Is posture related craniospinal compliance shift caused by jugular vein collapse? A theoretical analysis

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1 DOI /s Fluids and Barriers of the CNS RESEARCH Open Aess Is posture related raniospinal ompliane shift aused by jugular vein ollapse? A theoretial analysis Manuel Gehlen 1,2*, Vartan Kurtuoglu 2,3 and Marianne Shmid Daners 4 Abstrat Bakground: Postural hanges are related to hanges in erebrospinal fluid (CSF) dynamis. While sitting up leads to a derease in ranial CSF pressure, it also auses shifts in the raniospinal CSF volume and ompliane distribution. We hypothesized that jugular vein ollapse in upright posture is a major ontributor to these shifts in CSF volume and ompliane. Methods: To test this hypothesis, we implemented a mathematial lumped-parameter model of the CSF system and the relevant parts of the ardiovasular system. In this model, the CSF and the venous system are eah divided into a ranial and a spinal part. The pressures in these ranial and spinal portions differ by the posture-dependent hydrostati pressure olumns in the onneting vessels. Jugular ollapse is represented by a redution of the hydrostati pressure differene between ranial and spinal veins. The CSF pressure volume relationship is implemented as a funtion of the loal CSF to venous pressure gradient. This implies that an inrease in CSF volume leads to a simultaneous displaement of blood from adjaent veins. CSF pulsations driven by the ardiovasular system are introdued through a pulsating ranial arterial volume. Results: In upright posture, the implemented CSF pressure volume relationship shifts to lower ranial CSF pressures ompared to the horizontal position, leading to a derease in ranial CSF pressure when sitting up. Conurrently, the ompliane of the spinal ompartment dereases while the one of the ranial ompartment inreases. With this, in upright posture only 10% of the CSF system s ompliane is provided by the spinal ompartment ompared to 35% in horizontal posture. This redution in spinal ompliane is aompanied by a audal shift of CSF volume. Also, the ability of the spinal CSF ompartment to ompensate for erebral arterial volume pulsations redues in upright posture, whih in turn redues the alulated raniospinal CSF flow pulsations. Conlusion: The mathematial model enabled us to isolate the effet of jugular ollapse and quantify the indued shifts of ompliane and CSF volume. The good onordane of the modelled hanges with linially observed values indiates that jugular ollapse an be onsidered a major ontributor to CSF dynamis in upright posture. Keywords: Craniospinal, Compliane, Cerebrospinal fluid dynamis, Posture Bakground Several pathologies of the entral nervous system, like hydroephalus and syringomyelia, are aused or haraterized by altered erebrospinal fluid (CSF) dynamis. Therefore, the treatment of these onditions typially *Correspondene: mgehlen@ethz.h 1 Institute for Dynami Systems and Control, Department of Mehanial and Proess Engineering, ETH Zurih, Zurih, Switzerland Full list of author information is available at the end of the artile aims at restoring physiologial irulation of CSF and requires profound knowledge of the underlying pathophysiology. However, CSF dynamis are mostly studied in horizontal posture, even though we spend most of our time upright and CSF dynamis fundamentally hange with posture. For example, sitting up not only leads to hanges in intraranial pressure (ICP), but also to a audal shift of CSF volume and an inversion of the ompliane distribution between the ranial and the The Author(s) This artile is distributed under the terms of the Creative Commons Attribution 4.0 International Liense ( whih permits unrestrited use, distribution, and reprodution in any medium, provided you give appropriate redit to the original author(s) and the soure, provide a link to the Creative Commons liense, and indiate if hanges were made. The Creative Commons Publi Domain Dediation waiver ( publidomain/zero/1.0/) applies to the data made available in this artile, unless otherwise stated.

