a model of the cerebral and cerebrospinal fluid circulations to examine asymmetry in cerebrovascular...

A Model of the Cerebral and Cerebrospinal Fluid Circulations toExamine Asymmetry in Cerebrovascular Reactivity

Stefan K. Piechnik, Marek Czosnyka, Neil G. Harris, Pawan S. Minhas, and John D. Pickard

Wolfson Brain Imaging Center, Medical Research Council Center for Brain Repair and Academic Neurosurgery Unit,Addenbrooke’s Hospital, Cambridge, U.K.

Summary: The authors examined the steal phenomenon us-ing a new mathematical model of cerebral blood flow and thecerebrospinal fluid circulation. In this model, the two hemi-spheres are connected through the circle of Willis by an ante-rior communicating artery (ACoA) of varying size. The righthemisphere has no cerebrovascular reactivity and the left isnormally reactive. The authors studied the asymmetry of hemi-spheric blood flow in response to simulated changes in arterialblood pressure and carbon dioxide concentration. The hemi-spheric blood flow was dependent on the local regulatory ca-pacity but not on the size of the ACoA. Flow through the ACoA

and carotid artery was strongly dependent on the size of thecommunicating artery. A global interhemispheric “steal effect”was demonstrated to be unlikely to occur in subjects with non-stenosed carotid arteries. Vasoreactive effects on intracranialpressure had a major influence on the circulation in both hemi-spheres, provoking additional changes in blood flow on thenonregulating side. A method for the quantification of thecrosscirculatory capacity has been proposed. Key Words: Ce-rebral blood flow—Mathematical model—Simulation—Stealeffect—Collateral blood supply.

In focal head injury, a major therapeutic aim is themaintenance of cerebral blood flow (CBF) to the affectedregions. This is not a straightforward problem becausethe distribution and maintenance of CBF depends onmany factors. Potentially beneficial therapeutic maneu-vers are poorly understood and can have undesirableconsequences (Roberts et al., 1998). Intracerebral steal, aterm popularized by Symon (Symon, 1968), refers to theparadoxical decrease of flow in the ischemic areas inresponse to vasodilator stimuli (Brawley et al., 1967;Hoedt et al., 1967). This behavior is attributed to shunt-ing of blood flow away from nonautoregulating ischemicareas by the action of normally reactive vessels. The term“steal” has been suggested to be misleading because the

underlying mechanism is rooted in a decrease in perfu-sion pressure (Wade and Hachinski, 1987). The conceptof steal has been questioned and attributed to othersources such as a decrease in arterial pressure (Gogolaket al., 1985). Clinical studies have shown that the phe-nomenon has an ephemeral nature and can be docu-mented only in a small proportion of patients and only inlimited areas (Nariai et al., 1998; Olesen and Paulson,1971). Furthermore, the demonstration of the effective-ness of the “inverse steal,” by attempting to increase theflow in ischemic areas by vasoconstriction on normallyautoregulating brain regions, has not been straightfor-ward in head-injured patients (Darby et al., 1988; Rob-erts et al., 1998).

Mathematical modeling can provide a valuable insightinto this complex physiologic phenomenon (Cassot et al.,1995; Gao et al., 1997; Hudetz et al., 1982, 1993). Pre-vious authors have concentrated on the vascular com-partment neglecting the influence of the dynamics ofthe cerebrospinal fluid (CSF), which exerts pressure onthe cerebral vasculature. In this article, the authors pre-sent an original model of bihemispheric CBF includingcompensatory mechanisms at all levels—anastomoticupstream compensation, autoregulation, and CSF pres-sure dynamics.

This model examines the potential effect of a largedisparity between autoregulation in the cerebral hemi-

Received June 5, 2000; final revision received August 18, 2000;accepted November 1, 2000.

S. Piechnik received the ‘Fees Studentship‘ toward his PhD researchfrom the Cambridge Overseas Trust, the “Overseas Research Student-ship” awarded by the Committee of Vice-Chancellors and Principals ofthe Universities of U.K., and the maintenance grant from NeurosurgicalResearch Fund, St Catharine’s College. This work has been supportedby Technology Foresight Challenge Fund (Grant FCA 234/95) and byMRC program grant (MRC G9439390). Dr. M. Czosnyka and Mr. S.Piechnik are on unpaid leave from Warsaw University of Technology,Poland. Dr. N. Harris is in receipt of a Merck, Sharp & Dohne ResearchFellowship.

