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Mathematical models of the behavior of fine particles in gaseous environments

ISSN 2223-6775 Ukrainian journal of occupational health Vol.19, No 3, 2023

Mathematical models of the behavior of fine particles in gaseous environments

Kashuba M.O., Melnyk N.A.
I. Horbachevsky Ternopil National Medical, Ternopil, Ukraine

Full article (PDF): ENG / UKR

Introduction. In contemporary hygienic science, a paramount focus is the examination of nanotechnology's impact on human health. Particularly concerning are nanoparticles of anthropogenic origin, for which living organisms lack adaptive mechanisms. These nanoparticles constitute a novel and potentially hazardous factor increasingly widespread in the environment, posing potential risks to human well-being. Consequently, the development of methods for calculating key parameters to reduce aerosol concentrations to permissible levels is of great importance.

Objective. This study aims to construct and substantiate a mathematical model capable of calculating the time required to reduce aerosol concentrations to safe levels, specifically the maximum permissible concentration for the given aerosol, in occupational settings. This calculation will be based on parameters such as aerosol concentration, dispersed composition, physical properties of the substance, and characteristics of the surrounding air.

Materials and methods. Mathematical modeling of nanoparticle and microparticle behavior in gaseous environments was conducted using software applications within SPSS 16, Statistica 12.0, and EXCEL.

Results. This article introduces novel mathematical models that facilitate the description of aggregation and sedimentation processes of nanoparticles and microparticles of varying sizes in given space-time coordinates, depending on their concentration, dispersed composition, and other physical attributes of both the particles and the surrounding air.

Conclusions. In conclusion, our study has yielded mathematical models capable of quantitatively elucidating the processes of aggregation and sedimentation of particles with diverse size distributions. These models enable the calculation of the time required for concentrations of nano- and microparticles, varying in both quantity and weight dispersion, to reach specified levels at predefined spatial coordinates, frequently within an individual worker's breathing zone.

Keywords: mathematical model; fine particles; nanoparticles; aggregation; sedimentation.


  1. Grin Kh, Lein B.[Aerosols – dust, smoke and mists]. Leninhrad; 1972. 420 p. Russian.
  2. Craig M, Jenner AL, Namgung B, Lee LP, Goldman A. Engineering in Medicine To Address the Challenge of Cancer Drug Resistance: From Micro- and Nanotechnologies to Computational and Mathematical Modeling. Chemical Reviews. 2021;121(6):3352-89. DOI:
  3. Shulenburg M. [Nanoparticles are tiny particles with enormous potential. Opportunities and risks]. Bonn, Berlin: Federal Ministry of Education and Research (BMBF), Department "Nanomaterials; New substances"; 2008. Russian.
  4. Grattoni A, Cooke JP. Emerging nanotechnologies in cardiovascular medicine. Nanomedicine. 2022;39:102472. DOI:
  5. Kouzov PA. [Fundamentals of analysis of the dispersed composition of dusts and crushed materials]. Leninhrad: Khimiia; 1987. 264 p. Russian.
  6. Angeli E, Bousquet GA. Boom in Nanotechnologies for a High Level of Precision Medicine. Cancers (Basel). 2023;15(9):2522. DOI:
  7. Contera S, Bernardino de la Serna J, Tetley TD. Biotechnology, nanotechnology and medicine. Emerging Topics in Life Sciences. 2020;4(6):551-4. DOI:
  8. Grattoni A, Han A. Biomedical microdevices: the next phase of highlighting scientific discoveries in the field of micro-nanotechnologies for medicine. Biomedical microdevices. 2021;24(1):1. DOI:
  9. Nile SH, Baskar V, Selvaraj D, Nile A, Xiao J, Kai G. Nanotechnologies in Food Science: Applications, Recent Trends, and Future Perspectives. Nano-Micro Letters. 2020;12(1):45. DOI:
  10. Kubinová S, Syková E. Nanotechnologies in regenerative medicine. Minimally Invasive Therapy & Allied Technologies. 2020;19(3):144-56. DOI: