Statistical Approach to Describing the Processes of Transport of Lipoproteins and Other Components in Blood Vessels – I

To describe atherosclerotic processes in the intima of blood vessels, a statistical approach to describing non-equilibrium processes of blood component transport in the lumen-endothelium-intima system of blood vessels has been proposed, which involves taking into account the nature of interactions between blood components.  Using the method of non-equilibrium statistical operator for the parameters of the abbreviated description, a system of transport equations has been obtained, which, within the framework of the selected model of component interaction, can describe non-Markovian in time and non-local in space processes of blood transport in vessels, taking into account possible reaction-diffusion processes in the vessel walls.  A viscous reaction-diffusion description was used in the lumen, and a reaction-diffusion description in the endothelium-intima subsystem.

Atherosclerosis
blood vessels
erythrocytes
lipoproteins
low-density lipoproteins
diffusion processes of ions and water molecules
non-equilibrium statistical operator
рівняння Максвелла
peroxide free radicals
nitric oxide
macrophages
  1. Boggs S., Huang L., Stuehr D. J.  Formation and Reactions of the Heme-Dioxygen Intermediate in the First and Second Steps of Nitric Oxide Synthesis As Studied by Stopped-Flow Spectroscopy under Single-Turnover Conditions.  Biochemistry.  39 (9), 2332–2339 (2000).
  2. Davignon J., Ganz P.  Role of endothelial dysfunction in atherosclerosis.  Circulation. 109 (23_suppl_1), III-27–III-32 (2004).
  3. Wei C.-C., Wang Z.-Q., Durra D., et al.  The three nitric-oxide synthases differ in their kinetics of tetrahydrobiopterin radical formation, heme-dioxy reduction, and arginine hydroxylation.  Journal of Biological Chemistry.  280 (10), 8929–8935 (2005).
  4. Chen K., Popel A. S.  Theoretical analysis of biochemical pathways of nitric oxide release from vascular endothelial cells.  Free Radical Biology & Medicine.  41 (4), 668–680 (2006).
  5. Buerk D. G.  Can we model nitric oxide biotrasport? A survey of mathematical models for a simple diatomic molecule with surprisingly complex biological activities.  Annual Review of Biomedical Engineering. 3, 109–143 (2001).
  6. Sakellarios A. I., Siogkas P., Exarchos T., et al.  Modelling LDL accumulation in the case of endothelial dysfunction.  Journal of the Serbian Society for Computational Mechanics.  5 (2), 90–100 (2011).
  7. Sriram K., Laughlin J. G., Rangamani P., Tartakovsky D. M.  Shear-Induced Nitric Oxide Production by Endothelial Cells.  Biophysical Journal.  111 (1),  208–221 (2016).
  8. Vanhoutte P. M., Shimokawa H., Feletou M., Tang E. H. C.  Endothelial dysfunction and vascular disease – a 30th anniversary update.  Acta Physiologica.  219 (1), 22–96 (2017).
  9. Mahmoudi M., Farghadan A., McConnell D. R., Barker A. J., Wentzel J. J., Budoff M. J., Arzani A.  The story of wall shear stress in coronary artery atherosclerosis: biochemical transport and mechanotransduction.  Journal of Biomechanical Engineering.  143 (4), 041002 (2021).
  10. Gou Z., Zhang H., Abbasi M., Misbah C.  Red blood cells under flow show maximal ATP release for specific hematocrit.  Biophysical Journal.  120 (21), 4819–4831 (2021).
  11. Nayak A. K., Canepari M., Das S. L., Misbah C.  Nitric oxide modelling and its bioavailability influenced by red blood cells.  Journal of The Royal Society Interface.  21 (221),  20240458  (2024).
  12. Forsyth A. M., Wan J., Owrutsky P. D., Abkarian M., Stone H. A.  Multiscale approach to link red blood cell dynamics, shear viscosity, and ATP release.  Proceedings of the National Academy of Sciences of the United States of America.  108 (27), 10986–10991 (2011).
