Methods for establishing a sorption isotherm of a powder, for determining the water activity (AW) of a powder sample, for establishing the sorption isotherm of a solid composition and for determining the water activity (AW) of a mixture of powders, and use of the method

Description

26 METHODS FOR DETERMINING WATER ACTIVITY, TO ESTABLISH THE SORPTION ISOTHERM OF A COMPOSITION SOLID AND TO DETERMINE WATER ACTIVITY, AND, USE OF THE METHOD FIELD OF TECHNIQUE

[001] The present invention relates to the field of powders, in particular in mixtures, and in particular to the option of determining ideal mixtures to obtain a determined and desired value of the water activity (Aw) of the mixture. PREVIOUS TECHNIQUE

[002] Sorption isotherms allow the estimation of water activity (Aw) from the moisture content of a substance. Water activity is a known and important factor for the stability of material mixtures, especially when the mixture includes an active compound such as a biotic, whose activity must be preserved and stability must be ensured.

[003] An example is related to the formulation of probiotics in dry food mixtures. The water activity (Aw) of the mixture, unlike its moisture content, is a determining factor for microbial growth, toxin production, and the shelf life of probiotics. To protect probiotics, it is necessary to manage the Aw of food mixtures. Monitoring the Aw of complex mixtures forms the basis for probiotic formulation. In fact, Aw has a significant impact on the viability of ingredients over time and on the development of contaminating flora.In a mixture, the moisture captured by the powder is distributed unevenly among the components, despite the water activity being equal throughout the mixture. Excess water, and therefore high water activity, will allow the growth of altered flora and simultaneously negatively impact the number of colony-forming units (CFU) of probiotics.

[004] In the particular case of probiotic formulation, to protect Petition 870250087588, dated 09 / 26 / 2025, page 56 / 85 / 26 To prevent deterioration and avoid the growth of contamination, water activity should be kept as low as possible. Thus, mixtures that combine an excipient with low water activity with a probiotic with a higher Aw have a protective effect on the probiotic, decreasing its Aw. By mixing low Aw excipients with probiotics, the overall water activity of the mixture can be better controlled in order to reduce stability problems.

[005] Numerous mathematical models of moisture isotherms are available to understand how probiotics interact with excipients, so that the behavior of mixtures can be determined with the greatest possible precision. These mathematical models of moisture isotherms are described in the specialized literature to estimate the evolution of water activity according to moisture content. Some have theoretical foundations (for example, considering the moisture monolayer as a model parameter), others are simply empirical. These models allow the generation of virtual mixtures of isotherms, useful for identifying optimal formulations of ingredient mixtures. However, their accuracy varies according to the products, and they can be more or less approximate. Some models work very well for some products, but not for others. There are even products that cannot be correctly represented by sorption models due to their very complex inflection points.

[006] Thus, there is a clear need to define a new tool for modeling sorption isotherms, applicable to any type of product in powder form, including mixtures. DETAILED DESCRIPTION OF THE INVENTION

[007] The present invention is based on the fact that a local LOESS-type regression allows modeling the water activity (Aw) of a compound in solid form, advantageously in powder form, but also of a mixture of at least two compounds in solid form, Petition 870250087588, dated 09 / 26 / 2025, p. 57 / 85 / 26 advantageously in powder form.

[008] In example I it is demonstrated that this mathematical method is very efficient and far superior to the models used in the previous technique, such as the Guggenheim-Anderson-deBoer model (GAB; Van den Berg, C. (1984), In BM McKenna (Ed.), Engineering and Foods, pp. 311-321, London Elsevier) with a correlation between the moisture content of the compound and the relative hygrometry for the experimentally measured points (for example, at a temperature of 25°C), and the curve obtained by modeling, equal to 1.

[009] This method presented very good results and proved to be robust for all ingredients tested, particularly for ingredients with complex isotherms (with inflection points), which are poorly considered by existing physical or empirical models.

[0010] In the description below, the term moisture adsorption isotherm, hereinafter referred to as sorption isotherm, refers to the equilibrium relationship between moisture content and relative humidity of the medium, or relative hygrometry (P / P0), at a given temperature. In practice, this relationship can be represented by a curve with moisture content on the y-axis and the quantity P / P0 on the x-axis.

[0011] Water activity (Aw) represents the ratio between the water vapor pressure of a wet substance (P) and the saturated vapor pressure (P0) at the same temperature, that is, the relative hygrometry (P / P0).

[0012] The moisture content of a product (or MC) is the ratio between the mass of moisture and the mass of dry matter (or DM). It allows quantifying the amount of water in a product.

[0013] The moisture content (X) of a product depends on the total mass of wet matter Mh and the mass of dry matter Ms according to the relationship: X = (Mh - Ms) / Ms.

[0014] In a known manner, sorption isotherms are determined experimentally using static or dynamic methods: Petition 870250087588, dated 09 / 26 / 2025, p. 58 / 85 / 26

[0015] In the static or gravimetric method, the product to be analyzed is placed in an enclosure that is maintained at a fixed temperature and with constant relative humidity. The sample is left to reach equilibrium (with its mass unchanged) for several days, up to weeks.

[0016] In the dynamic method, the product is placed in an air current (dryer, for example) at constant temperature and relative humidity. Equilibrium (unchanged total mass) is usually reached after a few hours.

[0017] In the context of this application, the moisture content (MC) is expressed in g per 100g of dry matter (DM).

[0018] In this description, the terms substance, compound, component, material, ingredient, or product will be used interchangeably. Thus, these terms, in particular the term compound, can also refer to a pure molecule, as well as to a molecule already mixed with other molecules. In practice, it refers to a product available individually, for example, commercially available.