2 Page 2 of 11 spinal part of the CSF system. This inversion of the raniospinal ompliane was first observed by Magnaes in a small number of subjets [1]. In a reent study by Alperin et al. [2], the pulse amplitude of raniospinal CSF flow reorded with magneti resonane imaging (MRI), dereased in sitting posture, whih supports the findings of Magnaes. In CSF shunts, anti-siphon devies are used to ounterat posture-related hanges in pressures. However, the diversity of funtional priniples on whih these devies are based, indiates that the mehanisms of the posture-related hanges in CSF dynamis and their link to hemodynamis are largely unknown [2]. Knowing the ausalities of these interations would ontribute to the understanding of individual pathologies and to the hoie of the most appropriate treatment option, espeially in the ontext of various omorbidities typially seen in these patients. What we do know is that CSF pressure in equilibrium onditions is a funtion of venous pressure through Davson s equation [3], and that at least ranial venous pressure hanges with posture due to hydrostati gradients along the blood vessels. Also, ranial venous pressure hanges with the state of the jugular veins: when they ollapse in upright posture venous resistane inreases, reduing both the posture-related derease in ranial venous pressure and in CSF pressure [4 6]. We hypothesized that the ollapse of the jugular veins when upright not only affets mean ICP, but that it also auses the aforementioned audal shift of CSF volume: interruption of the venous hydrostati pressure olumn dereases the ranial CSF to venous pressure gradient by diminishing the redution in ranial venous pressure when sitting up. Due to the exponential nature of the CSF system s pressure volume relationship [7], this in turn, inreases ranial ompliane in upright posture. At the same time, the non-interrupted hydrostati pressure olumn leads to an inreased CSF to venous pressure gradient below the level of the jugular veins, ausing the observed audal shift of CSF volume. Consequently, the spinal dural sa volume inreases, reduing the ompliane of the spinal CSF spae [1]. We aimed at testing this hypothesis by implementing a mathematial model of the CSF system and the relevant parts of the ardiovasular system. This has enabled us to isolate the effet of jugular vein ollapse and quantify the indued shifts of ompliane and CSF volume. These estimated hanges in CSF dynamis were then ompared to the measurements of Magnaes [1]. Testing the hypothesis without a mathematial model would be diffiult, as jugular ollapse an hardly be avoided in vivo. To allow for further model validation, we omputed hanges in raniospinal CSF flow seondary to hanges in raniospinal ompliane distribution. Unlike the distribution of ompliane itself, hanges in CSF flow an be easily measured with MRI and used as surrogate for hanges in ompliane distribution. With this, we were able to validate the model by omparing the raniospinal flow rates estimated by the model to reported flow rates reorded in supine and sitting posture [2, 8, 9]. Methods We used a lumped parameter desription of the interation between the CSF and ardiovasular systems as shown in Fig. 1. As most of the CSF system s ompensatory reserve is provided by onurrent venous volume adaptation, the CSF pressure volume relationship was implemented as a funtion of the loal CSF to venous pressure gradient [10, 11]. This implies that an inrease in CSF volume leads to a simultaneous displaement of Fig. 1 Model shemati: interation between CSF and ardiovasular systems in upright posture. In the mathematial model, CSF and venous blood are divided into ranial and spinal ompartments. The orresponding pressures at the ranial (p CSF and p v ) and spinal (p CSF and p v ) level differ by hydrostati pressure olumns that are haraterized by the distanes l s and l jug. The interation of CSF and venous blood is determined by the loal pressure volume relationships ( V and V s ). The pulsating arterial blood flow Q a leads to a pulsating hange in the ranial arterial blood volume V a, whih is ompensated by raniospinal flows of CSF and venous blood (Q CSF and Q v ). CSF formation (Q form ) and absorption (Q abs ) are also indiated

