appropriate excipients (see also below) for the tablet to be acceptable to the patient.

Granulated diosmin

To improve the proper cohesion of the diosmin, instead of introducing a native diosmin into the mixture to be compressed, the latter can be subjected to a preliminary treatment of wet granulation. This step is carried out in a conventional manner by mixing the diosmin with an appropriate granulating agent, then granulating proper and finally drying the powder so as to adjust the residual water content.

The granulation is an aqueous granulation with the addition of a granulating agent, preferably a hydrophilic cellulose derivative such as hydroxypropylmethylcellulose. It is also possible to use, as a variant or in addition, hydrophilic granulation polymers such as polyvinylpyrrolidones, native or modified starches, dextrins, oses, polyols and their mixtures. The granulation is carried out so as to increase the particle size of the particles up to an average diameter of the order of 50 to 150 μm, preferably approximately 50 μm. A larger, technically possible mean diameter would, however, introduce a grainy, crunchy feeling, generally perceived less pleasant by patients than with a finer grain size. This granulation, in addition to the increase in the average diameter of the particles, is accompanied by a densification of the active principle (d = 0.5 to 0.55), a slight improvement in its wettability, an increase in the cohesion of the particles between them and makes the powder a little less electrostatic. Finally, the drying step after the actual granulation makes it possible to adjust the residual water content to a higher value than with a native diosmin, typically to a value of the order of 4 to 8%. excipients

The choice of excipients is dictated by a double purpose:

- meet a certain number of technical manufacturing constraints, avoiding:

• cleavage of the tablets after the direct compression step. In fact, the studies carried out by the inventors have shown that, for many qualitative and / or quantitative combinations of excipients, the tablets tend to cleave, that is to say to open in a perpendicular plane. to that of the direction of compression, phenomenon which occurs when the tablet is too hard, not hard enough, and / or lack of cohesion; compression in fact creates a certain number of stresses inside the tablet, and the excipients must be chosen to allow these stresses to be absorbed, without making the tablet too brittle.

The risk of a phenomenon, in particular of sticking of the powder, which would prevent a tablet from being produced, that is to say a unit dose obtained by compression of a powder, this method of preparation possibly leading to certain technical difficulties if the excipients are not selected appropriately.

- on the other hand, when it comes to making a chewable tablet, unlike the dry forms directly swallowed, in the case of a chewable tablet it is necessary to take into account:

• taste, the tablet must have acceptable taste and flavor, preferably pleasant for the patient,

• consistency, that is to say, the general impression in the mouth when chewing, an impression which must not be chalky or abrasive, for example, and

• hardness, which should not be too high to allow easy chewing, but must be sufficient to prevent the tablet from breaking in its packaging (for example in the event of the tube containing the tablets falling) or between patient's fingers. Essentially, the tablet of the invention comprises as excipients one or more diluents-sweeteners, a compression agent, a lubricating agent, a flavoring, and optionally an acidifier and / or an additional sweetener.

Thinners-sweeteners: one or more polyols, advantageously two different polyols, advantageously a mixture of xylitol and sorbitol, are advantageously used as diluent-sweeteners. These polyols have the advantage of offering a sweet flavor and providing a pleasant sensation of freshness on the palate. They are more acariogenic and very low in calories. They also have binder properties and excellent compressibility properties which make them suitable for manufacture by direct compression. The ratio of the mass of the polyol (if only one polyol is used) or of all the polyols, relative to the titrated mass of diosmin is preferably between 2 and 3.

In the case of a mixture of xylitol and sorbitol, the sorbitol / xylitol ratio is adjusted to meet the different technical and taste constraints mentioned above (absence of cleavage and sticking phenomenon, pleasant taste, masking of the earthy consistency of diosmin, etc.). More specifically, this sorbitol / xylitol ratio is greater than 1, preferably between 1.2 and 1.5.

