Journal of Biomechanics 47 (2014) 3361–3372

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Enhanced global mathematical model for studying cerebral venous blood flow Lucas O. Müller n, Eleuterio F. Toro Laboratory of Applied Mathematics, University of Trento, Via Mesiano 77, 38123 Trento, Italy

art ic l e i nf o

a b s t r a c t

Article history: Accepted 1 August 2014

Here we extend the global, closed-loop, mathematical model for the cardiovascular system in Müller and Toro (2014) to account for fundamental mechanisms affecting cerebral venous haemodynamics: the interaction between intracranial pressure and cerebral vasculature and the Starling-resistor like behaviour of intracranial veins. Computational results are compared with flow measurements obtained from Magnetic Resonance Imaging (MRI), showing overall satisfactory agreement. The role played by each model component in shaping cerebral venous flow waveforms is investigated. Our results are discussed in light of current physiological concepts and model-driven considerations, indicating that the Starling-resistor like behaviour of intracranial veins at the point where they join dural sinuses is the leading mechanism. Moreover, we present preliminary results on the impact of neck vein strictures on cerebral venous hemodynamics. These results show that such anomalies cause a pressure increment in intracranial cerebral veins, even if the shielding effect of the Starling-resistor like behaviour of cerebral veins is taken into account. & 2014 Elsevier Ltd. All rights reserved.

Keywords: Global circulation Mathematical model Cerebral blood flow Intracranial pressure Starling resistor

1. Introduction Recent developments on the potential link between extracranial venous anomalies and neurological conditions have increased interest on the physiology of cerebral venous return, currently, a poorly understood subject (Stoquart-ElSankari et al., 2009). A prominent example is Chronic CerebroSpinal Venous Insufficiency (CCSVI) and its potential link to Multiple Sclerosis (MS) (Zamboni et al., 2009). Interest in this area has been further increased by very recent findings on altered cerebrospinal fluid dynamics in MS patients (Magnano et al., 2012) and by reported improvements of such dynamics, as well the clinical course of the disease, after treatment of MS/CCSVI patients with percutaneous transluminal angioplasty (PTA) (Zivadinov et al., 2013). In this paper we are concerned with the development of a mathematical model to study cerebral haemodynamics. To this end we construct an extension of the closed-loop model for the cardiovascular system reported in Müller and Toro (2014) to account for two relevant factors. First we deal with the interaction of the cerebral vasculature with the pulsating intracranial pressure, for which we adopt the model proposed in Ursino (1988) and Ursino and Lodi (1997); this model describes the variation of

n

Corresponding author. E-mail address: [email protected] (L.O. Müller).

http://dx.doi.org/10.1016/j.jbiomech.2014.08.005 0021-9290/& 2014 Elsevier Ltd. All rights reserved.

intracranial pressure in time in terms of the variation of cerebral blood volume. Second, we incorporate a model to account for the Starling-resistor like behaviour of cerebral veins. This is supported by experimental evidence that shows that pressure in cerebral veins is always higher than intracranial pressure, for a wide range of intracranial pressures, independently of the pressure in downstream vessels, such as the dural sinuses (Johnston et al., 1974; Luce et al., 1982). The underlying mechanism is still the subject of debate, with some researchers speaking in favour of a purely hydraulic mechanism (Anile et al., 2009) and others hypothesizing a control mechanism (Dagain et al., 2009). Moreover, there is evidence of distinct morphological and mechanical properties of the terminal portion of cerebral veins, in the vicinity of dural sinuses (Vignes et al., 2007; Dagain et al., 2008, 2009; Chen et al., 2012). In order to model this behaviour, we have added cerebral veins to the venous network in Müller and Toro (2014) and implemented a simple model for Starling-resistor elements, proposed in Mynard (2011). These elements are placed in the vicinity of the point where cerebral veins drain into the dural sinuses. Our model is partially validated by comparing our computational results with MRI-derived flow measurements, obtaining satisfactory agreement. Computational results are discussed in light of current knowledge of the physiology of cerebral venous haemodynamics. As an example, we include here some preliminary results on the impact of neck vein strictures on cerebral venous haemodynamics. These results show that such anomalies cause a

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pressure increase in intracranial cerebral veins. The proposed mathematical model constitutes a computational tool suitable for the study of pathologies related to extracranial venous anomalies and their interaction with intracranial haemodynamics. This will be the subject of future investigations. The rest of the paper is structured as follows. Section 2 describes the extensions made to the model reported in Müller and Toro (2014). Section 3 contains computational results, comparison with MRI-derived measurements and a discussion of computational results. Section 4 contains an example to illustrate the applicability of the present model, while Section 5 includes a summary and conclusions.

