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Reduction of Dimensionality in Monte Carlo Simulation of Diffusion in Extracellular Space Surrounding Cubic Cells

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Abstract

The real-time iontophoretic method has measured volume fraction and tortuosity of the interstitial component of extracellular space in many regions and under different conditions. To interpret these data computer models of the interstitial space (ISS) of the brain are constructed by representing cells as Basic Cellular Structures (BCS) surrounded by a layer of ISS and replicating this combination to make a 3D ensemble that approximates brain tissue with a specified volume fraction. Tortuosity in such models is measured by releasing molecules of zero size into the ISS and allowing them to execute random walks in the ISS of the ensemble using a Monte Carlo algorithm. The required computational resources for such simulations may be high and here we show that in many situations the 3D problem may be reduced to a quasi-1D problem with consequent reduction in resources. We take the simplest BCS in the form of cubes and use MCell software to perform the Monte Carlo simulations but the analysis described here may be extended in principle to more complex BCS and an ISS that has a defined viscosity and an extracellular matrix that interacts with diffusing molecules. In the course of this study we found that the original analytical description of the relation between volume fraction and tortuosity for an ensemble of cubes may require a small correction.

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References

  1. Nicholson C, Phillips JM (1981) Ion diffusion modified by tortuosity and volume fraction in the extracellular microenvironment of the rat cerebellum. J Physiol 321:225–257

    Article  CAS  Google Scholar 

  2. Odackal J, Colbourn R, Odackal NJ, Tao L, Nicholson C, Hrabêtová S (2017) Real-time iontophoresis with tetramethylammonium to quantify volume fraction and tortuosity of brain extracellular space. J Vis Exp 125:e55755. https://doi.org/10.3791/55755

    Article  CAS  Google Scholar 

  3. Nicholson C, Syková E (1998) Extracellular space structure revealed by diffusion analysis. Trends Neurosci 21(5):207–215

    Article  CAS  Google Scholar 

  4. Syková E, Nicholson C (2008) Diffusion in brain extracellular space. Physiol Rev 88(4):1277–1340. https://doi.org/10.1152/physrev.00027.2007

    Article  PubMed  PubMed Central  Google Scholar 

  5. Nicholson C, Hrabêtová S (2017) Brain extracellular space: the final frontier of neuroscience. Biophys J 113(10):2133–2142. https://doi.org/10.1016/j.bpj.2017.06.052

    Article  CAS  PubMed  PubMed Central  Google Scholar 

  6. Zheng K, Jensen TP, Savtchenko LP, Levitt JA, Suhling K, Rusakov DA (2017) Nanoscale diffusion in the synaptic cleft and beyond measured with time-resolved fluorescence anisotropy imaging. Sci Rep 7:42022. https://doi.org/10.1038/srep42022

    Article  CAS  PubMed  PubMed Central  Google Scholar 

  7. Thorne RG, Lakkaraju A, Rodriguez-Boulan E, Nicholson C (2008) In vivo diffusion of lactoferrin in brain extracellular space is regulated by interactions with heparan sulfate. Proc Natl Acad Sci USA 105(24):8416–8421. https://doi.org/10.1073/pnas.0711345105

    Article  PubMed  Google Scholar 

  8. Nicholson C, Kamali-Zare P, Tao L (2011) Brain extracellular space as a diffusion barrier. Comput Visual Sci 14(7):309–325. https://doi.org/10.1007/s00791-012-0185-9

    Article  Google Scholar 

  9. Nicholson C (2018) Brain interstitial structure revealed through diffusive spread of molecules. In: Bunde A, Caro J, Kärger J, Vogl G (eds) Diffusive spreading in nature, technology and society. Springer International Publishing, Cham, pp 93–114. https://doi.org/10.1007/978-3-319-67798-9_6

    Chapter  Google Scholar 

  10. Nicholson C (1993) Ion-selective microelectrodes and diffusion measurements as tools to explore the brain cell microenvironment. J Neurosci Methods 48(3):199–213

    Article  CAS  Google Scholar 

  11. Tao L, Nicholson C (2004) Maximum geometrical hindrance to diffusion in brain extracellular space surrounding uniformly spaced convex cells. J Theor Biol 229(1):59–68

    Article  CAS  Google Scholar 

  12. Hrabe J, Hrabêtová S, Segeth K (2004) A model of effective diffusion and tortuosity in the extracellular space of the brain. Biophys J 87(3):1606–1617

    Article  CAS  Google Scholar 

  13. Tao A, Tao L, Nicholson C (2005) Cell cavities increase tortuosity in brain extracellular space. J Theor Biol 234(4):525–536

