Contribution of Mathematical Modelling to the Interpretation of Bedside Tests of Cerebrovascular Autoregulation

Home , Autoregulation, Cerebral autoregulation

Journal of Neurology, Neurosurgery, and Psychiatry 1997;63:721–731 721 J Neurol Neurosurg Psychiatry: first published as 10.1136/jnnp.63.6.721 on 1 December 1997. Downloaded from Contribution of mathematical modelling to the interpretation of bedside tests of cerebrovascular autoregulation

Marek Czosnyka, Stefan Piechnik, Hugh K Richards, Peter Kirkpatrick, Piotr Smielewski, John D Pickard

Abstract The final outcome after injury to the brain Objectives—Cerebral haemodynamic re- depends not only on the initial trauma but also sponses to short and longlasting episodes on secondary insults including, for example, of decreased cerebral perfusion pressure raised intracranial pressure, arterial hypoten- 1 contain information about the state of sion, and hypoxia. Autoregulation of cerebral blood flow (CBF) is an important factor in the autoregulation of cerebral blood flow. 23 Mathematical simulation may help to elu- brain’s capacity to respond to such insults. cidate which of the indices, that can be Disordered autoregulation is a prelude to refractory intracranial hypertension and corre- derived using transcranial Doppler ultra- lates with a poor outcome.4–6 The capacity for sonography and trends of intracranial autoregulation diVers between patients and pressure and blood pressure, are useful in varies in the same patient from day to day.6 clinical tests of autoregulatory reserve. Therefore, reliable and repeatable bedside Methods—Time dependent interactions techniques are required to monitor the varia- between pressure, flow, and volume of tions in autoregulatory reserve if this is to be cerebral blood and CSF were modelled useful guide to therapy.4–9 However, the tests using a set of non-linear diVerential equa- that can be used clinically explore only a small tions. The model simulates changes in part of the range of the autoregulatory response

arterial blood inflow and storage, arteri- and also depend on techniques such as copyright. olar and capillary blood flow controlled by waveform analysis of transcranial Doppler cerebral autoregulation, venous blood ultrasonography (TCD) and intracranial storage and venous outflow modulated by pressure610 that have a complex relation with changes in ICP,and CSF storage and rea- CBF. The data can be diYcult to interpret, bsorption. The model was used to simu- especially in deciding if autoregulation is intact late patterns of blood flow during either or impaired. short or longlasting decreases in cerebral Experimental studies are advancing the abil- ity to understand findings in patients but can- perfusion pressure. These simulations can not replicate all aspects of the clinical be considered as clinically equivalent to a problem.11–14 short compression of the common carotid We have therefore developed a mathematical Wolfson Brain Imaging artery, systemic hypotension, and intrac- model of cerebrovascular flow which embeds http://jnnp.bmj.com/ Centre, the MRC ranial hypertension. Simulations were previous models of CSF circulation15–17 to Cambridge Centre for performed in autoregulating and non- Brain Repair and simulate interactions between CBF and intrac- 18 19 Academic autoregulating systems and compared ranial pressure. The model incorporates Neurosurgical Unit, with recordings obtained in patients. realistic values for known components includ- Addenbrooke’s Results—After brief compression of the ing cerebrovascular resistance, intracranial and Hospital, Cambridge, common carotid artery, a subsequent arterial compliance, and CSF outflow UK transient hyperaemia can be interpreted resistance.20 21 This paper presents our