Address correspondence and reprints requests to Mr. Stefan Piech-nik, Academic Neurosurgery Unit, P.O. Box 167, Addenbrooke’s Hos-pital, Hills Road, Cambridge CB2 2QQ, U.K.

Journal of Cerebral Blood Flow and Metabolism21:182–192 © 2001 The International Society for Cerebral Blood Flow and MetabolismPublished by Lippincott Williams & Wilkins, Inc., Philadelphia

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spheres that can exist in clinical scenarios such as later-alized head injury, space occupying lesions, or stroke(Schmidt et al., 1999). The authors investigated the de-pendence of blood flow patterns on the extent of inter-hemispheric arterial communication. For the purpose ofthis study, the authors assumed normal carotid arteryanatomy without stenosis. The aim was to determine theextent to which manipulations in arterial blood pressureand Paco2 produce large-scale steal effects.

MATERIALS AND METHODS

Model descriptionAll simulations have been performed using dedicated soft-

ware encapsulating a new model of bilateral CBF and circula-

tion of CSF (Piechnik, 2000). The model has been created onthe basis of the authors’ earlier theoretical work on global CBF(Czosnyka et al., 1992, 1993, 1997). The diagrammatic repre-sentation of the model (Fig. 1) shows the simplified cerebro-vascular anatomy for the left hemisphere and its electricalequivalent for the right hemisphere. The detailed electricalschematic is given in the Appendix with the mathematical de-scription of its elements and the method for derivation of theresults. The electric representation is a convenient basis onwhich mathematical equations are derived (Agarwal et al.,1969).

The input for the model is generated as an arterial bloodpressure (ABP) pulse waveform that acts on two simulated,symmetrical neck arteries (internal carotid artery [ICA]) feed-ing into the circle of Willis. At the entrance to the cerebrospinalspace, the arterial compartments of each hemisphere commu-nicate through the anterior communicating artery (ACoA). This

FIG. 1. Schematic representation of the hydrodynamic structure of the model (left) and its electrical equivalent (right). The arterial,venous, and cerebrospinal fluid compartments are denoted by shading.

MODEL STUDY OF STEAL 183

J Cereb Blood Flow Metab, Vol. 21, No. 2, 2001

reflects the overall function of the anastomotic structure of thecircle of Willis.

The large intracranial arteries downstream from the circle ofWillis are represented as a single middle cerebral artery (MCA)on each side. The next element of the blood flow pathwayreflects the largest vascular compartment of the model, simu-lating the whole regulating arterial and arteriolar networkdownstream from the distributing arteries. It also incorporatespart of the draining venous system down to the level of the pialveins. The arterial part of the cerebrovascular system containsthe myogenic and metabolic mechanisms responsible for cere-bral autoregulation and these have been incorporated into themodel. These mechanisms control the vascular diameter andhence determine the cerebrovascular resistance (CVR) consti-tuted by this compartment. The regulating capacity was mod-eled differently for each hemisphere to simulate the cerebro-vascular asymmetry (Fig. 2). The left hemisphere was assumedto be normally autoregulating, whereas the right was passive.The resistance values were selected to simulate symmetric,normal CBF for a cerebral perfusion pressure (CPP) of 90 mmHg and Paco2 of 30 mm Hg.

Venous outflow from the brain has been divided into intra-and extracranial compartments. The intracranial venous out-flow, symmetrical for each hemisphere, is modeled by takinginto account the compression of the lacunae and bridging veinsby the surrounding CSF. This mechanism assures that the dif-ference between cerebral venous and intracranial pressures(ICP) is positive and nearly constant, as indicated by experi-mental studies (Johnston and Rowan, 1974; Nakagawa et al.,1974; Yada et al., 1973). At the exit from the cranium, thevenous flow from the hemispheres joins at the venous sinuses.Finally, the extracranial venous return is represented by a singlejugular vein.