3 Page 3 of 11 venous blood from adjaent veins. In upright posture, hydrostati pressure gradients between different loations in the CSF spae and in the venous system beome relevant. In our model, the CSF spae and the venous ompartment were divided into a ranial and a spinal portion, and CSF and venous pressures were evaluated at these two loations. While in horizontal posture these ranial and spinal pressures are approximately equal, they differ in upright posture due to the hydrostati gradients in the onneting vessels. If the jugular veins did not ollapse in upright posture, spinal and ranial venous pressures would differ by the same hydrostati pressure olumn as spinal and ranial CSF pressure, respetively. However, in upright posture and for reasonably low entral venous pressures, this hydrostati pressure gradient is interrupted by the ollapsing jugular veins [4, 5]. The ollapsed segment of the jugular veins ats as a differential pressure valve with opening pressure equal to ambient pressure. Therefore, in upright posture, erebral venous outflow is partly redireted through the highresistane pathway presented by the vertebral veins, and venous pressure at the site of the ollapse, is regulated to ambient pressure by the jugular veins [4]. Thus, erebral venous pressure is only determined by the hydrostati pressure gradient above the site of the ollapse [4, 5]. The referene level of the spinal portion was hosen suh that the spinal venous pressure is independent of posture. In ontrast to the venous blood vessels, the hydrostati olumn in the CSF system was assumed to be uninterrupted. Cerebrospinal fluid pulsations driven by the ardiovasular system were aounted for through a pulsating ranial arterial volume. The pulsations of this arterial volume were based on reorded flow rates in the internal arotid and vertebral arteries. As CSF ompetes with the arterial pulsations for the available ompliane, the arterial volume was added to the ranial CSF volume. Instantaneous flow rates for CSF and venous blood between ranial and spinal ompartments were alulated based on a volume balane, assuming onstant ranial volume (Monroe Kelly dotrine). Model derivation Loal pressure volume relationships The pressure gradient p between the CSF pressure p CSF and the venous pressure p v required for the displaement of venous blood was desribed by an exponential funtion: p = p CSF p v = p 1 e E V + p 0, where V is the CSF volume inrease from baseline (supine equilibrium), and E, p 1, and p 0 are onstants (Table 1). This exponential funtion was derived from the exponential pressure volume relation originally desribed by Marmarou [7, 12, 13]: p CSF = p 1 e E V + p M 0, where the referene pressure p0 M is the sum of the postural pressure omponent p 0 and the venous pressure pv s [11, 14]. Desribing the pressure volume relation as a funtion of the CSF to venous pressure gradient, p, allows hanges in venous pressure to be aounted for [10, 11]. In the model, the CSF spae and the venous ompartment were divided into a ranial and a spinal part. Thus, an inrease in CSF volume an be ompensated by a displaement of venous blood from the ranium ( V ) (1) (2) Table 1 Parameters, distintive for normal pressure hydroephalus Parameter Symbol Value Unit Referene Elastane E 0.23 ml 1 [9, 16]* Exponential parameter p 1 4 mmhg [16]* Offset pressure p mmhg [16]* Rate of CSF formation Q form 0.35 ml/min [13] Total CSF outflow resistane Rabs tot 20.6 mmhg/(ml/min) [13, 16]* Relative spinal ompliane k V 0.35 [17] Spinal venous pressure p s v 5.3 mmhg [4] Distane between spinal and ranial referene points l s 33.8 m [4] Distane between jugular veins and ranial referene point l jug 11.0 m [4] Density of CSF ρ CSF 1000 kg/m 3 Density of blood ρ blood 1060 kg/m 3 Gravitational aeleration g 9.81 m/s 2 Relative spinal outflow ondution k R n/a # * Computed from data given in the referened work: E = ln (RPPC + 1)/ V a [18], p 1 = ICP r p 0 p s v, p 0 = p M 0 ps v, Rtot abs = (ICP r p s v )/Q form [3] # The value of this parameter is unknown. It is estimated in this study

4 Page 4 of 11 or from venous vessels adjaent to the spinal anal and the spinal theal sa ( V s ): V tot = V + V s. (3) In horizontal posture, where ranial and spinal CSF as well as venous pressures an be assumed equal (pcsf = ps CSF and p v = ps v ), the ombined pressure volume relationship V tot must be equal to the established relation (Eq. 2). Thus, the loal pressure volume relationships in the spinal and the ranial ompartment were implemented as in Eq. 1, but as funtions of the respetive loal CSF to venous pressure gradients: V ( pcsf ) = (1 kv ) 1 ( p E ln CSF pv p ) 0 V s( pcsf s ) = kv 1 E ln Here, k V is a onstant that desribes the portion of the total ompensatory reserve of the CSF system attributed to the spinal ompartment. In horizontal posture, k v is the spinal ompliane ontribution as measured by Magnaes [1]. Hydrostati pressure gradients In upright posture, spinal CSF pressure pcsf s is higher than the ranial CSF pressure pcsf due to the hydrostati pressure olumn of length l s : pcsf s = p CSF + ρ CSF g l s, (6) where l s is the vertial distane between the referene points of the spinal and the ranial ompartments (Fig. 1), ρ CSF is CSF density, and g is gravitational aeleration. pcsf is often referred to as ICP. If lumbar CSF pressure is measured, Eq. 6 is aounted for by sensor alibration [4]. For the spinal part of the model, the hydrostati indifferene point of the venous system was hosen as the referene loation. Thus, the spinal venous pressure pv s was assumed to be independent of posture. Without ollapse of the jugular veins, ranial and spinal venous pressures