Modeling of CSF Dynamics: Legacy of Professor Anthony Marmarou

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1 Modeling of CSF Dynamics: Legacy of Professor Anthony Marmarou Marek Czosnyka, Zofia Czosnyka, Kiran J. Agarwal-Harding, and John D. Pickard Astract The mathematical model of cererosinal fluid (CSF) ressure volume comensation, introduced y Anthony Marmarou in 973 and modified in later studies, rovides a theoretical asis for differential diagnosis in hydrocehalus. The Servo-Controlled Constant Pressure Test (Umea, Sweden) and Comuterised Infusion Test (Camridge, UK) are ased on this model and are designed to comensate for inadequate accuracy of estimation of oth the resistance to CSF outflow and elasticity of CSF ressure volume comensation. Dr. Marmarou s further works introduced the ressure volume index (PVI), a arameter used to descrie CSF comensation in hydrocehalic children and adults. A similar technique has een also utilized in traumatic rain injury (TBI). The resence of a vascular comonent of intracranial ressure (ICP) was a concet roosed in the 9s. Marmarou demonstrated that only around 3% of cases of elevated ICP in atients with TBI could e exlained y changes in CSF circulation. The remaining 7% of cases should e attriutale to vascular comonents, which have een roosed as equivalent to raised rain venous ressure. Professor Marmarou s work has had a direct imact in the field of contemorary clinical neurosciences, and many of his ideas are still eing investigated actively today. Keywords Cererosinal fluid Intracranial ressure Hydrocehalus Mathematical modeling M. Czosnyka ( ) Deartment of Clinical Neuroscience, Neurosurgical Unit, University of Camridge, Camridge, UK and Academic Neurosurgery, Addenrooke s Hosital, Camridge, UK mc@medschl.cam.ac.uk Z. Czosnyka and J.D. Pickard Deartment of Clinical Neuroscience, Neurosurgical Unit, University of Camridge, Camridge, UK K.J. Agarwal-Harding Harvard Medical School, Harvard University, Camridge, MA, USA Introduction Models of cererosinal fluid (CSF) circulation usually differ from models simulating rain tissue dislacement. Anatomical structure and distriution of stress-strain in the tissue are not of interest here comared with the hydrodynamics of CSF flow. Dynamics of intracranial ressure (ICP) may e monitored invasively in clinical ractice with a ressure transducer, and dynamics of CSF flow can e measured noninvasively with hase-coded magnetic resonance imaging (MRI). Therefore, models of CSF dynamics have an estalished clinical alication in diagnosis and management of several diseases such as hydrocehalus, idioathic intracranial hyertension, and syringomyelia. Many theoretical/modeling studies on CSF dynamics were ulished efore the 97s [ 3,,, ]. However, Professor Anthony Marmarou was one of the first [, 3 ] who integrated all comonents CSF roduction, circulation, asortion, and storage in one elegant theoretical structure exressed as an electrical circuit (see Fig. ). He analyzed theoretically three asic maneuvers: olus CSF withdrawal, addition, and constant rate infusion. This model has withstood the test of time and, with only a very few cosmetic modifications, it is still used today. Consequently, hydrocehalus and other disorders of CSF circulation are now characterized using arameters from this model such as resistance to CSF outflow, elasticity, and ressure volume index (PVI). These arameters were introduced into clinical ractice y Marmarou et al. in 97 [ ]. He also roosed a mathematical exlanation of the linear relationshi etween ulse amlitude and mean ICP [ ], which was later elaorated y Avezaat and Eijndhoven [ ]. In 97, he descried the vascular comonent of ICP [ ]. In atients with traumatic rain injury (TBI), only 3% of cases of elevated ICP can e exlained y changes in CSF circulation. Therefore, Marmarou concluded that the remaining 7% of cases of elevated ICP are derived from changes in the intracranial vascular comonent. All three of these milestone achievements in the area of CSF dynamics are used today. The mathematical model of CSF dynamics will e resented riefly in the next section followed y a synosis of the legacy of Marmarou s works in contemorary clinical neuroscience. G.A. Aygok and H.L. Rekate (eds.), Hydrocehalus, Acta Neurochirurgica Sulementum, Vol. 3, DOI:.7/ _, Sringer-Verlag/Wien 9