In addition to cerebral blood circulation, the system for theproduction and reabsorption of CSF has also been representedin this model. From the ventricles, the CSF flows through thecerebral aqueduct to the cisterns, which are represented by thelumped compliance of the CSF space. The subsequent part ofthe circulation consists of CSF pathways that bifurcate towardsthe left and right subarachnoid spaces. They terminate at thearachnoid villi located in the dural sinuses, the site of CSFreabsorption into the blood circulation.

Experimental protocolTwo experiments were performed. Progressive systemic hy-

potension was obtained by programming a decrease in ABP,from hypertension to deep hypotension (120 mm Hg to 40 mmHg, respectively) at a rate of change of 3 mm Hg/min. Thesecond experiment consisted of a progressive CO2 challenge,changing PaCO2 in steps of 5 mm Hg every 2.5 minutes from 10to 85 mm Hg.

Both experiments were repeated for six different sizes ofACoA (Table 1). This corresponded to changing the resistanceby an order of magnitude each time. A resistance of 0.02 mmHg � min/mL corresponds to a normally sized ACoA. The lowestvalue of 0.0002, a decrease by 2 orders of magnitude, repre-sents an extremely efficient crosscirculation. However, thelargest resistance value of 20 corresponds to a practically ab-sent ACoA.

The results of the simulation were averaged in 30-secondepochs. For the CO2 challenge, only data from the final epochwas used when the response was stable.

RESULTS

Arterial blood pressure changeBlood flow in the large intracranial arteries (MCA) on

the left side was regulated within the ABP limits, 70 to110 mm Hg (Fig. 3A). The flow on the right side waspressure-passive for the whole range of ABP. The size ofthe ACoA had no effect on either left or right side MCAflow, as indicated by the closely overlapping results ofthe simulation (Fig. 3A). In contrast, flow in the neckarteries varied significantly depending on the resistanceof ACoA (Fig. 3B). At the smallest ACoA diameter (Ø� 0.2 mm), the hemispheres were practically isolatedfrom each other; the ICA flow closely resembled theMCA flow on the ipsilateral side. At larger ACoA di-ameters, the ICA flow converged towards the averageflow of the left and right MCA. The equalization of ICA

FIG. 2. Cerebrovascular resistance (CVR) in the model as a function of cerebral perfusion pressure (CPP) and changing partial pressureof Paco2. The left side is autoregulating. The right side has a flat response corresponding to vascular paralysis.

S. K. PIECHNIK ET AL.184


flow took place through the ACoA. Similarly, the flowthrough the ACoA was also significantly dependanton its size and on ABP (Fig. 4A). During hypotensionthe ACoA flow was diverted towards the regulating side.For larger than normal ACoA, the flow rates were upto 40 mL/min. The pressure difference at the level ofthe circle of Willis between the regulating and nonregu-lating side was minimal; at the level of MCA it neverexceeded 1.2 mm Hg, even for the smallest ACoA size(Fig. 4B).

PaCO2 changeAt the level of the MCA, the left side responded physi-

ologically to increases in Paco2, with an initial exponen-tial increase in the flow and saturation at greater PaCO2

values (Fig. 5A). The right side was unresponsive, in factthe flow decreased slightly with increasing PaCO2. Thesize of the ACoA only minimally influenced flow inMCA. The larger the ACoA size, the greater the CO2

response of MCA blood flow on the regulating side (upto 7 mL/min difference). There was a corresponding, butsmaller, decrease in flow on the nonregulating right side(up to 4 mL/min).

When the communication between the sides was poor,the ICA flow was different between the left and right(Fig. 5B). On each side ICA resembled the ipsilateralMCA flow (Fig. 5B and 5A, respectively). However,when the size of the ACoA increased, ICA flow on leftand right almost equalized, converging towards the av-erage of the downstream flow on both sides. This equal-ization of ICA flow was provided by the ACoA. TheACoA flow was directed towards the regulating hemi-sphere and away from the nonregulating side when PaCO2

was increased. The ACoA flow increased with the size ofACoA and with PaCO2, and its absolute values were up to120 mL/min.