[0019] In the description below, the term powder(s) refers to a fractional state of the material. It is a divided solid, present in the form of particles or granules, generally with a size (D) smaller than one tenth of a millimeter (100 μm). A powder is generally distinguished by its particle size distribution curve, which expresses the mass fraction of the particles in relation to each size class, that is, the statistical distribution of grain size.

[0020] A powder may or may not contain water.

[0021] The term powder mixture refers to at least two powders. It may be necessary to produce powder mixtures for several reasons: each powder may have a specific activity of interest; for economic reasons, the powder with the activity of interest may be mixed with one or more inactive but cheaper powders; some powders may help to formulate and even Petition 870250087588, dated 09 / 26 / 2025, page 59 / 85 / 26 to stabilize the powder(s) that have an activity of interest. In the last two scenarios, the inactive powders are chosen so as not to affect the activity of the powder of interest.

[0022] The present invention has numerous applications, particularly in fields where the material in powder form is commonly used, for example in the fields of cosmetics, food (for human or animal use), medicines (pharmaceuticals and veterinary), paints, metals, chemicals (fertilizers, polymers, plastics), etc.

[0023] Thus, certain compounds in the mixture exhibit an activity of interest in the field of application, for example, a therapeutic activity in the medical field or a pigmentation activity in the field of paints.

[0024] Other compounds do not have an activity per se, but may be useful for the formulation of active ingredients (excipients and additives) or for economic reasons. It is important that their presence does not affect, and especially does not deteriorate, the activity of the active compound, in particular, its water activity.

[0025] According to a first aspect, the present invention relates to the use of a non-parametric regression, for example, a LOESS-type local regression, to determine the water activity (Aw) of a compound in solid form, advantageously in the form of a powder, or of a mixture of at least two compounds in solid form, advantageously in the form of powders.

[0026] Therefore, and according to the invention, the modeling of sorption isotherms is based on the use of weighted local polynomial regression or LOESS (Locally Estimated Scatter Plot Smoothing) regression, and more generally, on the use of non-parametric regression approaches. Non-parametric regression methods are known to a person skilled in the art and can, for example, be chosen from the following Petition 870250087588, dated 09 / 26 / 2025, p. 60 / 85 / 26 group: LOESS regression (Locally Estimated Scatter Plot Smoothing); regressogram (Battery Smoother); mobile media; weighted mobile media (Nadaraya and Watson regression); kernel regression, multivariate adaptive regression spline (MARS), advantageously LOESS regression.

[0027] LOESS regression was developed by W.S. Cleveland in the 1980s (“Robust Locally Weighted Regression and Smoothing Scatterplots”, Journal of the American Statistical Association, Vol. 74, No. 368, 1979, pp. 829-836). However, to the best of the inventors' knowledge, its use has never been considered in the context of the present application, nor have other non-parametric regression models.

[0028] LOESS modeling fits a polynomial regression function at each point in the dataset, considering only points located in the surroundings. LOESS regression is a completely empirical approach, which takes into account all local regressions for all points. Two hyperparameters influence the smoothing features of LOESS modeling: the proportion of surrounding points used for a local regression and the degree of polynomials.

[0029] To optimize this application, the LOESS model can be configured so that only the two points in the vicinity of a given point are considered with a degree of 2 for the polynomial. This configuration makes LOESS modeling very flexible in cases where the sorption isotherm is dense.

[0030] More specifically, the present invention relates to a method for establishing a sorption isotherm of a powder comprising the following steps: - obtain a set of points corresponding to the moisture content (MC) of the powder according to the relative hygrometry (P / P0); - apply a non-parametric regression, for example, a Petition 870250087588, dated 09 / 26 / 2025, page 61 / 85 / 26 LOESS regression, at these points to establish the dust sorption isotherm.

[0031] In the scope of this application, the terms “establishment” (“establish”), “obtaining” (“obtain”) and “generation” (“generate”) are used interchangeably.

[0032] The points corresponding to the moisture content (MC) of the powder according to the relative hygrometry (P / P0) can be found in the literature or determined experimentally.

[0033] As is known, such points can be obtained by gravimetry or by dynamic vapor sorption (DVS) measurement. Appropriately, the measurements are carried out at a constant temperature, preferably at room temperature, for example, at 25°C.

[0034] According to an advantageous embodiment, each sorption isotherm is established from at least five, up to ten moisture content (MC) measurement points according to relative hygrometry (P / Po).

[0035] According to another aspect, the present invention relates to a method for determining the water activity (Aw) of a powder sample, comprising: - Establish the sorption isotherm of the powder using the method described above; - measure the moisture content of the sample; - Determine the water activity of the sample using the measured moisture content of the sample and the established powder sorption isotherm model.

[0036] In the very particular case where the established sorption isotherm is represented by a curve, it is possible to graphically determine the water activity (Aw) of the powder sample by plotting the measured moisture content on the y-axis of the curve and reading the corresponding value on the x-axis, this value being the water activity of the sample.

[0037] As shown in the examples, these methods also Petition 870250087588, dated 09 / 26 / 2025, page 62 / 85 / 26 can be implemented in powder mixtures. Below, the expressions "composition comprising at least two compounds" and "mixture of at least two compounds" are used interchangeably.

[0038] Thus, the invention also relates to a method for establishing the sorption isotherm of a solid composition comprising at least two compounds, advantageously in the form of a powder mixture, comprising: - obtain the sorption isotherms of each of the compounds, the isotherm of at least one of the compounds, advantageously of all the compounds, being established using the method described above; - Establish the sorption isotherm of the composition according to the weight ratio of each compound in the composition.

