A comparison of two mathematical models of the cerebrospinal fluid dynamics

Donatella Donatelli; Pierangelo Marcati; Licia Romagnoli; Donatella Donatelli; Pierangelo Marcati; Licia Romagnoli

doi:10.3934/mbe.2019140

Mathematical Biosciences and Engineering

2019, Volume 16, Issue 4: 2811-2851. doi: 10.3934/mbe.2019140

Previous Article Next Article

Research article

A comparison of two mathematical models of the cerebrospinal fluid dynamics

1.
Department of Information Engineering, Computer Science and Mathematics, University of L’Aquila, 67100 L’Aquila, Italy
2.
GSSI - Gran Sasso Science Institute, Viale F. Crispi 7, 67100 L’Aquila, Italy

Received: 20 June 2018 Accepted: 06 March 2019 Published: 02 April 2019

In this paper we provide the numerical simulations of two cerebrospinal fluid dynamics models by comparing our results with the real data available in literature (see Section 4). The models describe different processes in the cerebrospinal fluid dynamics: the cerebrospinal flow in the ventricles of the brain and the reabsorption of the fluid. In the appendix we show in detail the mathematical analysis of both models and we identify the set of initial conditions for which the solutions of the systems of equations do not exhibit blow up. We investigate step by step the accuracy of these theoretical outcomes with respect to the real cerebrospinal physiology and dynamics. The plan of the paper is provided in Section 1.5.
- cerebrospinal fluid,
- fluid dynamics,
- one-dimensional flow,
- physiological flows,
- hyperbolic equations,
- global well posedness
Citation: Donatella Donatelli, Pierangelo Marcati, Licia Romagnoli. A comparison of two mathematical models of the cerebrospinal fluid dynamics[J]. Mathematical Biosciences and Engineering, 2019, 16(4): 2811-2851. doi: 10.3934/mbe.2019140

Related Papers:

Abstract

In this paper we provide the numerical simulations of two cerebrospinal fluid dynamics models by comparing our results with the real data available in literature (see Section 4). The models describe different processes in the cerebrospinal fluid dynamics: the cerebrospinal flow in the ventricles of the brain and the reabsorption of the fluid. In the appendix we show in detail the mathematical analysis of both models and we identify the set of initial conditions for which the solutions of the systems of equations do not exhibit blow up. We investigate step by step the accuracy of these theoretical outcomes with respect to the real cerebrospinal physiology and dynamics. The plan of the paper is provided in Section 1.5.