Effect of CO2 and arterial blood pressure onintracranial pressures

There was no dependence of ICP on ACoA size (Fig.6A). However, there was a proportional change in ICPin response to the change in ABP. Over the whole range

of ABP, the change in ICP influenced local cerebral per-fusion pressure considerably more than any change inthe upstream MCA pressure because of varying ACoAdiameter (approximately 7 mm Hg compared with ap-proximately 1 mm Hg, respectively). Also, CO2 had asignificant effect on ICP (Fig. 6B). The PaCO2-inducedvasodilatation increased ICP by increasing intracranialblood volume. This response was more pronounced forlower PaCO2 levels and diminished when PaCO2 increasedover 50 mm Hg. The overall ICP change was approxi-mately 4 mm Hg over the whole range of PaCO2 (10 to 80mm Hg).

FIG. 3. Effects of arterial blood pressure (ABP) changes onblood flow in (A) intracranial arteries (middle cerebral artery[MCA]) and (B) extracranial arteries (internal carotid artery [ICA]).(A) The response of the downstream flow to hypotension on theleft and right side depended on autoregulatory capacity but not onanterior communicating artery (ACoA) size. Data points for thelargest and smallest ACoA almost completely overlap. The dif-ference between the remaining simulations is even smaller. Forclarity, the symbols for intermediate ACoA sizes are not shown.(B) The flow in the neck arteries depends strongly on the size ofthe ACoA. When the ACoA is small, ICA flows on both sidesreflect the ipsilateral downstream flows. When communicationbetween the hemispheres improves, they both converge towardsthe average flow between the regulating and nonautoregulatingside (arrows).

TABLE 1. Dimensions and resulting hydrodynamicresistance of the idealized anterior communicating artery

used in the model to simulate interhemisphericarterial crosscirculation

Length(cm)

Diameter(mm)

Resistancemm Hg � min/mL

0.3 4.18 — 0.00020.3 2.35 — 0.0020.3 1.32 Normal 0.020.3 0.74 — 0.20.3 0.42 — 20.3 0.24 — 20



DISCUSSION

Clinical application of the modelThe model assumes complete unilateral vasoparalysis

with preserved autoregulation in the contralateral hemi-sphere. Although this may not be completely represen-tative of the situation in the majority of head traumacases, it does provide a convenient platform to observethe exaggerated effect of two large brain regions com-peting for blood delivery. Arterial stenosis is not mod-eled and this reflects the number of young subjects thatcontribute to the head trauma patient group. The aim ofthe model was to simulate a range of clinical conditionsthat might result in abolition of autoregulation in onehemisphere. These include volume-occupying lesions(hematomas or tumors), brain edema, or vasospasm re-sulting from subarachnoid hemorrhage. These conditions

are likely to increase the vascular resistance in one seg-ment of the cerebral circulation and cause vasodilatationin the regulating arteries. In the worst case, if the vaso-dilatory capacity was completely exhausted, the overallresult would be pressure-passive as indicated by the un-changing CVR on the right side (Fig. 2).

Influence of the circle of Willis on the steal effectThe blood flow steal effect is produced through

changes in the upstream perfusion pressure (Fig. 7).The vasodilatory stimulus creates additional demandon flow in the autoregulating brain, whereas the non-regulated areas remain passive. Because they share a

FIG. 5. Response of the flow on the left and right side to a CO2

challenge. (A) The result is similar to the blood pressurechange—the downstream flow depends mainly on the autoregu-lating capacity. However, the overlap of data points for differentsizes of anterior communicating artery (ACoA) is not as completeas in Fig. 3A. A marginal effect of steal on the right can be noticedas a slight decrease in right middle cerebral artery (MCA) flowwhen PaCO2 increases. This trend is present even when ACoA isvery small and increases with ACoA size. For clarity, the symbolsfor intermediate ACoA sizes are not shown. (B) Flow in the ar-teries of the neck depends strongly on the size of the ACoA.When the ACoA is small, internal carotid artery (ICA) flow equalsthe downstream flow. When communication between the hemi-spheres is good, the flows converge towards the average flowbetween the regulating and nonautoregulating side (arrows).