[0039] In practice, the sorption isotherm of the composition is established using the sorption isotherm models of each of the compounds, weighted according to the weight proportion of each compound in the composition.

[0040] According to the invention, the isotherm of at least one of the compounds is established using the method described above, i.e., using a non-parametric regression. Advantageously, this is / are the compound(s) whose modeling using a parametric regression is not satisfactory. According to a particular embodiment, an isotherm of all the compositions of the composition or mixture is established using the method described above, i.e., using a non-parametric regression. However, and as demonstrated in the present application, it is possible to combine isotherms obtained by non-parametric and parametric regressions.

[0041] In fact, and in a known way, it is possible to combine or accumulate two or more regressions, including parametric and non-parametric regressions. Thus, Peleg and Norman (Trends in Food Science & Technology, July 1992 [Vol. 31]) described a strategy for Petition 870250087588, dated 09 / 26 / 2025, page 63 / 85 / 26 microcomputer estimation of sorption isotherms of complex formulations containing ingredients whose isotherms are known or unknown.

[0042] Again, the sorption isotherm of the composition thus established can be visualized in the form of a curve.

[0043] The weight or mass ratio refers to the relative proportion of each of the compounds in the mixture, expressed as weight / weight (w / w).

[0044] Consequently, and according to another aspect, the invention relates to a method for determining the water activity (Aw) of a powder mixture comprising: - Establish the sorption isotherm of the mixture using the method described above; - Measure the moisture content of the mixture or calculate the moisture content of the mixture using the moisture content of each of the compounds, weighted according to the weight proportion of each compound in the mixture; - Determine the water activity of the mixture using the measured or calculated moisture content of the mixture and the established sorption isotherm model of the mixture.

[0045] In the very particular case where the established sorption isotherm is represented by a curve, it is possible to graphically determine the water activity (Aw) of the powder mixture by plotting the measured or calculated moisture content on the y-axis of the curve and reading the corresponding value on the x-axis, this value being the water activity of the mixture.

[0046] The moisture content (MC) of the mixture can be determined experimentally or by calculation, as illustrated in the examples.

[0047] As already stated, a method according to the invention makes sense when at least one compound possesses an activity of interest, which must be preserved, respecting its water activity. Thus, and Petition 870250087588, dated 09 / 26 / 2025, pp. 64 / 85 / 26, as illustrated in the examples, for a probiotic, an Aw value of less than 0.3 or even less than 0.1 should be maintained, according to the strain and its drying method. In the specific case where the active ingredient is hygroscopic chondroitin (e.g., chondroitin sulfate), a mixture with an Aw value around 0.3 is recommended to ensure its stability. Ideal values ​​or ranges of values ​​are provided only as examples, but are known to a person skilled in the art.

[0048] According to a particular embodiment, the composition or mixture of powders comprises at least one active compound. An active compound may be a microorganism (yeast, bacteria or phage), living or dead, or an active molecule (vitamins, chondroitin, etc.). Therefore, and by way of example, the active compound may belong to the following categories: animal feed additives, prebiotics, proteins, enzymes, etc.

[0049] According to a particular embodiment, such a composition or mixture of powders is intended for the development of a food supplement or a medicine.

[0050] As illustrated in the examples, the method according to the invention makes it possible to determine the nature and / or weight content of a compound other than the active compound, in order to obtain a mixture having a water activity compatible with that of the active compound.

[0051] In practice, the compound other than the active compound may be an excipient or an additive, for example, maltodextrin or microcrystalline cellulose.

[0052] It becomes apparent that, in view of the above description, a person skilled in the art is capable of implementing an invention as defined in the claims below. EXAMPLES OF MODALITIES

[0053] The invention and the advantages derived from it will become more evident from the following examples of embodiments supported by Petition 870250087588, dated 09 / 26 / 2025, pp. 65 / 85 / 26 attached figures. They do not limit the scope in any way. DESCRIPTIONS OF THE FIGURES

[0054] Figure 1: This figure illustrates the correlation between the moisture content of the Vitamin K2 product (MC), expressed in g per 100 g of dry matter (DM), and the relative hygrometry (P / P0) for the points measured experimentally at a temperature of 25°C, and the curve obtained by modeling using the GAB model.

[0055] Figure 2: This figure illustrates the correlation between the moisture content of the Vitamin K2 (MC) product, expressed in g per 100 g of dry matter (DM), predicted by the GAB model and observed experimentally.

[0056] Figure 3: This figure illustrates the correlation between the moisture content of the Vitamin K2 product (MC), expressed in g per 100 g of dry matter (DM), and the relative hygrometry (P / P0) for the points measured experimentally at a temperature of 25°C, and the curve obtained by modeling using LOESS regression.

[0057] Figure 4: This figure illustrates the correlation between the moisture content of the anhydrous Dextrose (MC) product, expressed in g per 100 g of dry matter (DM), and the relative hygrometry (P / P0) for the points measured experimentally at a temperature of 25°C, and the curve obtained by modeling using the GAB model.

[0058] Figure 5: This figure illustrates the correlation between the moisture content of the anhydrous Dextrose (CM) product, expressed in g per 100 g of dry matter (DM), predicted by the GAB model and observed experimentally.