M Czosnyka on September 28, 2021 by guest. Protected S Piechnik as evidence of intact autoregulation. Dur- experience in using the model to enhance the H K Richards ing longlasting sustained hypoperfusion, a interpretation of bedside tests of cerebrovascu- P Kirkpatrick gradual increase in the systolic value of lar autoregulation. P Smielewski the blood flow velocity waveform along J D Pickard with a decrease in the diastolic value is Materials and methods specific for an autoregulating cerebrovas- MODEL Correspondence to: The first models of the mechanoelastic proper- Dr Marek Czosnyka, cular system. Academic Neurosurgical ties of the brain were related to the CSF circu- —Modelling studies help to Unit, PO Box 167, Conclusion lation and included the non-linear pressure- Addenbrooke’s Hospital, interpret both clinical and experimental Cambridge CB2 2QQ, UK. volume relation of the intracranial cerebral haemodynamic phenomena and 15 18 20 22 telelephone +44 1223 compartment. These studies laid the 336933; fax: +44 1223 their dependence on the state of autoregu- foundation for techniques used in the diagnosis 216926; email: lation. 17 23 24 MC141@MEDSCHL. of hydrocephalus today. Advances in CAM.AC.UK computing technology have prompted at- (J Neurol Neurosurg Psychiatry 1997;63:721–731) tempts to integrate the intracranial CSF circu- Received 29 April 1996 lation with CBF.25–27 and in final revised form 19 May 1997 Keywords: autoregulation, trascranial Doppler, intrac- The pressure of CSF results from the Accepted 19 June 1997 ranial pressure, mathematical modelling dynamic equilibrium between the volumetric 722 Czosnyka, Piechnik, Richards, Kirkpatrick, Smielewski, Pickard J Neurol Neurosurg Psychiatry: first published as 10.1136/jnnp.63.6.721 on 1 December 1997. Downloaded from

the circle of Willis (ﬁg 1). Arterial blood is Sagittal sinus contained in a high pressure arterial compli-

ance Ca. Forward ﬂow through the cerebrovascular resistance vessels is inﬂuenced by cerebral autoregulation. Capillary and venous

MODEL blood is contained in a compliance of Cv. Finally, venous blood ﬂows out to the sagittal

sinus through the bridging veins Rb. The CSF pathway encompasses CSF formation (I ), PCoAl PCoAr f storage in the distensible ﬂuid structures formed by the ventricles and basal cisterns

MCAl PCAl PCAr MCAr (Ci), and reabsorption through the arachnoid

granulations (Rcsf) to the sagittal sinus. ACAl ACAr All parameters and the mathematical description of the model are presented in greater detail in the appendix. ACoA BA CLINICAL RECORDINGS The results of simulations were visually com- Jugular vein l CAl CAr Jugular vein r pared with recordings chosen from the studies performed in 82 patients who had been admit- VAI VAr ted to Addenbrooke’s Hospital after moderate Figure 1 The anatomically equivalent location of the model is distal to the circle of Willis and severe head injury (median Glasgow coma and proximal to the jugular veins. PCoA=posterial communicating artery; PCA=posterior scale after resuscitation 6, range 3 to 13). Each cerebral artery; VA=vertebral artery; CA=carotid artery; BA=basilar artery; ACA=anterior cerebral artery; ACoA=anterior communicating artery. patient was mechanically ventilated to main- tain a PaCO2 between 3.5 and 4.5 kPa. Intrac- changes in brain tissue, CSF volume, and cer- ranial pressure was monitored continuously ebral blood volume.15 19 20 Any deviation may using a fibreoptic transducer (Camino direct generate a pressure response aVecting the cer- pressure monitor, Camino Laboratories, US), ebral circulation with several stabilising feed- inserted into brain perenchyma in the frontal back loops involved. These include cerebrovas- region. Arterial pressure was measured directly cular autoregulation/reactivity, neurogenic in the radial or dorsalis pedis artery and moni- innervation, intraparenchymal water fluxes, tored using a bedside monitor (System 8000, S copyright. and secretion/reabsorption of CSF. The model and W Vickers Ltd, UK). The middle cerebral we propose contains two major flow pathways artery (MCA) was insonated daily for periods (figs 1 and 2). The CBF pathway starts with the up to two hours using a PCDop 842 Doppler arterial blood inflow to the brain through the ultrasound unit (Scimed, Bristol, UK) to

resistance of large intracranial arteries (Ra). At record maximal blood flow velocity waveform present, the model is unable to simulate (FV). Cortical blood flow was monitored in 26 phenomena resulting from the complex topo- patients using a laser blood flow monitor graphy of the basal cerebral arteries included in (MBF3D, Moor Instruments, Axminster, http://jnnp.bmj.com/

Big cerebral Resistance Bridging arteries vessels veins Pa Pv Carotid and Dural basilar ABP Arterial blood Capillary and Pss sinuses arteries Ra storage venous blood

CVRstorage Rb on September 28, 2021 by guest. Protected Ca Cv CSF production If CSF reabsorption (RCSF) Pi (Intracranial pressure)

CSF storage Ci

Figure 2 Hydrodynamic equivalent of the model, comprising pathways of CBF and the CSF circulation. A rigid skull is

represented by the outer box, with a compensatory reserve Ci associated with the compliant dural sac within the lumbar channel. Mathematical modelling of interpretation of cerebrovascular autoregulation 723 J Neurol Neurosurg Psychiatry: first published as 10.1136/jnnp.63.6.721 on 1 December 1997. Downloaded from copyright.