FIG. 4. Side-to-side interaction during blood pressure changes.(A) Flow through the communicating vessel in the case of arterialhypotension is small (<40 mL/min) and nearly all of it reflects thedisparity of carotid, rather than downstream, cerebral flow. (B)Downstream pressure difference between the sides is minimal(<1.2 mm Hg). Note that for the decreased arterial blood pressure(ABP), the pressure is greater on the nonautoregulated side. Anincrease in anterior communicating artery (ACoA) size reducesthis small difference, in effect slightly reducing the middle cere-bral artery (MCA) flow.



common arterial supply, characterized by its proximalresistance, the additional demand decreases perfusionpressure upstream. Although this has no effect on theautoregulating side, which can respond with further va-sodilatation, a pressure-passive flow decrease will de-velop on the nonautoregulating side—this is the so-called steal phenomenon (Wade and Hachinski, 1987). Inthe authors’ model, perfusion pressure decreases faster inthe reactive left hemisphere during hypotension-inducedvasodilatation, resulting in an increased left-to-rightpressure difference (Fig. 4B). The ACoA is a potentialchannel for the steal and at larger diameters greater trans-mission of this pressure drop will result. As a conse-quence, a greater reduction in flow will occur on thepressure-passive side (Figs. 3A and 5A). However, theauthors found that under the assumptions of this model,the influence of the circle of Willis on downstream flowis minimal. The authors have shown that it is only thedistribution of flow in the neck arteries that is influencedby the size of the ACoA and not downstream flow in the

MCA (Figs. 3 and 5). The ACoA only acts to equalizeflow between the arteries of the neck without affectingdownstream flow.

There was greater evidence of steal during hypercap-nia than in hypotension, as indicated by the greater sen-sitivity of flow to the size of the ACoA. This may beattributed to the much larger flow changes produced bythe CO2 challenge on the regulating side. However, de-spite this difference, the effect was still only visible as asmall dissociation of the data points (compare Figs. 3Aand 5A).

The normal arteries of the neck have minimal resis-tance compared with the downstream system. The un-equal distribution of vascular resistances in patients withnormal, nonstenosed, large arteries may explain the ab-sence of studies demonstrating a steal effect in this pa-tient group. Literature reports of the steal phenomenonare mainly limited to those patients with more severecerebrovascular pathology, for example, carotid arterystenosis associated with transient ischemic attacks andmoyamoya disease (Nariai et al., 1998; Olesen and Paul-son, 1971). Furthermore, only rare examples of stealhave been observed in animal models (Dettmers et al.,1993). It appears that a combination of adverse factors isrequired to propagate the “steal” effect. The results fromthis study suggest that the “steal” lacks a significant im-pact on side-to-side differences in the patient group with-out carotid artery stenosis, for example, in young sub-jects. Similarly, in the same patient group, the gains fromhyperventilation therapy are unlikely to have a majoreffect through “inverse steal,” whereas it may have apotentially detrimental effect on the rest of the brain(Darby et al., 1988; Geraci and Geraci, 1996; Menon etal., 1999; Moore and Flood, 1993).

“Downstream” source of steal effectThe inclusion of CSF dynamics in the authors’ model

provided a much greater understanding of the effects ofcerebrovascular maneuvers on the downstream side ofthe cerebral circulation. Both hypercapnia and arterialhypotension influenced the ICP. Despite both maneuversbeing vasodilatory in nature, their effect was opposite.During gradual arterial hypotension, ICP steadily de-creased, contrary to what is observed in the normal au-toregulating brain. This presumably occurred because thevolume-reducing effect of decreased arterial filling pres-sure exceeded the opposing effect of the volume-increasing vasodilatation on the regulated side. This ef-fect is most likely propagated through the entire brainbecause of the connectivity of the CSF space. This had acompensatory effect on the unregulated side by effec-tively reducing the ABP-induced drop in CPP. Increasein CO2 increased ICP, which resulted in an additionalflow reduction on the passive side. However, this slightflow reduction is not a steal effect because it occurred

FIG. 6. Plots of intracranial pressures (ICP) versus changes in(A) arterial blood pressure (ABP) and (B) PaCO2 for a normalanterior communicating artery (ACoA) size. Data obtained forother ACoA diameters were the same and were excluded fromthe plot for clarity.



