Neurological Research A Journal of Progress in Neurosurgery, Neurology and Neurosciences
ISSN: 0161-6412 (Print) 1743-1328 (Online) Journal homepage: http://www.tandfonline.com/loi/yner20
A non-linear haemodynamic model for the arterial pulsatile component of the intracranial pulse wave William D. Lakin & Cordell E. Gross To cite this article: William D. Lakin & Cordell E. Gross (1992) A non-linear haemodynamic model for the arterial pulsatile component of the intracranial pulse wave, Neurological Research, 14:3, 219-225, DOI: 10.1080/01616412.1992.11740057 To link to this article: http://dx.doi.org/10.1080/01616412.1992.11740057
Published online: 20 Jul 2016.
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Citing articles: 4 View citing articles
Full Terms & Conditions of access and use can be found at http://www.tandfonline.com/action/journalInformation?journalCode=yner20 Download by: [Australian Catholic University]
Date: 23 August 2017, At: 05:41
A non-l inear haem odyn amic model for the arterial pulsatile comp onen t of the intracranial pulse wave William D. Lakin* and Cordell E. Grosst
Downloaded by [Australian Catholic University] at 05:41 23 August 2017
• Department of Mathematics and Statistics,. and
of Neurosurgery, University of Vermont, Burlington, VT 05405, USA
t Division
An indic~tion that pressure pulses in cerebral arteries may play a role in the configuration of intracranial pressure pulsations is given by the observation that vasospasm of cerebral' arteries narrows the amplitude of the intracranial pressure wave. The present work develops a mathematical model for the transmission of arterial pressure pulses across the compliant arterial wall to the surrounding intracranial space. Compliance of both the arterial segment and the intracranial space are considered. So as to retain accuracy at higher values of the mean intracranial pressure, a physiological range in which pulse transmission is enhanced due to lower pressure gradients but intracranial compliance is not necessarily decreased, a logistic fit is used to model the intracranial pressure- volume relationship. A sequence of approximations (with error bounds) is obtained for the induced intracranial pressure pulse amplitude as a function of arterial pulse amplitude, mean transmural pressure, and mean intracranial pressure. It is found that at higher mean intracranial pressures, where the usual exponential assumption for the intracranial pressure-v olume curve loses validity, the amplitude of the transmitted arterial pressure pulse depends non-linearly on the mean intracranial pressure. Keywords: Mathematical modelling; compliances; cerebral vasospasm; pressure- volume relationships; logistic fits; arterial pulsewave; intracranial pulsewave INTRODUCTION A recent investigation involving mathemat ical modelling coupled with experimental simulation 8 has demonstrated that arterial pulse pressure variations are capable of producing alterations in the configurat ion of the intracrania l pulse wave (ICPW). The mathemat ical model developed in that study is based on an assumptio n that an exponentia l function may be used to describe the intracrania l pressure- volume relationshi p over the full range of pressures. While an exponentia l does provide a good fit to this relationshi p over much of the normal range, this assumptio n is known to degrade significantiJ' at larger values of the mean intracrania l pressure 3 •1 . In particular, at higher intracranial pressures, the intracrania l complianc e does not necessarily decrease as pressure increases. However, it is precisely in the case of high intracrania l pressures that the lower pressure gradients between the artery and the surroundin g intracrania l space enable enhanced transmission of the pressure pulse across the arterial wall. Conseque ntly, at higher intracrania l pressures, the contributio n of the arterial pulse to the ICPW will be determine d by an interplay of different physical mechanisms. There is thus a need to develop a mathemat ical model based on a functional relationshi p which more accurately reflects the behaviour of the intracrania l pressure- volume curve over a larger range of pressures. The present work utilizes an intrinsicall y non-linear 'logistic' fit of the intracranial pressure-v olume Corresponde nce and reprint requests to: Prof W.O. Lakin, Department of Mathematics and Statistics, University of Vermont, Burlington, VT 05405, USA. Accepted for publication january 1992.
© 1992 Forefront Publishing Croup 0161-6412/92/ 030219-07
relationshi p. This fit remains valid in the entire region between resting pressure and the plateau demonstra ted experimen tally at high intracrania l pressures.