Figure 3 Example of two compression tests performed in patients after head injury. Test (A) shows a positive hyperaemic response in MCA blood ﬂow velocity (FV) and laser Doppler ﬂow meter (LDF) placed on ipsilateral hemisphere observed at CPP>70 mm Hg. Test (B) was performed when ICO increased to 40 mm Hg , decreasing CPP to around 50 mm Hg: there was no hyperaemia. Arterial pressure (ABP) was measured using a central line.

UK); the clinical technique for insertion of the Results 28 probe has been described. Short term com- SHORT TERM HYPOPERFUSION: TRANSIENT http://jnnp.bmj.com/ mon carotid artery compression (ﬁve to six HYPERAEMIC RESPONSE TEST seconds) was performed to assess cerebral The recordings made in patients after head autoregulation.29 In seven patients we com- injury were compared with the mathematical pared the results of two tests performed at simulations. In every clinical instance when either high (>80 mm Hg) or low (<60 mm Hg) CPP was >70 mm Hg (ﬁg 3 (A)) after four to levels of cerebral perfusion pressure (CPP). six seconds CCA compression the hyperaemic The events of Lundberg plateau waves of response was visible on TCD (laser Doppler

intracranial pressure (ICP) were recorded in recordings were performed in two patients). By on September 28, 2021 by guest. Protected eight patients and episodes of transient arterial contrast, in the same patients when CPP fell hypotension (arterial blood pressure below 50 mm Hg due to intracranial hyperten- (ABP)<70 mm Hg) in 15 patients. sion (ICP>30 mm Hg) the positive autoregulatory response disappeared (ﬁg 3 (B)). COMPARISON BETWEEN MODELLING SIMULATIONS A response to compression was simulated by AND CLINICAL RECORDINGS a short term increase in the resistance of the big

Changes in transcranial Doppler MCA blood cerebral arteries (Ra) from 0.01 to 0.08 flow velocity (FV) correspond to simulated mmHg/(ml/min). Simulation was conducted changes in flow through the great cerebral with intact autoregulation and with autoregula- arteries (CBF+). Changes in laser Doppler tion impaired (by equalising the levels of maxi- capillary red cell flux (LDF) may be compared mal vasodilatation and maximal vasoconstric- with simulated changes in blood flow through tion to the value of 0.2 mm Hg/(ml/min)). In resistance vessels (mCBF). Cerebral perfusion both cases a significant reduction of simulated

pressure (CPP= ABP−ICP) corresponds to the cerebral perfusion pressure (CPP= Pa−Pv)was

simulated gradient ABP−Pi. However “real” recorded during compression. In the autoregu-

cerebral perfusion pressure (Pa−Pv) cannot be lating system the cerebral vessels dilated, which measured clinically. was shown as a gradual decrease in CVR with 724 Czosnyka, Piechnik, Richards, Kirkpatrick, Smielewski, Pickard J Neurol Neurosurg Psychiatry: first published as 10.1136/jnnp.63.6.721 on 1 December 1997. Downloaded from

Figure 4 Dynamic response to a simulated seven second compression of the common carotid artery. CBF=flow through small cerebral vessels; CBF+=flow through big cerebral vessels; CVR=the resistance of small cerebral vessels. Compression causes transient hyperaemia in the autoregulating system (A) and a passive recovery of the flow to the baseline in the non-autoregulating system (B)