2. Methods The model presented in this paper is an extension of the closed-loop model for the cardiovascular system presented in Müller and Toro (2014) to which the reader is referred to for further details.

system we adopt a tube law relating pðx; tÞ to Aðx; tÞ and other parameters, namely 

  Aðx; tÞ m Aðx; tÞ n pðx; tÞ ¼ pe ðx; tÞ þ KðxÞ  ð2Þ þ P0 ; A0 ðxÞ A0 ðxÞ here pe ðx; tÞ is the external pressure. K(x), m, n, A0 ðxÞ and P0 are parameters that account for mechanical and geometrical properties of the vessel. For a discussion on the choice of these parameters for both arteries and veins, see Müller and Toro (2014) and references therein. Fig. 1 depicts a schematic representation of the global model. Fig. 2 shows the venous network for the head and neck used here. Cortical cerebral veins draining into the superior and inferior sagittal sinuses have been added to the venous network in Müller and Toro (2014). Note that there is high variability in the number of veins draining into the superior sagittal sinus; our choice is in line with the numbers reported in Vignes et al. (2007). Moreover, a Starling-resistor element was added to the end of each cerebral vein (see Section 2.3). Table 1 shows geometrical and mechanical properties of vessels added or changed with respect to the venous network of Müller and Toro (2014). As a consequence of the venous network extension, the lumped-parameter compartment linking middle and anterior cerebral arteries to the inferior and superior sagittal sinuses has also been modified. The new configuration of this lumped compartment is shown in Fig. 3 and coefficients for lumped elements are found in Table 2. 2.2. Intracranial pressure

2.1. Mathematical model of the cardiovascular system Our mathematical model includes a one-dimensional description of the networks of major arteries and veins, and lumped-parameter models for the heart, the pulmonary circulation and capillary beds linking arteries and veins, see Fig. 1. Geometrical information for major head and neck veins is obtained from segmentation of MRI data. This patient-specific characterization allows us to compare computational results versus patient-specific MRI-derived flow quantification data, see Müller and Toro (2014). One-dimensional blood flow in elastic vessels is described by the following system of non-linear, first-order hyperbolic equations: 8 q ¼ 0; > < ∂t A þ ∂x
 q2 A ð1Þ > : ∂t q þ ∂x α^ A þ ρ ∂x p ¼  f ; here x is the axial coordinate of the vessel; t is the time; Aðx; tÞ is the cross-sectional area of the vessel; qðx; tÞ is the flow rate; pðx; tÞ is the average internal pressure over a cross-section; f ðx; tÞ is the friction force per unit length of the tube; ρ is the fluid density and α^ is a coefficient related to the assumed velocity profile. In what follows we assume ρ ¼ constant and α^ ¼ 1 (blunt velocity profile). To close the

The cranial cavity is conventionally regarded as a space of fixed volume containing the brain parenchyma, the cerebrospinal fluid (CSF) and the cerebral vasculature. The cranial cavity is then connected to the spinal cavity, which exhibits elastic behaviour, allowing for volume changes. Variations in intracranial blood volume produce fluctuations of intracranial pressure and, consequently, exchange of CSF between the intracranial and spinal subarachnoid spaces. For intracranial pressure pic, here we adopt the model proposed in Ursino (1988) and Ursino and Lodi (1997) given by C ic

dpic dV cv pc  pic pic  psss ¼ þ  ; dt dt Rf R0

ð3Þ

here pc and psss are capillary and superior sagittal sinus pressures, respectively. Vcv is the volume of the cerebral vasculature, given by the sum of the volume occupied by arteries, arterioles, capillaries, venules and veins inside the cranium. Cic is the intracranial compliance, given as C ic ¼

1 ; ke pic

ð4Þ

where ke is the elastance coefficient of the craniospinal system. Here we use 1 ke ¼ 0:15 ml (Ursino, 1988). Rf and R0 are CSF filtration and re-absorption resistances. CSF filtration from the subarachnoid space towards the dural sinuses

Fig. 1. Global, closed-loop model in Müller and Toro (2014). Major arteries and veins are modelled one-dimensionally; lumped-parameter models are used for heart, pulmonary circulation, arterioles, capillaries and venules. Left frame: major arteries; middle frame: lumped-parameter models; and right frame: major veins.

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Fig. 2. Head and neck veins of the present model. Numbers refer to those used in Table 1, where geometrical and mechanical parameters for each vessel are reported. has been shown to vary linearly with the pressure difference pic  psss (Ekstedt, 1 1978). CSF outflow resistance is taken as R0 ¼ 15 mmHg min ml (Ekstedt, 1978) throughout this paper. Following Ursino (1988), Ursino and Lodi (1997), in order to obtain an average CSF filtration flow rate of 400 μl min  1 , Rf is set to Rf ¼ 1 48:33 mmHg min ml . Note that since physiological cerebral blood flow is approximately 720 ml min  1, the time scale of intracranial pressure variations due to an imbalance of CSF filtration and absorption will be several orders of magnitude bigger than a cardiac cycle. Eq. (3) is solved with the forward Euler scheme and requires the computation of dV cv =dt, which is approximated as dV cv V ncv  V ncv 1 ¼ ; dt Δt

 The initialization pressures for arteries P 0a ¼ 70 mmHg and for veins P 0v ¼   

5 mmHg define the volumes of intracranial arteries and veins, V a ¼ 3:52 ml and V v ¼ 20:36 ml. Then, we set V al ¼ 16 ml þ ð30 ml  Va Þ, V cap ¼ 20 ml and V ven ¼ 70 ml þ ð13 ml  V v Þ. These volumes correspond to V 0cv ¼ 162 ml, with a distribution of cerebral blood volume (CBV) between arteries–arterioles and capillaries–venules–veins over total volume of 30% and 70%, respectively. The volume of each compartment is updated at each time step by the mass conservation principle.

n

where superscript n refers to the current time step of the simulation.