  13. Turpin C., Catan A., Meilhac O., Bourdon E., Canonne-Hergaux F., Rondeau P.  Erythrocytes: Central Actors in Multiple Scenes of Atherosclerosis.  International Journal of Molecular Sciences.  22 (11), 5843 (2021).
  14. Kostrobij P., Markovych B., Ryzha I., Viznovych O., Tokarchuk M.  Generalized equations of hydrodynamics for viscoelastic fluids in fractional derivatives.  Physics of Fluids.  37 (7), 073118 (2025).
  15. Fumagalli I., Pagani S., Vergara C., et al.  The role of computational methods in cardiovascular medicine: a narrative review.  Translational Pediatrics.  13 (1), 146–163 (2024).
  16. Mc Auley M.  Modelling cholesterol metabolism and atherosclerosis.  Wiley Interdisciplinary Reviews (WIREs): Mechanisms of Disease.  14 (3), e1546 (2021).
  17. Avgerinos N. A., Neofytou P.  Mathematical Modelling and Simulation of Atherosclerosis Formation and Progress: A Review.  Annals of Biomedical Engineering.  47 (8), 1764–1785 (2019).
  18. Chalmers A. D., Cohen A., Bursill C. B., Myerscough M. R.  Bifurcation and dynamics in a mathematical model of early atherosclerosis.  Journal of Mathematical Biology.  71, 1451–1480 (2015).
  19. Owen D. G.  Multiphase cardiovascular haemodynamics: impact of surface roughness and platelet activation.  Doctoral dissertation, University of Birmingham (2022).
  20. Abi Younes G., El Khatib N.  Mathematical modeling of inflammatory processes of atherosclerosis.  Mathematical Modelling of Natural Phenomena.  17,  5 (2022).
  21. Ford H. Z., Byrne H. M., Myerscough M. R.  A lipid-structured model for macrophage populations in atherosclerotic plaques.  Journal of Theoretical Biology.  479, 48–63 (2019).
  22. Chambers K. L., Watson M. G., Myerscough M. R.  A lipid-structured model of atherosclerosis with macrophage proliferation.  Bulletin of Mathematical Biology.  86, 104 (2024).
  23. Watson M. G., Chambers K. L., Myerscough M. R.  ALipid  Structured Model of Atherosclerotic Plaque Macrophages with Lipid-Dependent Kinetics.  Bulletin of Mathematical Biology.  85, 85 (2023).
  24. Cobbold C. A., Sherratt J. A., Maxwell S. R. J.  Lipoprotein Oxidation and its Significance for Atherosclerosis: a Mathematical Approach.  Bulletin of Mathematical Biology.  64 (1),  65–95 (2002).
  25. Yang N., Vafai K.  Modeling of low-density lipoprotein (LDL) transport in the artery–effects of hypertension.  International Journal of Heat and Mass Transfer.  49 (5–6), 850–867 (2006).
  26. Wang S., Vafai K.  Analysis of Low Density Lipoprotein (LDL) Transport Within a Curved Artery.  Annals of Biomedical Engineering.  43 (7), 1571–1584 (2015).
  27. Cilla M., Peña E., Martínez M. A.  Mathematical modelling of atheroma plaque formation and development in coronary arteries.  Journal of The Royal Society Interface.  11 (90), 20130866 (2014).
  28. Pozzi S., Redaelli A., Vergara C., Votta E., Zunino P.  Mathematical Modeling and Numerical Simulation of Atherosclerotic Plaque Progression Based on Fluid-Structure Interaction.  Journal of Mathematical Fluid Mechanics.  23, 74 (2021).
  29. Kneller G. R., Hinsen K.  Fractional Brownian dynamics in proteins.  Journal of Chemical Physics.  121 (20), 10278–10283 (2004).
  30. Weiss M., Elsner M., Kartberg F., Nilsson T.  Anomalous Subdiffusion Is a Measure for Cytoplasmic Crowding in Living Cells.  Biophysical Journal.  87 (5), 3518–3524 (2004).