a time constant equivalent to the delay of toregulating (fig 6 left or non-autoregulating autoregulation. The model indicated that (fig 6 right). In an autoregulating system, the copyright. blood flow through the big cerebral arteries value of arterial CBF averaged in time (CBF+) and arterioles (CBF) decreased im- (mCBF) was constant until mean CPP de- mediately during the compression, and then creased below the lower limit of autoregulation started to increase, as the cerebral arterioles (this limit was programmed in the model at 40 dilated (fig 4 (A)). After a release of compres- mm Hg). A gradual decrease in CPP towards sion, the CPP increased abruptly towards the this lower limit translated into a gradual baseline. As the vessels remained predilated, a increase in ICP. The systolic flow in large cer- transient hyperaemia was recorded until CVR ebral arteries (CBF+) increased whereas di- returned to the baseline. When autoregulation astolic CBF+ decreased; thus the net eVect was was impaired, the reduction in CPP did not an increase in the pulse amplitude of CBF+. induce vasodilatation (CVR remained con- The cerebral arterioles dilated, as indicated by stant). After the release of compression, CBF a decrease in CVR, until the vessels at the lower http://jnnp.bmj.com/ and CBF+ returned passively to the baseline limit of autoregulation became dilated maxi- (fig 4 (B)) without any overshoot. Hence we mally at the time point A—corresponding to can conclude that the postcompression hyper- time point A in fig 5. Below this point, further aemia is a reflection of intact autoregulation. decrease in ABP led to a decrease in mean CBF and ICP. Coincidentally with the mean CPP ARTERIAL HYPOTENSION reaching the lower limit of autoregulation, the During episodes of spontaneous arterial hypo-

diastolic CPP started to fall below the critical on September 28, 2021 by guest. Protected tension, when MAP dropped from > 80 mm Hg to below 70 mm Hg for 10 minutes or more we closing pressure. This was indicated by a found a specific haemodynamic response in 15 momentary pulsatile increase in CVR, initially patients. Figure 5 shows an example in which for a short part of each pulse, and then during ABP fell after discontinuation of an infusion of most of the pulse cycle. The diastolic CBF+ dopamine. Whereas the diastolic flow velocity fell to zero as the arterioles started to close decreased from the very beginning, the systolic intermittently. Finally, when the systolic CPP flow velocity increased until point A and then fell below the lower limit of autoregulation, the started to decrease. The changes in ICP paral- systolic CBF+ started to decrease. leled changes in systolic flow velocity; thus ICP By contrast, when we modelled non- increased until point A and then decreased. autoregulating cerebrovascular systems, all Point A, which indicated the break in systolic variables behaved more uniformly, without vis- flow velocity and the beginning of reduction of ible breakpoints. Thus when ABP decreased, ICP, was detected below CPP of 55 mm Hg in both arteriolar CBF and ICP fell. Systolic and every instance (range 55 mm Hg to 37 mm Hg). diastolic CBF+ each decreased and did not We used our model to simulate the eVect of show the divergent behaviour seen in the a gradual decrease in arterial blood pressure autoregulating system. The pulse amplitude of according to the cerebrovascular system au- CBF+ did not change until it began to fall Mathematical modelling of interpretation of cerebrovascular autoregulation 725 J Neurol Neurosurg Psychiatry: first published as 10.1136/jnnp.63.6.721 on 1 December 1997. Downloaded from copyright.

Figure 5 An episode of transient arterial hypotension recorded in a patient with head injury. Arterial pressure (ABP), intracranial pressure (ICP), and the MCA blood flow velocity were recorded. Point A denotes switching point below systolic FV and ICP started to decrease. when the diastolic CPP reached the critical pliance of the big cerebral arteries increased as closing level. both the arteriolar resistance and the ampli- Using both clinical observation and model- tude of the CPP pulsations decreased. The ling data we can conclude that in patients with time average CBF+ was constant reflecting the gradual reduction of ABP, leading to a decrease systolic value increasing and the diastolic in CPP, maintained systolic FV, decreasing slightly decreasing. The arteriolar blood flow diastolic FV, and the increase in ICP indicates (mCBF) was maintained constant until the http://jnnp.bmj.com/ preserved autoregulation. mean CPP decreased below the lower limit of autoregulation at time point C—corresponding INTRACRANIAL HYPERTENSION to the breakpoint A on figure 7. Just below the Episodes of spontaneous intracranial hyperten- lower limit of autoregulation, the diastolic CPP sion were recorded in seven patients. Figure 7 decreased below the critical closing level. This shows a provoked haemodynamic response was reflected in the CVR starting to pulsate, that was similar in each patient. An increase in