[0059] Figure 6: This figure illustrates the correlation between the moisture content of the anhydrous Dextrose (MC) product, expressed in g per 100 g of dry matter (DM), and the relative hygrometry (P / P0) for the points measured experimentally at a temperature of 25°C, and the curve obtained by modeling using LOESS regression. Petition 870250087588, dated 09 / 26 / 2025, page 66 / 85 / 26

[0060] Figure 7: This figure illustrates the correlation between the moisture content (MC), expressed in g per 100 g of dry matter (DM), of a vitamin K2 formulation also containing trehalose and an emulsifying starch (E1450 sodium starch octenyl succinate) and the relative hygrometry (P / P0) for the points measured experimentally at a temperature of 25°C, and the curve obtained by modeling using LOESS regression.

[0061] Figure 8: This figure illustrates the correlation between moisture content expressed in g per 100 g of dry matter (DM) and relative hygrometry (P / P0) through modeling using LOESS regression: - a Lactobacillus rhamnosus GG probiotic (solid line); - maltodextrin with 10% water (large dots); - a formulation comprising, by weight / weight (w / w), a 20 / 80 mixture of Lactobacillus rhamnosus GG / maltodextrin containing 10% water (dotted line).

[0062] Figure 9: This figure illustrates the correlation between moisture content expressed in g per 100 g of dry matter (DM) and relative hygrometry (P / P0) through modeling using LOESS regression: - a Lactobacillus rhamnosus GG probiotic (solid line); - maltodextrin with 5% water (large dots); - a formulation comprising, by weight / weight (w / w), a 20 / 80 mixture of Lactobacillus rhamnosus GG / maltodextrin containing 5% water (dotted line).

[0063] Figure 10: This figure illustrates the correlation between moisture content expressed in g per 100 g of dry matter (DM) and relative hygrometry (P / P0) through modeling using LOESS regression: - a Lactobacillus rhamnosus GG probiotic (solid line); - maltodextrin with 3.9% water (large dots); - a formulation comprising, by weight / weight (w / w), a 20 / 80 mixture of Lactobacillus rhamnosus GG / maltodextrin containing Petition 870250087588, dated 09 / 26 / 2025, p. 67 / 85 / 26 3.9% water (dotted line).

[0064] Figure 11: This figure illustrates the correlation between moisture content expressed in g per 100 g of dry matter (DM) and relative hygrometry (P / P0) through modeling using LOESS regression: - chondroitin sulfate (light gray spots); - maltodextrin with 10% water (dark gray specks); - Microcrystalline cellulose or CMC (continuous line); - a formulation comprising, by weight / weight (w / w), a 10 / 80 / 10 mixture of chondroitin / 10% water maltodextrin / CMC (dotted line).

[0065] Figure 12: This figure illustrates the correlation between moisture content expressed in g per 100 g of dry matter (DM) and relative hygrometry (P / P0) through modeling using LOESS regression: - chondroitin sulfate (light gray spots); - maltodextrin with 10% water (dark gray specks); - Microcrystalline cellulose or CMC (continuous line); - a formulation comprising, by weight / weight (w / w), a 10 / 45 / 45 mixture of chondroitin / 10% water maltodextrin / CMC (dotted line).

[0066] Figure 13: This figure illustrates the correlation between moisture content expressed in g per 100 g of dry matter (DM) and relative hygrometry (P / P0) through modeling using LOESS regression: - chondroitin sulfate (light gray spots); - maltodextrin with 10% water (dark gray specks); - Microcrystalline cellulose or CMC (continuous line); - a formulation comprising, by weight / weight (w / w), a 10 / 36 / 54 mixture of chondroitin / 10% water maltodextrin / CMC (dotted line).

[0067] Figure 14: This figure illustrates the correlation between the content of Petition 870250087588, dated 09 / 26 / 2025, page 68 / 85 / 26 moisture expressed in g per 100 g of dry matter (DM) and relative hygrometry (P / P0): - Vitamin K2, modeled using LOESS (loseless squares) regression; - a yeast fraction, by modeling using the GAB model (circles); - dry potato starch, molded using the GAB model (triangles); - a mixture comprising these 3 ingredients (5 / 25 / 70 by weight / weight), modeled using LOESS regression (light gray lines); - a mixture comprising these 3 ingredients (5 / 25 / 70 by weight / weight), modeled using the compartment smoother regression (light gray traces of varying thickness); - a mixture comprising these 3 ingredients (5 / 25 / 70 by weight / weight), modeled using MARS regression (dark gray traces). I / Superiority of a LOESS-type local regression for modeling sorption isotherms: 1 / Example with vitamin K2:

[0068] 1-1. Ingredient: Vitamin K2 MK-7 Matrix (2000 ppm maltodextrin powder; Gnosis Bioresearch)

[0069] The sorption isotherm is determined experimentally using a static method.

[0070] 1-2. GAB Model: The correlation between the product moisture content (CM), expressed in g per 100 g of dry matter (DM), and the relative hygrometry (P / P0) is illustrated in Figure 1.

[0071] Figure 1 shows the experimentally measured points at Petition 870250087588, dated 09 / 26 / 2025, page 69 / 85 / 26 a temperature of 25°C, as well as the curve obtained by modeling using the GAB model.

[0072] The correlation obtained is equal to 0.988.

[0073] However, as illustrated in figure 2, the correlation graph reveals a correct overall correlation, but significant deviations for values ​​less than 5% moisture content.

[0074] Figure 2 shows the correlation between the product moisture content (CM), expressed in g per 100 g of dry matter (DM), predicted by the GAB model and observed experimentally.

[0075] 1-3.LoESS Regression: The data in Figure 1 were obtained by plotting the curve between the experimental points using a LOESS-type local regression.

[0076] As illustrated in Figure 3, the correlation in this case is equal to 1, that is, perfect and much better than that of the GAB model.