The computed intracranial pressure will act on intracranial onedimensional vessels, as well as on lumped-parameter models, as a prescribed external pressure pe ðx; tÞ in (2).

The initial volume of each compartment is calculated as follows: 2.3. Starling resistor

 The volume of intracranial vascular compartments is V a ¼ 30 ml, V al ¼ 16 ml, V cap ¼ 20 ml, V ven ¼ 70 ml and V v ¼ 13 ml, for arteries, arterioles, capillaries, venules and veins, respectively (Linninger et al., 2009).

The pressure difference that governs cerebral blood flow (CBF) in physiological conditions is the Cerebral Perfusion Pressure (CPP), defined as the difference between

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Table 1 Geometrical and mechanical parameters for veins added, or modified, to the venous network of Müller and Toro (2014). L: length; r0: inlet radius; r1: outlet radius; c0: wave speed for A ¼ A0 ; Type: vessel type (1: dural sinus, 9: cerebral vein); Ref: bibliographic source or MRI imaging segmented geometry. No.

Vessel name

L ðcmÞ

r0 (cm)

r1 (cm)

c0 (m/s)

Type

Ref.

103 105 106 149 150 151 152 153 157 158 159 161 162 165 236 237 238 239 241 245 248 249 260 261 262 263 264 265 266 267 268 269 270 271 272 273

Sup. sagittal sinus Inf. sagittal sinus Vein of Galen Sup. sagittal sinus R. Labbe v. L. Labbe v. Sup. sagittal sinus Sup. sagittal sinus Sup. sagittal sinus Cerebral vein Cerebral vein Cerebral vein Cerebral vein Sup. sagittal sinus Sup. sagittal sinus Cerebral vein Cerebral vein Cerebral vein Inf. sagittal sinus Cerebral vein Inf. sagittal sinus Cerebral vein Cerebral vein Terminal cerebral vein Terminal cerebral vein Terminal cerebral vein Terminal cerebral vein Terminal cerebral vein Terminal cerebral vein Terminal cerebral vein Terminal cerebral vein Terminal cerebral vein Terminal cerebral vein Terminal cerebral vein Terminal cerebral vein Terminal cerebral vein

2.50 3.67 0.60 4.33 5.00 5.00 4.33 2.50 5.00 5.00 5.00 5.00 5.00 4.33 2.00 5.00 5.00 5.00 3.67 3.00 3.67 3.00 3.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00

0.350 0.160 0.366 0.229 0.150 0.150 0.258 0.334 0.300 0.150 0.150 0.150 0.150 0.200 0.287 0.150 0.150 0.150 0.160 0.150 0.160 0.150 0.150 0.150 0.150 0.150 0.150 0.150 0.150 0.150 0.150 0.150 0.150 0.309 0.150 0.150

0.367 0.160 0.400 0.258 0.150 0.150 0.287 0.350 0.334 0.150 0.150 0.150 0.150 0.229 0.300 0.150 0.150 0.150 0.160 0.150 0.160 0.150 0.150 0.150 0.150 0.150 0.150 0.150 0.150 0.150 0.150 0.150 0.150 0.366 0.150 0.150

5.000 5.000 1.703 5.000 2.376 2.376 5.000 5.000 5.000 2.376 2.376 2.376 2.376 5.000 5.000 2.376 2.376 2.376 5.000 2.376 5.000 2.376 2.376 5.000 5.000 5.000 5.000 5.000 5.000 5.000 5.000 5.000 5.000 1.804 5.000 5.000

1 1 9 1 9 9 1 1 1 9 9 9 9 1 1 9 9 9 1 9 1 9 9 1 1 1 1 1 1 1 1 1 1 9 1 1

MRI MRI MRI MRI MRI MRI MRI MRI MRI Vignes Vignes Vignes Vignes MRI MRI Vignes Vignes Vignes MRI Vignes MRI Vignes Vignes Vignes Vignes Vignes Vignes Vignes Vignes Vignes Vignes Vignes Vignes MRI MRI MRI

et et et et

al. al. al. al.

(2007) (2007) (2007) (2007)

et al. (2007) et al. (2007) et al. (2007) et al. (2007) et et et et et et et et et et et et

al. al. al. al. al. al. al. al. al. al. al. al.

(2007) (2007) (2007) (2007) (2007) (2007) (2007) (2007) (2007) (2007) (2007) (2007)

Fig. 3. Lumped-parameter compartment G resulting from incorporating cortical veins. Numbers on the left refer to feeding arteries, see Müller and Toro (2014). Numbers on the right refer to veins draining into sagittal sinuses and are those in Table 1.