  31. Banks D. S., Fradin C.  Anomalous Diffusion of Proteins Due to Molecular Crowding.  Biophysical Journal.  89 (5), 2960–2971 (2005).
  32. Szymanski J., Weiss M.  Elucidating the Origin of Anomalous Diffusion in Crowded Fluids.  Physical Review Letters.  103, 038102 (2009).
  33. Jeon J.-H., Tejedor V., Burov S., et al.  In vivo anomalous diffusion and weak ergodicity breaking of lipid granules.  Physical Review Letters.  106, 048103 (2011).
  34. Metzler R., Jeon J.-H., Cherstvya A. G., Barkai E.  Anomalous diffusion models and their properties: non-stationarity, non-ergodicity, and ageing at the centenary of single particle tracking.  Physical Chemistry Chemical Physics.  16,  24128 (2014).
  35. Jeon J.-H., Javanainen M., Martinez-Seara H., Metzler R., Vattulainen I.  Protein Crowding in Lipid Bilayers Gives Rise to Non-Gaussian Anomalous Lateral Diffusion of Phospholipids and Proteins.  Physical Review X.  6, 021006 (2016).
  36. Longeville S., Stingaciu L.-R.  Hemoglobin diffusion and the dynamics of oxygen capture by red blood cells.  Scientific Reports.  7 (1), 10448 (2017).
  37. Morris J. F., Brady J. F.  Self-diffusion in sheared suspensions.  Journal of Fluid Mechanics.  312, 223–252 (1996).
  38. Leshansky A. M., Brady J. F.  Dynamic structure factor study of diffusion in strongly sheared suspensions.  Journal of Fluid Mechanics.  527, 141–169 (2005).
  39. Leshansky A. M., Morris J. F., Brady J. F.  Collective diffusion in sheared colloidal suspensions.  Journal of Fluid Mechanics.  597, 305–341 (2008).
  40. Johnstona B. M., Johnstona P. R., Corneyb S., Kilpatrick D.  Non-Newtonian blood flow in human right coronary arteries: steady state simulations.  Journal of Biomechanics.  37 (5), 709–720 (2004).
  41. Olgac U., Kurtcuoglu V., Poulikakos D.  Computational modeling of coupled b lood-wall mass transport of LDL: effects of local wall shear stress.  American Journal of Physiology-Heart and Circulatory Physiology.  294, H909–H919 (2008).
  42. Al-Azawy M. G., Kadhim S. K., Hameed A. S.  Newtonian and Non-Newtonian Blood Rheology Inside a Model of Stenosis.  CFD Letters.  12 (11), 27–36 (2020).
  43. Owen D. G., Schenkel T., Shepherd D. E. T., Espino D. M.  Assessment of surface roughness and blood rheology on local coronary haemodynamics: a multi-scale computational fluid dynamics study.  Journal of the Royal Society Interface.  17, 20200327 (2020).
  44. Vilchevskaya A. E., Vilchevskaya E. N., Müller W. H., Eremeyev V. A.  Modeling of blood flow in the framework of micropolar theory.  Continuum Mechanics and Thermodynamics.  35, 2337–2359 (2023).
  45. Zhang J., Johnson P.C., Popel A. S.  Effects of erythrocyte deformability and aggregation on the cell free layer and apparent viscosity of microscopic blood flows.  Microvascular Research.  77 (3), 265–272 (2009).
  46. Higashi T., Yamagishi A., Takeuchi T., Kawaguchi N., Sagawa S., Onishi S., Date M.  Orientation of erythrocytes in a strong static magnetic field.  Blood.  82 (4), 1328–1334 (1993).
  47. Haik Y., Pai V., Chen C.-J.  Apparent viscosity of human blood in a high static magnetic field.  Journal of Magnetism and Magnetic Materials.  225 (1–2), 180–186 (2001).
