Mathematical Modeling of Bread Baking: A Bibliometric Review of PDE-Based Multiphysics Approaches
Abstract
This study examines how partial differential equations serve as the mathematical foundation for modeling the coupled heat and mass transfer processes occurring during bread baking. Through a systematic bibliometric approach, data from major scientific databases were analyzed using mapping techniques to identify dominant themes, research clusters, and developmental trends in mathematical modeling for bread baking. The findings indicate that heat transfer, moisture migration, crust formation, and dough thermomechanics constitute the core of this field, with diffusion and convection–diffusion equations forming the principal modeling framework. Network, overlay, and density visualizations reveal a growing adoption of advanced numerical schemes and multiphysics simulations to predict temperature and moisture distributions within the bread matrix. This study underscores the essential role of mathematics in enhancing process understanding and modeling accuracy, and highlights future research opportunities that integrate computational approaches into food engineering.
References
Acevedo, C., Allendes, A., Campaña, G., Jaques, A., Naranjo, C., & Sepúlveda, G. (2025). Fick and Fokker–Planck problems: A finite element comparison. Physics of Fluids, 37(9). https://doi.org/10.1063/5.0280039
Afriansyah, D., Perdhana, F. F., Zahra, K., & Adriani, I. R. (2024). Tren, Inovasi, dan Keberlanjutan dalam Mathematical Modelling untuk Food Science: Analisis Bibliometrik 2014–2024. Mandalika Mathematics and Educations Journal, 6(2), 639–652. https://doi.org/10.29303/jm.v6i2.8078
Ajani, C. K., Zhu, Z., & Sun, D.-W. (2023). Microscale Modelling of Flow, Heat and Mass Transport During Vacuum Cooling of Porous Foods: Effective Property Computation. Transport in Porous Media, 148(3), 433–458. https://doi.org/10.1007/s11242-023-01942-4
Alpers, T., Becker, T., & Jekle, M. (2023). Strain-dependent assessment of dough’s polymer structure and functionality during the baking process. PLOS ONE, 18(3), e0282670. https://doi.org/10.1371/journal.pone.0282670
Amini, D., Haghighat, E., & Juanes, R. (2022). Physics-Informed Neural Network Solution of Thermo–Hydro–Mechanical Processes in Porous Media. Journal of Engineering Mechanics, 148(11). https://doi.org/10.1061/(ASCE)EM.1943-7889.0002156
Baas, J., Schotten, M., Plume, A., Côté, G., & Karimi, R. (2020). Scopus as a curated, high-quality bibliometric data source for academic research in quantitative science studies. Quantitative Science Studies, 1(1), 377–386. https://doi.org/10.1162/qss_a_00019
Barzegari, M., & Geris, L. (2022). Highly scalable numerical simulation of coupled reaction–Diffusion systems with moving interfaces. The International Journal of High Performance Computing Applications, 36(2), 198–213. https://doi.org/10.1177/10943420211045939
Brun, M. K., Ahmed, E., Berre, I., Nordbotten, J. M., & Radu, F. A. (2020). Monolithic and splitting solution schemes for fully coupled quasi-static thermo-poroelasticity with nonlinear convective transport. Computers & Mathematics with Applications, 80(8), 1964–1984. https://doi.org/10.1016/j.camwa.2020.08.022
Carvalho, O., Charalambides, M. N., Djekić, I., Athanassiou, C., Bakalis, S., Benedito, J., Briffaz, A., Castañé, C., Della Valle, G., de Sousa, I. M. N., Erdogdu, F., Feyissa, A. H., Kavallieratos, N. G., Koulouris, A., Pojić, M., Raymundo, A., Riudavets, J., Sarghini, F., Trematerra, P., & Tonda, A. (2021). Modelling Processes and Products in the Cereal Chain. Foods, 10(1), 82. https://doi.org/10.3390/foods10010082
Chakraborty, S., & Dash, K. K. (2023). A comprehensive review on heat and mass transfer simulation and measurement module during the baking process. Applied Food Research, 3(1), 100270. https://doi.org/10.1016/j.afres.2023.100270
Chakraborty, S., Routray, W., & Dash, K. K. (2022). Numerical Study of Baking. In Advanced Computational Techniques for Heat and Mass Transfer in Food Processing (pp. 247–274). CRC Press. https://doi.org/10.1201/9781003159520-12
Çi̇loğlu, P., & Yücel, H. (2023). Stochastic discontinuous Galerkin methods for robust deterministic control of convection-diffusion equations with uncertain coefficients. Advances in Computational Mathematics, 49(2), 16. https://doi.org/10.1007/s10444-023-10015-5
Cuomo, S., Di Cola, V. S., Giampaolo, F., Rozza, G., Raissi, M., & Piccialli, F. (2022). Scientific Machine Learning Through Physics–Informed Neural Networks: Where we are and What’s Next. Journal of Scientific Computing, 92(3), 88. https://doi.org/10.1007/s10915-022-01939-z
D’Autilia, M. C., Sgura, I., & Simoncini, V. (2020). Matrix-oriented discretization methods for reaction–diffusion PDEs: Comparisons and applications. Computers & Mathematics with Applications, 79(7), 2067–2085. https://doi.org/10.1016/j.camwa.2019.10.020
