Moscow, Russian Federation
Moscow, Russian Federation
Moscow, Russian Federation
Moscow, Russian Federation
Moscow, Russian Federation
Moscow, Russian Federation
Moscow, Russian Federation
The dairy industry needs new methods for food quality assessment at all stages of production and storage. Mathematical methods offer efficient tools of product quality control. This article introduces a mathematical approach to assessing the kinetics of protein and carbohydrate profiles of milk powder to predict their transformations throughout production and storage. The study featured milk at different processing temperatures and exposure times. The analysis focused on the concentrations of compounds that mark the changes in the carbohydrate and protein profiles of milk during heat treatment. The equation of pseudo-zero-order and the Arrhenius equation were applied to carbohydrates while a first-order equation made it possible to assess the changes in proteins. The calculation algorithm was processed in Python. The lactulose, hydroxymethylfurfural, and furosine accumulations proved to depend on the integral heat load. The activation energies that provided the linearity maximum were as follows: 130 kJ/mol for lactulose and furosine; 85 kJ/mol for hydroxymethylfurfural. The calculation model failed to assess the accumulation of carbohydrate profile products during prolonged thermal exposure due to more profound product changes. β-lactoglobulin appeared to be more susceptible to thermal effects than α-lactalbumin. The activation energies for whey proteins were 72.1 and 85.1–85.7 kJ/mol, respectively. The new mathematical approach provided a reliable quantitative assessment of thermal stress that can be used for food quality control and shelf-life adjustments, e.g., in milk powder production.
Mathematical modeling, milk quality, milk powder, heat treatment markers
1. Rosa E, Prudencio ES. A comprehensive approach about comparison between drying technologies and powdered dairy products. Food Research International. 2023;173(Part 1):113326. https://doi.org/10.1016/j.foodres.2023.113326
2. Kourkouta L, Frantzana A, Koukourikos K, Iliadis C, Papathanasiou IV, et al. Milk nutritional composition and its role in human health. Journal of Pharmacy and Pharmacology. 2021;9(1):10–15. https://doi.org/10.17265/2328-2150/2021.01.002
3. McCarthy NA, Magan JB, Kelleher CM, Kelly AL, O’Mahony JA, et al. Heat treatment of milk: Effect on concentrate viscosity, powder manufacture and end-product functionality. International Dairy Journal. 2022;128:105289. https://doi.org/10.1016/j.idairyj.2021.105289
4. Song X, Shen S, Dong G, Ding H, Xie Z, et al. A comprehensive review of direct, indirect, and AI-based detection methods for milk powder. Frontiers in Sustainable Food Systems. 2025;9:1571317. https://doi.org/10.3389/fsufs.2025.1571317
5. Ryabova AE, Semipyatnyi VK, Galstyan AG. Effects of storage conditions on milk powder properties. Journal of Dairy Science. 2023;106(10):6741–6758. https://doi.org/10.3168/jds.2022-23094
6. Ding Y, Han F, Xie Z, Li G, Zhuang Y, et al. Dairy fortification as a good option for dietary nutrition status improvement of 676 preschool children in China: A simulation study based on a cross-sectional diet survey (2018–2019). Frontiers in Nutrition. 2022;9:1081495. https://doi.org/10.3389/fnut.2022.1081495
7. Hamed AM, Galli B, Hogan SA, Abdel-Hamid M, Romeih E. Adaptive and predictive approaches to mitigate the impact of milk seasonality on composition, processing technologies and quality of milk powders. International Journal of Dairy Technology. 2025;78(1):e13148. https://doi.org/10.1111/1471-0307.13148
8. Qin Y, Pillidge C, Harrison B, Adhikari B. Pathways in formulating foods for the elderly. Food Research International. 2024;186:114324. https://doi.org/10.1016/j.foodres.2024.114324