would also differ by a hydrostati olumn of length l s : pv = ps v ρ blood g l s. (7) However, sine venous pressure is equal to ambient pressure at the loation where the jugular veins ollapse, erebral venous pressure is determined by the hydrostati pressure gradient of length l jug instead [4, 5]: p v = ρ blood g l jug p 1 ( p s CSF p s v p 0 p 1 ). (4) (5) (8) l jug is the distane between the upper end of the jugular ollapse and the referene point of the ranial ompartment. Using Eqs. 6 and 8, the CSF pressure volume relationships (Eqs. 4, 5) an also be written, for the upright posture, as a funtion of only the ranial CSF pressure (pcsf ): V ( p ) up CSF = (1 k V ) 1 ( p ) E ln CSF + ρ blood g l jug p 0 p 1 (9) V s( pcsf ) up = k V 1 E ln ( p CSF + ρ CSF g l s p s v p 0 (10) Compliane Compliane C is defined as hange of volume relative to the orresponding hange in pressure [7]. It was alulated analytially as the slope of the pressure volume urves (Eqs. 4, 5, 9, 10): C ( p ) hor CSF = d V dpcsf = 1 k V 1 hor E pcsf ps v p 0 (11) C s( pcsf ) hor = d V s = k V hor E 1 pcsf ps v p 0 dp CSF C ( pcsf ) up = d V dpcsf = 1 k V up E CSF formation and absorption The rate of CSF formation Q form was implemented as posture independent and onstant [13] (Table 1). The ranial and the spinal CSF absorption rates were assumed to be proportional to the loal CSF to venous pressure gradient, p i : ( ) Qabs i ( pi ) = pcsf i pi v /Rabs i (15) where Rabs i is the loal CSF outflow resistane. While the overall outflow resistane Rabs tot an be determined linially [15], its raniospinal distribution haraterized by the oeffiient, k R, is generally unknown. p 1 ). (12) 1 pcsf + ρ blood g l jug p 0 (13) C s( p ) up CSF = d V s dpcsf = k V up E 1 pcsf + ρ CSF g l s pv s p. 0 (14) 1 R tot abs = k R /Rabs tot }{{} 1/Rabs s + (1 k R )/R tot abs }{{} 1/R abs (16)

5 Page 5 of 11 Model parameters The parameters used for the alulations in this study (Table 1) are harateristi for patients with normal pressure hydroephalus (NPH). They desribe a patient with 12.5 mmhg resting intraranial pressure (ICP r ). Sensitivity analysis To analyze the sensitivity of the investigations with respet to the employed parameter values, a three step sensitivity analysis was performed. First, all alulations were repeated with a seond parameter set (E = 0.1/mL, p 1 = 10 mmhg, p 0 = 5.3 mmhg, Rabs tot = 13.4 mmhg/(ml/min)) that desribes physiologial CSF dynamis [19]. Seond, the parameters determining the hydrostati gradients within the CSF and the venous system (l s, l jug, and pv s ) were varied within reported standard deviations (l s = 33.8 ± 2.5 m, pv s = 5.3 ± 2.5 mmhg) [4] one at a time. Third, the ompliane distribution assumed in horizontal position was varied by ±50% (k V = 0.35 ± 0.175). Cranial arterial volume Given that arterial pressure is substantially higher than CSF pressure in all but the most extreme pathologi onditions, arterial blood flow rate to the ranium Q a was assumed unaffeted by CSF dynamis. Therefore, the hange in ranial arterial volume an be derived from in vivo measurements of Q a. We used flow rates reorded by phase-ontrast MRI in the internal arotid and vertebral arteries as arterial blood flow, Q a. These flow rates were obtained from the average of 16 NPH patients [9]. Additionally, the flow rates of a healthy volunteer in supine and sitting position [2] were applied to validate the predited hanges in raniospinal CSF flow. The flows leaving the ranial arterial ompartment are the apillary blood flow and the rate of CSF formation, both of whih were assumed non-pulsatile. Therefore, the volume balane in the ranial arterial ompartment redues to V a (t) = t 0 Q a (t) Q a dt, (17) where Q a is the mean arterial flow rate over one ardia yle. Evaluation Sitting up Immediately upon sitting up, total CSF volume is idential to the equilibrium volume in horizontal position. In other words, the hange in total CSF volume is initially zero. The orresponding CSF pressure in upright posture was determined by numerially solving V tot( pcsf )! up = V tot( pcsf ) hor = 0 for the ranial CSF pressure pcsf. While total CSF volume will not hange immediately after hanging posture, a rapid audal shift of CSF volume through the unrestrited CSF pathways an be expeted. This shifted volume orresponds to the hange in spinal CSF volume V s (Eq. 10) evaluated at the above alulated CSF pressure. Upright equilibrium The upright equilibrium is reahed when CSF absorption and formation rates are equal. However, the pressuredependent CSF absorption rate an only be alulated for a known raniospinal absorption distribution (k R ). Thus, Magnaes observation [1] of unhanged total ompliane in upright posture was used to determine CSF pressure in upright equilibrium: C tot( pcsf )! up = C tot( pcsf ) hor (18) (19) Then, the ratio k R that leads to equal CSF formation and absorption at this CSF pressures was alulated:! Q form = p CSF p v Rabs + ps