2 M. Czosnyka et al. Fig. Electrical model of cererosinal fluid (CSF) dynamics according to ( a ) Marmarou. Uer anel : Current source reresents formation of CSF, resistor, and diode unilateral asortion to sagittal sinus (voltage source ss reresents sagittal sinus ressure). Caacitor nonlinear comliance of CSF sace. Lower anel : Extended model showing hydrodynamic consequence of shunting Formation of CSF Intracranial ressure P Resistance to CSF outflow Pss Po Comliance of CSF sace Intracranial ressure P Valve Formation of CSF Comliance of CSF sace Resistance to CSF outflow Resistance of shunt system Po Pss Oening - sihoning ressure CSF circulation Hydrocehalus shunt Marmarou s Model of CSF Dynamics The mathematical model of CSF ressure volume comensation, introduced y Marmarou [, 3 ] and modified in later studies [, ], rovides a theoretical asis for differential diagnosis in hydrocehalus. Under normal conditions, without long-term fluctuations of the cereral lood volume, roduction of CSF is alanced y its storage and reasortion in the sagittal sinus: Production of CSF = Storage of CSF + Reasortion of CSF Production of CSF is assumed to e constant, although it may not always e the case. Reasortion is roortional to the gradient etween CSF ressure ( ) and ressure in the sagittal sinuses ( ss ): - Reasortion = R ss is considered to e a constant arameter determined y central venous ressure. However, it is not certain whether an interaction etween changes in CSF ressure and ss exists in all circumstances: in atients with enign intracranial ss () () hyertension ss is frequently elevated due to fixed or variale stenosis of transverse sinuses, and a similar situation can e seen in venous sinus thromosis. The coefficient R (symol R CSF is also used) refers to the resistance to CSF reasortion or outflow (units: mmhg/ (ml/min)). Storage of CSF is roortional to the cererosinal comliance C (units: ml/mmhg) and the rate of change of CSF ressure d /dt : d Storage = C dt The comliance of the cererosinal sace is inversely roortional to the gradient of CSF ressure and the reference ressure ( ): C = E ( - ) Some authors suggest that relationshi ( ) is valid only aove a certain ressure level called the otimal ressure [ ]. The coefficient E is termed the cereral elasticity (or elastance coefficient) (unit: ml ). Elevated elasticity (>. ml ) signifies a oor ressure volume comensatory (3) ()

3 Modeling of CSF Dynamics: Legacy of Professor Anthony Marmarou a c ICP [mm Hg] ICP-Po (P-Po) :3 :3 : : : : : 3 7 Volume [ml] 9 AMP [mm Hg].... :3 :3 : : : : : d AMP [mm Hg] ICP [mm Hg] Fig. Methods of identification of the model of cererosinal fluid (CSF) circulation during constant rate infusion study. ( a ) Recording of CSF ressure ( ICP ) versus time increasing during infusion with interolated modeling curve (7) Infusion of constant rate of. ml/min starts from vertical line. ( ) Recording of ulse amlitude ( AMP ) during infusion. Rise in AMP is usually well correlated with rise in ICP. ( c ) Pressure volume curve. On the x -axis, effective volume increase is lotted (i.e., infusion and roduction minus reasortion of CSF). On y -axis, the increase in ressure is measured as a gradient of current ressure minus reference ressure, relative to aseline ressure. ( d ) Linear relationshi etween ulse amlitude and mean ICP. Intercet of the line with x -axis (ICP) theoretically indicates the reference ressure reserve [ ]. This coefficient has recently een confirmed to e useful in redicting a atient s resonse to third ventriculostomy [7 ]. Relationshi ( ) exresses the most imortant law of the cererosinal dynamic comensation: When the CSF ressure increases, the comliance of the rain decreases. A comination of ( ) with ( ) and ( ) gives a final equation ( ): d - E ( - ) dt R + = It ( ) where I (t ) is the rate of external volume addition and is a aseline CSF ressure. The model descried y this equation may e resented in the form of its electric circuit equivalent [ ] (Fig. ). () Equation ( ) can e solved for various tyes of external volume additions I (t ). The most common in clinical ractice is (a) A constant infusion of CSF ( I (t ) = for t < and I (t ) = I inf for t > ) see Fig. : é - ù êiinf + [ ] R ú - Pt () = ë û + é- ù E Iinf t - é - ê + R ú ù ë û + Iinf êe ú R êë úû The analytical curve ( ) can e matched to the real recording of the ressure during the test, which results in an estimation of the unknown arameters: R, E, and (see Fig. a ). () A olus injection of CSF (volume D V ): ()

4 M. Czosnyka et al. a ICP [mmhg] AMP ICP [mmhg] ICP [mmhg] c AMP ICP [mmhg] ICP [mmhg] AMP ICP [mmhg] ICP [mmhg] AMP ICP [mmhg] d Fig. 3 Examles of a constant rate infusion test. ICP mean ICP (-s average), AMP ulse amlitude of ICP. The lue section is the duration of infusion. ( a ) Normal ressure hydrocehalus (NPH): Although the ase line ressure is normal, the resistance to cererosinal fluid (CSF) outflow increased, there are lots of strong vasogenic waves, and changes in ulse amlitude are fairly well correlated with changes in mean ICP. ( ) Acute hydrocehalus ost-suarachnoid hemorrhage (SAH): The t () é - ù EêD V+ t R ú ë û ( - ) e = + - é E t ù EDV t + e êe -ú ë û The olus injection can e used for calculation of the PVI, defined as the volume added externally to roduce a tenfold increase in the ressure [ ] : def DV PVI = ; æ - ö.3 E log ç è - ø (7) (a,) in the formula (a) is eak ressure recorded just after addition of the volume Δ V. The PVI is theoretically roortional to the inverse of the rain elastance coefficient normal aseline ressure was measured, ut the resistance to CSF outflow is high. Good resonse of shunt surgery was exected. ( c ) Cereral rain atrohy: the ase line ressure is low, ut the resistance to CSF outflow is low. No vasogenic waves were recorded, and ulse amlitude does not resond. ( d ) Normal: the ase line ressure, the resistance to CSF outflow, and other arameters are normal, and thus the result demonstrates normal CSF circulation E. The ressure volume comensatory reserve is insufficient when PVI <3 ml, and a PVI value aove ml signifies an over-comliant rain. These norms are valid for the PVI calculated as an inverse of E (according to ) using slow infusion. If the olus test is used, norms for PVI are higher (the threshold equivalent to 3 ml is around ml [ ] ). The formula ( 7 ) for time t = descries the shae of the relationshi etween the effective volume increase D V and the CSF ressure, called the ressure volume curve (Fig. c ): ( ) = - e + EDV Finally, Eq. ( 7 ) can e helful in the theoretical evaluation of the relationshi etween the ulse wave amlitude of ICP and the mean CSF ressure. If we resume that the rise in lood volume after a heart contraction is equivalent to a raid (9)

5 Modeling of CSF Dynamics: Legacy of Professor Anthony Marmarou olus addition of CSF fluid at the aseline ressure, the ulse amlitude (AMP) can e exressed as: ( ) ( E V ) AMP + - = - e D - () In almost all cases, when CSF ressure is eing increased y the addition of an external volume, the ulse amlitude rises [, ] see Fig., d. The gradient of the regression line etween AMP and is roortional to the elasticity. The intercet, theoretically, marks the reference ressure. Synosis of Clinical Alications of the Model The Servo-Controlled Constant Pressure Infusion Test [ 7 ] is used for assessment of CSF disorders. Its aim is to evaluate the resistance to CSF outflow in a reetitive and reliale way. Full identification of the model, including