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Department of Electronics, Computer Science, and Systems, University of Bologna, I-40136 Bologna, Italy
ABSTRACT
INTRODUCTION
QUALITATIVE MODEL DESCRIPTION
ASSIGNMENT OF PARAMETER BASAL VALUES
RESULTS
DISCUSSION
APPENDIX
FOOTNOTES
REFERENCES
ABSTRACT 
Ursino, Mauro, and Carlo Alberto Lodi. A simple
mathematical model of the interaction between intracranial pressure and
cerebral hemodynamics. J. Appl.
Physiol. 82(4): 1256-1269, 1997.
A simple
mathematical model of intracranial pressure (ICP) dynamics oriented to
clinical practice is presented. It includes the hemodynamics of the
arterial-arteriolar cerebrovascular bed, cerebrospinal fluid (CSF)
production and reabsorption processes, the nonlinear pressure-volume
relationship of the craniospinal compartment, and a Starling resistor
mechanism for the cerebral veins. Moreover, arterioles are controlled
by cerebral autoregulation mechanisms, which are simulated by means of
a time constant and a sigmoidal static characteristic. The model is
used to simulate interactions between ICP, cerebral blood volume, and
autoregulation. Three different related phenomena are analyzed: the
generation of plateau waves, the effect of acute arterial hypotension
on ICP, and the role of cerebral hemodynamics during pressure-volume index (PVI) tests. Simulation results suggest the following:
1) ICP dynamics may become unstable
in patients with elevated CSF outflow resistance and decreased
intracranial compliance, provided cerebral autoregulation is efficient.
Instability manifests itself with the occurrence of self-sustained
plateau waves. 2) Moderate acute
arterial hypotension may have completely different effects on ICP,
depending on the value of model parameters. If physiological compensatory mechanisms (CSF circulation and intracranial storage capacity) are efficient, acute hypotension has only negligible effects
on ICP and cerebral blood flow (CBF). If these compensatory mechanisms
are poor, even modest hypotension may induce a large transient increase
in ICP and a significant transient reduction in CBF, with risks of
secondary brain damage. 3) The ICP
response to a bolus injection (PVI test) is sharply affected, via
cerebral blood volume changes, by cerebral hemodynamics and
autoregulation. We suggest that PVI tests may be used to extract
information not only on intracranial compliance and CSF circulation,
but also on the status of mechanisms controlling CBF.
intracranial hemodynamics; cerebral autoregulation; pressure-volume
index; plateau waves; mathematical modeling
INTRODUCTION INTRACRANIAL PRESSURE (ICP) changes may have a serious
impact in the course of various neurosurgical disorders, such as brain injury, subarachnoid hemorrhage, hydrocephalus, and brain tumor. Providing a quantitative description of intracranial dynamics in these
cases is of the greatest clinical value for several reasons. First,
uncontrolled pressure increases in the craniospinal cavity are a
frequent cause of morbidity and mortality; hence, the choice of
appropriate treatment cannot ignore its possible effect on ICP. Second,
analysis of the ICP time pattern may provide fundamental information as
to the status of cerebral hemodynamics, cerebral perfusion, and
autoregulation reserve. Recent studies particularly emphasize the
existence of strict relationships between ICP changes and brain
hemodynamic perturbations (7, 12, 13, 17) and between ICP changes and
patient outcome (21).
Despite its great clinical importance, information on the ICP time
pattern is insufficiently utilized in the clinical setting. The
complexity of the relationships among physiological quantities and the
presence of significant nonlinearities make qualitative approaches
often inadequate to grasp important aspects of experimental and
clinical results. Mathematical models may represent a new tool for
improving our comprehension of ICP time patterns, inasmuch as they are
able to outline the main relationships among quantities in rigorous
quantitative terms. Indeed, several models have been presented in the
past decades, starting from the pioneering works by Marmarou and
co-workers (22, 23). Some are especially aimed at analyzing individual
aspects of intracranial dynamics, such as nonlinear cerebrospinal fluid
(CSF) production and reabsorption processes (8, 20), nonlinear
intracranial elasticity (4), and venous collapse (9). More
comprehensive multicompartmental approaches that simultaneously embody
the relationships between intracranial elasticity, CSF circulation, and
some features of cerebral hemodynamics have also been proposed (10, 16,
36).
Recently, we developed a model of craniospinal dynamics that
incorporates most of the phenomena mentioned above (38, 40). With this
model we were able to reproduce various clinical results, such as the
pattern of ICP pulsatile changes (39), the ICP response to vasodilatory
or vasoconstrictory stimuli (40), the origin of pathological
self-sustained ICP waves (40), and the different ICP responses to fluid
injection into or fluid removal from the craniospinal space. The model
has recently been applied to the quantitative analysis of these
responses in patients with acute brain disease with encouraging results
(41).
Most of the multicompartmental models, however, are still too complex
for routine use in real clinical environments. In other words, although
the studies mentioned above are of significant importance to improve
our physiological knowledge, they risk being insufficiently utilized
for the solution of real clinical problems because of their excessive
mathematical and formal intricacy. Thus there is also a need for some
simplified models that are able to describe certain clinical
aspects of ICP dynamics with sufficient accuracy and, at the same time,
incorporate a minimum of mathematics.
The aim of this work is to present a drastically simplified model of
ICP dynamics useful for the study of patients with severe brain damage.
When building the model we took advantage of the experience gained when
working with the complex model presented in a previous study (40)
applied to the analysis of real clinical cases (41). This experience
suggested to us that certain physiological phenomena might be described
in a much simpler way, without appreciable deterioration in model
performance. The reduced model presented here cannot adequately
describe all the clinical and physiological events concerning
intracranial hypertension. Limitations must be clearly specified and
recognized to avoid the inappropriate use of the model beyond its
actual capacity.
First, the main simplifications introduced in the model are explicitly
stated. Then a qualitative description of the model is presented.