MATHEMATICAl MODEl DEVElOPMENT Following Klassen et al. 8 , a cerebral artery in the craniospinal cavity will be modelled as a compliant vessel enclosed in a fluid-filled box with elastic walls. The vessel is assumed to have a wall of constant thickness, and to be homogene ous, linearly elastic, viscoelastically inert, and isotropic. For the elastic box which models craniospinal confines, linear elasticity is not assumed. In particular, the complianc e is not required to be a linear function of pressure. Outflow of cerebrosp inal fluid ( CSF) from the craniospinal cavity to the venous-si nus is assumed to be proportion al to the pressure difference P0 -P55 , where P0 is the intracrania l pressure (ICP) and P55 is the sagittal sinus pressure. If R0 is the outflow resistance and the absolute formation rate of CSF is given by:
Po - pss
lf=---
(1)
Ro then there will be a static balance between CSF production and absorption . For the present purposes, the volume Vr of fluid in the extravascu lar space will be assumed constant.
Pressure- volume relationships
The efficacy of transmission of the intralumina l pressure pulses to the fluid surroundin g the vessel segment will depend on the elastic properties of both
Neurological Research, 1992, Volume 14, june
219
Arterial pulsatile component of intracranial pulse wave: W.O. Lakin and
the vessel walls and the intracranial space. These parameters are reflected in the corresponding pressure-volume relationships and compliances for the arterial and intracranial spaces. If Prr denotes the transmural pressure, then the arterial wall becomes less distensible as Prr increases1.4·7 •10·11 . Thus, if V1 is the volume inside the vessel segment within the elastic CSF-filled box, the compliance C1 of the arterial space, defined as: dV1
- = C1(Ptr)
(2)
dPtr is a decreasing function of Prr· This, in turn, implies that the derivative of the function C1(Ptr) with respect to Pw must be negative. The simplest nontrivial relationship satisfying these constraints is
c; (Ptr),
Downloaded by [Australian Catholic University] at 05:41 23 August 2017
dC1 - = -2a dPtr
(3)
where a is a positive constant. Integration of (3) gives : C1(Ptr) = - 2aPtr
+b
(4)
with b a constant. Using this expression in (2), we get dV1 I dPtr = - 2aPtr + b, and integrating this equation gives a static pressure-volume relationship for the arterial vessel segment in the form:
V, = -a(Pr/
+ bPtr + C
(5)
where c is an additional constant of integration. This is a quadratic formula for the arterial segment volume V1 as a function of transmural pressure Prr· The three constants in (5) are determined by the specific quadratic fit chosen for the pressure-volume relationship of the vessel (not the intracranial space ). By (2), C1(Prr) is the slope of that fit at a given value of Prr· The constant c in (5) may be interpreted as the volume of the vessel segment at zero transmural pressure pressure, c = V1(0 ): Similarly, the constant b is the slope of the volume-pressure relationship (5) at zero transmural pressure, and hence b = C1 (0) is the compliance of the vessel segment when the arterial pressure equals the pressure of the surrounding intracranial space. For the model's intracranial space, i.e. the elastic box filled with fluid which encloses the vessel segment, the intracranial pressure-volume relationship wll have the general form : V0
=
F0 (P0 )
(6 )
where P0 is the ICP, V0 is the total volume of the box (including the volume of the enclosed vessel segment), and · dV0 = Co(Vo)
dP0
(7)
Note that the subscript '0' in (6) and (7) refers to the intracranial space in the model, whereas the subscript '1' in (2)-(5) referred to the vascular space. The function C0 = F~(P0 ) in (7) is the compliance of the intracranial space, i.e. the slope of the curve F0 (P0 ) which models the pressure-volume relation (6). To accord more easily with experimental pressure-volume curves obtained by techniques involving injection or withdrawal of fluid, (6) and (7) may be inverted, so that pressure is a
220
Neurological Research, 1992, Volume 14, june
C.E.