even for the smallest ACoA size (Fig. 5A). This mayexplain the wide variation of flow reactivity with a stealeffect after the administration of papaverine in patientswith intracranial tumors (Olesen and Paulson, 1971). Insuch cases, the dynamics of CSF would initially be chal-lenged resulting in decreased volume compensation andan even greater increase in ICP and decrease in CPP.Under such circumstances, the loss of autoregulation inmaximally dilated foci would result in a flow decrease.At the same time, the preserved side, although alreadychallenged by vasodilator, still would be within the regu-latory limits to compensate for the same CPP change onthe basis of myogenic mechanism of autoregulation. Inneurosurgical care after head injury, such a situationcould be detected using standard ICP monitoring andtreated within standard CPP-directed therapy (Paulson etal., 1972; Rosner et al., 1995).

Noninvasive detection of collateral flow capacityIn the current study, the authors found that with nor-

mal neck arteries, collateral flow through the ACoAcompensates upstream (ICA) flow rather than down-stream (MCA). It is possible that detecting differences inMCA flow reactivity may enable us to determine theefficiency of collateral flow at the level of the circle ofWillis. This detection is based on comparing the afferent(ICA) and efferent (MCA) branch reactivity in bothhemispheres. If downstream asymmetry is reproducedupstream, then it can be concluded that collateral flow ispoor. When the ICA reactivity is similar on both sidesand close to the average MCA reactivity, there must be agood anastomotic connection between the sides. Thismay be expressed as a side-to-side index of relativeasymmetry (RAI). For example, it may be defined as:

RAI =RICA

L − RICAR

RMCAL − RMCA

R

where Rsidevessel describes the reactivity of blood flow to a

vascular challenge. The RAI value is independent of thevascular challenge used because the reactivity indexescancel out. However, although the RAI values were simi-lar between the indexes, the authors found a greatervariation with the hypercapnic challenge when comparedwith hypotension (Fig. 8). One explanation for this is thatat high Paco2 levels, CO2 reactivity approached zero inboth hemispheres (Fig. 5A and 5B) so that the differencein reactivity between the hemispheres becomes too smallto estimate ACoA size. The model does not enable one tostudy the influence of the posterior anastomoses and thevertebrobasilar arterial system on blood flow in the ce-rebral hemispheres. Nevertheless, the clinical interest ismore concerned with the anterior circulation carryingmajority of cerebral blood flow, which can be studied inthis aspect using commercially available 4-channel trans-cranial Doppler monitors and specialized probe holders(Makinaga et al., 1992).

ConclusionBlood flow through the brain depends on pressure

changes both upstream and downstream from the lesionarea. For most neurotrauma patients, with normal, non-stenosed arteries of the neck, the influence of the up-stream pressure changes because of “steal” was predictedas not essential for downstream flow when large brainareas are in question. The downstream pressure changesbecause of CSF dynamics were shown to possess thesame ability to modulate the blood flow and this abilitymay increase further in head trauma patients. Closemonitoring of ICP and CPP in response to vasoreactivemaneuvers is the most straightforward way to resolvethis problem.

Finally, the authors propose a noninvasive method fortesting of the efficiency of the collateral circulation that

FIG. 7. The steal effect results from additional flow demand cre-ated by the reactive part of the brain.

FIG. 8. Relative asymmetry index calculated using either arterialblood pressure or PaCO2 flow reactivity correlates with the size ofthe anterior communicating artery (ACoA).



is based on the observation of both the afferent (ICA) andefferent (MCA) flow from the circle of Willis.

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APPENDIX

The detailed electric schematic is shown in Fig. 9. Asshown on the simulated ICA, the compartments throughwhich cerebral blood flow passes are represented asserial connections of two resistances that constitutethe total compartmental resistance to flow. The lumpedcompliance of the vessel is connected to the middle ofthe vessel.

Table 2 lists the values of the static elements of themodel. The resistance value of the ICA calculated byassuming the resting diameter of the ICA � 3 mm, thevessel length (18 cm) to represent the entire length fromthe aorta to the circle of Willis. The resistance was re-calculated for the transmural pressure of 100 mm Hg,taking into account the elastic stretch of the vascularwall. This was achieved by assuming an exponentialpressure–area relation (Drzewiecki et al., 1997; Drzew-iecki and Pilla, 1998). The resulting resistance value wascalculated from the Hagen-Poiseuille’s formula assum-ing a blood viscosity of 0.4 poise and by doubling theresult to account for significant losses of energy becauseof secondary flow effects in the neck arteries (Cherif etal., 1999; Slawomirski et al., 1999). Compliance wascalculated as the pressure derivative of volume changeresulting from the stretch of vascular wall.