Computer simulation results are subsequently shown, demonstrating how
the model, despite its simplifications, is able to describe many
important phenomena concerning ICP changes. Finally, a sensitivity
analysis on model parameters is performed. All the model equations are
presented in APPENDIXES A-C,
together with parameter numerical values and some guidelines to
facilitate practical implementation of the model using a computer. In
the companion article (42), the reduced model, together with automatic
parameter identification techniques, is employed to analyze real ICP
tracings in patients with severe brain damage.
QUALITATIVE MODEL DESCRIPTION To build a simplified model of ICP dynamics aimed at clinical purposes,
a few simplifications were introduced with respect to the more accurate
mathematical model presented elsewhere (40, 41). The two main
simplifications are discussed below. Other minor simplifications are
introduced in the presentation of the qualitative model.
1) The model does not distinguish
between proximal and distal segments of the arterial-arteriolar
cerebrovascular bed; i.e., only one arterial-arteriolar segment,
extending from large intracranial arteries down to cerebral
capillaries, is included.
2) Pressure at the terminal point of
the large cerebral veins is assumed equal to ICP. This assumption is
justified, since the venous cerebrovascular bed behaves as a Starling
resistor (27). According to this mechanism, a primary ICP increase
provokes a collapse or narrowing of the terminal intracranial veins
(bridge veins and lateral lacunae or lakes), which, in turn, causes
pressure in the upstream large cerebral veins to rise to ICP. In this
condition, cerebral blood flow (CBF) depends on the difference between
arterial pressure and ICP; i.e., it is independent of the downstream
pressure at the venous sinuses.
The main consequences and possible shortcomings introduced by these
simplifications are critically considered in the
DISCUSSION.
Despite its limitations, the use of a reduced model presents some
significant advantages. First, it exhibits a small number of
parameters, each able to account for an entire physiological and
clinical phenomenon in a concise way. The presence of a restricted number of parameters and equations improves the clinical meaning of the
results obtained and facilitates the process of parameter identification. Furthermore, the reduced model is of the second order,
with only two state (or memory) variables. Hence, as shown in
APPENDIX Ac, computation of equilibrium
points and stability properties (eigenvalues) can be carried out using
rather simple algorithms, and model dynamics can be presented as
trajectories in the phase plane. This is a plane describing the mutual
dependence of the two memory variables: the first is plotted as an
independent variable in the x-axis and
the second as the dependent variable in the
y-axis. The loci of points in the
plane covered by the model during the simulations represent the
"trajectories" of the system, which account for its time dynamics
in a simple straightforward way. In particular, in a second-order
system with constant input quantities, the trajectories can converge
toward a stable equilibrium point or approach a closed curve (the
"limit cycle"). The first behavior occurs when the system settles
at a steady-state condition, the second when it exhibits self-sustained
periodic oscillations.
First, the main aspects of the intracranial hemo- and hydrodynamics are
presented. Subsequently, attention is focused on the action of
cerebrovascular control mechanisms.
Fig. 1.
Electric analog (A)
and corresponding mechanical analog
(B) of intracranial dynamics
according to present model. Cerebral blood flow (CBF, q) enters skull
at pressure approximately equal to systemic arterial pressure
(Pa). Arterial-arteriolar
cerebrovascular bed consists of a regulated capacity
(Ca), which stores a certain amount of blood volume, and a regulated resistance
(Ra), which accounts for
pressure drop to capillary pressure
(Pc). At capillary level,
cerebrospinal fluid (CSF) is produced through a CSF formation resistance (Rf). CBF then passes
through venous cerebrovascular bed, mimicked as series arrangement of
proximal venous resistance (Rpv)
and resistance of collapsing lateral lacunae and bridge veins
(Rdv). Model assumes that,
because of collapse of last section, cerebral venous pressure
(Pv) is always approximately
equal to intracranial pressure
(Pic). Finally, CSF is
reabsorbed at venous sinus pressure
(Pvs) through CSF outflow
resistance (Ro). Intracranial pressure is determined by amount of volume stored in nonlinear intracranial compliance (Cic).
This volume results from a balance between CSF inflow
(qf), CSF outflow
(qo), blood volume changes in
arterial capacity, and mock CSF injection rate
(Ii).
[View Larger Version of this Image (16K GIF file)]
where
Pa and
Pic denote systemic arterial
pressure and ICP, respectively. Ra
and Ca are actively adjusted by
the action of cerebrovascular control mechanisms (see
Action of cerebrovascular autoregulation
mechanisms).
(1)
|
(2) |
. For the comprehension of model performance, it is important to
underline that the static autoregulation characteristic in Fig. 2
requires the specification of several parameters:
1) the basal value of arterial
compliance (Ca n),
2) a maximum autoregulation gain
(G), which is the slope of the static autoregulation curve at its
central point, and 3) the saturation
levels Ca max and Ca min.
qn)/qn.
First block represents central autoregulation gain, G. Second block
describes static sigmoidal autoregulation response, with lower and
upper limits. Third block simulates low-pass dynamics of
autoregulation, with time constant .
Equation 1, with Ca regulated as in the block diagram in Fig. 2, permits the simulation of passive and active blood volume changes. By differentiating Eq. 1, the following equation is obtained
|
(3) |
dPic/dt),
and the second takes into account active blood volume changes caused by
autoregulation. In particular, vasoconstriction is simulated through a
reduction in compliance (dCa/dt < 0), which contributes to a reduction in blood volume (Eq. 3); vasodilation is simulated
through a compliance increase (dCa/dt > 0) with a consequent increase in blood volume.
In the model the variations in arterial resistance are strictly related
to the variations in compliance and blood volume. By roughly
considering the arterial-arteriolar cerebrovascular bed as the parallel
arrangement of several equal microvessels and applying the
Hagen-Poiseuille law (25), resistance is inversely proportional to the
fourth power of inner radius, hence, to the second power of blood
volume. It can thus be written
|
(4) |
R is a constant
parameter.