9. Zanini B, Simonetto A, Zubani M, Castellano M, Gilioli G. The effects of cow-milk protein supplementation in elderly population: Systematic review and narrative synthesis. Nutrients. 2020;12(9):2548. https://doi.org/10.3390/nu12092548
10. Schons PF, Colet R, Pinto SS, Verruck S. Powdered milk. In: da Cruz AG, Pimentel TC, Esmerino EA, Verruck S, editors. Dairy Foods Processing. NY: Springer US; 2024. pp. 179–192. https://doi.org/10.1007/978-1-0716-4144-6_13
11. Ryabova AE. Influence of storage conditions on the quality of powder milk. Dairy industry. 2021;(7):56–57. (In Russ.) https://doi.org/10.31515/1019-8946-2021-07-56-57
12. Duan J, Gao S, Diao Y, Zhang Q, Gu Y, et al. Effects of heat treatment on liquid milk quality and safety: Nutritional changes, hazardous substances and control strategies. Food Control. 2025;176:111377. https://doi.org/10.1016/j.foodcont.2025.111377
13. Riazantseva KA, Sherstneva NE. Traditional and innovative uses of ultraviolet treatment in the dairy industry. Food Processing: Techniques and Technology. 2022;52(2):390–406. (In Russ.) https://doi.org/10.21603/2074-9414-2022-2-2372
14. Bista A, Murphy EG, O’Donnell CP, O’Shea N. The effect of heat treatment on physicochemical properties of skim milk concentrate and spray-dried skim milk powder. International Journal of Dairy Technology. 2022;75(3):690–700. https://doi.org/10.1111/1471-0307.12876
15. Alkadour MI, Pryanichnikova NS, Yurova EA, Petrov AN. Effect of thermal treatment and pasteurization on milk powder quality. Food Processing: Techniques and Technology. 2024;54(2):275–284. (In Russ.) https://doi.org/10.21603/2074-9414-2024-2-2506
16. Alkadour MI, Petrov AN, Pryanichnikova NS. Influence of the class of thermal treatment of dry milk on the structure and properties of the fermented curd. Food industry. 2024;(11):82–89. (In Russ.) htt ps://doi.org/10.52653/PPI.2024.11.11.015
17. Kalugina DN, Yurova EA. Rationale for determining the nitrogen index of whey protein to assess the protein composition of milk powder. Dairy industry. 2022;(7):35–37. (In Russ.) https://doi.org/10.31515/1019-8946-2022-07-35-37
18. Meldenberg DN, Polyakova OS, Semenova ES, Yurova EA. Development of a comprehensive milk protein composition assessment from raw materials of various farm animals for the functional products production. Storage and Processing of Farm Products. 2020;(3):118–133. (In Russ.) https://doi.org/10.36107/spfp.2020.352
19. Paul A, Martin F, Simard B, Scher J, Gaiani C, et al. Deciphering the segregation of proteins in high-protein dairy powders after spray-drying. Journal of Dairy Science. 2023;106(2):843–851. https://doi.org/10.3168/jds.2022-22133
20. Ryabova AE, Kondratenko VV. Modeling of the self-pressing process of powdered milk during storage. Food industry. 2024;(9):65–69 (In Russ.) https://doi.org/10.52653/PPI.2024.9.9.013
21. Fan X, Wang C, Cheng M, Wei H, Gao X, et al. Markers and mechanisms of deterioration reactions in dairy products. Food Engineering Reviews. 2023;15:230–241. https://doi.org/10.1007/s12393-023-09331-9
22. Mahmood N, Muhoza B, Huang Y, Munir Z, Zhang Y, et al. Effects of emerging food pretreatment and drying techniques on protein structures, functional and nutritional properties: An updated review. Critical Reviews in Food Science and Nutrition. 2024;64(26):9365–9381. https://doi.org/10.1080/10408398.2023.2212302
23. Rauh V, Xiao Y. The shelf life of heat-treated dairy products. International Dairy Journal. 2022;125:105235. https://doi.org/10.1016/j.idairyj.2021.105235
24. Clarke HJ, McCarthy WP, O’Sullivan MG, Kerry JP, Kilcawley KN. Oxidative quality of dairy powders: Influencing factors and analysis. Foods. 2021;10(10):2315. https://doi.org/10.3390/foods10102315