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Table 2 Parameters for artery–vein connection shown in Fig. 3, derived from Liang et al. (2009). Rda is resistance for distal arteries, in mmHg s ml  1, R is resistance in mmHg s ml  1, L is inductance in mmHg s2 ml  1 and C is capacitance in ml mmHg  1, for arterioles, capillaries and venules. Venule capacitance is divided into the superior sagittal sinus (SSS) and inferior sagittal sinus (ISS). Vessel numbers refer to Fig. 2 for veins and to network of Müller and Toro (2014) for arteries. Parent/daughter vessel Lumped model G 63 65 61 ISS 249 245 260 SSS 158 159 161 162 237 238 239

Rda

Ral

Lal

Cal

Rcp

Lcp

Ccp

Rvn

Lvn

Cvn

20.6859 – 20.6859 12.6609 – – – – – – – – – – – – –

24.4500 – 24.4500 10.5700 – – – – – – – – – – – – –

0.0179 – 0.0179 0.0078 – – – – – – – – – – – – –

0.0070 – 0.0070 0.0140 – – – – – – – – – – – – –

18.8100 18.8100 18.8100 18.8100 4.0700 – – – – – – – – – – – –

0.0059 0.0059 0.0059 0.0059 0.0013 – – – – – – – – – – – –

0.0007 – 0.0007 0.0014 – – – – – – – – – – – – –

– – – – – – 11.3400 11.3400 11.3400 – 4.8300 4.8300 4.8300 4.8300 4.8300 4.8300 4.8300

– – – – – – 0.0199 0.0199 0.0199 – 0.0109 0.0109 0.0109 0.0109 0.0109 0.0109 0.0109

– – – – – 0.0210 – – – 0.1050 – – – – – – –

Table 3 Location of Starling resistor elements in the venous network shown in Fig. 2. No.

Left vessel index

Right vessel index

1 2 3 4 5 6 7 8 9 10 11 12 13

158 159 161 162 237 238 239 249 245 260 271 150 151

261 262 263 264 265 266 267 268 269 270 106 272 273

Table 4 Initial pressure values for all compartments of the closed-loop model. Compartment

pðx; 0Þ (mmHg)

Arteries Veins Heart chambers and pulmonary compartments Arterioles Capillaries Venules Intracranial pressure

70.0 5 10.0 45.0 25.0 10.0 9.0

arterial and intracranial pressure (Johnston et al., 1974; Luce et al., 1982; Rossitti, 2013). This behaviour of CBF, where the downstream pressure is the intracranial pressure and not the central venous pressure, was the reason for the medical community to consider CBF as a flow phenomenon governed by a Starling resistor (Knowlton and Starling, 1912). The Starling resistor is an experimental device in which a collapsible tube is connected to two rigid tubes at its extremities and located in a chamber of fixed volume with variable ambient pressure (Conrad, 1969). One can change inlet, outlet and chamber pressure (often called external pressure) and observe the resulting flow patterns. The collapsible nature of the tube produces peculiar effects, such as flow limitation. In practice, there is a range of pressure values for which the flow rate through the tube becomes practically independent of the downstream pressure and the driving pressure difference is given by the upstream and external pressures. This happens when the external pressure is much higher than the outlet pressure. For a thorough analysis of the mathematical background of flow limitation the reader is referred to Shapiro (1977), for example.

Fig. 4. Cardiac-cycle averaged flow rate in head and neck veins: computational results versus MRI flow quantification data. SSS: superior sagittal sinus; StS: straight sinus; TS: transverse sinus; IJV: internal jugular vein. Vessel numbers refer to Fig. 2 and Table 1.

There is undeniable experimental evidence suggesting that for physiological conditions the brain as a whole behaves as a Starling resistor (Johnston et al., 1974; Luce et al., 1982; Rossitti, 2013). Intracranial veins are flexible thin-walled tubes surrounded by the brain parenchyma or CSF, depending on their location, that drain into the various dural sinuses (superior sagittal, inferior sagittal, straight and transverse sinuses). Dural sinuses are located in the dura matter and are therefore considerably more rigid than cerebral veins. Moreover, there is evidence of drastic variations of the morphology and mechanical properties of cerebral veins in the vicinity of their confluence with the dural sinuses (Vignes et al., 2007; Dagain et al., 2008, 2009; Yu et al., 2010; Chen et al., 2012). These considerations have led to the hypothesis that the terminal portion of cerebral veins acts as a Starling resistor (Johnston et al., 1974; Luce et al., 1982; Anile et al., 2009). These observations are supported by pressure measurements in animals (Johnston et al., 1974; Luce et al., 1982) that indicate that the relation pa Z pcv Z pic Z pds

ð5Þ

holds for a wide range of pic values. Here pa, pcv and pds are arterial, cerebral venous and dural sinuses pressures, respectively. As a first attempt to model this phenomenon, we introduce a Starling resistor (SR) in the terminal portion of cerebral veins of our venous network. We follow Mynard (2011), who made use of this concept to model the zero-flow pressure in