indicating transient arteriolar closure during on September 28, 2021 by guest. Protected ICP above 30 mm Hg was followed by a diastole. The diastolic CBF+ decreased to gradual decrease in diastolic FV and an zero. The compliance C was maintained increase in systolic FV. After point A ( when a ICP was from 30 to 45 mm Hg, and CPP was because the pressure in the large cerebral <50 mm Hg) systolic FV ceased to increase. arteries was greater than in the cerebral arteri- After point B (CPP below 35 mm Hg) systolic oles. Beyond point B, the CPP became too low FV started to decrease. The average separation to keep the big arteries open thus the between ICP level corresponding to points A compliance Ca started to fall, causing a steep and B was 17 mm Hg (range 10 to 22 mm Hg). decrease in the systolic CBF+ and the pulse An increase in intracranial pressure was amplitude of the ICP waveform started to modelled as a response to an external infusion decrease. of CSF with a stepwise increase in rate. When In the non-autoregulating system (fig 8 we simulated this change in the model, in addi- right) the divergence of the systolic and diasto- tion to the parameters analysed for systemic lic CBF+ specific for intact autoregulation was hypotension, we included the compliance of not seen. The arteriolar CBF, and the systolic, the large cerebral arteries (Ca) (fig 8). and diastolic CBF+ each decreased as CPP The CPP fell with increasing ICP. The decreased and the cerebrovascular resistance amplitude of the ICP pulsations and the com- and compliance of the big arteries did not 726 Czosnyka, Piechnik, Richards, Kirkpatrick, Smielewski, Pickard J Neurol Neurosurg Psychiatry: first published as 10.1136/jnnp.63.6.721 on 1 December 1997. Downloaded from copyright.

Figure 6 EVect of stepwise decrease in ABP was modelled in autoregulating (left) and non-autoregulating (right) cerebrovascular systems. Vertical line A shows limit of autoregulation; other lines show moments of step changes in arterial pressure (ABP); mCBF=time average cortical blood ﬂow; ICP=intracranial pressure; CPP=cerebral perfusion pressure; CBF+=pulsatile blood ﬂow through big cerebral arteries; CVR=cerebrovascular resistance.

change. A breakpoint was detected, (similar to SHORT TERM HYPOPERFUSION: CAROTID ARTERY the time point B in the autoregulating system) COMPRESSION TEST below which transient closure of arterioles and Carotid artery compression was introduced as a gradual decrease in the compliance of big a test of cerebral autoregulation many years ago 7293233 arteries were found. and has been recently revived. Patients http://jnnp.bmj.com/ During gradual intracranial hypertension with atheroma of the carotid artery should not rising systolic flow velocity with stable or be examined using this method, therefore we decreasing diastolic flow velocity indicates pre- performed a careful carotid artery Doppler served autoregulation. study before the test, to exclude patients not suitable for compression. Discussion A transient hyperaemia after compression These results of mathematical simulations help and release of the common carotid artery is the interpretation, in terms of preservation of regarded as evidence of intact on September 28, 2021 by guest. Protected autoregulatory capacity, of the haemodynamic autoregulation.732 The extent of the hyperae- responses to short and long lasting episodes of mic overshoot is proportional to the extent hypoperfusion that have been seen 18 19 30 decrease in CVR during compression, and can clinically and produced experiment- be related to the gradient of the autoregulation ally.11–14 31 Changes in CBF through the big curve (fig 9 (C)). However, the hyperaemic arteries can be measured indirectly using response is also related to the decrease in TCD,8 assuming that the cross sectional area of the artery insonated by the probe is constant. cerebral perfusion pressure during compres- Cortical blood flow can be measured using sion, which cannot be measured in clinical laser Doppler flowmetry in the laboratory1–14 practice because pressure distal to the com- and in clinical practice28 with very good time pression is not known. Therefore, quantifica- resolution, but at only one small location. More tion of transient hyperaemia may be mislead- reliable ICP transducers are now available. ing, even if complex computer support is With more refined techniques for clinical used.33 The main diYculty in interpretation of measurement, mathematical modelling of the the test is that it produces “yes or no” results, data becomes an important supplement for whereas autoregulation is not an “all or none” interpretation of the phenomena recorded. phenomenon. Mathematical modelling of interpretation of cerebrovascular autoregulation 727 J Neurol Neurosurg Psychiatry: first published as 10.1136/jnnp.63.6.721 on 1 December 1997. Downloaded from copyright.

Figure 7 Example of recording of arterial pressure (ABP), blood ﬂow velocity (FV), and intracranial pressure (ICP) during ascending slope of ICP plateau wave. Lines A and B represent the points where systolic FV stopped increasing and started to decrease respectively.