According to Eq. 4, any decrease in
the arterial-arteriolar blood volume caused by a passive drop or active
vasoconstriction is reflected in a resistance increase, and vice versa.
The final model is of the second order; i.e., it contains only two
state (or memory) variables: ICP, which reflects the volume in the
craniospinal pressure-volume curve, and the arterial-arteriolar compliance, which is influenced by the action of cerebrovascular control mechanisms.
Finally, the model, despite its simplicity, includes several distinct
feedback loops. These are summarized in the block diagram of Fig.
3. Three feedback loops are negative, and
so they tend to stabilize ICP. They are imputable to the CSF
circulation (feedback I), to the
effect of passive CBV changes on ICP (feedback
II), and to the effect of autoregulation on CBF
(feedback III). However, one can
also note the existence of a positive-feedback loop brought about by
active arterial-arteriolar blood volume changes
(feedback IV). This is analogous to
a qualitative model developed by Rosner (32) called "vasodilatory
cascade." According to Rosner, when cerebral perfusion pressure (CPP = SAP ICP, where SAP is systemic arterial pressure) is reduced,
vasodilation occurs and is accompanied by an increase in cerebral blood
volume (CBV). This, in turn, causes an increase in ICP, which further
reduces CPP. This cascade of events, or self-sustained cycle, can then
continue until vasodilation is maximal.
The existence of a positive-feedback loop in the model may cause instability of intracranial dynamics in particular pathological conditions. As is well known (11), instability in a second-order nonlinear system may manifest itself through the occurrence of self-sustained oscillations, the so-called limit cycles. Another significant consequence of the positive-feedback loop is the possible occurrence of paradoxical responses, i.e., responses characterized by a delayed amplification of a small initial perturbation. Both phenomena are described in the clinical literature (19, 31) and are analyzed in depth in RESULTS.
ASSIGNMENT OF PARAMETER BASAL VALUES
Here we aim to determine a basal value for all model parameters to reproduce the intracranial dynamics of healthy humans. We require the model to settle at a stable equilibrium level when all model parameters and input quantities are at their basal value. The value of a quantity in this condition is denoted with the subscript n.
The basal value of arterial and venous resistances was computed using the pressure distribution reported in Table 1 (see also Refs. 38 and 40) and assuming that normal CBF in humans is qn = 12.5 ml/s = 0.75 l/min.
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The basal value of arterial-arteriolar blood volume (Va n) was assigned starting from data reported by Tomita (37). According to Tomita, overall CBV in humans is 45-100 ml. About 80% is contained in the venular and venous segments, whereas the remaining 20% (9-20 ml) is in arterial-arteriolar vessels. In this model we assumed that Va n = 13.5 ml, a value that is in the range reported above. Starting from the previous assignments, the basal value of arterial compliance (Ca n) and the arterial resistance proportionality coefficient (kR; see Eq. A11), can easily be calculated.
Values for CSF outflow resistance (Ro) and CSF formation resistance (Rf) were computed assuming that, with the basal pressure distribution of Table 1, CSF production rate and CSF reabsorption rate are equal to 400 µl/min (38).
Normal values for
kE
in healthy humans range from 0.05 to 0.15 ml
1 (6). We assumed a
normal
kE
of 0.11 ml1, which is in
the range reported by Avezaat and van Eijndhoven. (6).
Finally, we must assign a value to the parameters describing the action
of the feedback autoregulation mechanisms. Studies performed in animals
(18) and humans (1) demonstrate that the brain vessel autoregulatory
response is quite fast, completing its action within 0.5-1 min
from the beginning of a perfusion pressure change. Consequently, we
assumed a time constant for autoregulation () of 20 s.
The upper and lower saturation values in the autoregulation curve (Ca max and Ca min, respectively) were given, starting from knowledge of the percent increase and percent reduction in arteriolar caliber observed experimentally. Various authors report that pial arterioles can increase their caliber up to 200% of baseline during autoregulation (18, 24) and up to 250% of baseline after other vasodilatory stimuli, such as hypoxia, hypercapnia, and functional hyperemia (24). Hence, we can assume a sixfold increase in volume during massive vasodilation, i.e., Ca max = 6 · Ca n. Similarly, arterioles decrease their caliber by ~25% after vasoconstrictory stimuli (24), which corresponds to a 50% reduction in blood volume (Ca min = 0.5 · Ca n).
Finally, the value of the maximum autoregulation gain, G
(Eqs. A6 and A9), was given to ensure that CBF
remains quite constant within the autoregulation range (i.e., when CPP
is 60-140 mmHg). A satisfactory pattern of CBF vs. CPP was
obtained by using the value G = 11.25 × 104
ml · cm2 · dyn1 · 100%
CBF change1 = 1.5 ml · mmHg1 · 100%
CBF change1.
Examples of the effect of G on the autoregulation curve are shown in
Fig. 4. As clearly shown in Fig. 4,
patients with progressively impaired autoregulation can be simulated by
decreasing the value of G.
1 · 100%
CBF change1, which is
assumed as normal autoregulation (CBF quite constant in SAP range
50-150 mmHg). Other 2 curves were obtained using G = 0.45 ml · mmHg1 · 100%
CBF change1 (dashed line,
autoregulation partly impaired) and G = 0.15 ml · mmHg1 · 100%
CBF change1 (dot-dashed
line, autoregulation severely impaired). In normal case, maximum CBV
increase occurs close to autoregulation lower limit, in accordance with
data by Kontos et al. (18).
The basal value of all parameters is shown in Table 1. In the following, basal parameter values are denoted with the subscript 0.
RESULTS
The results of a few simulations aimed at validating the model and at clarifying its potential applicability in the analysis of patients with severe brain disease are illustrated by 1) a study of intracranial stability, 2) a consideration of the effect of sudden SAP changes on ICP, and 3) a sensitivity analysis of the ICP response to classical clinical maneuvers (PVI tests).