Furthermore, it is possible to substitute another polyol, in particular mannitol, for sorbitol or xylitol, which has similar physicochemical and gustatory properties. It is also possible to substitute for one of the two polyols a dare such as fructose or sucrose. The preferred choices of diluents-sweeteners are as follows:

- two different polyols, in particular xylitol + sorbitol (preferential choice), or xylitol + mannitol, or mannitol + sorbitol; - a single polyol, which is then preferably sorbitol;

- three different polyols, for example xylitol + sorbitol + mannitol;

- single polyol + additional synthetic sweetener (see below);

- single polyol + dare, for example sorbitol + fructose or sucrose, or xylitol + fructose or sucrose. Compression agent: this excipient brings hardness to the tablet and prevents post-compression cleavage. Dicalcium phosphate is preferably used (for example of the Emcompress® type from Pennwest Pharmaceuticals Co.). Other compression agents can be used as a variant or as a supplement, for example lactose.

Lubricant: this excipient has the function of avoiding the seizing phenomenon at the level of the dies of the presses to compress and has non-stick properties; it is chosen from conventional lubricants such as stearic acid, magnesium stearate and talc or (preferably) a combination of two or three of these products.

Flavor: this excipient is chosen from conventional additives according to the taste that one wishes to give to the tablet, for example an orange flavor.

Acidifier: it is possible to add a small amount to the preparation of an acidifier such as citric acid to improve its taste qualities.

Additional synthetic sweetener: to increase the sweetness when it is not deemed sufficient, especially when using a single diluent-sweetener such as sorbitol, it is possible to add a small synthetic sweetener powerful such as aspartame, potassium cyclamate, potassium acesulfame or sodium saccharinate.

Example

From the various constituents described above, a tablet is produced by direct compression. The manufacturing operating protocol is conventional, by simple mixing of the active principle (native diosmin or granulated diosmin) with the various excipients then compression on a press. The (non-limiting), qualitative and quantitative, formula of this example pre- is as follows, for a tablet dosed with 600 mg of titrated diosmin:

- diosmin: 600 mg of titrated mass, corresponding in this example to a mass of native diosmin of 683.45 mg if this native diosmin has a residual water content of 2.95% and a titer of 90.46% over dry pure diosmin,

- xylitol (type Xylisorb® 300, Roquette): 700 mg,

- sorbitol (type Neosorb® P60W, Roquette): q.s. a tablet with a unit mass of 2400 mg, or in this example 871, 55 mg, - dicalcium phosphate (type Emcompress®, Pennwest): 25 mg,

- orange flavor: 25 mg,

- citric acid: 5 mg,

- talc: 30 mg,

- stearic acid: 15 mg, - magnesium stearate: 45 mg.

These values can be modified around the values indicated, the preferred value ranges being, always a 2400 mg tablet dosed with 600 mg of titrated diosmin:

- xylitol: 600 to 800 mg, - sorbitol: 750 to 950 mg (q.s. a 2400 mg tablet)

- dicalcium phosphate: 10 to 60 mg,

- orange flavor: 15 to 40 mg,

- citric acid: 3 to 8 mg,

- talc: 20 to 50 mg, - stearic acid: 10 to 30 mg,

- magnesium stearate: 30 to 60 mg.

The tablet obtained has a total mass of 2400 mg per 600 mg of titrated diosmin. In general, the unit mass of the tablet is preferably between 2000 and 3000 mg, advantageously between 2200 and 2600 mg, very advantageously 2400 mg, for 600 mg of titrated diosmin.

Tablet hardness, measured according to the analytical methods described in the European Pharmacopoeia, ^4th edition, the title 2.9.8 "Tensile strength of the tablets" is 150 N. In order generalized stripe, the hardness is preferably between 150 and 250 N, very advanced tagging between 150 and 200 N.

It can be noted that this formulation is a formulation without sugar (without dare), without sodium and without additional synthetic sweetener (most of them, for example aspartame, having certain contraindications).

Various quantitative adaptations can be made to this formulation.

It is also possible to provide other unit doses of diosmin, the dose typically varying between 300 and 1200 mg of titrated diosmin.

It will be noted that it thus becomes possible to produce tablets making it possible to administer in a single unit dose 1200 mg of diosmin, a quantity which it has not been possible to achieve with the other types of tablets hitherto proposed.

Claims

1. A dry oral formulation for phlebotonic and vasculoprotective purposes for the treatment of venous insufficiency, capillary fragility and hemorrhoidal crisis, this formulation being in pharmaceutical form of tablet comprising, in mixture:

- at least one active ingredient containing diosmin, and

- a combination of excipients, characterized in that said pharmaceutical form is a form of chewable, suckable or chewable tablet.