INTERPRETATION OF TCD PULSATILITY INDEX blood flow velocities during spontaneous fluc- When there is sustained hypoperfusion due tuations in CPP may enable detection of either to intracranial hypertension or to arterial breakpoints of autoregulation (point A figs hypotension, our simulations predict that, 5–8). A negative correlation between changes when autoregulation is intact, the pulsatile in CPP and systolic flow velocity, accompanied component of flow through the big cerebral by a stable time average flow velocity is an indi- arteries will increase. Data from clinical cation of “good autoregulation”. A pressure observations34 and laboratory studies11–13 are passive decrease in time averaged and systolic http://jnnp.bmj.com/ each consistent with these predicted from our flow velocities with decreasing CPP signifies modelling simulations. They have proved that impaired autoregulation. An intermediate there is need for caution in equating an situation—that is, decreasing CPP accompa- increasing transcranial Doppler pulsatility nied by a falling time averaged flow velocity but index (defined as the ratio of the amplitude of a constant systolic flow velocity— marks the pulsations to the time average value of the state when autoregulation is compromised but blood flow velocity) with exhaustion of au- still intact.11 The results of previous studies in toregulatory reserve.30 Our simulations showed patients with head injury36 are compatible with on September 28, 2021 by guest. Protected that an increase in pulse amplitude of CBF+ such a conclusion. Such a correlation of occurs in an autoregulating system when CPP fluctuations of CPP with changes in the blood is decreasing toward the lower limit of flow velocity waveform can be expressed, using autoregulation. However, the pulsatility index computer waveform analysis, as a time depend- may also increase in non-autoregulating sys- ent index of the cerebral autoregulatory tems, when the CBF + pulse amplitude reserve. remains constant and the time average CBF+ 35 decreases passively. Recent experimental INTRACRANIAL PRESSURE WAVEFORM work31 has confirmed that the rise in the pulsa- Our simulations illustrate that fluctuations in tility index when CPP was decreasing could be ABP produce inverse changes in ICP when recorded both with intact autoregulation and autoregulation is intact.37 The correlation in cases when autoregulation was disturbed by becomes positive—that is, ABP and ICP previous ischaemic insults. change simultaneously when autoregulation is impaired. Hence, a running correlation coef- WAVEFORM ANALYSIS OF TCD ficient between the time averaged ICP and The results of simulation suggest that analysis arterial pressure could be of clinical utility as of changes in the systolic and time average an index of autoregulatory capacity.38 728 Czosnyka, Piechnik, Richards, Kirkpatrick, Smielewski, Pickard J Neurol Neurosurg Psychiatry: first published as 10.1136/jnnp.63.6.721 on 1 December 1997. Downloaded from copyright.

Figure 8 EVect of stepwise increase in intracranial pressure (ICP) was modelled in autoregulating (left) and non-autoregulating (right) cerebrovascular systems. Line A shows limit of autoregulation, line B critical closing of cerebral arteries. Other vertical lines denote changes in the rate of the simulated external infusion. ICP=intracranial pressure; CPP=cerebral perfusion pressure; mCBF=time average cortical blood ﬂow; CBF+=pulsatile blood ﬂow through big cerebral arteries; CVR=cerebrovascular resistance; Ca=compliance of big cerebral arteries.

Finally, the relation between the pulse lar reactivity to changes in cerebral perfusion amplitude of the ICP waveform and the time pressure. averaged ICP can be used in patients with intracranial hypertension to detect the critical

We thank the Cambridge Overseas Trust for funding the PhD http://jnnp.bmj.com/ closing phenomenon. When the time average projects for PS and SP and the Raymond Sackler Foundation ICP is increasing the decrease in amplitude of for supporting PS. MC, SP, and PS are on unpaid leave from the Warsaw University of Technology, Poland. We are very grateful the ICP pulsations signiﬁes that the cerebral to Professor G Teasdale for his helpful comments. perfusion pressure has reached the critical closing threshold (point B, ﬁg 8 (B)). This technique has been utilised already in methods for continuous analysis of the intracranial pres- Appendix: mathematical description of sure waveform.39 40

