Journal of the Neurological Sciences, 27 (1976) 4 8 5 4 9 8

485

~(5~ Elsevier Scientific Publishing Company, Amsterdam - Printed in The Netherlands

A M A T H E M A T I C A L M O D E L OF CEREBROSPINAL F L U I D DYNAMICS

RALPH BLOCH and ANDREW TALALLA

Department of Clinical Neurosciences, MeMaster University, Hamilton, Ont. (Canada) (Received 4 August, 1975)

SUMMARY

The ability to solve systems of simultaneous non-linear differential equations by a combination of analytical and computational techniques has encouraged the development of valid mathematical models of biological phenomena. The dynamics of the cerebrospinal fluid (CSF) system has been the subject of closer scrutiny in recent years since the recognition of symptomatic low-pressure hydrocephalic states in man. A mathematical model has been derived from 7 assumptions: (1) That the brain is a spherical shell. (2) That CSF is secreted at a constant rate. (3) That CSF absorption is linearly dependent on pressure. (4) That flow between the CSF compartments is proportional to the pressure difference. (5) That Laplace's Law holds for the visco-elastic properties of the brain. (6) That there is compliance in the spinal compartment of the CSF system. (7) That vascular pulsations in the cranial and spinal compartments are capacitatively coupled. Using known data (and estimates of as yet unknown values) for the several parameters, the validity of the model has been successfully tested against 3 clinical conditions. This model extends our understanding of derangements of CSF dynamics and suggests where further research may yield data at present lacking.

INTRODUCTION

The non-linear characteristics of most biological systems present a formidable Address for reprints: Andrew Talalla, M.D., F.R.C.S., Clinical Neurosciences, McMaster University, Hamilton, Ont., Canada.

486 obstacle to tile quantitative analysis of even tile seemingly simple physioJogicai mechanisms. With the ability to solve complex systems of equations ~,ing corn putational techniques, interest has been increasingly focussed on the development of valid mathematical models of biological phenomena. The altered dynamics of the cerebrospinal fluid (CSF) system in maa.. clinicai syndromes has been, until relatively recently, the subject of qualitative observations leading to descriptive hypotheses. Towards a more rigorous analysis of the various derangements ,~t the CSI-: system, an abstract but well-disciplined model has been derived. THE MODEl.

The following assumptions define the proposed model: ( 1) Spherical shell brain It is assumed that the brain can be represented as a homogenous spherical shell comprising 3 compartments (,Fig. 1), viz. the ventricle, a spherical space with radius R, corresponding to the lateral and third ventcicles; the spinal subarachnoid compartment which under normal circumstances would include the fourth ventricle and the cisterna magna; and the supratentorial subarachnoid compartment. The volumes and pressures in the 3 compartments are designated V1, V,, V3 and pl, p~, p's, respectively. (2) Constant secretion of CSF It is assumed that CSF is secreted at a constant rate in all 3 compartments even

Fig. 1. Diagrammatic representation of CSF compartments.

487 though the bulk of CSF will be secreted in the first compartment. The rates of secretion are G1, G2, and Ga respectively.

(3) £,inear absorption of CSF It is assumed that absorption is linearly dependent on pressure in each compartment (L6fgren, Von Essen and Zwetnow 1973). The proportionality constants (absorption admittances) are designated ya, y'~, ya.

(4) Linear pressure-flow relationships It is assumed that the flow between the compartments is proportional to the pressure difference between the compartments; the proportionality constants (flow admittances) are ym and y2a, respectively.

(5) Laplace's law It is assumed that the wall tension T in the brain shell is related to the pressure difference between the first and third compartments pl - - P3 and the ventricular radius R by Laplace's law: 2T pl - - P 3

R

(1)

where T is a function of both R and time t because of the visco-elastic properties of the brain. This is due to the fact that cerebral matter can flow, if subjected to sustained stress (Schettini and Walsh 1973). This relationship obviously does not hold for the true wall tension, applying only to infinitesimally thin spherical shell membranes. Since this quantity is practically impossible to measure in brain, we can replace it by a pseudo-wall tension T which is defined by equation (1). Equation (1) thus becomes a tautologism. This function can then be substituted by an expression:

T where a

H~a'R dT dR

(2)

The quantity a is obviously also related to the time course of the charge in radius R. The viscous flow properties of the brain can be described by a time constant rv. For time intervals large compared with ~v, a will be a static constant and will reflect the change in wall tension with radius after equilibration. We call this a 0 (R). On the other hand, for changes of radius in time intervals short compared to ~v, a different constant will describe the change of wall tension dynamically; we call this (~D.