    Article  CAS  Google Scholar 

  14. Kinney JP, Spacek J, Bartol TM, Bajaj CL, Harris KM, Sejnowski TJ (2013) Extracellular sheets and tunnels modulate glutamate diffusion in hippocampal neuropil. J Comp Neurol 521(2):448–464. https://doi.org/10.1002/cne.23181

    Article  PubMed  PubMed Central  Google Scholar 

  15. Xiao F, Hrabe J, Hrabêtová S (2015) Anomalous extracellular diffusion in rat cerebellum. Biophys J 108(9):2384–2395. https://doi.org/10.1016/j.bpj.2015.02.034

    Article  CAS  PubMed  PubMed Central  Google Scholar 

  16. Lacks DJ (2008) Tortuosity and anomalous diffusion in the neuromuscular junction. Phys Rev E 77(4 Pt 1):041912

    Article  Google Scholar 

  17. Nicholson C, Tao L (1993) Hindered diffusion of high molecular weight compounds in brain extracellular microenvironment measured with integrative optical imaging. Biophys J 65(6):2277–2290

    Article  CAS  Google Scholar 

  18. Fenstermacher JD, Kaye T (1988) Drug “diffusion” within the brain. Ann NY Acad Sci 531:29–39

    Article  CAS  Google Scholar 

  19. Nicholson C (2001) Diffusion and related transport properties in brain tissue. Rep Prog Phys 64:815–884

    Article  CAS  Google Scholar 

  20. Chen KC, Nicholson C (2000) Changes in brain cell shape create residual extracellular space volume and explain tortuosity behavior during osmotic challenge. Proc Natl Acad Sci USA 97(15):8306–8311

    Article  CAS  Google Scholar 

  21. Berg HC (1983) Random walks in biology. Princeton University Press, New Jersey

    Google Scholar 

  22. Crank J (1975) The mathematics of diffusion, 2nd edn. Clarendon Press, Oxford

    Google Scholar 

  23. Stiles JR, Bartol TM (2001) Monte Carlo methods for simulating realistic synaptic microphysiology using MCell. In: De Schutter E (ed) Computational neuroscience: realistic modeling for experimentalists. CRC Press, London, pp 87–127

    Google Scholar 

  24. Thorne RG, Nicholson C (2006) In vivo diffusion analysis with quantum dots and dextrans predicts the width of brain extracellular space. Proc Natl Acad Sci USA 103(14):5567–5572

    Article  CAS  Google Scholar 

  25. Sokolov IM (2012) Models of anomalous diffusion in crowded environments. Soft Matter 8(35):9043–9052. https://doi.org/10.1039/C2SM25701G

    Article  CAS  Google Scholar 

  26. Mathias RT (1983) Effect of tortuous extracellular pathways on resistance measurements. Biophys J 42(1):55–59

    Article  CAS  Google Scholar 

  27. El-Kareh AW, Braunstein SL, Secomb TW (1993) Effect of cell arrangement and interstitial volume fraction on the diffusivity of monoclonal antibodies in tissue. Biophys J 64(5):1638–1646

    Article  CAS  Google Scholar 

  28. Kamali-Zare P, Nicholson C (2013) Editorial: brain extracellular space: Geometry, matrix and physiological importance. Basic Clin Neurosci 4(4):4–8

    Google Scholar 

  29. Tucker JL, Wen R, Oakley B 2nd (1991) A deconvolution technique for improved estimation of rapid changes in ion concentration recorded with ion-selective microelectrodes. IEEE Trans Biomed Eng 38(2):156–160

    Article  CAS  Google Scholar 

  30. Bell GE, Crank J (1974) Influence of imbedded particles on steady-state diffusion. J Chem Soc-Faraday Trans II 70:1259–1273

    Article  CAS  Google Scholar 

Download references

Acknowledgement

Supported in part by National Institutes of Health Grant R01-NS28642 from the National Institute of Neurological Disorders and Stroke.

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  1. Padideh Kamali-Zare

    Present address: Darmiyan Inc, 1 Sansome Street, Suite 3500, San Francisco, CA, 94104, USA

Authors and Affiliations

  1. Department of Neuroscience and Physiology, New York University School of Medicine, 550 First Avenue, New York, NY, 10016, USA

    Charles Nicholson & Padideh Kamali-Zare

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  1. Charles Nicholson
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  2. Padideh Kamali-Zare
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Correspondence to Charles Nicholson.

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Special Issue: In honor of Prof. Eva Syková.

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Nicholson, C., Kamali-Zare, P. Reduction of Dimensionality in Monte Carlo Simulation of Diffusion in Extracellular Space Surrounding Cubic Cells. Neurochem Res 45, 42–52 (2020). https://doi.org/10.1007/s11064-019-02793-6

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