the model on September 28, 2021 by guest. Protected

Figure 2 shows the general structure of the Conclusion model presented in and discussed in the Meth- Our mathematical model helps interpretation ods section. This appendix includes a more of the phenomena that occur when the cerebral detailed description of the model parameters haemodynamic reserve is becoming exhausted. and presents the diVerential equations used to A transient hyperaemia after a short term com- describe the time dependent relation and mon carotid artery compression, a divergence internal variables. between the systolic and diastolic blood ﬂow velocity when the cerebral perfusion pressure is decreasing, and/or a negative correlation be- ELEMENTS OF THE MODEL

tween time average intracranial and arterial Resistance of great cerebral arteries (Ra) pressures can each be interpreted as indicating There are no clear data about any non-linear intact cerebral autoregulation. Conversely, behaviour of this resistance. It can be perma- each of an absent transient hyperaemic re- nently increased in the course of arterio- sponse, a pressure-passive behaviour of TCD sclerotic stenotic disease, during cerebral waveform, and coherent changes in ABP and vasospasm, or increased temporarily—for ICP slow waves indicate reduced cerebrovascu- example, by external compression of the Mathematical modelling of interpretation of cerebrovascular autoregulation 729 J Neurol Neurosurg Psychiatry: first published as 10.1136/jnnp.63.6.721 on 1 December 1997. Downloaded from

Figure 9 Graphical representation of the most important parameters of the model. (A) Averaged relation between the ratio of the pulse amplitude of ICP to the pulse amplitude of arterial pressure plotted against CPP in 55 patients with head

injury. The ratio is theoretically proportional to the compliance of big cerebral arteries Ca .(B) Modelling relation between

Ca and CPP.(C) Modelling relation between the resistance of small cerebral vessels (CVR) and the CPP,plotted for two

diVerent values of arterial CO2 tension. Region between the lower (LL) and upper (UL) limits of autoregulation represents copyright. linear dependence of CVR on CPP.(D) Relation of compliance of CSF space (Ci ) with intracranial pressure (Pi ). Popt represents the so called optimum pressure, above which compliance decreases when ICP increases. common carotid artery—a manipulation which modate changes in CPP and respond to the 2343 is discussed in this paper. arterial concentration of CO2. Figure 9 shows the relation between the CVR and Compliance of the great cerebral arteries (Ca ) cerebral perfusion pressure. The lower and Temporary storage of the inﬂowing arterial upper limits of the range where CVR is blood in the compliant big arteries is repre- proportional to CPP correspond to the lower sented by the container C a. This non-linear (LL) and upper (UL) limits of cerebral compliance plays an important part in the autoregulation. Outside these limits the CVR transmission of the arterial blood pulse pres-

converges to the horizontal asymptotic values http://jnnp.bmj.com/ sure to the pulsatile pattern of intracranial representing the limits of maximal vasodilata- pressure.10 41 The ratio of the pulse wave of ICP tion and maximal vasoconstriction. When arte- to the pulse wave of ABP is theoretically 40 rial CO2 increases the CVR decreases, and both proportional to the compliance Ca. The 23 transmission coeYcient recorded clinically in LL and UL increase. A programmable delay of the autoregulatory response (range five to patients with head injury and plotted against 38 the cerebral perfusion pressure presents a nine seconds) is included in the model. When the CPP decreases below the critical gradual increase of Ca when CPP decreases, with a maximum at a CPP of around 30 mm closing pressure (CCP), the CVR starts to on September 28, 2021 by guest. Protected Hg (fig 9 (A)). This can be explained by a increase as arterioles “collapse” with the low gradual decrease of the basal tone of the big transmural pressure.44 In the modelling soft- cerebral arteries as CPP decreases (fig 9 (B)). ware, the autoregulatory curve may be changed Below the critical closing pressure (CCP) cer- graphically by alteration of the maximal ebral arteries tend to collapse.42 The pulsatile vasodilatation and vasoconstriction levels, the blood volume flowing into the brain decreases, upper and lower limits of autoregulation, the while the pulsatile pressure drive remains critical closing pressure, and the tension of 25 almost unchanged; thus Ca decreases to zero. arterial blood CO2 (fig 9 (C)).The resistance

The dependence of the Ca on arterial gases is CVR constitutes the functional and anatomical

poorly documented. In hypercapnia, Ca prob- boundary between the high pressure (arterial), ably increases but much less than with arterial and the low pressure (venous) parts of the cer- hypotension.43 44 ebrovascular bed. Cerebral resistance vessels

The next narrowing of the blood ﬂow pathway Compliance of capillaries and small veins (Cv ) simulates the CRV constituting the hydro- There are no data about its non-linear dynamic resistance CVR that is able to accom- character.25 730 Czosnyka, Piechnik, Richards, Kirkpatrick, Smielewski, Pickard J Neurol Neurosurg Psychiatry: first published as 10.1136/jnnp.63.6.721 on 1 December 1997. Downloaded from

Ra CVR Rb CBF+Pa CBF Pv

If Ca Cv

ABP R CSF Pss Pi

Figure 10 Electrical structure equivalent to the hydrodynamic model (ﬁg 2). All components are described in the text of the appendix.