For the calculations of the model we will use numerically given static equilibrium wall tension functions, together with a dynamic derivative, an.

488 (6) Compliance oJ'the second compartment It has been pointed out by Weed and Flexner (1932) that while in the adult the skull represents a rigid compartraent, the spinal compartment shows a compliance, that is :

d V~ dpe This is probably due to the compressibility of the extradural structures, parucularly in the lumbar canal. (7) Vascular pulsations it is assumed that in compartments i and 2, vascular pulsations are capacitatively coupled :

d I/

dp z~

.'. . . .

dp2,

c,.. ( dt ~"

dtJ' 2

1, 2

(3t

that is, the pulsatile change in volumes I or 2 is proportional to the difference in change between pressure in the compartment P2 and the driving pressure p ~ . The proportionality constant C a is called the capacity in analogy to electric network theory. It is not necessary to include a pulsatile source in compartment 3, since volume changes in compartments 1 and 3 are coupled by the Monroe-Kelly law. Tile driving pressures pao are assumed to be harmonic functions and can be best described as p~. =p°.cosm0.t, where o)0 -- 2~ :~ heartrate, 2 -= 1, 2. Based upon the above 7 assumptions the model can be represented as a system of 4 simultaneous non-linear first order differential equations (Fig. 2):

()'le ! )'1)" p l

dR O'] t 4xt? ~

.!"12"pe

d,t

/ dplq ('1

" {\

dt

,IV l ", dI

LAPLACE'S LAW

[I (~p,.sinoa.t o t I(c'3'o (~P2-sin~-t G1 / .LC 1

"Ventricle"

G

-L'C2

G3

I[ !

"Spinal "Supratentorial Subarachnoid Subarachnoid Space" Space"

Fig. 2. Mathematical representation o f C S F dynamics, based on the seven assumptions.

)

(4a)

489 )qz'pl

- 0'12 !-yz by2a)'pe + yzzp3 = ---G2 -~

dp2

/ dp~2

dt

dt

)'23"p2 - - (3'23~ y~)'P3 . . . .

pl .... p~

dp2)

dR Gs--4:r R 2 dt

2T R

(4b)

(4c)

(4d)

The system has been solved by a combination of analytical and computational techniques (see Appendix)*. Instead of ventricular radius R, the relative radial increment per heartbeat x(2~/~o0), is calculated. This quantity is defined by equation (8d) in the appendix. For the system to be stable, 2 conditions have to be fulfilled: (a) the relative radial increment per heartbeat has to be zero; (b) with increasing radius, x(2n/o~0) has to become negative, while decreasing radius should lead to a positive x(2x/(,m). These two conditions define an equilibrium radius R0 eq.

A P P L I C A T I O N A N D RESULTS

The model has been used to simulate three clinical situations. (a) Normal intracranial hydrodynamics In man, the only values available for the parameters used are heart-rate, brain volume, intracranial volume, CSF secretion rate and transmitted vascular pulse pressures. For the other parameters we have perforce to choose somewhat arbitrary values that will result in CSF pressures that conform to the known normal. The parameters and results are showp in Tables 1 and 2 and in Fig. 3. The solid ctwve in Fig. 3 represents wall tension function used as input, the vertical arrow identifying the resulting stable radius at 2.08 cm. It will be seen in Table 2 that the CSF pressures are within normal limits.

(b) Normal pressure hydrocephalus ( NPH) Parameter values identical to the normal condition are used, the simulation of the clinical state being achieved by altering the visco-elastic property of the brain (wall tension function). Fig. 4 shows that even a minimal flattening of the wall tension curve results in ventricular enlargement without alteration of CSF pressures (Table 2).