elasticity, can e made using a comuterized constant rate infusion test [] suorted y the dedicated software ICM+ ( htt:// ). Use of the constant rate infusion test in many centers contriuted to a definition of rofiles of ICP and its ulse amlitude in different ossile clinical scenarios, including normal ressure hydrocehalus, rain atrohy, acute hydrocehalus, and in nondistured CSF circulation (see Fig. 3 ). Analysis of the constant rate infusion test in shunted atients can e helful in shunt assessment in vivo. The electrical circuit model roosed y Professor Marmarou, sulemented y a ranch-defining nonlinear ressure flow erformance curve, is resented in Fig.. Direct knowledge of the curve, as assessed in shunt evaluation laoratories [, ] allows in vivo identification of the model and sensitive rediction whether the shunt is working roerly, underdraining, or overdraining. The roortional increase of the ulse waveform of ICP with mean ICP, has een exlained y Marmarou [ ] as a consequence of the exonential ressure volume curve. Although further works demonstrated that in a system with a good ressure volume comensatory reserve, the ressure volume curve is linear [, ], at higher ressures, the curve ecomes exonential [ ] This led to analysis of a moving correlation coefficient (-s to -min eriod) etween mean ICP and AMP. The resulting RAP coefficient indicates the state of comensatory reserve. RAP = suggests good comensatory reserve; RAP =, oor comensatory reserve [ 9 ]. The idea of a vasogenic comonent of ICP led to modification of Davson s equation: ICP = R CSF * CSF formation + ss + Arterial vasogenic comonent The arterial vasogenic comonent is a comonent of ICP which is derived y detection of ulsatile lood flow in nonlinear comonents of cererosinal sace (intracranial and arterial ed comliance, resistance of collasile ridging veins, and autoregulation-controlled main cererovascular resistance). Conclusion When Professor Marmarou was terminally ill and was asked y his coworkers what they should do in future years, he simly said Continue (Dr. G. Aygok, ersonal communication). There are certainly a lot of directions to continue in and many questions initiated y Anthony Marmarou in the field of clinical neurosciences that remain unanswered. Acknowledgements This work was suorted y the National Institute of Health Research, Biomedical Research Centre, Camridge University Hosital Foundation Trust Neurosciences Theme, and a Senior Investigator Award (to J. D. P.). Disclosure ICM+ is a software for rain monitoring in clinical/exerimental neurosciences (htt// ). It is licensed y the University of Camridge (Camridge Enterrise Ltd). M.C. has a share in a fraction of the licensing fee. Conflicts of interest statement what is in Disclosure may e in Conflict of interest. References 3. Aschoff A, Kremer P (99) Determining the est cererosinal fluid shunt valve design: the ediatric valve design trial. Neurosurgery ():99 9. Avezaat CJJ, Eijndhoven JHM (9) Cererosinal fluid ulse ressure and craniosinal dynamics. Ph.D. thesis, The Jongloed an Zoon Pulishers, The Hague 3. Benaid AL (97) Contriution a l etude de l hyertension intracranienne modele mathematique. M.D. thesis, Grenole University. Czosnyka M, Whitehouse H, Smielewski P et al (99)Testing of cererosinal comensatory reserve in shunted and non-shunted atients: a guide to interretation ased on an oservational study. J Neurol Neurosurg Psychiatry :9. Czosnyka Z, Czosnyka M, Richards HK et al (99) Posture-related overdrainage: comarison of the erformance of hydrocehalus shunts in vitro. Neurosurgery (): Davson H, Hollingsworth JR, Segal MD (97) The mechanism of drainage of the cererosinal fluid. Brain 93: 7 7. Eklund A, Lundkvist B, Koskinen LO, Malm J () Infusion technique can e used to distinguish etween dysfunction of a hydrocehalus shunt system and a rogressive dementia. Med Biol Eng Comut (): 9

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