The resistance of venous outﬂow (R ) b dP 1 ABP−P P −P P −P The venous outﬂow to the dural sinuses is i = .( a − i ss − v ss) through the collapsible bridging veins and is dt Ci Ra Rcsf Rb

represented by a Starling resistor labelled Rb , dP dP 1 P −P P −P controlled by a gradient between intracranial v = 1 + .( a v − v ss) pressure and cerebral venous pressure (Pv-Pi ). dt dt Cv CVR Rb

If the gradient becomes negative, Rb increases and blocks the venous outﬂow. If the gradient is dPa dP1 1 ABP−Pa Pa−Pv positive, R remains low and relatively con- = + . − −If b dt dt C ( R CVR ) stant. This mechanism (which has been a a questioned in experimental observations45), In the above equations t is a time and dx/dt together with the CSF circulation, preserves denotes the ﬁrst order derivative of the variable the physiological gradients between the cer- x versus time. copyright. ebral venous, intracranial, and sagittal sinus The “state variables” Pa,Pi , and Pv together 39 46 pressures. with input ABP, Pss, and model parameters per- mit calculation of the other simulated variables. P >P >P The program for the IBM PC compatible v i ss computer has been designed.35 Numerical solving of these equations requires an IBM PC CSF circulation The pathway of the CSF circulation is computer (486 or Pentium processor) with a represented by constant rate secretion of CSF hard disk space of at least 5MB. 46 47 (If) from the arterial blood, storage of CSF

inside a cerebrospinal compliance Ci,, and its Selected abbreviations and acronyms: reabsorption into the dural sinuses through a ABP = arterial blood pressure (measured or http://jnnp.bmj.com/ zero pressure valve of an internal resistance simulated)

Rcsf. Figure 9 (D) shows the relation between ICP, Pi = intracranial pressure measured the intracranial pressure and CSF (first) and simulated (second) compliance.15 16 20 It results from the so-called CPP = cerebral perfusion pressure CSF pressure-volume curve. Below a lower FV = transcranial Doppler maximal blood limit, often called the optimal pressure,16 17 the flow velocity in the MCA changes in intracranial pressure are propor- LDF = laser Doppler cortical red cell flux tional to changes in the intracerebral volume. CBF = simulated blood flow through the on September 28, 2021 by guest. Protected The compliance, equal to the inverse of the cerebral resistance vessels gradient of this curve, is constant. Above the CBF+ = simulated blood flow through the optimal pressure, the increase in the intracer- great cerebral arteries

ebral volume causes an exponential increase in Pa = cerebral arterial pressure in the small ICP.20–22 Hence, the inverse of its gradient is arteries

proportional to the inverse of intracranial Pss = sagittal sinus pressure

pressure measured relative to the reference Pv = pressure in the cortical veins

pressure Po (fig 9 (D)). Ra = resistance of great cerebral arteries The electrical circuit (fig 10) corresponds CVR = cerebrovascular resistance controlled directly to the hydrodynamic structure (fig 2). by autoregulation

It is used for simulation of the phenomena Ca = compliance of the great cerebral arteries

induced by changes in ABP. The two input Ci = compliance of the CSF containers

voltage sources represent ABP and venous Rb = resistance of the cortical veins and

pressure in the dural sinuses Pss. For further bridging veins

analysis of the response to deﬁned excitations, Rcsf = resistance to CSF outﬂow

the electrical circuit was described by a set of PaCO2 = tension of CO2 in arterial blood diVerential non-linear equations: CCP = critical closing pressure Mathematical modelling of interpretation of cerebrovascular autoregulation 731 J Neurol Neurosurg Psychiatry: first published as 10.1136/jnnp.63.6.721 on 1 December 1997. Downloaded from

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