[0077] Figure 3 shows the points measured experimentally at a temperature of 25°C, as well as the curve obtained by modeling using LOESS regression. 2 / Example with anhydrous dextrose:

[0078] 2-1. Ingredient: Anhydrous dextrose (or anhydrous D-glucose)

[0079] The sorption isotherm is determined experimentally in the static method.

[0080] 2-2. GAB Model: The equivalent of Figure 1 is shown in Figure 4.

[0081] The correlation obtained is equal to 0.966.

[0082] The equivalent of Figure 2 is shown in Figure 5. This correlation plot shows a correct overall correlation, but significant deviations in the lowest and highest values.

[0083] 2-1. LOESS Regression: The data in Figure 4 were obtained, but by plotting the curve Petition 870250087588, dated 09 / 26 / 2025, page 70 / 85 / 26 between experimental points using a LOESS type local regression.

[0084] As illustrated in figure 6, the correlation in this scenario is again equal to 1, that is, perfect and much better than with the GAB model. 3 / Application to a mixture: Example with a mixture of vitamin K2, trehalose and emulsifying starch (E1450 sodium starch octenyl succinate)

[0085] Figure 7 represents the sorption isotherm observed for a vitamin K2 formulation that also contains trehalose and an emulsifying starch (E1450 sodium starch octenyl succinate), which is particularly complex, with non-monotonic growth and a sharp inflection point. This product cannot be satisfactorily represented by any conventional nonlinear parametric model, while the curve generated by LOESS modeling fits the observed points very well. CONCLUSION:

[0086] These examples reveal the superiority of a LOESS-type local regression in correlating the moisture content of a product (MC), expressed in g per 100 g of dry matter (MS), and its relative hygrometry (P / P0), and thus determining its water activity (Aw).

[0087] As seen above, the LOESS model limits errors. Consequently, based on this, water activity (Aw) calculations are more accurate and closer to reality than generic models, such as the GAB model, recommended in the state of the art.

[0088] Therefore, it is no longer necessary to specify a deterministic function, a priori, to model the data of a sample. The curve fit according to the invention is perfect, with excellent correlation coefficients between observed and predicted values. This is particularly interesting for predicting the results of mixtures, where approximations about different constituents risk being accumulated. On the other hand, the underlying method is relatively simple, which makes it easy to Petition 870250087588, dated 09 / 26 / 2025, page 71 / 85 / 26 to understand and interpret. II / Use of a LOESS-type local regression to determine the nature of an excipient in a two-component system

[0089] The reliability of the LOESS model can, for example, be used to identify a suitable excipient, that is, one that can be used in combination with an active ingredient without deleteriously altering the water activity (Aw) of said active ingredient.

[0090] Therefore, and in the particular case where the active ingredient is a probiotic, in this case the LifeinU™ Lactobacillus rhamnosus GG strain ((ATCC 53103); Gnosis), the stability of said probiotic is guaranteed only for an Aw value less than or equal to 0.1. In other words, and in this case, the Aw of the mixture (probiotic + excipient) must be < 0.1.

[0091] Maltodextrin is an excipient known for its compatibility with probiotics in formulations intended for ingestion.

[0092] It is considered a formulation comprising a 20 / 80 weight / weight (w / w) mixture of Lactobacillus rhamnosus GG / maltodextrin.

[0093] Knowing that there are several qualities of maltodextrin, i.e., with varying moisture contents, the LOESS model allows us to verify if maltodextrin is an acceptable excipient in this context, and which maltodextrin to choose to respect a final Aw of the mixture of, at most, 0.1. It should be noted that the maximum moisture content of maltodextrin is given with an accuracy of 0.1%.

[0094] Thus, the correlation between the moisture content of the mixture (MC), expressed in g per 100 g of dry matter (MS), and the relative hygrometry (P / P0) of the mixture was established using LOESS regression.

[0095] This correlation is illustrated below with 3 qualities of maltodextrin: Petition 870250087588, dated 09 / 26 / 2025, pp. 72 / 85 18 / 26 a) Maltodextrin with 10% water Product % in the mixture MC in g / 100g of product MC in g / 100g of DM % of DM in g / 100g of product Active probiotic = Lactobacillus rhamnosus GG 20 0.1 0.1 / (100-0.1) X 100 = 0.1001 0.1 / 0.1001 X 100 = 99.9 Excipient = maltodextrin with 10% water GLUCIDEX 12D] 80 10 10 / (100-10) X 100 = 11.1111 10 / 11.1111 x 100 = 90 TOTAL 100 (20x0.1) + (80x10) / 100 = 8.02 8.02 / (100-8.02) x 100 = 8.7193 8.02 / 8.7193 x 100 = 91.98 Table 1: Features of the implemented products

[0096] Figure 8 illustrates the correlation between moisture content expressed in g per 100 g of dry matter (DM) and relative hygrometry (P / P0) by modeling using LOESS regression.

[0097] The isotherm of the probiotic Lactobacillus rhamnosus GG (known or experimentally measured) is represented by a solid line.

[0098] The maltodextrin isotherm with 10% water (known or experimentally measured) is represented by large dots.

[0099] The mixture isotherm, resulting from modeling each of the isotherms using the LOESS method and weighting the isotherm of each ingredient according to the weight proportion of those same ingredients, is represented by a dashed line.

[00100] According to the calculations in Table 1, the MC (Moisture content) in g / 100g of DM (dry matter) is equal to 8.7193. It is deduced, from the reading on the mixture curve, a final Aw of the mixture equal to 0.445, with a confidence interval between 0.433 and 0.457 (i.e., only 0.024 units).