CSF ps v Rabs s + 1 k R ( p s Rabs tot CSF pv s ). = k R ( p Rabs tot CSF pv) (20) Simulation of raniospinal flow rates Any inrease or derease in volume of one entity (fluids and tissue) within the ranium has to be ompensated, respetively, by an equivalent volume derease or inrease of the other entities (Monroe Kelly dotrine) [20]. Therefore, the pulsating volume inrease of arterial blood in the ranium V a (Eq. 17) has to be ompensated by a redution in CSF or venous blood volume. Similar to hanges in ranial CSF volume, a hange in the ranial arterial volume an be ompensated by either a shift of CSF from or to the spinal ompartment or a redution or inrease of ranial venous blood volume. With this, ranial arterial blood diretly ompetes with CSF for the available ompliane, and the total amount of displaed venous blood V tot is equal to the sum of hanges in CSF volume V CSF and arterial volume V a (t): V CSF (t) + V a (t) =! V tot( pcsf ) (21) Solving this equation for the ranial CSF pressure p CSF allows for determining the urrent pressure-dependent CSF absorption rates and alulating the CSF volume V CSF, whih may flutuate throughout a ardia yle:

6 Page 6 of 11 d dt V CSF (t) = Q form Qabs ( ) ( ) p CSF Q s abs p CSF. (22) The CSF volume and the ranial CSF pressure during a ardia yle were omputed by solving this system of differential algebrai equations (Eqs. 21, 22) using the Matlab (The MathWorks, In., Natik, MA, USA) variable-order solver ode15s. Based on these omputations, the CSF flow rate into the spinal ompartment was alulated as the hange in spinal CSF volume: Q CSF (t) = d dt V s( p CSF (t)), (23) and the raniospinal venous flow rate was alulated based on a volume balane in the ranium: Q v (t) = Q a (t) Q CSF (t). (24) Results After alulating the loal and total pressure volume relationships of the CSF spae in horizontal and upright posture, these orrelations were used to derive the loal and total omplianes. Based on this, CSF volume and pressure in upright posture were determined under the assumption of unhanged CSF volume (Eq. 18) or under the assumption of unhanged total ompliane (Eq. 19). The posture-related volume and ompliane shifts were then evaluated under these two onditions. Finally, the model output was alulated (Eqs. 21, 22) for one ardia yle and the raniospinal flow rates of blood and CSF were derived (Eqs. 23, 24) as a basis for disussion of model validity. Pressure volume relationships In horizontal position, ranial and spinal CSF and venous pressures are equal. The overall pressure volume relationship was thus desribed by Eq. 1. In upright posture, the loal pressure volume relationships shifted to lower ranial CSF pressures (Eqs. 9, 10) ompared to the horizontal position (Fig. 2a). Hereby, the shift of the spinal pressure volume relationship V s( p CSF ) was determined by the distane between the spinal and the ranial referene point, l s. With the assumption that the jugular veins ollapse in upright posture, the shift of the ranial pressure volume relationship ( V ( p CSF ) ) is redued to a hydrostati pressure olumn of length l jug. The overall pressure volume relationship V tot( p CSF ) was found by summation of these two loal pressure volume relationships (Eq. 3). Compliane In Fig. 2b, the loal omplianes derived analytially from the orresponding pressure volume relationships (Eqs ) were plotted along with the ombined total ompliane for horizontal and upright posture. Similar to the total pressure volume relationship, the total ompliane shifted towards lower ranial CSF pressures in upright posture. Due to a steep inrease of the ranial ompliane at low CSF pressures, the ranial ompartment beame the dominant soure of ompliane at ranial CSF pressures below approximately 0 mmhg. Posture hange The resulting ranial CSF pressure in upright posture without any hange in CSF volume, whih orresponds Fig. 2 CSF pressure volume relationships and omplianes in horizontal and upright posture. a The ranial and spinal pressure volume relationships V (p CSF ) and V s (p CSF ) are plotted along with the ombined overall pressure volume relationship, V tot( p CSF). b Visualizes the derivatives of these orrelations (Eqs ), whih represent the respetive loal and total omplianes of the CSF system. Equilibrium onditions in both postures are indiated by blak dots