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Fig. 5. Computed pressure and flow rate in dural sinuses. PC-MRI flow quantification data is shown with symbols and dashed lines.

capillary beds. Flow across the SR element is 8 p p up ic > if pdown r pic ; > < R SR qSR ¼ p  p up down > > if pdown 4 pic ; : RSR

ð6Þ

where pup and pdown are pressures upstream and downstream of the SR element with respect to flow towards the heart, respectively. The resistance to flow in the SR element is taken as ( 0 RSR if pup Z pic ; RSR ¼ ð7Þ 1 if pup o pic ; R0SR

where is a reference resistance; considering that flow rate in individual cerebral 1 veins is always below 1 ml s  1, we choose R0SR ¼ 1 mmHg ml s. By doing so, we create a moderate pressure drop across the SR element, when open. Note that if the external pressure pic is higher than the venous pressure pup, the SR element collapses completely and does not allow for flow. Moreover, if the downstream

pressure pdown is lower than the external pressure pic, flow across the SR element is limited to that given by the pressure difference between the cerebral vein pup and the external pressure pic. SR elements are placed in the vicinity of where cerebral veins connect to dural sinuses. Table 3 shows the pairs of vessels between which an SR element has been placed. 2.4. Model setting One-dimensional vessels are discretised into computational cells of mesh size

Δx ¼ 1 cm, imposing that a single vessel should contain at least three computa-

tional cells. Eqs. (1) and (2) are solved numerically by first reformulating them (Toro and Siviglia, 2013) and then deploying a modern numerical method (Müller and Toro, 2013). For background on modern numerical methods for hyperbolic equations see Toro (2009). In this work, valves are modelled using the approach proposed in Mynard et al. (2012). The friction loss term f is taken as f ¼ 8 π μu=ρ, with μ ¼0.0045 Pa s and ρ ¼1060 kg/m3. At time t ¼0, uðx; 0Þ ¼ 0 m=s is imposed in the entire network and

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Fig. 6. Computed pressure and flow rate in internal jugular veins. PC-MRI flow quantification data is shown with symbols and dashed lines.

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pressure pðx; 0Þ in each model compartment is set as specified in Table 4. A periodic solution is obtained after 15–16 cardiac cycles. Results shown in the following sections are for the time interval 18 s r t r 19 s.

3. Results Here we describe numerical results focusing our attention on head and neck veins. We perform a set of simulations in which different components of the model are switched on and off in order to assess their influence on the shaping of cerebral venous flow waveforms.

effect of including both the intracranial pressure and the SR elements. Pressure in cerebral veins is always higher than intracranial pressure, while pressure in the superior sagittal sinus is governed by downstream conditions. If the SR elements are not considered, as shown in Fig. 7(b), cerebral veins and the superior sagittal sinus are directly connected, resulting in lower cerebral venous pressures. Another consequence of not considering SR elements is the lower intracranial pressure values, see Fig. 7(b), compared to those when SR elements are present, see Fig. 7(a). These lower values for intracranial pressure are a consequence of a reduction in the volume of cerebral veins. Results excluding the time variation of intracranial pressure over time are analogous to those previously discussed, see Fig. 7(c) and (d).

3.1. Computational results versus MRI measurements 3.3. Discussion on physiological aspects Fig. 4 shows a comparison of computed results against MRI measurements of cardiac-cycle averaged flow rates, for a selected set of head and neck veins. The agreement between computed and measured quantities is satisfactory. Flow distribution amongst cerebral vessels is reproduced well, though these quantities do not contain information on waveforms. Such description is represented in Figs. 5 and 6, which show a comparison of flow rate waves obtained by computation and MRI measurements, together with computed pressure waves. As for time-averaged quantities, the agreement is satisfactory. 3.2. Starling resistor and intracranial pressure effects on cerebral venous haemodynamics Fig. 7 shows the computed pressure in the superior sagittal sinus and in a cerebral vein, together with the intracranial pressure for four model settings depicted in Fig. 7(a) to (d). In Fig. 7(a) we observe the

The flow distribution between superior sagittal and straight sinus in this model is mainly determined by the configuration of the microcirculation network and affluent cerebral veins. These networks were constructed taking into account the rather constant ratio of flow distribution between superior and straight sinuses (Stoquart-ElSankari et al., 2009). The same inter-subject homogeneity observed for flow distributions ratios between straight and superior sagittal sinuses is not found between left and right transverse sinuses and, consequently, between left and right internal jugular veins (Stoquart-ElSankari et al., 2009). The agreement of flow distribution in these vessels between computational results and MRI-derived data, as shown in Fig. 4, is a result of the patient-specific characterization performed for major head and neck veins, as shown in Müller and Toro (2014). An interesting effect of SR elements on cerebral venous haemodynamics is the reduction of pulsatility of velocity in cerebral vessels, as

Fig. 7. Effect of a Starling resistor element and intracranial pressure on computed pressures. Results obtained using: (a) variable intracranial pressure and SR elements; (b) variable intracranial pressure and no SR elements; (c) constant intracranial pressure and SR elements; and (d) constant intracranial pressure and no SR elements. CV: cerebral vein (No. 158); ICP: intracranial pressure; and SSS: superior sagittal sinus (No. 165).