* Access to the digital computer programme is available through the authors.

490 TABLE 1 I N P U T P A R A M E T E R VALUES USED TO S I M U L A T E THE N O R M A L S T A I I A N t ) THF L O W - P R E S S U R E H Y D R O C E P H A L U S STATE Name of parameter

iJnits

Heart rate Brain volume Cranial volume Absorption resistance ventricle Absorption resistance spinal Absorption resistance subarachnoid Flow resistance ventr.--spinal Flow resistance spinal - subarachnoid Secretion rate ventricle Secretion rate spinal Secretion rate subarachnoid Ventricle pulse pressure Ventricle pulse capacity Spinal pulse pressure Spinal pulse capacity Lumbar compliance Dynamic wall tension

60 1,450 1,500 10 (i.~ia 5 0.05 0.055 0.05 20 3 2 ,~ 4 40 1 3.2 21)

per rain cm :~ cm ;~ mm Hgl cm:~/hr mm Hg/cm:~/hr mm Hg/cm:~/hr m m Hg/cm:~/hr mm Hg/cm:~/hr cm~/hr cm:~/hr cm:~/hr mm Hg cma/mm Hg m m Hg cm:3/mm Hg cm:~/mm Hg mm Hg

'~ The value for absorption resistance is reduced to 1.6 units to simulate CSF diversion from the ventricle. TABLE 2 R E S U L T I N G VALUES F R O M T H E T H R E E S I M U L A T E D C L I N I C A L STA'[ES Name of parameter

Normal

N PH

NPHshunted

Units

Equilibrium radius Ventricle pressure Spinal pressure Subarachnoid pressure

2.08 3.4 2.32 1.21

2.28 3.4 2.32 1.21

1,8/) 3.15 2.16 1.13

cm m m Hg m m Hg mm Hg

(c ) Normal pressure hydrocephahts after CSF diversion (shunt) The simulation of a shunting procedure is achieved by increasing the absorption admittance of CSF in the ventricle, all other parameters of the normal pressure hydrocephalic state including the visco-elastic property of the brain beingunchanged. Fig. 5 shows a reduction in ventricular size (lowering of equil!brium radius) after simulated CSF diversion. DISCUSSION

The report (Adams, Fisher, Hakim, Ojemann and Sweet 1965) of the beneficial effects of a CSF-diverting procedure for an unusual group of patients whose distinctive clinical syndrome is associated with hydrocephalus and normal intracranial

491

•

1,]

E'~',. "18I//fl'*~**~ee

11 ~ ~--"

_~ × -.,fl~

i

-.42[

ee

-.72~ 1.7

.,..,~

I 1.8

I 1.9

l 2.0

Ventricular Radius

>

I 2.1 R [cm]

I 2.2

m

Fig. 3. Computer print-out of the normal intracranial state. Code: The solid curve ( - - ) represents wall tension T as a function of ventricular radius R. It is an assumed curve used to calculate the corresponding relative radial increment per heartbeat ( ~ ' ~ ' ' O ) . The vertical arrow ( I ' ) indicates the resulting equilibrium radius. It corresponds to the zero transition of the relative radial increment and was determined by a Gaussian zero-finding algorhythm.

i"" .,8

t

..

, |~

Resulting Radius

Equilil~ium

i

f

n.

-.72 1 I Ventricular Radius

I

m

R [cmj

Fig. 4. Computer print-out of a normal pressure hydrocephalic state. For code, see Fig. 3.

pressures, g a v e t h e i m p e t u s to c u r r e n t a t t e m p t s to e x p l a i n d e r a n g e m e n t s o f C S F d y n a m i c s o n the basis o f p h y s i c a l laws. H a k i m a n d A d a m s (1965) w e r e the first to p o i n t o u t that, in a s p h e r i c a l c o n tainer, t r a n s m u r a l p r e s s u r e p, wall t e n s i o n T a n d r a d i u s R, a r e c o n n e c t e d by the relationship :

492

g

T

,74

.46 Z

•

• ~ '"

~ E

.17

- ' ~ - -.11 Ig er

-.~1

~,.mbrk,m ~

o

i

2,2 Ventricular Radi.s R [era ]

Fig. 5. Computer print-out of a shunted normal-pressure hydrocephalus "'patienl", I-~o~code, see Fig. 3.