References

[1]	M. Bulat and M. Klarica, Recent insights into a new hydrodynamics of the cerebrospinal fluid, Brain Res. Rev., 112 (2011), 65–99.
[2]	P. D. Brown, S. L. Davies, T. Speake, et al., Molecular mechanisms of cerebrospinal fluid production, Neuroscience, 70 (2004), 129–957.
[3]	C. E. Johanson, J. A. Duncan, P. M. Klinge, et al. , Multiplicity of cerebrospinal fluid functions: New challenges in health and disease, Cerebrospinal Fluid. Res., (2008), 5–10.
[4]	T. Brinker, E. Stopa, J. Morrison, et al. , A new look at cerebrospinal fluid circulation, Fluids Barriers CNS, 11 (2014), 10.
[5]	D. Orešković and M. Klarica, The formation of cerebrospinal fluid: Nearly a hundred years of interpretations and misinterpretations, Brain Res. Rev., 64 (2010), 241–262.
[6]	M. Bulat, V. Lupret, D. Orehković, et al. , Transventricular and transpial absorption of cerebrospinal fluid into cerebral microvessels, Coll. Antropol., 50 (2008), 32 Suppl. 1–43.
[7]	R. Spector, S. R. Snodgrass and C. E. Johanson, A balanced view of the cerebrospinal fluid composition and functions: Focus on adult humans, Exp. Neurol., 68 (2015), 273–257.
[8]	R. Roales - Buján, P. Páez, M. Guerra, et al. , Astrocytes acquire morphological and functional characteristics of ependymal cells following disruption of ependyma in hydrocephalus, Acta Neuropathol., 46 (2012), 124–531.
[9]	Z. Czosnyka, M. Czosnyka, A. Lavinio, et al. , Clinical testing of CSF circulation, Eur. J. Anaesthesiol. Suppl., 5 (2008), 42–142.
[10]	A. A. Linninger, K. Tangen, C. Hsu, et al. , Cerebrospinal fluid mechanics and its coupling to cerebrovascular dynamics, Annu. Rev. Fluid Mech., 48 (2016), 219–257.
[11]	I. Cherian, M. Beltran, E. M. Kasper, et al. , Exploring the Virchow-Robin spaces function: A unified theory of brain diseases, Surg. Neurol. Int., 7 (2016), S711–S714.
[12]	J. Mack, W. Squier and J. T. Eastman, Anatomy and development of the meninges: Implications for subdural collections and CSF circulation, Pediatr. Radiol., 10 (2009), 39–200.
[13]	A. A. Linninger, C. Tsakiris, D. C. Zhu, et al. , Pulsatile cerebrospinal fluid dynamics in the human brain, IEEE T. Bio-Med. Eng., 52 (2005), 557–565.
[14]	J. Buishas, I. G. Gould and A. A. Linninger, A computational model of cerebrospinal fluid production and reabsorption driven by Starling forces, Croat. Med. J., 97 (2014), 55–481.
[15]	A. A. Linninger, M. Xenos, B. Sweetman, et al. , A mathematical model of blood, cerebrospinal fluid and brain dynamics, J. Math. Biol., 59 (2009), 729–759.
[16]	A. Marmarou, A theoretical model and experimental evaluation of the cerebrospinal fluid system, Ph.D thesis, Drexel University, Philadelphia, 1973.
[17]	A. Marmarou, K. Shulman and R. M. Rosende, A nonlinear analysis of the cerebrospinal fluid system and intracranial pressure dynamics, J. Neurosurg, 48 (1978), 332–344 .
[18]	D. Chou, J. C. Vardakis, L. Guo et al. , A fully dynamic multicompartmental poroelastic system: Application to aqueductal stenosis, J. Biomech., 49 (2016), 2306–2312.
[19]	L. Guo, J. C. Vardakis, T. Lassila, et al. , Subjectspecific multi-poroelastic model for exploring the risk factors associated with the early stages of Alzheimer's disease, Interface Focus, 8 (2018), 20170019.
[20]	D. Chou, J. C. Vardakis and Y. Ventikos, Multiscale modelling for cerebrospinal fluid dynamics: Multicompartmental poroelacticity and the role of AQP4, J. Biosci. Med., 2 (2014), 1–9.
[21]	M. Ursino, A mathematical study of human intracranial hydrodynamics. Part 1 - The cerebrospinal fluid pulse pressure, Ann. Biomed. Eng., 16 (1988), 379–401.
[22]	M. Ursino, A mathematical study of human intracranial hydrodynamics. Part 2 - Simulation of clinical tests, Ann. Biomed. Eng., 16 (1988), 403–416.
[23]	G. Gadda, A. Taibi, F. Sisini, et al. , A new hemodynamic model for the study of cerebral venous outflow, Am. J. Physiol. Heart Circ. Physiol., 308 (2015), H217–H231.
[24]	M. Gehlen, V. Kurtcuoglu and M. Schmid Daners, Is posture-related craniospinal compliance shift caused by jugular vein collapse? A theoretical analysis, Fluids and Barriers of the CNS, 14 (2017).
[25]	L. O. Müller and E. F. Toro, Enhanced global mathematical model for studying cerebral venous blood flow, J. Biomech., 47 (2014), 3361–3372.
[26]	D. Orešković, M. Radoš and M. Klarica, New concepts of cerebrospinal fluid physiology and development of hydrocephalus, Pediatr. Neurosurg., 52 (2017), 417–425.
[27]	A. A. Linninger, C. Xu, K. Tangen, et al. , Starling forces drive intracranial water exchange during normal and pathological states, Croat. Med. J., 58 (2017), 384–394.
[28]	D. Donatelli, P. Marcati and L. Romagnoli, Analysis of solutions for a cerebrospinal fluid model, Nonlinear Anal. Real World Appl., 44 (2018), 417–448.
[29]	H. Davson, K.Welch and M. B. Segal, Physiology and pathophysiology of the cerebrospinal fluid, Edinburgh: Churchill-Livingstone, 1987.
[30]	N. Alperin, S. H. Lee and A. M. Bagci, MRI Measurements of Intracranial Pressure in the Upright Posture: The Effect of the Hydrostatic Pressure Gradient, J. Magn. Reson. Imaging, 42 (2015), 1158–1163.
[31]	M. Klarica, M. Rados, G. Erceg, et al. , The Influence of Body Position on Cerebrospinal Fluid Pressure Gradient and Movement in Cats with Normal and Impaired Craniospinal Communication, Plos One, 9 (2014).
[32]	L. C. Piccinini, G. Stampacchia and G. Vidossich, Ordinary Differential Equations in Rn, Problems and Methods, Springer-Verlag, 1984.
[33]	A. A. Linninger, M. Xenos, D. C. Zhu, et al., Cerebrospinal Fluid Flow in the Normal and Hydrocephalic Human Brain, IEEE Trans. Biomed. Engng., 54 (2007), 291–302.
[34]	S. Gholampour, N. Fatouraee, A. S. Seddighi, et al. , A Hydrodynamical Study to Propose a Numerical Index for Evaluating the CSF Conditions in Cerebral Ventricular System, Intern. Clinical Neurosci. J., 1 (2014), 1–9.
[35]	K. J. Streitberger, E. Wiener, J. Hoffmann, et al. , In Vivo Viscoelastic Properties of the Brain in Normal Pressure Hydrocephalus, NMR Biomed., 24 (2011), 385–392.
[36]	I. K. Pople, Hydrocephalus and Shunts: What the Neurologist Should Know, J. Neurology, Neurosurgery Psychiatry, 73 (2002), 17–22.
[37]	A. K. Sharma, S. Gaikwad, V. Gupta, et al. , Measurement of peak CSF flow velocity at cerebral aqueduct, before and after lumbar CSF drainage, by use of phase-contrast MRI: Utility in the management of idiopathic normal pressure hydrocephalus, Clin. Neurol. Neurosur., 110 (2008), 363–368.
[38]	A. Majda, Compressible Fluid Flow and Systems of Conservation Laws in Several Space Variables, Springer Science+Business Media, LLC, Series: Applied Mathematical Sciences, 53, 1984.

Reader Comments

Your name:*

Email:*
© 2019 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)