25. Lu J, Zhu T, Dai Y, Xing L, Jinqi L, et al. The effect of heat treatment on the lactosylation of milk proteins. Journal of Dairy Science. 2023;106(12):8321–8330. https://doi.org/10.3168/jds.2023-23526
26. Li Y, Quan W, Jia X, He Z, Wang Z, et al. Profiles of initial, intermediate, and advanced stages of harmful Maillard reaction products in whole-milk powders pre-treated with different heat loads during 18 months of storage. Food Chemistry. 2021;351:129361. https://doi.org/10.1016/j.foodchem.2021.129361
27. Dong L, Yu Z, Zhao R, Peng B, Zhang Y, et al. The effect of lactulose thermal degradation products on β-lactoglobulin: Linear-, loop-, and cross-link structural modifications and reduced digestibility. Food Chemistry. 2023;403:134333. https://doi.org/10.1016/j.foodchem.2022.134333
28. Claeys WL, Ludikhuyze LR, Hendrickx ME. Formation kinetics of hydroxymethylfurfural, lactulose and furosine in milk heated under isothermal and non-isothermal conditions. Journal of Dairy Research. 2001;68(2):287–301. https://doi.org/10.1017/s0022029901004745
29. Andrews GR. Determining the energy of activation for the formation of lactulose in heated milks. Journal of Dairy Research. 1985;52(2):275–280. https://doi.org/10.1017/S0022029900024146
30. Wang Y, Xiao R, Liu S, Wang P, Zhu Y, et al. The impact of thermal treatment intensity on proteins, fatty acids, macro/micro-nutrients, flavor, and heating markers of milk—A comprehensive review. International Journal of Molecular Sciences. 2024;25(16):8670. https://doi.org/10.3390/ijms25168670
31. Abdulhameed AF, Memon QA. An improved trapezoidal rule for numerical integration. Journal of Physics: Conference Series. 2021;2090(1):012104. https://doi.org/10.1088/1742-6596/2090/1/012104
32. Kessler HG. Food and bio process engineering. In: Kessler VA. Dairy Technology. 2002. 694 p.
33. Li M, Shen M, Lu J, Yang J, Huang Y, et al. Maillard reaction harmful products in dairy products: Formation, occurrence, analysis, and mitigation strategies. Food Research International. 2022;151:110839. https://elibrary.ru/UGUAWT
34. Calvo MM, Olano A. Formation of galactose during heat treatment of milk and model systems. Journal of Dairy Research. 1989;56(5):737–740. https://doi.org/10.1017/S0022029900029307
35. Lan XY, Wang JQ, Bu DP, Shen JS, Zheng N, et al. Effects of heating temperatures and addition of reconstituted milk on the heat indicators in milk. Journal of Food Science. 2010;75(8):C653–C658. https://doi.org/10.1111/j.1750-3841.2010.01802.x
36. Marconi E, Messia MC, Amine A, Moscone D, Vernazza F, et al. Heat-treated milk differentiation by a sensitive lactulose assay. Food Chemistry. 2004;84(3):447–450. https://doi.org/10.1016/s0308-8146(03)00268-1
37. Olano A, Calvo MM. Kinetics of lactulose, galactose and epilactose formation during heat-treatment of milk. Food Chemistry. 1989;34(4):239–248. https://doi.org/10.1016/0308-8146(89)90101-5
38. Topnikova EV, Myagkonosov DS, Abramov DV, Kashnikova OG. Colorimetric method for estimating the intensity of heat load during milk pasteurization. Food Systems. 2024;7(3):481–490. (In Russ.) ht tps://doi.org/10.21323/2618-9771-2024-7-3-481-490
39. Chen Z, Yan X. Simultaneous determination of melamine and 5-hydroxymethylfurfural in milk by capillary electrophoresis with diode array detection. Journal of Agricultural and Food Chemistry. 2009;57(19):8742–8747. https://doi.org/10.1021/jf9021916
40. Morales FJ, Jiménez-Pérez S. HMF formation during heat-treatment of milk-type products as related to milkfat content. Journal of Food Science. 1999;64(5):855–859. https://doi.org/10.1111/j.1365-2621.199+9.tb15927.x
41. Morales FJ, Romero C, Jiménez-Pérez S. Characterization of industrial processed milk by analysis of heat-induced changes. International Journal of Food Science and Technology. 2000;35(2):193–200. https://doi.org/10.1046/j.1365-2621.2000.00334.x