Analysis of intracranial stability. Previously, using a more complex model, we were able to demonstrate that intracranial dynamics may become unstable in pathological conditions (40). The main alterations favoring intracranial instability were a decrease in intracranial compliance and an increase in the Ro, provided these changes occur in patients with preserved autoregulation mechanisms. Similar instability conditions may also occur in the present simplified model. Figure 5A shows the time pattern of ICP simulated using a high value for kE (2.1 · kE 0 = 0.23 ml1), a high value
for Ro
(12 · Ro 0 = 6.32 × 103
mmHg · s · ml1),
and a normal autoregulation gain. In this circumstance, ICP exhibits
self-sustained periodic waves that resemble, in amplitude, period, and
shape, the well-known Lundberg A waves (or plateau waves) (31). The
periodic oscillations of ICP are correlated with periodic oscillations
in arterial-arteriolar blood volume, as shown by the closed orbits in
Fig. 5B.
1)
and a significant increase in intracranial elastance coefficient (kE = 2.1 · kE 0 = 0.23 ml1). In this
condition, model becomes unstable, and self-sustained oscillations,
similar to plateau waves, develop. B:
limit cycle describing relationship between ICP and arteriolar volume
during these waves.
To better understand the conditions leading to the appearance of self-sustained oscillations, the so-called "bifurcation diagrams" can be drawn. The term "bifurcation" is currently used in the mathematical literature to denote those particular values of parameters at which a model exhibits a qualitative change in the topological structure of its trajectories. Among the different kinds of bifurcation described in mathematical textbooks, a particular role is played by the so-called Hopf bifurcation, which represents the condition where an equilibrium point loses its stability and a periodic oscillation appears (11). This is the bifurcation leading to the emergence of ICP plateau waves in patients with severe brain disease. Figure 6 shows two examples of bifurcation diagrams obtained with the present model (see APPENDIX Ac for mathematical details). The curves in Fig. 6 represent the locus of points in the parameter space (kE vs. Ro in Fig. 6A, G vs. Ro in Fig. 6B), where the model is exactly at the boundary between stability and instability (stability means that ICP settles down at a steady-state level; instability means that ICP exhibits periodic self-sustained oscillations). As clearly shown in Fig. 6, an increase in Ro, an increase in kE, and an increase in G are changes that may lead to intracranial instability. Moreover, there are several different combinations of these parameters, associated with a severe pathology, at which the model predicts the occurrence of plateau waves.
1). All
parameters are normalized to basal value of Table 1.
From a strictly mathematical point of view, instability is characterized by the loss of steady-state equilibrium and by the occurrence of self-sustained waves. From a clinical point of view, however, the principal interest may be in examining the situations where intracranial dynamics are still "stable" but operate close to the boundary between stability and instability. In these conditions, even a small external perturbation might cause uncontrolled or paradoxical ICP time patterns: the system, perturbed from its steady-state level, returns toward equilibrium only after a long, wide-amplitude transient response. Conditions of clinical relevance, characterized by paradoxical responses, may occur, among others, during acute hypotension (see ICP response to acute SAP changes) and PVI tests (see Sensitivity analysis to parameter changes during PVI tests). ICP response to acute SAP changes. Figure 7 shows the ICP response to a quick arterial pressure decrease in a patient having a modest increase in Ro and a progressive reduction in the intracranial compliance. In all the examples, the initial arterial pressure decrease causes arteriolar vasodilation and an increase in CBV, which triggers the positive-feedback loop in intracranial dynamics (feedback IV in Fig. 3). However, when the intracranial compliance is sufficiently high to buffer the blood volume increase, the system works far from its instability boundary, and only a small transient rise in ICP occurs. In contrast, if intracranial compliance worsens, the system moves toward its instability boundary and the positive-feedback loop becomes more influential in intracranial dynamics. In the poorest condition, even a small reduction in SAP is able to induce an abrupt transient rise in ICP to 50-60 mmHg, the shape of which is similar to that of a single plateau wave. However, in Fig. 7 the model is always stable. This means that the increase in ICP represents only an isolated, transient response to the arterial pressure perturbation, which is not followed by the production of self-sustained periodic waves. As clearly shown by the bifurcation diagram in Fig. 6, self-sustained plateau waves can occur only if Ro is further increased with respect to the value used in Fig. 7.
1 · 100%
CBF change1), a
moderately increased Ro
(5 · Ro 0 = 2.63 × 103
mmHg · s · ml1),
and 6 values for
kE
(0.11, 0.15, 0.18, 0.21, 0.24, and 0.27 ml1). The higher
kE
is, the greater the ICP increase to hypotension and, consequently, the
greater the reduction in CBF.
Sensitivity analysis to parameter changes during PVI tests. A maneuver frequently performed in clinical practice to estimate intracranial parameters is the PVI test first introduced by Marmarou and co-workers (22, 23). The maneuver consists in injecting or withdrawing a small amount of mock CSF into the craniospinal compartment and in monitoring the consequent ICP response. According to its original version, this test is normally used to estimate the PVI (defined as the volume, in ml, that should be added to the CSF space to produce a 10-fold increase in ICP), Ro, and CSF production rate. Recently, however, we suggested that the response to PVI tests may also contain information on the status and the dynamics of cerebral autoregulation (41). The existence of a strict correlation between PVI and autoregulation has been documented by other authors through clinical and experimental works (7, 12, 13), but the deep nature of this relationship is insufficiently understood. Figure 8 shows the simulation of a 2-ml bolus injection in a patient with efficient autoregulation and a modest increase in Ro. Four different sensitivity analyses have been performed concerning the effect on the response of small changes in kE (Fig. 8A), G (Fig. 8B), Ro (Fig. 8C), and the basal value of arterial-arteriolar compliance (Fig. 8D).