2T P

R

This was expanded by Geschwind (1968) who examined how the ~vatl tension T itself could be related to small changes in the radius R. Both papers describe a total force as the scalar product of pressure surface area of a spherical container. They reason that, for constant wall tension, this total force increases as the container expands. A further step was taken by Epstein (1974) who tried to relate the wall tension to the elasticity module o f cerebral matter and skull by applying Hooke's taw {Feyn,nan 1963). While these papers break new ground in our understanding of normal pressure hydrocephalus, they also proposed some misconceptions. Force is a vector. If pressure acts on a curved surface, therefore, one has to integrate the infinitesimal force vectors d F p d~ (d~ is an infinitesimal surface element whose direction ,s at right angles to the surface at each point): t-'--

,( dk: " X

j "

p act

~51

S

This equation can be transformed into a volume integral F vgrad p d e , using a well-known integral identity (Morse and Feshbach 1953). In tlae purely hydrostatic case, in the absence o f gravity, where p is constant within tile whole

* The relationship p 27]R is erroneously ascribed to Pascal. It should be credited to Laplace (1807). Walker (1922) accords priority even to Lagrange.

493 volume, grad p is zero, which therefore brings the total force to 0. I f however, the fluid is subjected to gravity, the pressure is then p - - Po - - z ' ~ . g where z is the coordinate pointing vertically up, (2 is the density of the fluid, g is the gravitational acceleration constant and pa is the pressure at the level z --- 0; grad p is then zero in the x and y direction, but gradzp -- 2.g, therefore: Fx--

Fy--O

and F~ = - - o ' g "

( dV J V

--9'g'V=--m'g

(6)

In other words, the negative z component of the total force is equal to the weight of the fluid in the ventricle. Furthermore, Laplace's law does not apply to the absolute pressure, but rather to the transmural pressure, that is pressure difference between ventricles and subarachnoid space. A n analysis of the forces present in the rheological system representing the CSF dynamics enables us to examine its stability in three clinical conditions. The model makes use of many parameters that have as yet not been experimentally determined. Our use of guessed values can be justified, in that the three clinical situations simulated can be achieved by only a minimal and readily obvious change in the otherwise somewhat arbitrary parameter values. A wide range of parameter values leading to a normal state results in similar qualitative changes if brain elasticity is changed or if CSF is diverted. The only change needed, to lead from the normal condition to a progressive dilatation of the ventricle is a slight flattening of the wall tension function. This corresponds to a minimal loss of elasticity of the cerebral matter, changes which occur as a result of ageing or degeneration. Once the dilatation has occurred, facilitation of removal of CSF from the ventricle leads to ventricular contraction. The wall tension functions chosen in the above calculations show total reversibility. The actual visco-elastic properties of cerebral matter have non-reversible features, leading to hysteresis. Thus, once dilatation has occurred, shunting will not necessarily result in a total restoration of original ventricular size. Extension of the present model to non-reversible processes is possible, but has not been attempted here. It is hoped that the arbitrarily chosen parameter values can be progressively replaced by those determined experimentally. This should stimulate a series of experiments designed to measure these quantities. It is obvious that the applicability of these results is contingent upon the validity of the assumptions and limited by the arbitrary choice of parameter values. Thus, it may be argued that the brain is not a spherical shell but that its true geometry is too complex for analysis. Secretion and absorption of CSF are not

494 necessarily zero and tirst order functions respectively of CSF p)es,~ure, t-,~arthermorc the pulsatile pressures are not harmonic and are subject to feedback reguialions and other variations. It becomes evident, however, that the rigor of the analysis surpasse~ , , , ) present knowledge of quantitative pressure-flow relationships and visco-elaslic properties o f the living brain. While the model as described does not as yet help us predict the t'alc ot'a gixe)~ patient, it enables us to pose more aecuralely the questions that require ,tn:,wers f)'on~ experiments• APPENDIX

Derivation