7 Page 7 of 11 to the onditions diretly after sitting up from horizontal position, was 3.3 mmhg. This derease in ranial CSF pressure was aompanied by a shift of CSF from the ranial to the spinal ompartment ( V s in Table 2). The ondition of equal ompliane in horizontal and upright posture was satisfied for a ranial CSF pressure of 2.5 mmhg as depited in Fig. 2b. It required a slight inrease in total CSF volume (Table 2). Despite this inrease in total CSF volume, the amount of ranial CSF was smaller than in horizontal equilibrium. Furthermore, in equilibrium, the rate of CSF absorption has to math the rate of formation. In upright posture, this was ahieved for k R = In other words, the spinal resistane to CSF outflow was around nine times the ranial resistane to CSF outflow (Eq. 16). While this meant that in horizontal position 10% of the CSF absorption ourred within the spinal ompartment, it equated to 24% spinal absorption in upright posture due to the inreased CSF to venous pressure gradient in the spinal ompartment. Compliane shift As mentioned before, in upright posture the importane of ranial ompliane inreased for low CSF pressures. In upright equilibrium, only 10% of the total ompliane were provided by the spinal ompartment. This orresponded to a 71% redution relative to the spinal ompartment s ontribution in upright posture (Table 2). Under the ondition of no hange in total CSF volume after sitting up, the total ompliane in upright posture strongly inreased due to the steep inrease in ranial ompliane at low CSF pressure. Consequently, the ontribution of the spinal ompartment towards overall ompliane beame even lower. Cerebral CSF pressure (pcsf ), total, ranial, and spinal hange in CSF volume ( V tot, V, and V s ), total ompliane (C tot ), and spinal ompliane (C s ) in upright posture are shown in omparison to their referene values in Table 2 Comparison of CSF pressure, volume and ompliane in horizontal and upright posture Posture ondition Horizontal Upright Eq. 18 Eq. 19 p CSF (mmhg) V tot (ml) V (ml) V s (ml) C tot (ml/mmhg) C s /C tot (%) (C s /C tot )/k V (%) horizontal position. The values were alulated under the two alternative assumed onditions of unhanged volume (Eq. 18) and unhanged total ompliane (Eq. 19) relative to the horizontal position. Sensitivity We analyzed the sensitivity of the reported results to hanges in the nominal parameter values (Table 1). This nominal parameter set desribes an NPH patient. The physiologi parameter set used to analyze the sensitivity of the model towards hanges in the parameters E, p 1, p 0, and Rabs tot desribes a subjet with slightly lower CSF pressure in horizontal position. Also, the alulated ranial CSF pressure in upright posture was lower in the physiologi ase ( 6.6 mmhg after sitting up and 5.9 mmhg in upright equilibrium) ompared to the NPH parameter set. The audal shift of CSF volume aused by sitting up ( V s in Table 2) was slightly higher (2.1 ml with the physiologi parameter set ompared to 1.8 ml in the NPH ase). The shift in ompliane was not as pronouned as for the NPH parameter set, but the ontribution of the spinal ompartment to the total ompliane still redued to 18% in upright posture. For a longer hydrostati pressure olumn in the CSF system (l s = 36.3 m), the effet of posture inreased as the initial volume shift inreased to 2.0 ml (not presented in Table 2), and the ontribution of the spinal ompliane in upright equilibrium dereased to 7%. Conversely, inreased spinal venous pressure pv s redued the effet of the jugular vein ollapse. Consequently, spinal ompliane in upright equilibrium was still 12% and the initial CSF volume shift was redued to 1.5 ml for 7.7 mmhg spinal venous pressure. When using different values for the ompliane ontribution of the spinal ompartment in horizontal position (k V ), the audal shift in CSF volume hanged almost proportionally. For example, V s redued to 1.0 ml when k V was redued by 50% (k V = 0.175) and inreased to 4.2 ml when k V was inreased by 50% (k V = 0.525). However, even for suh large variations in the ompliane distribution (±50%), the redution of the relative spinal ompliane remained between 70 and 82% of its value in horizontal position (1(C s /C tot )/k V ). Patent jugular veins Without the ollapse of the jugular veins (Eq. 7 instead of Eq. 8) only the differene in density an lead to shifts in CSF volume and ompliane distribution when hanging posture. In this modified model with patent jugular veins in upright posture, 0.4 ml of CSF flowed from the spinal into the ranial ompartment when sitting up from horizontal. Cranial CSF pressure in upright posture dereased further (to 13.3 mmhg) with patent jugular

8 Page 8 of 11 veins ompared to the ase with ollapsed jugular veins ( 3.3 mmhg). Cardia pulsations The pulsatile arterial inflow Q a measured in NPH patients [9] aused a ranial arterial volume pulsation with 1.8 ml stroke volume (differene between maximum and minimum arterial volume, V a ). This hange in ranial arterial volume was ompensated by raniospinal flows of CSF and venous blood with 0.6 ml and 1.2 ml stroke volume, respetively (Fig. 3, left olumn). This stroke volume of the raniospinal CSF flow was 35% of the arterial stroke volume, whih orresponds to the analytial value of k V. The total CSF volume hardly hanged during one ardia yle (less than 1 µl) due to negligible variations in CSF absorption during that short time frame. Nevertheless, the ardia pulsations aused substantial CSF pressure amplitudes (1.6 mmhg). The pulsation of the spinal CSF volume was diretly proportional to