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Theoretical Study of Cerebral Venous Haemodynamics Associated with the CCSVI Condition Eleuterio Toro, Ph.D. and Lucas Mueller University of Trento, Italy

By means of global mathematical model for the entire human circulation we study the physical aspects of cerebral venous haemodynamics resulting from anomalous extra---cranial venous malformations. The mathematical model: Starting from first principles, we have constructed a global, closed---loop, multi---scale mathematical model for the entire human circulation, including the arterial system, the venous system, the heart, the pulmonary circulation, the cerebrospinal fluid system and the microcirculation [1]. The model includes a detailed description of intracranial and extra---cranial veins. Medium to large vessels are described by partial differential equations and the remaining components by systems of differential---algebraic equations. State---of---the art numerical methods are used to solve the equations. Patient---specific characterization of major veins of the head and neck is carried out using MRI data. Thorough validation of the model is carried out through the use of published data for the arterial system and most regions of the venous system. For head and neck veins, validation is carried out through a detailed comparison of simulation results against patient---specific phase---contrast MRI flow quantification data. The mathematical model is then used to study the CCSVI condition as described by Zamboni et al. [2]. Results from our study will be presented in this talk.

Preliminary results: Our computations so far reveal that stenotic extra---cranial veins cause a pressure increase upstream of their location, with the pressure drop across the stenosis being around 1.5 mmHg. This pressure increase has a direct impact on the dural sinuses, effect that is strongly influenced by the specific configuration of the confluence of sinuses. However, due to Starling resistor mechanism, the impact of the pressure increase on intracranial---vein pressure appears to be indirect and proceeds as follows: increased dural sinus pressure reduces CSF reabsorption rates leading to an increase in intracranial pressure. Intracranial pressure will increase until a new balance between CSF generation and absorption is reached. Then, the increased intracranial pressure will be directly transmitted to intracranial veins.

Acknowledgements: The authors warmly thank Prof. E. M. Haacke (MR Research Facility Wayne State University, Detroit, USA) for providing MRI data used in this study. This work has been partially supported by CARITRO (Fondazione Cassa di Risparmio di Trento e Rovereto, Italy), project No. 2011.0214.

References [1] Lucas O. Mueller and Eleuterio F. Toro. A global multi---scale model for the human circulation with emphasis on the venous system. International Journal for Numerical Methods in Biomedical Engineering. In press, 2013. Also published on line as pre---print. Isaac Newton Institute for Mathematical Sciences, University of Cambridge, UK. http://www.newton.ac.uk/preprints2013.html [2] Zamboni P. Zamboni, R. Galeotti, E. Menegatti, A. M. Malagoni, G. Tacconi, S. Dall’Ara, I. Bar--- tolomei, and F. Salvi. Chronic cerebrospinal venous insufficiency in patients with multiple sclerosis. Journal of Neurology, Neurosurgery and Psychiatry, 80:392–399, 2009.

Update in computational fluid modelling of the brain

Mauro Ursino

Several aspects concur in making the cerebral circulation extremely complex: i) the presence of
diffuse anastomotical pathways among cerebral vessels (both in the intracranial arterial and
extracranial venous circulations); ii) the presence of sophisticate mechanisms which regulate
cerebral blood flow following pressure changes and changes in blood gas content; iii) the
occurrence of a portion of this circulation within a closed space (the skull and neuroaxis) with a
limited volume storage capacity. A deeper understanding of how these complex factors interact
reciprocally, and of their possible role in pathological conditions, may be attained with the use of
mathematical models and computer simulation techniques.

Aim of the presentation is to illustrate the complex mechanisms affecting the cerebral
hemodynamics, by making use of computational models developed in past years, and showing
some practical examples.

The first part of the presentation is focused on the intracranial circulation, laying emphasis on the
role of cerebrovascular regulatory mechanisms. A few pathological cases are simulated, to illustrate
the complexity of factors operating on brain hemodynamics. An example considers the case of
patients with reduced storage capacity and altered CSF circulation (a condition, for instance,
typically occurring in patients with severe head injury). In these cases, instability of intracranial
dynamics may lead to uncontrollable increase in intracranial pressure, with the development of
large ICP waves [1]. A further example simulates hemodynamics in patients with unilateral internal
carotid artery stenosis; in this case, local blood flow regulation is progressively lost in the
ipsilateral territory with the presence of a steal phenomenon, while the anterior communicating
artery plays the major role to redistribute the available blood flow [2].

The second part presents a very recent extension of this model, in which a detailed description of
the extracranial venous pathways (jugular veins, vertebral-azygos vein complex, collateral
anastomoses) are included. The model accounts for the changes in jugular vessels lumen occurring
when passing from supine to standing, and simulates how these changes can affect flows and
pressures in specific points of the system [3]. Furthermore, the model provides quantitative
predictions on how this redistribution can be altered by stenotic patterns, and how a failure of the
extracranial venous drainage may be reflected in the upstream intracranial circulation.

We claim these models may have a great perspective value, to help clinicians in reaching a deeper
understanding of the multiple mechanisms operating on the brain circulation, and to be acquainted
on the complex effects of pathological alterations in brain vessels.