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Fig. 8. Effect of a Starling resistor element and intracranial pressure on computed velocity in cerebral venous vessels. CV: cerebral vein (No. 158); BVR: basal vein of Rosenthal (No. 247); SSS: superior sagittal sinus (No. 165); and STS: straight sinus (No. 104).

Fig. 9. Effect of a Starling resistor element and intracranial pressure on computed flow rate in dural sinuses. TS-R: right transverse sinus (No. 101); TS-L: left transverse sinus (No. 102); SSS: superior sagittal sinus (No. 103); and STS: straight sinus (No. 104).

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shown in Fig. 8. The reduced pulsatility of intracranial pressure as compared to the pulsatility of pressure in the dural sinuses greatly reduces the pulsatility of flow in cerebral veins and, as a consequence, in dural sinuses. This pulsatility reduction improves the agreement between computational results and MRI-derived flow rate waveforms, as shown in Fig. 9. These elements guarantee a pressure difference between cerebral veins and dural sinuses that is favourable to flow towards the heart over the entire cardiac cycle. Removing the SR elements from the network results in highly pulsatile flow in all dural sinuses, see Fig. 9. The shape of the intracranial pressure wave is defined by the interaction of variations in CBV and CSF exchange between the cranial and spinal subarachnoid spaces. Based on 65 short-term pressure recordings on patients with a variety of intracranial disorders, Avezaat and Eijndhoven (1986) showed that mean pulsatile change in CBV is 1.6 ml (0.36–4.38 ml). In our model this value is around 1 ml. Moreover, cerebral arterial plus arteriolar blood volume is 30% of total CBV, while venous plus capillary fraction is around 70%, in agreement with measurements reported in Ito et al. (2001). This distribution pattern was assigned as initial condition, see Section 2.2, and is maintained throughout the cardiac cycle, as it can be concluded from Fig. 10. The role played by each vascular compartment in shaping intracranial pressure can be better understood by considering Fig. 10, which shows the time variation of blood volume V minus the average blood volume Vav occupied by each compartment. The leading compartment in terms of volume changes is the one representing distal arteries and arterioles (Val). On the other hand, veins seem to play a rather passive role, being compressed by high intracranial pressure and expanding during the diastolic phase. The shape of flow rate waveforms in dural sinuses is less pulsatile than venous flow in neck veins (Stoquart-ElSankari et al., 2009). This observation is also found in the MRI measurements reported in the present work and confirmed by our computational results. Some authors claim that the pulsatility of cerebral venous flow is due to the compression of the venous vasculature by intracranial pressure and use this concept to reproduce the jugular venous flow waveforms (Kim et al., 2007). The results presented in this work suggest that this phenomenon influences cerebral venous flow only marginally and could by no means be the explanation of venous flow waveforms at the level of internal jugular veins. In fact, volume changes in venules and veins over the cardiac cycle are rather modest, as shown in Fig. 10.

4. Example: impact of neck vein structures on intracranial haemodynamics Here, as an example of the applicability of the present model, we show computations from a preliminary study on the impact of extracranial venous anomalies on extra- and intracranial venous haemodynamics for one of the CCSVI patterns identified in Zamboni et al. (2009). Our CCSVI case study is modelled by introducing stenosis to the reference venous network introduced in Section 2, through a reduction of the vessel equilibrium crosssectional area A0 ðxÞ to a stenotic reference area As0 ðxÞ ¼ 0:1A0 ðxÞ in vessels 224, 225 and 244 of Fig. 2; these correspond to the right internal jugular vein, the left internal jugular vein and the azygous vein, respectively. In the case of a stenosis, an additional source term is included in the second equation of system (1), according to the expression proposed in Seeley and Young (1976). From now on we will refer to the reference configuration as the Healthy Control (HC). In Fig. 11(a) and (b) we show the computed pressure and flow rate for the HC and the CCSVI cases at post- and pre-stenotic locations. Pressure drops between these two locations are almost negligible in the case of the healthy subject, whereas in the case of stenotic vessels the mean pressure experiences a drop of around 1.5 mmHg, in agreement with measured pressure drops in stenotic internal jugular veins, Zamboni et al. (2009). Another direct consequence of the presence of a stenosis is the drastic reduction of blood flow in the corresponding vessel. For example, the pressure increment caused by the stenosis results in a reduction of average flow rate from 9.83 ml s  1 to 3.90 ml s  1, a reduction of 60%. In Fig. 11(c) we show pressure and flow rate in the superior sagittal sinus. As expected, pressure increments at extracranial locations are transmitted upstream and can be encountered in the dural sinuses. The next aspect to be investigated is whether the observed extracranial pressure increments actually affect the pressure field in cerebral veins. These vessels are located upstream of Starling resistors and consequently the pressure in those locations is higher than the intracranial pressure, which in turn is higher than that of the dural sinuses. We find that there is also a pressure increment in cerebral venous pressure as a consequence of extracranial stenoses. However, this effect is more involved and the mechanisms at play become apparent through the computational model, as is now explained. Pressure increments in dural sinuses affect CSF absorption, which depends linearly on the pressure difference between the intracranial pressure and the superior sagittal sinus pressure. In order to maintain a balance between CSF generation and absorption, intracranial pressure in the CCSVI case must be higher than that of the HC. In the study presented here, intracranial pressure increases from 12.8 mmHg to 14.1 mmHg, reaching a periodic solution after 2000 cardiac cycles. As a consequence of the observed intracranial pressure increment and the presence of the Starling resistor, pressure in cerebral veins for the CCSVI case increases, compared to that of the healthy control, see Fig. 11(d). The observed venous pressure increase is chronic but its clinical significance is uncertain.