Gross
function of volume, and written in the form :
P0 = C 0 (Vo)
(8)
and (9)
The function E0 (P 0 ) in (9) is now the elastance of the intracranial space. By (7) and (9), compliance and elastance are inversely related by: C0(V0 ) = [f0(P0)]- 1 (10) The pressure-volume relationship in (6) has been previously modelled in the literature as a linearlogarithmic function which ex,presses the elastance of the entire craniospinal axis . This corresponds to choosing E0 (P0 ) in (9) to be a linear function of P0 . The function C 0 ( V0 ) in (8) is an exponential in this case, and may involve addative constants .. For example, an exponential pressure-volume relationship which results from an initial condition in (9) that zero volume corresponds to zero pressure is given bl:
Po
=
ekVo -
1
(11 )
An exponential model of C 0 (V0), such as (11), is known to be appropriate over much of the normal ICP range. At higher values of ICP, however, the exponential assumption loses validity. This may be seen by taking the logarithm_ of (11 ) to produce the relation log (P 0 + 1) = kV0 where k = k log e. If the quantity P0 + 1 is now plotted against the volume V0 on semi-logarithmic graph [?aper, the result is simply a straight line with slope k. However, the slope of this line is simply the pressure-volume index (PVI ), the volume in millilitres which would raise ICP by ten times the opening pressure. Thus, if (11 ) holds for all pressures, the PVI would have a single constant value k, independent of V0 . However, as noted by Friden and Ekstedt3, this is clearly not the case. A typical measured experimental pressure-volume relationship for intracranial space, taken from Sullivan and Allison 14, is shown as the solid line in Figure 1. This curve has a lower pressure plateau at relatively low values of ICP and an upper pressure plateau where the ICP approaches the diastolic blood pressure. Both of these regions are associated with low elastances (or high compliances) of the intracranial space. The two plateaus are connected by a steep portion of the curve where elastances are large. An exponential fit of the typical experimental intracranial pressure-volume relationship is also shown as a dotted line in Figure 7. The exponential does provide a good fit for the solid curve in the pressure range co_rresponding to the lower plateau and part way up the ~Fl-ortion of the curve. However, the exponential fit rapidly degrades higher on the steep portion, and it is completely erroneous in the region of the upper pressure plateau. Use of an exponential fit in these regions would thus appear inappropriate. The deficiencies in an exponential fit at higher ICP can be addressed by using a 'logistic' fit to the intracranial pressure-volume relationship, shown as the dashed line in Figure 7. Logistic functions are approximately exponential over much of their range. However, they also exhibit a non-linear limiting
Arterial pulsatile component of intracranial pulse wave: W.O. Lakin and C.E. Gross
intracranial space by 11P1 . The amplitude of the corresponding transmural pressure pulsations is now:
90.00 Experimental Data
MO.OO -
j
11Ptr
70.00 upper prc!'>!lurc platcnn
Fit
E
50.00
11P1 and 11P0 , the chain rule for differentiation may be used to obtain the variation of P0 with Ptr· In particular,
~
40.00
using (9):
~ ~
1
~
,;; 30.00
j
20.00 lower prcs.urc platcnn
lll.OO -
I 0.00
I_.......1.. ... .. _ . .I ··· -10.00 -5.00 ti.OU
...I..._ .._I. _
• 1. ..... 5.00
IO.fXI
15.00
20.00
_
L.___j_j 25.00
30.00
Vuluuu: Chang(• (m l)
Downloaded by [Australian Catholic University] at 05:41 23 August 2017
(15)
As a first step in determining the relationship between
Logistic
60.00
= i1P1 -11P0 •
Figure 1: The pressure-volume relationship for the human intracranial compartment showing intracranial pressure relative to atmospheric as a function of volume change. The solid curve, taken from Sullivan and Allison 14, is a typical experimentally measured relationship. The exponential model of this relationship, shown as the dotted curve, fits the measured curve for low pressures, but increasingly deviates from the actual curve at higher pressures above 30 mmHg. The logistic model, shown as the dashed curve, provides a good fit to the measured relationship for the full range of pressures through the upper pressure plateau, and only deviates at very high pressures above diastolic
behaviour which inhibits continued exponential growth at higher values of their argument and provides a smooth approach to a final state, in this case the upper pressure plateau. The logistic fit is not valid in the very high pressure range above the upper plateau, i.e. for P0 greater than Pmax, where Pmax is the pressure value at the end of the upper plateau. For the present purposes, pmax can be replaced by the diastolic pressure Pdias· It is worth noting that while both exponential and logistic functions are non-linear, exponentials are solutions of linear differential equations. By contrast, the differential equation satisfied by a logistic function is intrinsically non-linear. For the logistic model of the intracranial pressurevolume relationship, E0 (P0 ) in (9) becomes: 0 dP0 -=kP 0 ( 1 -P-)
dVo
(12)
Pdias
Integrating (12), the intracranial pressure-volume relationship now has the form:
Po = ( 1