The intracranial large arteries of each hemisphere arerepresented by a single MCA. The selection of lengthand diameter of the equivalent artery is difficult becauseit needs to represent a bifurcating network of arteries.



The authors assumed a resistance of 0.01, which corre-sponds to a vessel 8-cm long and 2-mm resting diameter(Hillen et al., 1986). Compliance was increased four-fold greater than the value obtained from the singletube model (from 0.0024 to 0.01 mL/mm Hg). This re-flects the greater volume capacity of the branching treeof vessels compared with the single tube. The valuesof the remaining linear elements describing the bloodflow pathway—Rana, Rvo, and Cvo—were assumedarbitrarily.

The regulating arterial and arteriolar network down-stream from distributing arteries is represented by non-linear CVRl and CVRr. These elements incorporate partof the venous system of each hemisphere, up to the level

of the pial veins. The mathematical description of themyogenic mechanism has been adapted with changesfrom Hoffmann (1985, 1987), which has been thor-oughly tested in the authors’ previous work (Czosnyka etal., 1992, 1993, 1997). In the model, the following ef-fective formula was used:

CVR�CPPm� = CVRmax −CVRmax − CVRmin

1 + �CPPm − CPPoffset

CPPN�

6

In the formula above, CVR represents cerebrovascularresistance; CPPm represents the time averaged value ofcerebral perfusion pressure, (the filter time constant was

FIG. 9. Detailed electrical schematicfor the model. Blood pressure is pro-vided by the voltage source arterialblood pressure (ABP). There are twomajor flow pathways for left and righthemispheres denoted by postfix ‘l’ or‘r’. ICA and MCA represent the larg-est arteries of the neck and the cere-brum (internal carotid and middle ce-rebral artery, respectively). CVRstands for cerebrovascular resistancerepresenting here the majority of ce-rebral vascular bed. Rv and unpairedRvo correspond to intra- and extra-cranial venous outflow. The light greyelements represent the pathways ofCSF circulation with its formation (Ifand Rf), aqueduct, and subarachnoidspaces (Rac, Rs) and finally reab-sorption to venous blood stream (Rr).Crosscirculation is provided by vary-ing the size of ACoA. Downstreamanastomoses Ran were assigned alarge value of resistance makingthem ineffective.



6 seconds); CVRmax represents maximal CVR for upperautoregulation limit (� 0.35(left)/0.25(right) mm Hg �min/mL); CVRmin represents minimal CVR for lowerautoregulation limit (� 0.1(left)/0.24(right) mm Hg �min/mL); CPPN represents constant (120 mm Hg) equalto 1.5 � (CPPmax − CPPmin); CPPmax, CPPmin representsasymptotic upper (120 mm Hg) and lower (40 mm Hg)pressure limits of autoregulation; CPPoffset representsconstant offset enabling horizontal repositioning ofCVR-CPP curve equal to CPPN − 0.5 � (CPPmax − CPPmin).

The metabolic mechanism was set to modify theCPPmax, CPPmin in such a way that the exponential flowresponse could be obtained (Aaslid et al., 1989; Czos-nyka et al., 1993). The compliance of the cerebrovascularcompartment was assumed to change inversely withCVR (Czosnyka et al., 1997). This reflects the fact thatthe downstream compliance is a function of the activevascular tone rather than a purely mechanical effect.

Ca�CPPm� =CAmax

CVR �CPPm��CVRmin

The scaling coefficient CAmax was 0.1 and 0.05 mL/mmHg on the left and right side, respectively, so that thecompliance of the cerebrovascular compartments wasapproximately 0.04 mL/mm Hg on both sides at 90 mmHg of CPP.