1 · 100%
CBF change1), a
moderately increased Ro
(5 · Ro 0 = 2.63 × 103
mmHg · s · ml1),
and a moderately increased
kE
(0.13 ml1).
A: sensitivity to changes in
kE (dashed line,
0.11 ml1; solid line, 0.13 ml1; dot-dashed line,
0.15 ml1).
B: sensitivity to changes in G (dashed
line, 0.66 ml · mmHg1 · 100%
CBF change1; solid line,
1.5 ml · mmHg1 · 100%
CBF change1; dot-dashed
line, 2.66 ml · mmHg1 · 100%
CBF change1).
C: sensitivity to changes in
Ro (dashed line,
4 · Ro 0 = 2.1 × 103
mmHg · s · ml1;
solid line, Ro = 5 · Ro 0 = 2.63 × 103
mmHg · s · ml1;
dot-dashed line, Ro = 6 · Ro 0 = 3.15 × 103 × 103
mmHg · s · ml1).
D: sensitivity to changes in basal
arteriolar-arterial compliance, hence, in basal arterial-arteriolar
blood volume (dashed line, 0.075 ml/mmHg; solid line,
0.15 ml/mmHg; dot-dashed line, 0.225 ml/mmHg).
The results in Fig. 8 suggest that, in a patient with efficient autoregulation, ICP does not always return monotonically toward baseline after the maneuver but often exhibits a paradoxical response characterized by a delayed increase. We define the ICP increase "delayed" or "paradoxical" when it occurs after the termination of the injection phase. Because in all simulations of Fig. 8 the injection was carried out between 10 and 12 s, we consider paradoxical the ICP increases occurring after 12 s. Because CSF is not augmenting in this period, but rather lessening because of CSF reabsorption, the paradoxical response is imputable to a progressive rise in CBV induced by the action of cerebral autoregulation mechanisms. According to Fig. 8, the conditions that may favor the occurrence of paradoxical responses are a high kE value (i.e., a steeper pressure-volume relationship, Fig. 8A), a high G value (Fig. 8B), and a high basal value of arterial-arteriolar compliance (i.e., a high basal value of the arterial-arteriolar blood volume, Fig. 8D). Furthermore, according to Fig. 8C, increasing Ro causes an increase in the ICP equilibrium level (i.e., the level before the maneuver). This means that the equilibrium point is located high on the exponential intracranial pressure-volume relationship, which corresponds to a zone of reduced intracranial compliance and to a high risk of paradoxical response (Fig. 8C). Figure 9 shows a few examples of the ICP response to a 2-ml bolus injection in a patient with defective autoregulation. In these simulations the gain of the autoregulation mechanisms was set at zero; hence only passive arterial-arteriolar blood volume changes may occur after the maneuver. Consequently, as in the classic model of Marmarou et al. (22, 23), ICP exhibits an initial peak followed by a monotonic return toward baseline. The results of three distinct simulations are shown characterized by different basal values of the arterial-arteriolar compliance, hence, of different arterial-arteriolar blood volume. In Fig. 9, as in Fig. 8D, the peak of ICP after the injection, and so PVI, is significantly affected by arterial-arteriolar blood volume. There is, however, a profound difference between Fig. 9 and Fig. 8D. In the simulations in Fig. 8D, blood volume progressively increases after the maneuver because of arteriolar vasodilation; hence, a high basal level of blood volume determines a sharp ICP increase and a reduced PVI. In contrast, in Fig. 9, where autoregulation is absent, blood volume decreases passively after the rise in ICP. The passive blood volume reduction, in turn, buffers the effect of fluid injection into the craniospinal compartment and contributes to increasing PVI.
1),
a moderately increased
kE
(0.13 ml1), and complete
impairment in G (0). Dashed line,
Ca n = 0.075 ml/mmHg; solid
line, Ca n = 0.15 ml/mmHg;
dot-dashed line, Ca n = 0.225 ml/mmHg.
Finally, Fig. 10A shows the ICP response to various PVI tests performed at different rates. The rates were chosen to have a total fluid injection as great as 2 ml accomplished in 2, 10, 20, 40, and 80 s. The time pattern of the arterial-arteriolar blood volume is presented in Fig. 10B.
1 · 100%
CBF change1), a
moderately increased Ro
(5 · Ro 0 = 2.63 × 103
mmHg · s · ml1),
and a moderately increased
kE
(0.15 ml1). All tests
concern a 2-ml injection from 10 s at 1, 0.2, 0.1, 0.05, and 0.025 ml/s. Vertical dotted lines, end of injection period in 5 trials.
The simulation results suggest that the paradoxical response, defined as the ICP increase after the termination of the injection phase, becomes progressively less important the slower the maneuver. At high rates (which are those commonly adopted in clinical practice) the arterial-arteriolar blood volume shows a passive fall during the injection period, which significantly limits the initial rise in ICP. Only subsequently active cerebral vasodilation develops, causing a delayed ICP increase up to a level more than three times the ICP increase caused by the injection. In contrast, if the maneuver is performed at a low rate, active vasodilation develops almost entirely within the injection period, and so the paradoxical rise in ICP becomes small. However, the peak of the ICP response and the terminal monotonic return toward baseline are scarcely influenced by the injection rate.
DISCUSSION
This study is aimed at analyzing the relationship between cerebral autoregulation, arteriolar blood volume changes, and ICP by means of a simple mathematical model. The need to include cerebral hemodynamics, besides CSF dynamics and intracranial elasticity, in the analysis of ICP has recently been stressed by several authors (7, 12, 41) and is becoming a subject of increasing clinical and physiological importance. It is now well accepted that treatment of patients with severe brain injury cannot neglect the role of CBV changes and their possible detrimental effect on ICP and CBF. Among others, the relationship between CBV changes and ICP is thought to play a pivotal role in the genesis of ICP plateau waves (31), in the ICP response after acute SAP changes (7, 17), and in the ICP time pattern during PVI tests (12, 13). In the following, results obtained with the model on each of these aspects are considered separately and compared with the clinical and physiological data available.
All the main results presented here agree quite closely with those achieved with the more complex model described previously (40, 41); the main advantage of the new model is that the same kinds of behavior can be obtained with a smaller number of parameters and less mathematical complexity. Differences between the two models are further discussed at the end of the DISCUSSION.
Generation of A waves. The present simple model predicts the occurrence of ICP A waves (or plateau waves) in conditions similar to those documented in the clinical literature. According to the bifurcation diagrams in Fig. 6, the main pathological conditions leading to intracranial instability and ICP self-sustained oscillations are a reduction in intracranial compliance (i.e., an increase in kE) and an impairment in CSF circulation, provided these alterations occur in patients with preserved autoregulation mechanisms. Similar alterations during A waves have been documented (5, 14, 28). The mechanism leading to ICP waves in our model is similar to that described by Rosner (32) by the term "vasodilatory cascade." This mechanism may be understood by looking at the positive-feedback loop in Fig. 3. According to the Starling resistor hypothesis, any increase in ICP causes a parallel increase in cerebral venous pressure, hence, a reduction in CPP and CBF. The latter, in turn, triggers the autoregulation mechanisms, thus causing a vasodilation, with a consequent increase in CBV and a delayed rise in ICP. In physiological conditions this vasodilatory cascade does not lead to permanent ICP oscillations, since active blood volume increases are first rapidly accommodated by the high intracranial compliance and then compensated by CSF outflow mechanisms (see "escape ways" in Fig. 3). The mathematical model, however, predicts a threshold in the values of these parameters, after which the system loses its stability, the vasodilatory cascade cannot be further neutralized by the physiological compensatory mechanisms (elasticity and CSF outflow), and large ICP oscillations develop. In the model, oscillations are self-sustained; i.e., they may occur without any external perturbation, simply as a consequence of the intrinsic instability of system dynamics. This result disagrees with the thesis held by Rosner and Becker (33), who claimed that a decrease in SAP or any other vasodilatory stimulus is always necessary to evoke the start of a plateau wave, but agrees with data reported by Hayashi et al. (15). However, although self-sustained oscillations occur in the instability region without the need of arterial hypotension, a decline in SAP may favor the production of a single wave when the system is in the stable region close to the boundary between stability and instability (see ICP response to acute SAP changes). ICP response to acute SAP changes. The results in Fig. 7 point out that the ICP response to an acute arterial hypotension may exhibit different characteristics depending on the values of model parameters. If the system works far from the boundary between stability and instability (i.e., line in bifurcation diagrams in Fig. 6), a small alteration in SAP is able to cause only a modest transient increase in ICP, without appreciable effects on CBF. In contrast, if the system works close to the stability boundary, even a mild arterial hypotension may activate the vasodilatory cascade, leading to a significant increase in ICP. The latter may cause CPP to decline well below the autoregulation limit, with a reduction in CBF. According to Fig. 7, the magnitude of the ICP rise depends mainly on intracranial compliance, whereas its duration is largely affected by the status of CSF circulation. An impaired CSF outflow determines a longer intracranial hypertension, with the risk of cerebral ischemia and secondary brain damage. The results in Fig. 7 are substantially confirmed in the clinical literature. Bouma et al. (7) observed that, in patients with intact autoregulation, lowering SAP causes a steep increase in ICP. In some cases the increase in ICP was as high as +10 or +15 mmHg, whereas in others only modest ICP increments were evident (Fig. 1 in Ref. 7). In experiments in cats, Rosner and Becker (33) demonstrated that modest decreases in SAP (15.3 mmHg on average) are sometimes able to
elicit a disproportionate rise in ICP, the amplitude and shape of which
are analogous to those of a plateau wave.
Understanding the effect of SAP on ICP may be of very great value for
clinical practice. Indeed, the choice of the optimum arterial pressure
in the management of head-injured patients is still a matter of debate,
and controversial suggestions can be found in recent literature.
Whereas some authors advocated reducing SAP in patients to minimize the
risk of brain edema (2, 35), others proposed raising SAP to prevent
cerebral vasodilation and uncontrollable increase in ICP (26, 34). The
present model may constitute valuable support for more rigorous
management of arterial pressure in patients with severe brain disease.
We suggest that the effect of SAP on ICP depends significantly on model
parameters, especially on
kE,
Ro,
Rf, and G. Estimation of the value
of these parameters, widely varying in pathological conditions, may be of use in the treatment of patients in intensive care and may guide the
choice of an improved therapy. Basically, the model suggests that
arterial hypotension should be avoided as far as possible, since it may
have unpredictable and disproportionate effects on ICP in patients with
reduced compliance and reduced CSF outflow.
Simulation of PVI tests.
PVI tests are frequently used to estimate the intracranial compliance
and the status of CSF outflow. The basic assumption is that the peak of
the ICP response mainly depends on intracranial elasticity, whereas the
rate of ICP return toward baseline (i.e., the time constant of the
response) reflects the product of intracranial compliance and
Ro. Both these assumptions,
however, must be reconsidered according to the simulation results in
Figs. 8, 9, 10.
Figures 8-10 show that PVI not only contains information on
kE but is also
significantly affected by arterial-arteriolar hemodynamics and by the
status of cerebral autoregulation. In a patient with intact
autoregulation, PVI may be lessened by the occurrence of active CBV
expansion, secondary to the maneuver (Fig. 8). In contrast, in patients
with impaired autoregulation (Fig. 9), PVI may be increased as a result
of passive blood volume reduction, which attenuates the initial rise in
ICP.