[00101] This Aw value is much higher than recommended and allows us to exclude the use of this quality of maltodextrin in this quantity in the mixture. b) Maltodextrin with 5% water

[00102] The same approach was followed, but using maltodextrin with 5% water, still in the same quantity. Petition 870250087588, dated 09 / 26 / 2025, pp. 73-85 19 / 26 Product % in the mixture MC in g / 100g of product MC in g / 100g of DM % of DM in g / 100g of product Active probiotic = Lactobacillus rhamnosus GG 20 0.1 0.1 / (100-0.1) X 100 = 0.1001 0.1 / 0.1001 X 100 = 99.9 Excipient = maltodextrin with 5% water GLUCIDEX 12D] 80 5 5 / (100-5) x 100 = 5.2631 5 / 5.2631 x 100 = 95 TOTAL 100 (20x0.1)+ (80x5) / 100 = 4.02 4.02 / (100-4.02) x 100 = 4.1884 4.02 / 4.1884 x 100 = 95.98 Table 2: Features of the implemented products

[00103] Figure 9 illustrates the correlation between moisture content expressed in g per 100 g of dry matter (DM) and relative hygrometry (P / P0) by modeling using LOESS regression.

[00104] The isotherm of the probiotic Lactobacillus rhamnosus GG (known or experimentally measured) is represented by a continuous line and does not change.

[00105] The isotherm of maltodextrin with 5% water (known or experimentally measured) is represented by large dots.

[00106] The mixture isotherm, resulting from modeling each of the isotherms using the LOESS method and weighting the isotherm of each ingredient according to the weight proportion of those same ingredients, is represented by a dashed line.

[00107] According to the calculations in Table 2, the MC (Moisture content) in g / 100g of DM (dry matter) is equal to 4.1884. It can be deduced, from the reading on the mixture curve, a final Aw of the mixture equal to 0.162, with a confidence interval between 0.150 and 0.173 (i.e., only 0.023 units).

[00108] To obtain the desired Aw value, there are therefore 2 options: - Increase the proportion of maltodextrin in the mixture; - Use a maltodextrin with a lower moisture content. c) Maltodextrin with 3.9% water Petition 870250087588, dated 09 / 26 / 2025, pp. 74 / 85 20 / 26

[00109] The same approach was followed, but using maltodextrin with 3.9% water, still in the same quantity:_________________________ Product % in the mixture MC in g / 100g of product MC in g / 100g of DM % of DM in g / 100g of product Active probiotic = Lactobacillus rhamnosus GG 20 0.1 0.1 / (100-0.1) X 100 = 0.1001 0.1 / 0.1001 X 100 = 99.9 Excipient = maltodextrin with 3.9% water GLUCIDEX 12D] 80 3.9 3.9 / (100-3.9) x 100 = 4.0583 3.9 / 4.0583 x 100 = 96.1 TOTAL 100 (20x0.1)+ (80x3.9) / 100 = 3.14 3.14 / (100-3.14) x 100 = 3.2418 3.14 / 3.2418 x 100 = 96.86 Table 3: Features of the implemented products

[00110] Figure 10 illustrates the correlation between moisture content expressed in g per 100g of dry material (DM) and the relative hygrometry (P / P0) by modeling using LOESS regression.

[00111] The isotherm of the probiotic Lactobacillus rhamnosus GG (known or experimentally measured) is represented by a continuous line and does not change.

[00112] The maltodextrin isotherm with 3.9% water (known or experimentally measured) is represented by large dots.

[00113] The mixture isotherm, resulting from modeling each of the isotherms using the LOESS method and weighing the isotherm of each ingredient according to the weight proportion of those ingredients, is represented by a dashed line.

[00114] According to the calculations in Table 3, the MC (Moisture content) in g / 100g of DM (Dry material) is equal to 3.242. From reading the mixture curve, a final Aw of the mixture equal to 0.089 is deduced, with a confidence interval between 0.072 and 0.106 (i.e., only 0.034 units). This value is compatible with the range of values ​​suitable for good probiotic stability. This quality of maltodextrin has therefore proven to be adequate. CONCLUSION: Petition 870250087588, dated 09 / 26 / 2025, pp. 75 / 85 21 / 26

[00115] The three examples demonstrate that the use of a LOESS-type local regression helps in choosing a product in a mixture of two products, for example, an excipient in a mixture consisting of an active ingredient and an excipient, in order to obtain an optimized water activity (Aw) in relation to said active ingredient. III / Use of a LOESS-type local regression to determine the relative proportion of compounds in a three-component system

[00116] The reliability of the LOESS model can, for example, be used to determine the relative amount of two excipients that can be used in combination with an active ingredient, without deleteriously altering the water activity (Aw) of said active ingredient.

[00117] Therefore, and in the particular case where the active ingredient is chondroitin sulfate (Mythocondro®; Gnosis Bioresearch), it is recommended, to ensure its stability, that a mixture with an Aw value of around 0.3 be used.

[00118] The objective, in this case, is to formulate chondroitin sulfate with a moisture content equal to 5% in a composition containing 10% by weight of said chondroitin.

[00119] The excipients used, which are low-cost and whose addition aims to optimize the formula from an economic point of view without sacrificing the stability of the active ingredient, are as follows: - Excipient 1: maltodextrin (Glucidex 12D) - Excipient 2: microcrystalline cellulose or CMC (MCC Vivapure 103).