-p~='_kVo)
(13)
The constant a in (13) may be determined from an initial condition on (12). For example, if at volume V0 the ICP is specified to be P0 (e.g. P0 = ~ is the resting pressure with a corresponding volume V0 = V,), then a has the value:
a
= ( 1 _ P~:s
)ekVo
(14)
Relation of pulse amplitudes Let 11P, denote the amplitude of the intraluminal pressure pulsations and let 11P0 be the amplitude of the pressure pulsations induced in the surrounding
dP0 dP0 dV0 dV0 = - - = E0 (P0 ) dPtr dV0 dPtr dPtr
(16)
Since the total intracranial fluid volume is the extravascular fluid plus the intravascular fluid, or V0 = V1 + V1
(17)
and V1 is assumed constant, by (2):
dV0
-
dV1
= - = C1 (Pt,)
dPtr
(18)
dPtr
Consequently, dP0 = f 0 (P0 lC1 (Ptr) dPtr
(19)
Higher derivatives of P0 with respect to Ptr may be similarly derived. For example, if a prime denotes a derivative with respect to the argument of a function,
d 2 ~0 = c; (Ptr)E dPtr
0
(P0 )
+ ('C, (Ptr))2 E~(P0 )
In the present model, C1 (Pt,) and
(20)
c; (Ptr) are given by
(3) and (4), while E0 (P0 ) is given by the right hand side
of (12). Let Ptr be the mean transmural pressure and P0 be the corresponding mean ICP. If the amplitude of the arterial pressure pulse is 11P1 and the amplitude of the ICPW is 11P0 , then, as given by (15), the amplitude of the transmural pressure pulse is 11Ptr =::.i1P1 - 11P0 . The peak transmural pressure is now Ptr + 11Ptr· If the intracranial pressure function P0 is considered a function of the transmural pressure Pt, the peak ICP is now simply P0 evaluated at the peak transmural pressure. To relate the various mean values, amplitudes, and peak values, consider first the general situation in which a variable y is a function of x, e.g. f (x) = y. Then if x is some value of the independent variable, so that f (x) = 9, if x is incremented by an amount 11x, the corresponding increment 11y in the dependent variable is defined by the relation f (x + 11x) = 9 + 11y. In the present application, identifying the function f with the ICP function P0 , and 9with mean transmural pressure and mean ICP, and 11x and 11y with amplitudes of the transmural pressure pulse and ICPW, respectively,
x
P0 (~,
+ 11Ptr) = P0 + i1P0 .
(21)
Taylor series expansions provide a classical means for approximating the value of a function f(x) evaluated at a point + 11x in terms of values of the function and its derivatives evaluated at the reference point x. In the present context, relations between 11P0 and 11Ptr may be obtained by expanding the left hand side of (21) in a Taylor s~_ries e~ansio'}_,ilbout the mean transmural pressure Ptr. If P~ and P0 denote the first and second derivatives in (19) and (20) evaluated at the
x
Neurologica.t'Research, 1992, Volume 14, june
22"1
Arterial pulsatile component of intracranial pulse wave: W.O. Lakin and C.E. Cross
mean pressures Prr and in the Taylor series are:
Po(Prr
+ i'lPtr) =
Po
P0 ,
then the first three terms
+ P~ i'lPtr
+ ~P~(IlPtr)2 + 0(/lPtr)3
+ ~P~(IlPtY + 0(/lPtr)3 .
(23 )
Equation (23) is sufficient for deriving consistent linear and quadratic approximations relating mean ICP, intracranial pressure pulse amplitude, and arterial pulse amplitude.
Downloaded by [Australian Catholic University] at 05:41 23 August 2017
RESULTS Equation (15) may be used to eliminate the amplitude of the transmural pressure· pulse in (23) to obtain an implicit equation for the ICP pulse amplitude in terms of the arterial pulse amplitude. The resulting equation provides the basis for a sequence of approximate relations between the change in ICP and the change in arterial pressure at a given level of mean ICP. The linear approximation If only the first term on the right hand side of (23 ) is retained, the resulting implicit expression for the amplitude of the ICPW, or llP0 , will be linear in the arterial pressure pulse amplitude f:.P 1. In particular,
f:.P0 = P~(i'lP1 -llP0 )
+ 0(f:.P1 -
f:.P0 f.
(24)
As a first step towards analysing (24 ), it may be noted that the error term in this approximation expands as: (/lP1 -llP0 f
=
(i'lP1)2
-
2f:.P1 1lP0
-
(f:.P0 )2 . (25 )
On the lower . pressure plateau of the intracranial pressure-volume curve, the amplitude of the ICPW is small compared to the amplitude of the arterial pressure pulse8, i.e. llP0 « llP1. Because both the lower and upper pressure plateaus oare regions of high intracranial compliance, this relation also holds for high values of the ICP on the upper pressure plateau. Hence, for both high and low values of the ICP, the second and third terms on the right hand side of (25) are negligible compared to the first term, and (f:.P1 - f:.P 0 )2 = 0(/lP1)2 . . For mid-range values of ICP, the amplitude of the ICPW
ISSN: 0161-6412 (Print) 1743-1328 (Online) Journal homepage: http://www.tandfonline.com/loi/yner20
A non-linear haemodynamic model for the arterial pulsatile component of the intracranial pulse wave William D. Lakin & Cordell E. Gross To cite this article: William D. Lakin & Cordell E. Gross (1992) A non-linear haemodynamic model for the arterial pulsatile component of the intracranial pulse wave, Neurological Research, 14:3, 219-225, DOI: 10.1080/01616412.1992.11740057 To link to this article: http://dx.doi.org/10.1080/01616412.1992.11740057
Published online: 20 Jul 2016.