42. Morales FJ, Romero C, Jiménez-Pérez S. Chromatographic determination of bound hydroxymethylfurfural as an index of milk protein glycosylation. Journal of Agricultural and Food Chemistry. 1997;45(5):1570–1573. https://doi.org/10.1021/jf960930v
43. Morales FJ, Romero C, Jiménez-Pérez S. Evaluation of heat-induced changes in Spanish commercial milk: Hydroxymethylfurfural and available lysine content. International Journal of Food Science and Technology. 2003;31(5):411–418. https://doi.org/10.1046/j.1365-2621.1996.00357.x
44. Xing Q, Fu X, Liu Z, Cao Q, You C, et al. Contents and evolution of potential furfural compounds in milk-based formula, ultra-high temperature milk and pasteurised yoghurt. International Dairy Journal. 2021;120:105086. https://doi.org/10.1016/j.idairyj.2021.105086
45. Birlouez-Aragon I, Sabat P, Gouti N. A new method for discriminating milk heat treatment. International Dairy Journal. 2002;12(1):59–67. https://doi.org/10.1016/S0958-6946(01)00131-5
46. Sakkas L, Moutafi A, Moschopoulou E, Moatsou G. Assessment of heat treatment of various types of milk. Food Chemistry. 2014;159:293–301. https://doi.org/10.1016/j.foodchem.2014.03.020
47. Alkadour MI, Illarionova EE. Effect of heat treatment class of dry milk on protein transfer to whey. Food industry. 2025;(3):9 6–99. (In Russ.) https://doi.org/10.52653/PPI.2025.3.3.018
48. Ryazantseva KA, Sherstneva NE, Agarkova EYu, Zhizhin NA. Influence of ultraviolet modified whey proteins on the stability of the stability of the stabilized milk cloud. Izvestiya vuzov. Food technology. 2023;392(2–3):83–92. (In Russ.) https://doi.org/10.26297/0579-3009.2023.2-3.14
49. Kalugina DN, Yurova EA. The protein composition characteristics in the formation of ultra pasteurized milk shelf life. Bulletin of KSAU. 2021;(10):165–172. (In Russ.) https://doi.org/10.36718/1819-4036-2021-10-165-172
50. Wu J, Chen S, Van der Meeren P. Heat stability assessment of milk: A review of traditional and innovative methods. Foods. 2024;13(14):2236. https://doi.org/10.3390/foods13142236
51. Wu Y, Wu S, Sun M, Nie L, Zhang Y, et al. Reduction of the levels of 5-hydroxymethylfurfural and advanced glycation end products in milk by the combination of high pressure and moderate heat pre-incubation. European Food Research and Technology. 2023;249:923–937. https://doi.org/10.1007/s00217-022-04184-8
52. Zhang Y, Yi S, Lu J, Pang X, Xu X, et al. Effect of different heat treatments on the Maillard reaction products, volatile compounds and glycation level of milk. International Dairy Journal. 2021;123:105182. https://doi.org/10.1016/j.idairyj.2021.105182
53. Bolshakova EI, Strizhko MN. Influence of the protein profile on the technological properties of canned milk with sugar. Izvestiya vuzov. Food technology. 2023;391(1):107–113. (In Russ.) https://doi.org/10 .26297/0579-3009.2023.1.17
54. Mayer HK, Raba B, Meier J, Schmid A. RP-HPLC analysis of furosine and acid-soluble β-lactoglobulin to assess the heat load of extended shelf life milk samples in Austria. Dairy Science & Technology. 2010;90(4):413–428. https://doi.org/10.1051/dst/2009058
55. Gralevskaya IV, Ionova LV, Havrov IV, Barsukova LS. Monitoring of technological characteristics of soft acid-rennet cheese. Food Processing: Techniques and Technology. 2012;25(2). (In Russ.)
56. Riazantseva KA, Sherstneva NE. Approaches to increasing the heat stability of whey proteins. Food industry. 2021;(8):25–28. (In Russ.) https://doi.org/10.52653/PPI.2021.8.8.006
57. Čurlej J, Zajác P, Čapla J, Golian J, Benešová L, et al. The effect of heat treatment on cow’s milk protein profiles. Foods. 2022;11(7):1023. https://doi.org/10.3390/foods11071023