Several authors remarked that the determination of PVI may be
significantly affected by autoregulation and CPP. In particular, Gray
and Rosner (12, 13) in the cat and Bouma et al. (7) in patients
observed that PVI exhibits a positive correlation with CPP within the
autoregulation range and a negative correlation below the lower
autoregulation limit. Moreover, PVI is increased in the presence of
deep anesthesia, showing a negative correlation with CPP in the entire
autoregulation range (13). These results can be explained quite well by
the model. Computer simulations (Fig. 8) suggest that PVI decreases at
the lower autoregulation limit where the active blood volume expansion
reaches a maximum (see also Fig. 4). In contrast, during deep
anesthesia or below the lower autoregulation limit, PVI increases owing
to passive blood volume variations (Fig. 9).
Moreover, model simulation results point out that the rate at which ICP
returns toward baseline after a bolus injection (Fig. 8) not only
reflects intracranial elastance and CSF outflow but is also influenced
by cerebral hemodynamics and active arterial-arteriolar blood volume
changes. In patients with efficient autoregulation, the ICP decrease to
baseline is accompanied by CBV reduction as long as arterioles,
previously dilated, recover their basal caliber. The latter phenomenon
is superimposed on the effect of CSF circulation and may give the
misleading impression of a high CSF outflow. By way of an example, let
us consider the three curves in Fig. 8B. These correspond to the same
values of
kE
and Ro but exhibit significant
differences in the peak and rate of change of the ICP response,
depending on G.
Finally, simulation results show that the time pattern of ICP during
PVI tests depends on the duration of the injection phase compared with
. If the duration of the maneuver is much smaller than , we can
expect a significant delayed increase in ICP. The latter becomes
progressively less important if the injection period and become
comparable. We have used = 20 s. Other authors, however, on the
basis of transcranial Doppler measurements, suggested that the
autoregulation dynamics can be even more rapid in humans (1). In this
case, the paradoxical ICP increase would be less evident than in the
simulation of Fig. 8.
According to the previous considerations, we suggest that a more
accurate and rigorous estimation of intracranial parameters (not only
Ro and
kE, but also G
and ) during PVI tests may be achieved by performing a best fit
between the entire measured ICP time pattern and that simulated by the
model. Minimization should be performed via a suitable minimization
algorithm. This approach is substantially different from that commonly
adopted in the clinical literature. In the classic approach, only
information at two specific points of the ICP response (usually the
peak value or the value immediately after the injection and a value 1 or 2 min later) is used to derive information on intracranial dynamics. The new approach is used in the companion article (42), which is
devoted to a clinical application of the present model.
Finally, the present model aspires to be a compromise between two
opposite requirements: accuracy in the reproduction of the physiological reality, on the one hand, and simplicity, on the other.
Naturally, some imperfections unavoidably arise when a complex model is
simplified. These must be recognized to avoid the model being used
beyond its actual limits.
A first possible shortcoming derives from having neglected the role of
venous blood volume changes. This is a consequence of the Starling
resistor hypothesis, according to which large cerebral veins always
remain open during intracranial hypertension. The small decrease in
venous blood volume at the collapsing terminal veins (bridge veins and
lateral lakes) was considered a part of the intracranial
pressure-volume relationship. Indeed, the present model tries to
represent only alterations in the arterial-arteriolar blood
volume, which is the portion of the cerebral vasculature under the
control of autoregulation mechanisms. With this limitation, however,
the model cannot be used to simulate the rapid effect of perturbations
arising in the extracranial venous drainage pathways ( jugular
vein compression, Valsalva maneuver, effect of the respiration on ICP),
which are characterized by an increase in cerebral venous pressure
transmitted back to the cranial cavity.
A second shortcoming is that the model cannot distinguish between the
action of mechanisms working on large pial arteries and those working
on small arterioles. Several authors (18, 24) assert that CBF is
controlled mainly by a dilation of large pial arteries in the central
autoregulation range, whereas small arterioles exhibit a massive
vasodilation only when CPP approaches the lower autoregulation limit. A
distinction between mechanisms working on large and small arteries,
introduced by Ursino and Di Giammarco (40), permits more accurate
reproduction of the ICP response to arterial hypotension.
Finally, the model cannot be used to study ICP pulsating waves
synchronous with the cardiac beat or with respiration. ICP pulsatility
is significantly affected by large intracranial arteries and by the
venous circulation; hence, its analysis requires the use of more
complex multicompartmental models.
The more complete model presented in Ursino and Di Giammarco (40) did
not have all these limitations. Hence, that model should be preferred
to investigate the physiological bases of ICP dynamics from a
theoretical point of view or to design new experimental procedures.
However, the complete model exhibits some serious drawbacks when used
in a clinical setting. In particular, its structure is too complex to
be entirely identified starting from PVI tests or other routine
clinical measurements, the model is computationally quite onerous, and
a best fit with clinical data cannot be achieved in the short time
available for a diagnosis.
The present simplified model overcomes these limitations: it has the
virtues of simplicity and physiological reliability and aspires to be
directly usable in neurosurgery intensive care units (see Ref. 42 for a
test of model applicability to the analysis of real patients).
FOOTNOTES
Address for reprint requests: M. Ursino, Dept. di Elettronica, Informatica e Sistemistica, viale Risorgimento 2, I-40136 Bologna, Italy.
Received 21 June 1996; accepted in final form 29 October 1996.
APPENDIX a
|
(A1) |
|
(A2) |
|
(A3) |
|
(A4) |
|
(A5) |
![<FR><NU>dC<SUB>a</SUB></NU><DE>d<IT>t</IT></DE></FR> = <FR><NU>1</NU><DE>&tgr;</DE></FR> ⋅ [−C<SUB>a</SUB> + &sfgr;(G ⋅ <IT>x</IT>)]](/content/vol82/issue4/fulltext/1256/img010.gif)
|
(A6) |
|
(A7) |
is the time constant of the regulation, q is CBF, 
represents a
sigmoidal static function with lower and upper saturation, and G is the
maximum autoregulation gain (i.e., the gain at the central point of the
autoregulation curve). Finally, qn
represents the value of CBF required by tissue metabolism, and so
x denotes the CBF changes normalized
to the basal level.