Submit your article to this journal
View related articles
Citing articles: 4 View citing articles
Full Terms & Conditions of access and use can be found at http://www.tandfonline.com/action/journalInformation?journalCode=yner20 Download by: [Australian Catholic University]
Date: 23 August 2017, At: 05:41
A non-l inear haem odyn amic model for the arterial pulsatile comp onen t of the intracranial pulse wave William D. Lakin* and Cordell E. Grosst
Downloaded by [Australian Catholic University] at 05:41 23 August 2017
• Department of Mathematics and Statistics,. and
of Neurosurgery, University of Vermont, Burlington, VT 05405, USA
t Division
An indic~tion that pressure pulses in cerebral arteries may play a role in the configuration of intracranial pressure pulsations is given by the observation that vasospasm of cerebral' arteries narrows the amplitude of the intracranial pressure wave. The present work develops a mathematical model for the transmission of arterial pressure pulses across the compliant arterial wall to the surrounding intracranial space. Compliance of both the arterial segment and the intracranial space are considered. So as to retain accuracy at higher values of the mean intracranial pressure, a physiological range in which pulse transmission is enhanced due to lower pressure gradients but intracranial compliance is not necessarily decreased, a logistic fit is used to model the intracranial pressure- volume relationship. A sequence of approximations (with error bounds) is obtained for the induced intracranial pressure pulse amplitude as a function of arterial pulse amplitude, mean transmural pressure, and mean intracranial pressure. It is found that at higher mean intracranial pressures, where the usual exponential assumption for the intracranial pressure-v olume curve loses validity, the amplitude of the transmitted arterial pressure pulse depends non-linearly on the mean intracranial pressure. Keywords: Mathematical modelling; compliances; cerebral vasospasm; pressure- volume relationships; logistic fits; arterial pulsewave; intracranial pulsewave INTRODUCTION A recent investigation involving mathemat ical modelling coupled with experimental simulation 8 has demonstrated that arterial pulse pressure variations are capable of producing alterations in the configurat ion of the intracrania l pulse wave (ICPW). The mathemat ical model developed in that study is based on an assumptio n that an exponentia l function may be used to describe the intracrania l pressure- volume relationshi p over the full range of pressures. While an exponentia l does provide a good fit to this relationshi p over much of the normal range, this assumptio n is known to degrade significantiJ' at larger values of the mean intracrania l pressure 3 •1 . In particular, at higher intracranial pressures, the intracrania l complianc e does not necessarily decrease as pressure increases. However, it is precisely in the case of high intracrania l pressures that the lower pressure gradients between the artery and the surroundin g intracrania l space enable enhanced transmission of the pressure pulse across the arterial wall. Conseque ntly, at higher intracrania l pressures, the contributio n of the arterial pulse to the ICPW will be determine d by an interplay of different physical mechanisms. There is thus a need to develop a mathemat ical model based on a functional relationshi p which more accurately reflects the behaviour of the intracrania l pressure- volume curve over a larger range of pressures. The present work utilizes an intrinsicall y non-linear 'logistic' fit of the intracranial pressure-v olume Corresponde nce and reprint requests to: Prof W.O. Lakin, Department of Mathematics and Statistics, University of Vermont, Burlington, VT 05405, USA. Accepted for publication january 1992.
© 1992 Forefront Publishing Croup 0161-6412/92/ 030219-07
relationshi p. This fit remains valid in the entire region between resting pressure and the plateau demonstra ted experimen tally at high intracrania l pressures.