Davidson, I. (2023). Baking process. In Biscuit Baking Technology (pp. 45–56). Elsevier. https://doi.org/10.1016/B978-0-323-99923-6.00026-0
Erdogdu, F. (2023). Mathematical modeling of food thermal processing: current and future challenges. Current Opinion in Food Science, 51, 101042. https://doi.org/10.1016/j.cofs.2023.101042
Janssen, F., Wouters, A. G. B., & Delcour, J. A. (2021). Gas cell stabilization by aqueous‐phase constituents during bread production from wheat and rye dough and oat batter: Dough or batter liquor as model system. Comprehensive Reviews in Food Science and Food Safety, 20(4), 3881–3917. https://doi.org/10.1111/1541-4337.12761
Julius, R., Abd Halim, M. S., Abdul Hadi, N., Alias, A. N., Mohd Khalid, M. H., Mahfodz, Z., & Ramli, F. F. (2021). Bibliometric Analysis of Research in Mathematics Education using Scopus Database. Eurasia Journal of Mathematics, Science and Technology Education, 17(12), em2040. https://doi.org/10.29333/ejmste/11329
K, P. K. (2024). An Analytical Approach to Solving Partial Differential Equations in Mathematical Modeling. International Journal for Research in Applied Science and Engineering Technology, 12(12), 1651–1661. https://doi.org/10.22214/ijraset.2024.66100
Li, P., Chen, L., & Atroshchenko, E. (2023). Editorial: Moving boundary problems in multi-physics coupling processes. Frontiers in Physics, 11. https://doi.org/10.3389/fphy.2023.1219806
Liu, R., & Gerstoft, P. (2023). SD-PINN: Physics Informed Neural Networks for Spatially Dependent PDES. ICASSP 2023 - 2023 IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP), 1–5. https://doi.org/10.1109/ICASSP49357.2023.10095076
Lü, Q., & Zhang, X. (2024). Inverse problems for stochastic partial differential equations: Some progresses and open problems. Numerical Algebra, Control and Optimization, 14(2), 227–272. https://doi.org/10.3934/naco.2023014
Mansour, Y., Rahmé, P., El Hajj, N., & Rouaud, O. (2025). Thermal and Expansion Analysis of the Lebanese Flatbread Baking Process Using a High-Temperature Tunnel Oven. Applied Sciences, 15(15), 8611. https://doi.org/10.3390/app15158611
Mosalam, H. (2021). Digital Modeling of Heat Transfer during the Baking Process. Modelling and Simulation in Engineering, 2021, 1–8. https://doi.org/10.1155/2021/8957148
Parlina, A., Ramli, K., & Murfi, H. (2020). Theme Mapping and Bibliometrics Analysis of One Decade of Big Data Research in the Scopus Database. Information, 11(2), 69. https://doi.org/10.3390/info11020069
Petcu, M. A., Ionescu-Feleaga, L., Ionescu, B.-Ștefan, & Moise, D.-F. (2023). A Decade for the Mathematics: Bibliometric Analysis of Mathematical Modeling in Economics, Ecology, and Environment. Mathematics, 11(2), 365. https://doi.org/10.3390/math11020365
Priyadarshini, S. R., Reshmi, S. K., Moses, J. A., & Anandharamakrishnan, C. (2024). Computational Modeling of Food Processing Operations. In Emerging Technologies for the Food Industry (pp. 1–30). Apple Academic Press. https://doi.org/10.1201/9781003413660-1
Purlis, E., Cevoli, C., & Fabbri, A. (2021). Modelling Volume Change and Deformation in Food Products/Processes: An Overview. Foods, 10(4), 778. https://doi.org/10.3390/foods10040778
Quaghebeur, W., Nopens, I., & De Baets, B. (2021). Incorporating Unmodeled Dynamics Into First-Principles Models Through Machine Learning. IEEE Access, 9, 22014–22022. https://doi.org/10.1109/ACCESS.2021.3055353
Rosokhata, A., Minchenko, M., Khomenko, L., & Chygryn, O. (2021). Renewable energy: a bibliometric analysis. E3S Web of Conferences, 250, 03002. https://doi.org/10.1051/e3sconf/202125003002
Sajovic, I., & Boh Podgornik, B. (2022). Bibliometric Analysis of Visualizations in Computer Graphics: A Study. Sage Open, 12(1). https://doi.org/10.1177/21582440211071105
SOKMEN, O., SÖKMEN GÜRÇAM, Ö., DÜNDAR, A. N., & SARICAOĞLU, F. T. (2022). What have been highly pointed in bread and sourdough researches: using bibliometric analysis as a powerful tool. European Food Science and Engineering, 3(1), 36–43. https://doi.org/10.55147/efse.1121959
Solorzano, G., & Plevris, V. (2022). Computational intelligence methods in simulation and modeling of structures: A state-of-the-art review using bibliometric maps. Frontiers in Built Environment, 8. https://doi.org/10.3389/fbuil.2022.1049616
Szpicer, A., Bińkowska, W., Stelmasiak, A., Zalewska, M., Wojtasik-Kalinowska, I., Piwowarski, K., Piepiórka-Stepuk, J., & Półtorak, A. (2025). Computational Fluid Dynamics Simulation of Thermal Processes in Food Technology and Their Applications in the Food Industry. Applied Sciences, 15(1), 424. https://doi.org/10.3390/app15010424
Therdthai, N. (2021). Modeling and optimization of food processes. In Engineering Principles of Unit Operations in Food Processing (pp. 419–441). Elsevier. https://doi.org/10.1016/B978-0-12-818473-8.00004-9