  48. Ichioka S., Minegishi M., Iwasaka M., Shibata M., Nakatsuka T., Harii K., Kamiya A., Ueno S.  High-intensity static magnetic fields modulate skin microcirculation and temperature in vivo.  Bioelectromagnetics.  21 (3), 183–188 (2000).
  49. Javadzadegan A., Moshfegh A., Afrouzi H. H., Omidi M.  Magnetohydrodynamic blood flow in patients with coronary artery disease.  Computer Methods and Programs in Biomedicine.  163, 111–122 (2018).
  50. Wang Y., Zhao Z.-G., Chai Z., Fang J.-C., Chen M.  Electromagnetic field and cardiovascular diseases: A state-of-the-art review of diagnostic, therapeutic, and predictive values.  The FASEB Journal.  37 (10), e23142 (2023). 
  51. Alain T., Mallat Z.  Cytokines in Atherosclerosis: Pathogenic and Regulatory Pathways.  Physiological Reviews.  86 (2), 515–581 (2006).
  52. Ait-Oufella H., Taleb S., Mallat Z., Tedgui A.  Recent advances on the role of cytokines in atherosclerosis.  Arteriosclerosis, Thrombosis, and Vascular Biology.  31 (5), 969–979 (2011).
  53. Tousoulis D., Oikonomou E., Evangelos K. E., Crea F., Kaski J. C.  Inflammatory cytokines in atherosclerosis: current therapeutic approaches.  European Heart Journal.  37 (22), 1723–1732 (2016).
  54. Adak A., Devi A., Gupta P. K.  Influence of Pro- and Anti-inflammatory Cytokines during Pathogenesis Atherosclerosis: A Dynamical Approach.  Preprint available at Research Square (2022).
  55. Wootton D. M., Ku D. N.  Fluid Mechanics of Vascular Systems, Diseases, and Thrombosis.  Annual Review of Biomedical Engineering.  1, 299–329 (1999).
  56. Chiu J.-J., Chien S.  Effects of disturbed flow on vascular endothelium: Pathophysiological basis and clinical perspectives.  Physiological Reviews.  91 (1), 327–387 (2011).
  57. Chatzizisis Y. S., Coskun A. U., Jonas M., et al.  Role of Endothelial Shear Stress in the Natural History of Coronary Atherosclerosis and Vascular Remodeling: Molecular, Cellular, and Vascular Behavior.  Journal of the American College of Cardiology.  49 (25), 2379–2393 (2007).
  58. Stone P. H., Coskun A. U., Kinlay S., et al.  Effect of endothelial shear stress on the progression of coronary artery disease, vascular remodeling, and in-stent restenosis in humans: In vivo 6-month follow-up study.  Circulation.  108 (4), 438-444 (2003).
  59. Stangeby D. K., Ethier C. R.  Computational analysis of coupled bloodwall arterial LDL transport.  Journal of Biomechanical Engineering.  124 (1), 1–8 (2002).
  60. Himburg H. A., Grzybowski D. M., Hazel A. L., LaMack J. A., Li X.-M., Friedman M. H.  Spatial comparison between wall shear stress measures and porcine arterial endothelial permeability.  American Journal of Physiology-Heart and Circulatory Physiology.  286 (5), H1916–H1922 (2004).
  61. Traub O., Berk B. C.  Laminar Shear Stress Mechanisms by Which Endothelial Cells Transduce an Atheroprotective Force.  Arteriosclerosis, Thrombosis, and Vascular Biology.  18 (5), 677–685 (1998).
  62. Dimmeler S., Fleming I., Fisslthaler B., Hermann C., Buss R., Zeiher A. M.  Activation of nitric oxide synthase in endothelial cells by Akt-dependent phosphorylation.  Nature.  399 (6736),  601–605 (1999).
  63. Tamis A., Drapaca C. S.  Modeling NO Biotransport in Brain Using a Space-Fractional Reaction-Diffusion Equation.  Frontiers in Physiology.  12,  644149 (2021).