Key references
1. M. Ursino, A. Ter Minassian, C.A. Lodi, L. Beydon, “Cerebral hemodynamics during arterial
and CO2 pressure changes; in vivo prediction by a mathematical model”, Am. J. Physiol. Heart
Circ. Physiol Vol. 279: H2439-H2455, 2000.
2. Ursino M, Giannessi M. “A Model of Cerebrovascular Reactivity Including the Circle of Willis
and Cortical Anastomoses.”, Ann Biomed Eng. Vol. 38(3): 955-974, 2010.
3. G. Gadda, A. Taibi, F. Sisini, M. Gambaccini, P. Zamboni, M. Ursino. “A new hemodynamic
model for the study of cerebral venous outflow”, Am. J. Physiol. Heart Circ. Physiol, in press.

from twitter wrote:Ziv Haskal MD @ZHaskal ·
Physicists, biomech engineers & mathematicians may make head hurt,but confirm #CCSVI thru modeling of vein,art,CSF

Phlebology. 2015 Jun 2. pii: 0268355515586526. [Epub ahead of print]

Impact of CCSVI on cerebral haemodynamics: a mathematical study using MRI angiographic and flow data.

Müller LO1, Toro EF2, Haacke EM3, Utriainen D4.

BACKGROUND:
The presence of abnormal anatomy and flow in neck veins has been recently linked to neurological diseases. The precise impact of extra-cranial abnormalities such as stenoses remains unexplored.

METHODS:
Pressure and velocity fields in the full cardiovascular system are computed by means of a global mathematical model that accounts for the relationship between pulsating cerebral blood flow and intracranial pressure.

RESULTS:
Our model predicts that extra-cranial strictures cause increased pressure in the cerebral venous system. Specifically, there is a predicted pressure increase of about 10% in patients with a 90% stenoses. Pressure increases are related to significant flow redistribution with flow reduction of up to 70% in stenosed vessels and consequent flow increase in collateral pathways.

CONCLUSIONS:
Extra-cranial venous strictures can lead to pressure increases in intra-cranial veins of up to 1.3 mmHg, despite the shielding role of the Starling resistor. The long-term clinical implications of the predicted pressure changes are unclear.

© The Author(s) 2015 Reprints and permissions: sagepub.co.uk/journalsPermissions.nav.

Preliminary results: Our computations so far reveal that stenotic extra---cranial veins cause a pressure increase upstream of their location, with the pressure drop across the stenosis being around 1.5 mmHg. This pressure increase has a direct impact on the dural sinuses, effect that is strongly influenced by the specific configuration of the confluence of sinuses. However, due to Starling resistor mechanism, the impact of the pressure increase on intracranial---vein pressure appears to be indirect and proceeds as follows: increased dural sinus pressure reduces CSF reabsorption rates leading to an increase in intracranial pressure. Intracranial pressure will increase until a new balance between CSF generation and absorption is reached. Then, the increased intracranial pressure will be directly transmitted to intracranial veins.

I bolded the conclusion, but what does it mean?

The airways are arranged as a bifurcating network, with each parent airway splitting roughly symmetrically into two daughter irways at each bifurcation. In humans, there are roughly 23 generations of bifurcations from the largest airway (the trachea) down to the terminal air units (the alveoli, of which there are around 300 million in an adult lung). The lungs fit tightly inside a space confined by the rib-cage and diaphragm.

Brain venous haemodynamics, neurological diseases and mathematical modelling. A review

Eleuterio F. Toro, , , E-mail the corresponding author
Laboratory of Applied Mathematics, DICAM, University of Trento, Via Mesiano 77, I-38123 Trento, Italy
Received 31 January 2015, Revised 25 May 2015, Accepted 9 June 2015, Available online 10 August 2015

Abstract
Behind Medicine (M) is Physiology (P), behind Physiology is Physics (P) and behind Physics is always Mathematics (M), for which I expect that the symmetry of the quadruplet MPPM will be compatible with the characteristic bias of hyperbolic partial differential equations, a theme of this paper. I start with a description of several idiopathic brain pathologies that appear to have a strong vascular dimension, of which the most prominent example considered here is multiple sclerosis, the most common neurodegenerative, disabling disease in young adults. Other pathologies surveyed here include retinal abnormalities, Transient Global Amnesia, Transient Monocular Blindness, Ménière’s disease and Idiopathic Parkinson’s disease. It is the hypothesised vascular aspect of these conditions that links medicine to mathematics, through fluid mechanics in very complex networks of moving boundary blood vessels. The second part of this paper is about mathematical modelling of the human cardiovascular system, with particular reference to the venous system and the brain. A review of a recently proposed multi-scale mathematical model then follows, consisting of a one-dimensional hyperbolic description of blood flow in major arteries and veins, coupled to a lumped parameter description of the remaining main components of the human circulation. Derivation and analysis of the hyperbolic equations is carried out for blood vessels admitting variable material properties and with emphasis on the venous system, a much neglected aspect of cardiovascular mathematics. Veins, unlike their arterial counterparts, are highly deformable, even collapsible under mild physiological conditions. We address mathematical and numerical challenges. Regarding the numerical analysis of the hyperbolic PDEs, we deploy a modern non-linear finite volume method of arbitrarily high order of accuracy in both space and time, the ADER methodology. In vivo validation examples and brain haemodynamics computations are shown. We also point out two, preliminary but important new findings through the use of mathematical models, namely that extracranial venous strictures produce chronic intracranial venous hypertension and that augmented pressure increases the blood vessel wall permeability.