5. Concluding remarks

Fig. 10. Variation in time of V  V av , where V is the volume of a compartment and Vav is the average volume over the cardiac cycle of different compartments. Five compartments were considered, namely: A, arteries; V, veins; Al, distal arteries and arterioles; Cp, capillaries; and Vn, venules.

We have extended the closed-loop model for the cardiovascular system, with emphasis on the cerebral venous system, presented in Müller and Toro (2014) to account for the effect of intracranial pressure and Starling-resistor elements on cerebral venous haemodynamics. Computational results have been compared to MRIderived flow measurements, observing overall satisfactory agreement. Major determinants of cerebral venous flow waveforms have been studied and discussed. Our results indicate that the

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Fig. 11. Computed results for veins at different locations for two model configurations: the standard venous network (HC) and the CCSVI case. Left column shows location of the point of interest; middle column shows pressures; and right column shows flow rates. Numbers in square brackets represent averages over the cardiac cycle.

Starling-resistor like behaviour of intracranial veins at the point where they join dural sinus is the main determinant of intracranial venous waveforms. As an example of the applicability of our model, we have presented a preliminary study of the impact of neck vein strictures on intracranial venous haemodynamics; the results show that neck vein strictures do cause a pressure increase at the intracranial level. The present model is a suitable computational tool to study a wide range of pathologies associated to extracranial venous anomalies, which presumably have an impact on intracranial haemodynamics; these issues will be the subject of future investigations. It is appropriate to point out some limitations of the presented model. The intracranial pressure lumped-parameter submodel used here does not admit spatial variability. Consequently, pressure changes affect all vascular components instantaneously and in the same manner. In order to gain a deeper understanding on the effect of blood flow on CSF dynamics, the model should

include a more sophisticated description for intracranial pressure. Such model should be able to account for the space- and timedependent CSF dynamics.

Conflict of interest statement None declared.

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

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References Anile, C., De Bonis, P., Di Chirico, A., Ficola, A., Mangiola, A., Petrella, G., 2009. Cerebral blood flow autoregulation during intracranial hypertension: a simple, purely hydraulic mechanism. Child's Nerv. Syst. 25, 325–335. Avezaat, C.J., Eijndhoven, J.M., 1986. The role of the pulsatile pressure variations in intracranial pressure monitoring. Neurosurg. Rev. 9 (1–2), 113–120. Chen, J., Wang, X., Luan, L., Chao, B., Pang, B., Song, H., Pang, Q., 2012. Biological characteristics of the cerebral venous system and its hemodynamic response to intracranial hypertension. Chin. Med. J. 125, 1303–1309. Conrad, W.A., 1969. Pressure–flow relationships in collapsible tubes. IEEE Trans. Biomed. Eng. 16 (4), 284–295. Dagain, A., Vignes, J.R., Dulou, R., Dutertre, G., Delmas, J.M., Guerin, J., Liguoro, D., 2008. Junction between the great cerebral vein and the straight sinus: an anatomical, immunohistochemical, and ultrastructural study on 25 human brain cadaveric dissections. Clin. Anat. 21 (5), 389–397. Dagain, A., Vignes, R., Dulou, R., Delmas, J., Riem, T., Guerin, J., Liguoro, D., 2009. Study of the junction between the cortical bridging veins and basal cranial venous sinus. Neurochirurgie 55, 19–24. Ekstedt, J., 1978. CSF hydrodynamic studies in man. 2. Normal hydrodynamic variables related to CSF pressure and flow. J. Neurol. Neurosurg. Psychiatry 41 (4), 345–353. Ito, H., Kanno, I., Iida, H., Hatazawa, J., Shimosegawa, E., Tamura, H., Okudera, T., 2001. Arterial fraction of cerebral blood volume in humans measured by positron emission tomography. Ann. Nucl. Med. 15, 111–116. Johnston, I.H., Rowan, J.O., Harper, A.M., Jennett, W.B., 1974. Raised intracranial pressure and cerebral blood flow. IV. Intracranial pressure gradients and regional cerebral blood flow. J. Neurol. Neurosurg. Psychiatry 37, 392–402. Kim, J., Thacker, N.A., Bromiley, P.A., Jackson, A., 2007. Prediction of the jugular venous waveform using a model of CSF dynamics. Am. J. Neuroradiol. 28, 983–989. Knowlton, F.P., Starling, E.H., 1912. The influence of variations in temperature and blood pressure on the performance of the isolated mammalian heart. J. Physiol. 44, 206–219. Liang, F.Y., Takagi, S., Himeno, R., Liu, H., 2009. Biomechanical characterization of ventricular–arterial coupling during aging: a multi-scale model study. J. Biomech. 42, 692–704. Linninger, A.A., Xenos, M., Sweetman, B., Ponkshe, S., Guo, X., Penn, R., 2009. A mathematical model of blood, cerebrospinal fluid and brain dynamics. J. Math. Biol. 59, 729–759. Luce, J.M., Huseby, J.S., Kirk, W., Butler, J., 1982. A Starling resistor regulates cerebral venous outflow in dogs. J. Appl. Physiol. 53, 1496–1503. Magnano, C., Schirda, C., Weinstock-Guttman, B., Wack, D.S., Lindzen, E., Hojnacki, D., Bergsland, N., Kennedy, C., Belov, P., Dwyer, M.G., Poloni, G.U., Beggs, C., Zivadinov, R., 2012. Cine cerebrospinal fluid imaging in multiple sclerosis. J. Magn. Reson. Imaging 36, 825–834.