Venous outflow from the brain has been modeled tak-ing into account the compression of the lacunae andbridging veins. There are two basic models of collapsiblevenous outflow: the pressure follower (Hoffmann, 1985,1987) and the Starling resistor (Ursino, 1990). In thecurrent model, the authors used the formula more closelyrepresented by the pressure follower model by makingthe conductivity of venous outflow progressively largerwhen the venous transmural pressure increased. The ve-nous conductivity was calculated depending on the in-tracranial and venous pressure values.

Gv�Pv − ICPs� =1

Rv�Pv − ICPs�= �Pv − ICPs� � Gv0

In the formula above, Gv represents venous conductance,reciprocal of Rv; Rv represents venous resistance to flow;Pv represents venous pressure at the “middle” point ofvenous compartment, where the lumped venous compli-ance is connected; ICPs represents intracranial pressureat subarachnoid space surrounding the veins; Gv0 repre-sents scaling coefficient (100 mL/min . mm Hg).

The CSF system consists of elements responsible forthe production, movement, and reabsorption of CSF. Theproduction is lumped in the ventricles. Two sources ofproduction have been designed to reproduce both activeenergy-consuming excretion of CSF in the choroidplexus and passive bulk filtration of CSF from the pa-

renchyma. However, the proportion between the two isstill disputed (McComb, 1983). In the model, the lattermechanism was canceled by assuming conductances Gfl� Gfr � 0, it is the resistance being infinite. The activeproduction was assumed to equal 0.17 mL/min on eachside (0.34 mL/min total) and this is in agreement withliterature values (Cutler et al., 1968; Ekstedt, 1977; Sha-piro et al., 1980). The communication between the ven-tricles, cisterns, and the sagittal sinus is characterized byde facto arbitrary resistance values. The final resistancesRrl � Rrr � 14 mm Hg � min/mL represent the resis-tance of the arachnoid villi located on the dural sinuses,the final place of reabsorption of CSF into blood stream.In general, CSF reabsorption ceases at ICP values lessthan venous pressure measured at the sagittal sinus. Thishas been incorporated in the model by increasing reab-sorption resistance to infinity for negative ICP. Despitedefining an arbitrary distribution of the resistance alongCSF pathway, the authors kept normal values of theoverall resistance to flow of CSF. From the point of itsorigin in the ventricles to reabsorption at the arachnoidvilli, this resistance corresponds to the well-acceptedclinical average of 9 mm Hg � min/mL (Cutler et al.,1968; Ekstedt, 1977).

In the model, the authors used a lumped element Cirepresenting the overall compliance of CSF space.

Ci�ICPi� = �Ci0

ICPi − P0for: ICPi � Pz

Ci0

P0 − Pzfor: ICPi � Pz

where ICPi represents ICP in cisterns. Ci0 represents con-stant (� 5mL), inverse of elastance E. PZ representsoptimal pressure (� 5 mm Hg). P0 represents equilib-rium pressure (� 0 mm Hg).

Based on standard methods for numerical analysis ofelectronic circuits (Chua, 1975; Press et al., 1992), themodel has been implemented in the form of 10 stateequations using the values of the static parameters. Thenonlinear elements were evaluated using the solution

TABLE 2. Parameters of the static components of the model

ElementResistance

[mm Hg � min/mL]Compliance

[mL/mm Hg]

ICAl, ICAr 0.01 0.01ACoA see Table 1 —MCAl, MCAr 0.01 0.02Ran 100.0 —Rfl, Rfr infinite —Rac 1.0 —Rsl, Rsr 2.0 —Rvo 0.01Cvo — 1.0

ICA, internal carotid artery; ACoA, anterior communicating artery;MCA, middle cerebral artery.



from the previous simulation step or a starting point ob-tained by the Newton-Ralphson iteration procedure. Thestate equations were formulated for each branch contain-ing compliance as shown in the electrical schematic ofthe model (Fig. 9). Additional equations were used toderive pressures and flows at different elements of themodel. The solution of the state equations in the timedomain was performed using the 4th order Runge-Kuttamethod. Automatic step adjustment was used to keepstep-to-step accuracy less than 0.001%. The analysis wasperformed using specialized software written in C++(Borland C++Builder v.3.0 Professional, Borland Inter-national, Scotts Valley, CA, U.S.A.) running on a PCcomputer.

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a model of the cerebral and cerebrospinal fluid circulations to examine asymmetry in cerebrovascular...

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