the arterial waveform. By definition (Eq. 21), the remaining portion of the pulsatile arterial blood flow was ompensated by flutuations of raniospinal venous blood flow rate Q v. In upright posture, this piture hanged (Fig. 3, right olumn). While no hange in arterial blood flow was presribed, raniospinal CSF stroke volume was nevertheless redued to 10% of the arterial stroke volume (0.2 ml). However, despite these hanges in fluid dynamis and hanges in absolute pressures, CSF pulse pressure amplitudes remained onstant at 1.6 mmhg. The simulated raniospinal CSF flow pulsations of a healthy subjet in horizontal and upright position (Fig. 4) differed from the orresponding measurements in supine and sitting posture [2] by a mean absolute error of 22 and 21 ml/min, respetively. The physiologial parameter set (E = 0.1/mL, p 1 = 10 mmhg, p 0 = 5.3 mmhg, Rabs tot = 13.4 mmhg/(ml/min)) was used for these simulations. Fig. 3 Effet of arterial pulsation in horizontal and upright posture. During the ardia yle, the ranial arterial inflow Q a (soure [9]) leads to flutuations in ranial arterial volume, V a. Compensating raniospinal flows of CSF (Q CSF ) and venous blood (Q v ), raniospinal CSF distribution ( V versus V s ), and the orresponding ranial and spinal CSF pressures (p CSF and ps CSF, respetively) are shown for horizontal and upright posture. The orresponding measured CSF flow in supine position [9] is shown as dashed line. The flow rates are positive in the diretions indiated in (Fig. 1)

9 Page 9 of 11 Fig. 4 Comparison of simulated to measured CSF flow pulsation in horizontal and upright posture. Craniospinal CSF flow rates (Q CSF ) were simulated in supine and upright posture, based on arterial inflow measured by Alperin et al. [2] in supine and sitting posture. The orresponding measured CSF pulsations are plotted as dashed lines Disussion Volume and ompliane shifts Our model predits a posture-dependent shift of the raniospinal ompliane distribution aused by a audal displaement of CSF volume. As previously observed by Magnaes [1], this CSF volume displaement in upright posture redues the ompliane provided by the spinal ompartment inluding the spinal theal sa. It is indued by the hydrostati pressure olumn, whih is greater in the CSF system ompared to the veins, where it is interrupted by the ollapsing jugular veins. The estimated shifts of CSF volume and ompliane are in range of the observations of Magnaes [1], although he assumed a muh higher ontribution of the spinal ompartment to ompliane than in this study [17]. Furthermore, the posture-dependent shift of the raniospinal ompliane distribution was also observed for large variations of the employed parameter values, indiating that our analysis is robust. Jugular ollapse Without ollapsing jugular veins, the model showed neither a audal shift of CSF volume nor a ranial shift of the ompliane distribution. Furthermore, the fall in ranial CSF pressure was greater than that observed linially [4, 5]. As jugular ollapse redues this fall in pressure in upright posture, the jugular veins may be seen as serving a protetive funtion for the brain. In hydroephalus patients with ventriuloperitoneal or ventriuloatrial shunts, this protetive mehanism is partially bypassed so that, without appropriate siphon prevention, ICP an derease to levels as low as those predited by our model without jugular ollapse. Pressure volume relationship The exponential pressure volume relationship of the CSF system is well proven, at least for normal and reasonably inreased CSF pressures (relative to the sagittal sinus pressure). However, for suffiiently dereased CSF volume, it implies infinite ompliane. This attribute of the exponential pressure volume relationship beomes espeially problemati when applied to the ranial ompliane in upright posture, beause negative CSF to venous pressure gradients ould easily be reahed here. However, as suh gradients were not reahed in this study this limitation does not affet the results or onlusions reported herein. Nonetheless, extrapolation to low CSF pressures would be invalid (Fig. 2). Therefore, a more aurate desription of the pressure volume relationships would need to be used to study the effet of shunting on CSF dynamis [16, 21]. Only onsidering the mean venous pressure as ounter-pressure for the pressure volume relationship might seem simplisti, as venous pressure varies over the different generations of venous vessels. However, the implemented pressure volume relationship aptures this venous pressure variation and distribution with its exponential shape [22]. The only mehanism of ompliane inluded in the model is the displaement of venous blood. While this mehanism is aepted as the main ontributor to ompliane in the ranium [10], this is less lear for the rest of the raniospinal spae, espeially for the spinal theal sa. However, due to the high distensibility of venous vessels [23], tissue pressure strongly orrelates with venous pressure throughout the body. Therefore, it is reasonable to assume that venous pressure is the relevant