  64. Tamis A., Drapaca C. S.  A Mathematical Model of Nitric Oxide Mechanotransduction in Brain.  Mechanics of Biological Systems and Materials & Micro- and Nanomechanics & Research Applications.  River Publishers (2021).
  65. Upston J. M., Terentis A. C., Stocker R.  Tocopherol-mediated peroxidation of lipoproteins: implications for vitamin E as a potential antiatherogenic supplement.  FASEB Journal.  13 (9), 977–994 (1999).
  66. Doba T., Burton G. W., Ingold K. U.  Antioxidant and co-antioxidant activity of vitamin C. The effect of vitamin C, either alone or in the presence of vitamin E or a water-soluble vitamin E analogue, upon the peroxidation of aqueous multilamellar phospholipid liposomes.  Biochimica et Biophysica Acta (BBA) – Lipids and Lipid Metabolism.  835 (2), 298–303 (1985).
  67. Bendich A., Machlin L. J., Scandurra O., Burton G. W., Wayner D. D. M.  The antioxidant role of vitamin C.  Advances in Free Radical Biology & Medicine.  2, 419–444 (1986).
  68. Wayner D. D. M., Burton G. W., Ingold K. U., Barclay L. R. C., Locke S. J.  The relative contributions of vitamin~E, urate, ascorbate and proteins to the total peroxyl radical-trapping antioxidant activity of human blood plasma.  Biochimica et Biophysica Acta (BBA) – General Subjects.  924 (3),  408–419 (1987).
  69. Watanabe A., Noguchi N., Takahashi M., Niki E.  Rate Constants for Hydrogen Atom Abstraction by $\alpha$-Tocopheroxyl Radical from Lipid, Hydroperoxide and Ascorbic Acid.  Chemistry Letters.  28 (7), 613–614 (1999).
  70. Tokarchuk M. V., Kostrobii P. P., Humenyuk Y. A.  Generalized transport equations of diffusion-reaction processes. The nonequilibrium statistical operator method.  Journal of Physical Studies.  5 (2), 111–120 (2001), (in Ukrainian).
  71. Kostrobii P. P., Tokarchuk M. V., Humenyuk Y. A.  Nonequilibrium statistical operator method: Generalized transport equations of diffusion–reaction processes.  The European Physical Journal B – Condensed Matter and Complex Systems.  36, 555–565 (2003).
  72. Kostrobij P. P., Markovych B. M., Viznovych O. V., Tokarchuk M. V.  Methods of mathematical modeling of stochastic systems.  Lviv, Publishing house "Rastr-7" (2020), (in Ukrainian).
  73. Cukier R. I., Nocera D. G.  Proton-coupled electron transfer.  Annual Review of Physical Chemistry.  49 (1), 337–369 (1998).
  74. Tyburski R., Liu T., Glover S. D., Hammarstrom L.  Proton-coupled electron transfer guidelines, fair and square.  Journal of the American Chemical Society.  143 (2),  560–576 (2021).
  75. Stasyuk I. V., Stetsiv R. Ya., Sizonenko Yu. V.  Dynamics of charge transfer along hydrogen boud.  Condensed Matter Physics.  5 (4), 685–706 (2002).
  76. Stasyuk I. V., Stetsiv R. Ya., Yurechko R. Ya.  Dynamics of charge transfer in quasi-one-dimensional hyydrogen-bonded structures.  Journal of Physical Studies.  9 (4), 349–357 (2005), (in Ukrainian).
  77. Krasnoholovets V. V., Tomchuk P. M., Lukyanets S. P.  Proton transfer and coherent phenomena in molecular structures with hydrogen bonds.  Advances in Chemical Physics.  125, 351–548 (2003).
  78. Kostrobij P., Markovych B., Tokarchuk M., Ryzha I.  Generalized diffusion equation with nonlocality of space-time: Analytical and numerical analysis.  Journal of Mathematical Physics.  62, 103304 (2021).