[00120] It is therefore necessary to determine its relative proportion, knowing that it must represent 90% by weight of the formulation. a) Majority maltodextrin_____________________________________________ Product % in the mixture MC in g / 100g of product MC in g / 100g of DM % of DM in g / 100g of product Mythocondro® Chondroitin sulfate 10 5 5 / (100-5) x 100 = 5.2632 5 / 5.2632 x 100 = 95 Petition 870250087588, dated 09 / 26 / 2025, pp. 76 / 85 22 / 26 Excipient 1 = maltodextrin with 10% water GLUCIDEX 12D 80 10 10 / (100-10) x 100 = 11.1111 10 / 11.1111 x 100 = 90 Excipient 2 = microcrystalline cellulose (CM=C) MCC Vivapure 103] 10 0 0 100 TOTAL 100 (10x5) + (80x10) / 100 = 8.5 8.5 / (100-8.5) x 100 = 9.2896 8.5 / 9.2896 x 100 = 91.5 Table 4: Features of the implemented products

[00121] Figure 11 illustrates the correlation between moisture content expressed in g per 100g of dry material (DM) and relative hygrometry (P / PO) by modeling using LOESS regression.

[00122] The chondroitin sulfate isotherm (known or experimentally measured) is represented by light gray dots.

[00123] The isotherm of excipient 2, namely maltodextrin with 10% water (known or experimentally measured), is represented by dark gray dots.

[00124] The isotherm of excipient 1, namely microcrystalline cellulose or CMC (known or experimentally measured), is represented by a solid line.

[00125] The mixture isotherm, resulting from modeling each of the isotherms using the LOESS method and weighting each curve according to the weight proportion of each ingredient (10 / 80 / 10), is represented by a dashed line.

[00126] According to the calculations in Table 4, the MC (Moisture content) in g / 100g of DM (Dry material) is equal to 9.29. It can be deduced from the reading on the mixture curve that the final Aw of the mixture is equal to 0.522, with a confidence interval between 0.509 and 0.534. This results in an inexpensive formula, but with a very high water activity (Aw) to ensure the stability of the active ingredient. b) Balanced mixture of maltodextrin and microcrystalline cellulose

[00127] The same approach was followed, but balancing the Petition 870250087588, dated 09 / 26 / 2025, pp. 77 / 85 23 / 26 quantities of maltodextrin and microcrystalline cellulose. Product % in the mixture MC in g / 100g of product MC in g / 100g of DM % of DM in g / 100g of product Mythocondro® Chondroitin sulfate 10 5 5 / (100-5) x 100 = 5.2632 5 / 5.2632 x 100 = 95 Excipient 1 = maltodextrin with 10% water GLUCIDEX 12D 45 10 10 / (100-10) x 100 = 11.1111 10 / 11.1111 x 100 = 90 Excipient 2 = microcrystalline cellulose (CM=C) MCC Vivapure 103] 45 0 0 100 TOTAL 100 (10x5) + (45x10) / 100 = 5 5 / (100-5) x 100 = 5.2632 5 / 5.2632x 100 = 95 Table 5: Features of the implemented products

[00128] Figure 12 illustrates the correlation between moisture content expressed in g per 100g of dry material (DM) and relative hygrometry (P / PO) by modeling using LOESS regression.

[00129] The chondroitin sulfate isotherm (known or experimentally measured) is represented by light gray dots and does not undergo changes.

[00130] The isotherm of excipient 1, namely maltodextrin with 10% water (known or experimentally measured), is represented by dark gray dots and does not undergo changes.

[00131] The isotherm of excipient 2, namely microcrystalline cellulose or CMC (known or experimentally measured), is represented by a continuous line and does not change.

[00132] The mixture isotherm, resulting from modeling each of the isotherms using the LOESS method and weighting each curve according to the weight proportion of each ingredient (10 / 45 / 45), is represented by a dashed line.

[00133] According to the calculations in Table 5, the MC (Moisture content) in g / 100g of DM (Dry material) is equal to 5.26. From reading the mixture curve, a final Aw of the mixture equal to 0.350 can be deduced, with Petition 870250087588, dated 09 / 26 / 2025, pp. 78 / 85 24 / 26 a confidence interval between 0.342 and 0.357. This results in an improved formula, but with a water activity (Aw) that was still too high to guarantee the stability of the active ingredient. c) Adjusted formula

[00134] The previous formula was adjusted to within 1% to obtain the Aw target with very low uncertainty.

[00135] Ideal formula: Chondroitin sulfate: 10% Maltodextrin: 36% MCC: 54% Product % in the mixture MC in g / 100g of product MC in g / 100g of DM % of DM in g / 100g of product Mythocondro® Chondroitin sulfate 10 5 5 / (100-5) x 100 = 5.2632 5 / 5.2632 x 100 = 95 Excipient 1 = maltodextrin with 10% water GLUCIDEX 12D 36 10 10 / (100-10) x 100 = 11.1111 10 / 11.1111 x 100 = 90 Excipient 2 = microcrystalline cellulose (CM=C) MCC MCC Vivapure 103] 54 0 0 100 TOTAL 100 (10x5) + (36x10) / 100 = 4.1 4.1 / (100-4.1) x 100 = 4.2753 4.1 / 4.2753 x 100 = 95.9 Table 6: Features of the implemented products

[00136] Figure 13 illustrates the correlation between moisture content expressed in g per 100g of dry material (DM) and relative hygrometry (P / P0) by modeling using LOESS regression.

[00137] The chondroitin sulfate isotherm (known or experimentally measured) is represented by light gray dots and does not undergo changes.

[00138] The isotherm of excipient 1, namely maltodextrin with 10% water (known or experimentally measured), is represented by dark gray dots and does not undergo changes.