A value for CBF can be computed from the electric analog in Fig. 1
|
(A8) |
|
(A9) |
Ca are the central value and
the amplitude of the sigmoidal curve, respectively, and
k
is a constant parameter that permits us to set the central slope.
According to Eq. A9, any decrease in
CBF below the metabolic requirement (hence,
x < 0) causes a vasodilation with an
increase in compliance, and vice versa.
A value for
k
was computed to set the central slope of the static curve at G.
With this choice, G simply represents the maximum autoregulation gain.
We have
(d/dx)|x = 0 = G · (Ca/4k),
and so we choose
k = Ca/4.
However, the sigmoidal autoregulation curve (Eq. A9) is not symmetrical, since the increase in blood
volume induced by vasodilation is higher than the decrease in blood
volume induced by vasoconstriction. Hence, two different values must be
chosen for the parameter Ca in
Eq. A9, depending on whether
vasodilation or vasoconstriction is considered. We have
|
(A10) |
Ca 1/2 and Ca min = Ca n Ca 2/2.
Finally, autoregulation not only affects arterial-arteriolar compliance
and blood volume but also the hydraulic resistance, Ra. Moreover, changes in
Ra and
Ca are related through the
geometrical properties of arterioles. As a first approximation, one can
consider a microvascular bed consisting of the parallel arrangement of several microvessels with equal inner radius
r. Hence, blood volume is directly
proportional to
r2,
whereas resistance can be assumed to be inversely proportional to
r4
(Hagen-Poiseuille law). We can thus write
|
(A11) |
APPENDIX b
|
(B1) |
(t) by means of
Eqs. A7, A10, and
A9.
8) Compute the rate of change of the
arterial-arteriolar compliance
[dCa(t)/dt]
by means of Eq. A6.
9) Compute the rate of change in ICP
[dPic(t)/dt]
by rearranging Eqs. A1, A2, and
A5. In particular, by substituting
Eqs. A2 and A5 into Eq. A1, one obtains
|
(B2) |
APPENDIX c
![<FENCE> <AR><R><C><FR><NU>d<B>x</B></NU><DE><B>d</B><IT>t</IT></DE></FR> (<IT>t</IT>) = <B>f</B>[<B>x</B>(<IT>t</IT>),<B>v</B>(<IT>t</IT>),<B>u</B>(<IT>t</IT>);&THgr;]</C></R><R><C><B>g</B>[<B>x</B>(<IT>t</IT>),<B>v</B>(<IT>t</IT>),<B>u</B>(<IT>t</IT>);&THgr;] = 0</C></R></AR></FENCE>](/content/vol82/issue4/fulltext/1256/img018.gif)
|
(C1) |
]T
is the vector of auxiliary variables,
u(t) = [Pa,dPa/dt,Pvs,Ii]T
is the vector of input quantities, and 
denotes the vector of model
parameters. f() represents the system
of two nonlinear equations [Eqs.
B2 and A6], and
g(), the system of six nonlinear
equations (Eqs. A4, A11, B1, A8, A7,
and A9).
Given a set of parameters, , and a set of constant input quantities,
u0,
Eq. C1 exhibits only one equilibrium
value, [x0(u0,) v0(u0,)]T,
with an acceptable physiological meaning, satisfying the system of
equations
![<FENCE> <AR><R><C><B>f</B>[<B>x</B><SUB>0</SUB>(<B>u</B><SUB>0</SUB>,&THgr;),<B>v</B><SUB>0</SUB>(<B>u</B><SUB>0</SUB>,&THgr;),<B>u</B><SUB>0</SUB>;&THgr;] = 0</C></R><R><C><B>g</B>[<B>x</B><SUB>0</SUB>(<B>u</B><SUB>0</SUB>,&THgr;),<B>v</B><SUB>0</SUB>(<B>u</B><SUB>0</SUB>,&THgr;),<B>u</B><SUB>0</SUB>;&THgr;] = 0</C></R><R><C></C></R></AR></FENCE>](/content/vol82/issue4/fulltext/1256/img019.gif)
|
|
(C2a) |
|
(C3) |
|
|
(C4) |
|
(C5) |
) is
explicitly indicated on the left-hand side.
The characteristic equation for the computation of eigenvalues is
|
(C6) |
1,2 = ± j
. Hence, a necessary condition is
|
(C7) |
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 K. Lu, J. W. Clark Jr., F. H. Ghorbel, C. S. Robertson, D. L. Ware, J. B. Zwischenberger, and A. Bidani Cerebral autoregulation and gas exchange studied using a human cardiopulmonary model Am J Physiol Heart Circ Physiol, February 1, 2004; 286(2): H584 - H601. [Abstract] [Full Text] [PDF]
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J. J. Van Lieshout, W. Wieling, J. M. Karemaker, and N. H. Secher Syncope, cerebral perfusion, and oxygenation J Appl Physiol, March 1, 2003; 94(3): 833 - 848. [Abstract] [Full Text] [PDF]
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M. Ursino, A. Ter Minassian, C. A. Lodi, and L. Beydon Cerebral hemodynamics during arterial and CO2 pressure changes: in vivo prediction by a mathematical model Am J Physiol Heart Circ Physiol, November 1, 2000; 279(5): H2439 - H2455. [Abstract] [Full Text] [PDF]
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C. A. Lodi, A. Ter Minassian, L. Beydon, and M. Ursino Modeling cerebral autoregulation and CO2 reactivity in patients with severe head injury Am J Physiol Heart Circ Physiol, May 1, 1998; 274(5): H1729 - H1741. [Abstract] [Full Text] [PDF]
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M. Ursino, C. A. Lodi, S. Rossi, and N. Stocchetti Intracranial pressure dynamics in patients with acute brain damage J Appl Physiol, April 1, 1997; 82(4): 1270 - 1282. [Abstract] [Full Text] [PDF]
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