MATHEMATICAl MODEl DEVElOPMENT Following Klassen et al. 8 , a cerebral artery in the craniospinal cavity will be modelled as a compliant vessel enclosed in a fluid-filled box with elastic walls. The vessel is assumed to have a wall of constant thickness, and to be homogene ous, linearly elastic, viscoelastically inert, and isotropic. For the elastic box which models craniospinal confines, linear elasticity is not assumed. In particular, the complianc e is not required to be a linear function of pressure. Outflow of cerebrosp inal fluid ( CSF) from the craniospinal cavity to the venous-si nus is assumed to be proportion al to the pressure difference P0 -P55 , where P0 is the intracrania l pressure (ICP) and P55 is the sagittal sinus pressure. If R0 is the outflow resistance and the absolute formation rate of CSF is given by:
Po - pss
lf=---
(1)
Ro then there will be a static balance between CSF production and absorption . For the present purposes, the volume Vr of fluid in the extravascu lar space will be assumed constant.
Pressure- volume relationships
The efficacy of transmission of the intralumina l pressure pulses to the fluid surroundin g the vessel segment will depend on the elastic properties of both
Neurological Research, 1992, Volume 14, june
219
Arterial pulsatile component of intracranial pulse wave: W.O. Lakin and
the vessel walls and the intracranial space. These parameters are reflected in the corresponding pressure-volume relationships and compliances for the arterial and intracranial spaces. If Prr denotes the transmural pressure, then the arterial wall becomes less distensible as Prr increases1.4·7 •10·11 . Thus, if V1 is the volume inside the vessel segment within the elastic CSF-filled box, the compliance C1 of the arterial space, defined as: dV1
- = C1(Ptr)
(2)
dPtr is a decreasing function of Prr· This, in turn, implies that the derivative of the function C1(Ptr) with respect to Pw must be negative. The simplest nontrivial relationship satisfying these constraints is
c; (Ptr),
Downloaded by [Australian Catholic University] at 05:41 23 August 2017
dC1 - = -2a dPtr
(3)
where a is a positive constant. Integration of (3) gives : C1(Ptr) = - 2aPtr
+b
(4)
with b a constant. Using this expression in (2), we get dV1 I dPtr = - 2aPtr + b, and integrating this equation gives a static pressure-volume relationship for the arterial vessel segment in the form:
V, = -a(Pr/
+ bPtr + C
(5)
where c is an additional constant of integration. This is a quadratic formula for the arterial segment volume V1 as a function of transmural pressure Prr· The three constants in (5) are determined by the specific quadratic fit chosen for the pressure-volume relationship of the vessel (not the intracranial space ). By (2), C1(Prr) is the slope of that fit at a given value of Prr· The constant c in (5) may be interpreted as the volume of the vessel segment at zero transmural pressure pressure, c = V1(0 ): Similarly, the constant b is the slope of the volume-pressure relationship (5) at zero transmural pressure, and hence b = C1 (0) is the compliance of the vessel segment when the arterial pressure equals the pressure of the surrounding intracranial space. For the model's intracranial space, i.e. the elastic box filled with fluid which encloses the vessel segment, the intracranial pressure-volume relationship wll have the general form : V0
=
F0 (P0 )
(6 )
where P0 is the ICP, V0 is the total volume of the box (including the volume of the enclosed vessel segment), and · dV0 = Co(Vo)
dP0
(7)
Note that the subscript '0' in (6) and (7) refers to the intracranial space in the model, whereas the subscript '1' in (2)-(5) referred to the vascular space. The function C0 = F~(P0 ) in (7) is the compliance of the intracranial space, i.e. the slope of the curve F0 (P0 ) which models the pressure-volume relation (6). To accord more easily with experimental pressure-volume curves obtained by techniques involving injection or withdrawal of fluid, (6) and (7) may be inverted, so that pressure is a
220
Neurological Research, 1992, Volume 14, june
C.E.