10 Page 10 of 11 ounter-pressure to ompliane in the entire CSF system. If the surrounding tissue itself ould provide elasti reoil, part of the pressure volume relation would have to be modelled independent of venous pressure. This would only then derease the modelled ompliane shift, if the elasti tissue were loated intraranially, sine the ounter-pressure of the spinal ompartment is already assumed to be independent of posture due to its proximity to the venous hydrostati indifferene point [4]. CSF absorption Before Magnaes [1] determined the raniospinal ompliane distribution in some of his patients, similar experiments had been done in adult ats [7]. While in ats the spinal ompartment appeared to be less important for ompliane, it was still responsible for a signifiant portion of CSF absorption (16%). Similar proportions of the raniospinal CSF absorption distribution were predited by our model under the ondition of posture-independent total ompliane. While this result supports the hypothesis that there is spinal CSF absorption, the exat proportion predited by the model is sensitive to the employed equilibrium pressure in upright posture. Furthermore, the lengths of hydrostati pressure olumns were alulated based on the assumption of 100% ranial absorption [4]. Taking spinal absorption into aount, the estimated value of l jug would slightly inrease, whih would derease the ompliane shift predited by our model. Craniospinal flows Pulsatile arterial inflow into the ranium were ompensated by simultaneous raniospinal outflow of CSF and venous blood. Stroke volume and amplitude of the alulated CSF pulsations (Figs. 3, 4) were very lose to the respetive values measured in vivo [2, 8, 9]. Even the redution in CSF stroke volume was predited well (Fig. 4). These are strong indiations that the inreased resistane of the jugular veins in upright posture is responsible for the shift in ompliane observed in vivo. Jugular vein ollapse an thus be onsidered a major ontributor to CSF dynamis in upright posture. The alulated overall CSF volume hardly hanged within a ardia yle. Therefore, the ratio of the estimated CSF and the applied arterial stroke volumes was equal to the ontribution of the spinal ompartment to the overall ompliane. In MRI measurements, CSF and even more so the venous pulses are delayed ompared to the arterial input. At least some of this delay an be attributed to wave propagation due to vasular distensibility [23]. In the model, these phase shifts between the alulated raniospinal waves were ignored with the impliit assumption of instantaneous transmission of pressures throughout the raniospinal spae. However, when assuming that most of the phase shift originates from a wave propagation delay, it does not influene the ompliane distribution estimated from the ratio of CSF and arterial stroke volume. In addition to being delayed, reorded venous pulsations appear damped ompared to the modelled pulsations. This damping is probably aused by the Windkessel effet in the larger veins, whih is not inluded in our model. However, the raniospinal venous flow rate is not only diffiult to model, it is also diffiult to measure with MRI as, espeially in upright posture it is distributed over numerous small vessels. Conlusion Our results support the hypothesis that the jugular veins play an important role in posture-related hanges of CSF dynamis, as their ollapse in upright posture indues substantial hanges in CSF pressure and ompliane. Abbreviations CSF: erebrospinal fluid; ICP: intraranial pressure; MRI: magnet-resonane imaging; NPH: normal pressure hydroephalus. Authors ontributions All authors designed the study and interpreted the data. MG performed the alulations, analyzed the data, and drafted the manusript. VK and MSD ritially revised the manusript. All authors read and approved the final manusript. Author details 1 Institute for Dynami Systems and Control, Department of Mehanial and Proess Engineering, ETH Zurih, Zurih, Switzerland. 2 The Interfae Group, Institute of Physiology, University of Zurih, Zurih, Switzerland. 3 Neurosiene Center Zurih, and the Zurih Center for Integrative Human Physiology, University of Zurih, Zurih, Switzerland. 4 Produt Development Group Zurih, Department of Mehanial and Proess Engineering, ETH Zurih, Zurih, Switzerland. Aknowledgements None. Competing interests The authors delare that they have no ompeting interests. Availability of data and materials All data generated or analyzed during this study are inluded in this published artile. Funding The presented study was supported by grants from the Swiss Aademy of Engineering Sienes ( ), the 3R Researh Foundation (140-14), and the Swiss National Siene Foundation through NCCR Kidney.CH. Reeived: 9 Otober 2016 Aepted: 6 February 2017 Referenes 1. 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