2. The formulation of claim 1, wherein the tablet has a hardness of between 150 and 250 N.

3. The formulation of claim 2, wherein the tablet has a hardness between 150 and 200 N.

4. The formulation of claim 1, wherein the diosmin is a native diosmin.

5. The formulation of claim 4, wherein the residual water content of the diosmin is between 1 and 5%.

6. The formulation of claim 5, wherein the residual water content of the diosmin is between 2 and 4%.

7. The formulation of claim 6, wherein the residual water content of the diosmin is 3%.

8. The formulation of claim 4, in which the average diameter of the diosmin particles is between 20 and 50 μm.

9. The formulation of claim 1, wherein the diosmin is a granulated diosmin.

10. The formulation of claim 9, wherein the residual water content of the diosmin is between 4 and 8%.

11. The formulation of claim 9, in which the average diameter of the diosmin particles is between 50 and 150 μm.

12. The formulation of claim 1, wherein the combination of excipients comprises at least one polyol.

13. The formulation of claim 12, in which the ratio of the mass of the polyol, or all of the polyols, to the titrated mass of diosmin is between 2 and 3.

14. The formulation of claim 12, wherein the polyol is sorbitol.

15. The formulation of claim 12, wherein the combination of excipients comprises at least two polyols.

16. The formulation of claim 15, wherein the two polyols are xylitol and sorbitol.

17. The formulation of claim 16, wherein the mass ratio of sorbitol to xylitol is greater than 1.

18. The formulation of claim 17, wherein the mass ratio of sorbitol to xylitol is between 1.20 and 1.50.

19. The formulation of claim 12, wherein the combination of excipients comprises:

- a polyol,

- a sweetener,

- a compression agent,

- a lubricant, - an acidifier and a flavor.

20. The formulation of claim 12, wherein the combination of excipients comprises:

- a polyol, - a dare,

- a compression agent,

- a lubricant,

- an acidifier and an aroma.

21. The formulation of claim 12, wherein the combination of excipients comprises:

- sorbitol,

- a compression agent,

- a lubricant, - an acidifier and a flavor.

22. The formulation of claim 12, wherein the combination of excipients comprises:

- xylitol, - sorbitol,

- a compression agent,

- a lubricant,

- an acidifier and an aroma.

23. The formulation of one of claims 19 to 22, in which the compression agent is chosen from the group comprising dicalcium phosphate, lactose and their mixtures.

24. The formulation of claim 23, wherein the compression agent is dicalcium phosphate.

25. The formulation of one of claims 19 to 22, in which the lubricant is chosen from the group comprising talc, stearic acid, magnesium stearate and their mixtures.

26. The formulation of one of claims 19 to 22, in which, the active principle being a granulated diosmin, the combination of excipients further comprises:

- an additional granulating agent.

27. The formulation of claim 26, wherein the additional granulating agent is selected from the group comprising hydroxypropylmethylcellulose, polyvinylpyrrolidone, starches, dextrins, oses, polyols and mixtures thereof.

28. The formulation of one of claims 16 to 18 and 22 to 27, in which one of the sorbitol or xylitol polyols is replaced by mannitol.

29. The formulation of claim 12, comprising: - native diosmin,

- xylitol,

- sorbitol,

- dicalcium phosphate,

a lubricant chosen from the group comprising talc, stearic acid, magnesium stearate and their mixtures,

- an acidifier and an aroma.

30. The formulation of claim 29, the quantitative formula of which is:

- diosmin: mass titrated 600 mg, - xylitol: 600 to 800 mg

- sorbitol: 750 to 950 mg q.s. one 2400 mg unit mass tablet

- dicalcium phosphate: 10 to 60 mg

- talc: 20 to 50 mg - stearic acid: 10 to 30 mg

- magnesium stearate: 30 to 60 mg

- flavor: 15 to 40 mg

- citric acid: 3 to 8 mg, or a quantitative homothetic formula.

31. The formulation of claim 30, the quantitative formula of which is

- diosmin: mass titrated 600 mg,

- xylitol: 700 mg

- sorbitol: q.s. one tablet of unit mass 2400 mg - dicalcium phosphate: 25 mg

- talc: 30 mg

- stearic acid: 15 mg

- magnesium stearate: 45 mg

- flavor: 25 mg - citric acid: 5 mg, or a quantitative homothetic formula.