Gross
function of volume, and written in the form :
P0 = C 0 (Vo)
(8)
and (9)
The function E0 (P 0 ) in (9) is now the elastance of the intracranial space. By (7) and (9), compliance and elastance are inversely related by: C0(V0 ) = [f0(P0)]- 1 (10) The pressure-volume relationship in (6) has been previously modelled in the literature as a linearlogarithmic function which ex,presses the elastance of the entire craniospinal axis . This corresponds to choosing E0 (P0 ) in (9) to be a linear function of P0 . The function C 0 ( V0 ) in (8) is an exponential in this case, and may involve addative constants .. For example, an exponential pressure-volume relationship which results from an initial condition in (9) that zero volume corresponds to zero pressure is given bl:
Po
=
ekVo -
1
(11 )
An exponential model of C 0 (V0), such as (11), is known to be appropriate over much of the normal ICP range. At higher values of ICP, however, the exponential assumption loses validity. This may be seen by taking the logarithm_ of (11 ) to produce the relation log (P 0 + 1) = kV0 where k = k log e. If the quantity P0 + 1 is now plotted against the volume V0 on semi-logarithmic graph [?aper, the result is simply a straight line with slope k. However, the slope of this line is simply the pressure-volume index (PVI ), the volume in millilitres which would raise ICP by ten times the opening pressure. Thus, if (11 ) holds for all pressures, the PVI would have a single constant value k, independent of V0 . However, as noted by Friden and Ekstedt3, this is clearly not the case. A typical measured experimental pressure-volume relationship for intracranial space, taken from Sullivan and Allison 14, is shown as the solid line in Figure 1. This curve has a lower pressure plateau at relatively low values of ICP and an upper pressure plateau where the ICP approaches the diastolic blood pressure. Both of these regions are associated with low elastances (or high compliances) of the intracranial space. The two plateaus are connected by a steep portion of the curve where elastances are large. An exponential fit of the typical experimental intracranial pressure-volume relationship is also shown as a dotted line in Figure 7. The exponential does provide a good fit for the solid curve in the pressure range co_rresponding to the lower plateau and part way up the ~Fl-ortion of the curve. However, the exponential fit rapidly degrades higher on the steep portion, and it is completely erroneous in the region of the upper pressure plateau. Use of an exponential fit in these regions would thus appear inappropriate. The deficiencies in an exponential fit at higher ICP can be addressed by using a 'logistic' fit to the intracranial pressure-volume relationship, shown as the dashed line in Figure 7. Logistic functions are approximately exponential over much of their range. However, they also exhibit a non-linear limiting
Arterial pulsatile component of intracranial pulse wave: W.O. Lakin and C.E. Gross
intracranial space by 11P1 . The amplitude of the corresponding transmural pressure pulsations is now:
90.00 Experimental Data
MO.OO -
j
11Ptr
70.00 upper prc!'>!lurc platcnn
Fit
E
50.00
11P1 and 11P0 , the chain rule for differentiation may be used to obtain the variation of P0 with Ptr· In particular,
~
40.00
using (9):
~ ~
1
~
,;; 30.00
j
20.00 lower prcs.urc platcnn
lll.OO -
I 0.00
I_.......1.. ... .. _ . .I ··· -10.00 -5.00 ti.OU
...I..._ .._I. _
• 1. ..... 5.00
IO.fXI
15.00
20.00
_
L.___j_j 25.00
30.00
Vuluuu: Chang(• (m l)
Downloaded by [Australian Catholic University] at 05:41 23 August 2017
(15)
As a first step in determining the relationship between
Logistic
60.00
= i1P1 -11P0 •
Figure 1: The pressure-volume relationship for the human intracranial compartment showing intracranial pressure relative to atmospheric as a function of volume change. The solid curve, taken from Sullivan and Allison 14, is a typical experimentally measured relationship. The exponential model of this relationship, shown as the dotted curve, fits the measured curve for low pressures, but increasingly deviates from the actual curve at higher pressures above 30 mmHg. The logistic model, shown as the dashed curve, provides a good fit to the measured relationship for the full range of pressures through the upper pressure plateau, and only deviates at very high pressures above diastolic
behaviour which inhibits continued exponential growth at higher values of their argument and provides a smooth approach to a final state, in this case the upper pressure plateau. The logistic fit is not valid in the very high pressure range above the upper plateau, i.e. for P0 greater than Pmax, where Pmax is the pressure value at the end of the upper plateau. For the present purposes, pmax can be replaced by the diastolic pressure Pdias· It is worth noting that while both exponential and logistic functions are non-linear, exponentials are solutions of linear differential equations. By contrast, the differential equation satisfied by a logistic function is intrinsically non-linear. For the logistic model of the intracranial pressurevolume relationship, E0 (P0 ) in (9) becomes: 0 dP0 -=kP 0 ( 1 -P-)