Müller, L.O., Toro, E.F., 2013. Well-balanced high-order solver for blood flow in networks of vessels with variable properties. Int. J. Numer. Methods Biomed. Eng. 29, 1388–1411. Müller, L.O., Toro, E.F., 2014. A global multiscale mathematical model for the human circulation with emphasis on the venous system. Int. J. Numer. Methods Biomed. Eng. 30, 681–725. Mynard, J.P., 2011. Computer Modelling and Wave Intensity Analysis of Perinatal Cardiovascular Function and Dysfunction (Ph.D. thesis). Department of Paediatrics, The University of Melbourne. Mynard, J.P., Davidson, M.R., Penny, D.J., Smolich, J.J., 2012. A simple, versatile valve model for use in lumped parameter and one-dimensional cardiovascular models. Int. J. Numer. Methods Biomed. Eng. 28 (6–7), 626–641. Rossitti, S., 2013. Pathophysiology of increased cerebrospinal fluid pressure associated to brain arteriovenous malformations: the hydraulic hypothesis. Surg. Neurol. Int. 4, 42. Seeley, B.D., Young, D.F., 1976. Effect of geometry on pressure losses across models of arterial stenoses. J. Biomech. 4, 430–448. Shapiro, A.H., 1977. Steady flow in collapsible tubes. J. Biomech. Eng. 99, 126–147. Stoquart-ElSankari, S., Lehmann, P., Villette, A., Czosnyka, M., Meyer, M., Deramond, H., Baledent, O., 2009. A phase-contrast MRI study of physiologic cerebral venous flow. J. Cereb. Blood Flow Metab. 29, 1208–1215. Toro, E.F., 2009. Riemann Solvers and Numerical Methods for Fluid Dynamics: A Practical Introduction, third ed. Springer-Verlag, Berlin, Heidelberg, ISBN 978-3540-25202-3. Toro, E.F., Siviglia, A., 2013. Flow in collapsible tubes with discontinuous mechanical properties: mathematical model and exact solutions. Commun. Comput. Phys. 13 (2), 361–385. Ursino, M., 1988. A mathematical study of human intracranial hydrodynamics part 1. The cerebrospinal fluid pulse pressure. Ann. Biomed. Eng. 16 (4), 379–401. Ursino, M., Lodi, C.A., 1997. A simple mathematical model of the interaction between intracranial pressure and cerebral hemodynamics. J. Appl. Physiol. 82, 1256–1269. Vignes, J., Dagain, A., Guerin, J., Liguoro, D., 2007. A hypothesis of cerebral venous system regulation based on a study of the junction between the cortical bridging veins and the superior sagittal sinus. J. Neurosurg. 107, 1205–1210. Yu, Y., Chen, J., Si, Z., Zhao, G., Xu, S., Wang, G., Ding, F., Luan, L., Wu, L., Pang, Q., 2010. The hemodynamic response of the cerebral bridging veins to changes in ICP. Neurocrit. Care 12 (1), 117–123. Zamboni, P., Galeotti, R., Menegatti, E., Malagoni, A.M., Tacconi, G., Dall'Ara, S., Bartolomei, I., Salvi, F., 2009. Chronic cerebrospinal venous insufficiency in patients with multiple sclerosis. J. Neurol. Neurosurg. Psychiatry 80, 392–399. Zivadinov, R., Magnano, C., Galeotti, R., Schirda, C., Menegatti, E., WeinstockGuttman, B., Marr, K., Bartolomei, I., Hagemeier, J., Malagoni, A., Hojnacki, D., Kennedy, K., Carl, E., Beggs, C., Salvi, F., Zamboni, P., 2013. Changes of cine cerebrospinal fluid dynamics in patients with multiple sclerosis treated with percutaneous transluminal angioplasty: a case-control study. J. Vasc. Interv. Radiol. 24 (6), 829–838.