[00139] The isotherm of excipient 2, namely, microcrystalline cellulose Petition 870250087588, dated 09 / 26 / 2025, page 79 / 85 / 26 or CMC (known or experimentally measured), is represented by a continuous line and does not change.

[00140] The mixture isotherm, resulting from modeling each of the isotherms using the LOESS method and weighting each curve according to the weight proportion of each ingredient (10 / 36 / 54), is represented by a dashed line.

[00141] According to the calculations in Table 6, the MC (Moisture content) in g / 100g of DM (Dry material) is equal to 4.28. It is deduced, from the reading on the mixture curve, a final Aw of the mixture equal to 0.299, with a confidence interval between 0.293 and 0.305. Thus, a formula is obtained that guarantees the stability of the active ingredient, while having a maximum amount of excipient 1 (maltodextrin with 10% water). CONCLUSION:

[00142] The three examples demonstrate that the use of a LOESS-type local regression helps in choosing a proportion of products in a mixture of three products, for example two excipients in a mixture composed of an active ingredient and two excipients, in order to obtain an optimized water activity (Aw) with respect to the active ingredient and an optimized cost. IV / Combination of a non-parametric regression (of the LOESS, Compartment Smoother or MARS type) and a parametric regression (GAB) in a three-component system.

[00143] A mixture comprising 3 ingredients, namely, vitamin K2 (Tastetech) at 5%, a yeast fraction (LYNSIDE® basic wall) at 25% and dry potato starch (Extra dry potato starch) at 70%, was studied under the following conditions: - The behavior of the yeast fraction and dry potato starch was modeled using the GAB model; The behavior of vitamin K2 was modeled using Petition 870250087588, dated 09 / 26 / 2025, pages 80 / 85 / 26 LOESS, compartment smoother and MARS regression, respectively, since parametric models such as GAB do not allow for correctly modeling the observed drop in moisture content towards 65% P / P0; The behavior of the mixture is constructed based on the predictions of the models applied to the mixture's ingredients.

[00144] Figure 14 shows that the 3 curves related to the mixture can be superimposed. CONCLUSION:

[00145] Figure 14 shows that in a three-component system it is possible to use a non-parametric regression only for the ingredient whose modeling is not satisfactory using a parametric regression, in this case, vitamin K2 (see example I-1). Furthermore, it demonstrates that other non-parametric regressions, such as Compartment Smoother and MARS, can be implemented instead of LOESS regression. Petition 870250087588, dated 09 / 26 / 2025, pp. 81 / 85

Claims

1 / 3 CLAIMS 1. A method for determining the water activity (Aw) of a powder sample, characterized in that it comprises: - establishing the sorption isotherm of the powder using a method comprising: - obtaining from a set of points corresponding to the moisture content (MC) of the powder according to the relative hygrometry (P / P0); - applying a non-parametric regression on these points to establish the sorption isotherm of the powder; - measuring the moisture content of the sample; - determining the water activity of the sample using the measured moisture content of the sample and the established powder sorption isotherm model.

2. Method for establishing the sorption isotherm of a solid composition, characterized in that it comprises at least two compounds, advantageously in the form of a powder mixture, comprising: - obtaining the sorption isotherms of each of the compounds, the isotherm of at least one of the compounds being established using a method comprising: - obtaining from a set of points corresponding to the moisture content (MC) of the powder according to the relative hygrometry (P / P0); - applying a non-parametric regression on these points to establish the sorption isotherm of the powder; - establishing the sorption isotherm of the composition according to the weight ratio of each compound in the composition.

3. Method according to claim 2, characterized in that the isotherm of all compounds is established using the method comprising: Petition 870250087588, dated 09 / 26 / 2025, page 82 / 85 2 / 3 - obtaining from a set of points corresponding to the moisture content (MC) of the powder according to relative hygrometry (P / P0); - applying a non-parametric regression on these points to establish the sorption isotherm of the powder.

4. Method for determining the water activity (Aw) of a powder mixture, characterized in that it comprises: - establishing the sorption isotherm of the mixture using the method as defined in claim 2 or 3; - measuring the moisture content of the mixture or calculating the moisture content of the mixture using the moisture content of each of the compounds, measured according to the weight ratio of each compound in the mixture; - determining the water activity of the mixture according to the moisture content of the sample and the model of the sorption isotherm of the mixture using the measured or calculated moisture content of the mixture and the model of the established sorption isotherm of the mixture.

5. Method according to any one of claims 1 to 4, characterized in that the non-parametric regression is chosen from the following group: LOESS regression (Locally Estimated Scatter Plot Smoothing); regressogram (Compartment Smoother); moving average; weighted moving average (Nadaraya and Watson regression); Kernel regression; multivariate adaptive regression (MARS).

6. Method according to claim 5, characterized in that the non-parametric regression is a LOESS regression.

7. Method according to claim 5 or 6, characterized in that the LOESS regression is configured such that: - only the two points in the vicinity of a given point are considered; and / or - the polynomial is of degree 2.

8. Method according to any of claims 1 to Petition 870250087588, dated 09 / 26 / 2025, page 83 / 85 3 / 3 7, characterized in that the sorption isotherm is based on at least ten moisture content (MC) measurement points according to relative hygrometry (P / P0).

9. A method according to any one of claims 2 to 7, characterized in that the composition or mixture is intended for the development of a food supplement or a medicine.

10. Use of the method as defined in claim 9, characterized by being for determining the nature and / or weight content of a compound, in addition to the active compound, in order to obtain a mixture that has a water activity compatible with that of the active compound. Petition 870250087588, dated 09 / 26 / 2025, pp. 84 / 85

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