dVo
(12)
Pdias
Integrating (12), the intracranial pressure-volume relationship now has the form:
Po = ( 1
-p~='_kVo)
(13)
The constant a in (13) may be determined from an initial condition on (12). For example, if at volume V0 the ICP is specified to be P0 (e.g. P0 = ~ is the resting pressure with a corresponding volume V0 = V,), then a has the value:
a
= ( 1 _ P~:s
)ekVo
(14)
Relation of pulse amplitudes Let 11P, denote the amplitude of the intraluminal pressure pulsations and let 11P0 be the amplitude of the pressure pulsations induced in the surrounding
dP0 dP0 dV0 dV0 = - - = E0 (P0 ) dPtr dV0 dPtr dPtr
(16)
Since the total intracranial fluid volume is the extravascular fluid plus the intravascular fluid, or V0 = V1 + V1
(17)
and V1 is assumed constant, by (2):
dV0
-
dV1
= - = C1 (Pt,)
dPtr
(18)
dPtr
Consequently, dP0 = f 0 (P0 lC1 (Ptr) dPtr
(19)
Higher derivatives of P0 with respect to Ptr may be similarly derived. For example, if a prime denotes a derivative with respect to the argument of a function,
d 2 ~0 = c; (Ptr)E dPtr
0
(P0 )
+ ('C, (Ptr))2 E~(P0 )
In the present model, C1 (Pt,) and
(20)
c; (Ptr) are given by
(3) and (4), while E0 (P0 ) is given by the right hand side
of (12). Let Ptr be the mean transmural pressure and P0 be the corresponding mean ICP. If the amplitude of the arterial pressure pulse is 11P1 and the amplitude of the ICPW is 11P0 , then, as given by (15), the amplitude of the transmural pressure pulse is 11Ptr =::.i1P1 - 11P0 . The peak transmural pressure is now Ptr + 11Ptr· If the intracranial pressure function P0 is considered a function of the transmural pressure Pt, the peak ICP is now simply P0 evaluated at the peak transmural pressure. To relate the various mean values, amplitudes, and peak values, consider first the general situation in which a variable y is a function of x, e.g. f (x) = y. Then if x is some value of the independent variable, so that f (x) = 9, if x is incremented by an amount 11x, the corresponding increment 11y in the dependent variable is defined by the relation f (x + 11x) = 9 + 11y. In the present application, identifying the function f with the ICP function P0 , and 9with mean transmural pressure and mean ICP, and 11x and 11y with amplitudes of the transmural pressure pulse and ICPW, respectively,
x
P0 (~,
+ 11Ptr) = P0 + i1P0 .
(21)
Taylor series expansions provide a classical means for approximating the value of a function f(x) evaluated at a point + 11x in terms of values of the function and its derivatives evaluated at the reference point x. In the present context, relations between 11P0 and 11Ptr may be obtained by expanding the left hand side of (21) in a Taylor s~_ries e~ansio'}_,ilbout the mean transmural pressure Ptr. If P~ and P0 denote the first and second derivatives in (19) and (20) evaluated at the
x
Neurologica.t'Research, 1992, Volume 14, june
22"1
Arterial pulsatile component of intracranial pulse wave: W.O. Lakin and C.E. Cross
mean pressures Prr and in the Taylor series are:
Po(Prr
+ i'lPtr) =
Po
P0 ,
then the first three terms
+ P~ i'lPtr
+ ~P~(IlPtr)2 + 0(/lPtr)3
+ ~P~(IlPtY + 0(/lPtr)3 .
(23 )
Equation (23) is sufficient for deriving consistent linear and quadratic approximations relating mean ICP, intracranial pressure pulse amplitude, and arterial pulse amplitude.
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RESULTS Equation (15) may be used to eliminate the amplitude of the transmural pressure· pulse in (23) to obtain an implicit equation for the ICP pulse amplitude in terms of the arterial pulse amplitude. The resulting equation provides the basis for a sequence of approximate relations between the change in ICP and the change in arterial pressure at a given level of mean ICP. The linear approximation If only the first term on the right hand side of (23 ) is retained, the resulting implicit expression for the amplitude of the ICPW, or llP0 , will be linear in the arterial pressure pulse amplitude f:.P 1. In particular,
f:.P0 = P~(i'lP1 -llP0 )
+ 0(f:.P1 -
f:.P0 f.
(24)
As a first step towards analysing (24 ), it may be noted that the error term in this approximation expands as: (/lP1 -llP0 f
=
(i'lP1)2
-
2f:.P1 1lP0
-
(f:.P0 )2 . (25 )
On the lower . pressure plateau of the intracranial pressure-volume curve, the amplitude of the ICPW is small compared to the amplitude of the arterial pressure pulse8, i.e. llP0 « llP1. Because both the lower and upper pressure plateaus oare regions of high intracranial compliance, this relation also holds for high values of the ICP on the upper pressure plateau. Hence, for both high and low values of the ICP, the second and third terms on the right hand side of (25) are negligible compared to the first term, and (f:.P1 - f:.P 0 )2 = 0(/lP1)2 . . For mid-range values of ICP, the amplitude of the ICPW
