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1 Department of Electronics, Computer Science and Systems, University of Bologna, I-40136 Bologna, Italy; and 2 Departement d'Anesthésie, Centre Huniversitaire Hospitalier de Angers, 49033 Angers Cedex, France
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ABSTRACT |
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The aim of this work was to analyze changes in cerebral hemodynamics and intracranial pressure (ICP) evoked by mean systemic arterial pressure (SAP) and arterial CO2 pressure (PaCO2) challenges in patients with acute brain damage. The study was performed by means of a new simple mathematical model of intracranial hemodynamics, particularly aimed at routine clinical investigation. The model was validated by comparing its results with data from transcranial Doppler velocity in the middle cerebral artery (VMCA) and ICP measured in 44 tracings on 13 different patients during mean SAP and PaCO2 challenges. The validation consisted of individual identification of 6 parameters in all 44 tracings by means of a best fitting algorithm. The parameters chosen for the identification summarize the main aspects of intracranial dynamics, i.e., cerebrospinal fluid circulation, intracranial elastance, and cerebrovascular control. The results suggest that the model is able to reproduce the measured time patterns of VMCA and ICP in all 44 tracings by using values for the parameters that lie within the ranges reported in the pathophysiological literature. The meaning of parameter estimates is discussed, and comments on the main virtues and limitations of the present approach are offered.
carbon dioxide reactivity; intracranial pressure; cerebral autoregulation; severe brain damage; transcranial Doppler
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INTRODUCTION |
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CEREBRAL PERFUSION PRESSURE (CPP) and CO2 pressure have a strong impact on intracranial vessels through the action of cerebrovascular control mechanisms (7, 13, 32). Under normal circumstances, a fall in blood pressure causes a rapid dilatation of resistance vessels, whereas a rise in blood pressure causes vasoconstriction (23, 26). As a consequence, cerebral blood flow (CBF) is maintained at the preexisting level in healthy subjects in the CPP range of 50-150 mmHg [a phenomenon named "autoregulation" (11)].
Moreover, arterial PCO2 (PaCO2) is a strong vasodilator of the cerebral vasculature because it is able to increase CBF by 100% during severe hypercapnia (17, 37). The role of CO2 is mainly mediated by pH changes in the perivascular space (11, 40).
The previous mechanisms, however, not only affect cerebral hemodynamics but also may have a serious impact on intracranial pressure (ICP) via complex nonlinear relationships (7, 16). First, cerebral circulation occurs within a closed space (the skull and neuroaxis); hence, alterations in cerebral blood volume (CBV) may modify ICP through the craniospinal pressure-volume relationship. Furthermore, CBF variations modulate the cerebrospinal fluid (CSF) production rate and ICP through changes in intracranial capillary pressure.
Knowledge of the previous relationships is of paramount importance for the treatment of patients with severe head injury in neurosurgical intensive care units. Decreasing CO2 pressure through hyperventilation, in fact, is frequently used to reduce ICP in patients at risk of intracranial hypertension in whom cerebral vasculature is reactive (13, 35). Vasoconstriction, however, may induce an excessive fall in CBF, with the potential hazard of cerebral ischemia and secondary brain damage (32). Setting a correct level of mean systemic arterial pressure (SAP) is also controversial. Although some authors (2) in recent years proposed to maintain low SAP in head-injured patients to minimize the risk of cerebral edema, a more common opinion today is that CPP should be maintained >70-80 mmHg to avoid hypoperfusion and the consequent brain ischemia (7, 15, 36, 39). The danger of ICP instability (a phenomenon often called "vasodilatory cascade") should also be considered when CPP approaches the autoregulation lower limit (38, 50).
This scenario is complicated by the observation that cerebrovascular control mechanisms may be impaired in head injury. Severe head trauma disturbs autoregulation in most patients (7, 13, 21, 35). Impairment in cerebrovascular reactivity, in turn, may be a consequence of the trauma per se, may be secondary to alterations in ICP and craniospinal storage capacity, or may occur during episodes of arterial hypotension with reduced CPP (15, 36).
Because cerebral hemodynamics adjustments in head-injured patients are complex multifactorial phenomena, they may be better understood with the use of mathematical models and computer simulation techniques. Many such models have been presented in the previous decades, with the focus on different aspects of intracranial dynamics and cerebrovascular control (8, 20, 28, 49); however, none of them focused attention on the interaction among ICP, CBV, CSF circulation, and cerebrovascular control mechanisms.
In recent years we formulated a mathematical model of the relationships among ICP, CBV, and cerebrovascular control mechanisms in acute brain damage (50, 54). The model revealed itself able to reproduce ICP and cerebral hemodynamics in different conditions of clinical interest (50-52).
The previous model, however, is computationally too heavy to be routinely used in a clinical setting. The aim of this work is to describe and validate a simplified model that can incorporate the main mechanisms involved in intracranial dynamics and permit their quantitative individual assessment with a drastic reduction in mathematical complexity.
The paper is structured as follows. First, the qualitative aspects of the model are described. The clinical and computational methods used to test the validity of the model are then presented. The results concern the reproduction of ICP and middle cerebral artery (MCA) velocity tracings during SAP and PaCO2 challenges. Finally, the virtues and limitations of the obtained results are critically discussed.
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QUALITATIVE MODEL DESCRIPTION |
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The model presented in this paper is significantly simplified compared with that used in a previous recent work (54). The main simplifications are 1) that the model does not distinguish between the behavior of large and small pial arteries and 2) that the biomechanics of the pial arterial-arteriolar vascular bed are reproduced by means of a simple windkessel model (that is, the arrangement of a hydraulic resistance and a hydraulic capacity). By contrast, in previous works (50, 52, 54) the biomechanics of large and small pial arteries were described starting from the equilibrium of forces in the vessel wall, with wall elastic and active (muscular) characteristics taken into account.
The main aspects of the model are described in qualitative terms. All
model equations, with considerations of numerical integration methods,
are given in APPENDIX A. A biomechanical analog of the
model is shown in Fig. 1.
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Intracranial Hemodynamics and Hydrodynamics
Large intracranial arteries.
The first segment in the model represents circulation in the basal
intracranial arteries, down to and excluding the large pial arteries.
The hemodynamics of this segment are described by means of a single
hydraulic resistance (Rla). Because the impact of cerebrovascular regulation mechanisms on the basal intracranial arteries is quite small (14), Rla
has been maintained constant throughout the simulations. An approximate
estimation of blood flow velocity at an MCA
(VMCA), to be compared with the results of
transcranial Doppler (TCD), is computed by assuming that blood flow in
an MCA is about one-third of total CBF. We can thus write
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(1) |
ICP) through a monoexponential
pressure-radius relationship taken from Hayashi et al.
(19). This means that the MCA behaves passively and
becomes progressively more rigid when transmural pressure increases. We
are aware that the use of Eq. 1 may involve some errors.
First, data reported by Hayashi et al. (19) concern human
specimens in vitro, while the behavior of the vessel in vivo might be
different. Moreover, head trauma may modify the pressure-radius
characteristic of the MCA. Finally, changes in the velocity profile or
the presence of cerebrovascular heterogeneity may alter the
relationship between mean CBF and maximum velocity, thus making the
coefficient kV time dependent. All these
differences might induce some errors in Eq. 1 and, hence, in
the VMCA prediction by the model.
Pial arterial-arteriolar cerebrovascular bed. The second segment in Fig. 1 simulates the pial arterial circulation, extending from the large pial arteries down to and including small pial arteries and intraparenchymal arterioles. For the sake of simplicity, the model does not distinguish between hemodynamics in different-sized vessels; hence, we consider only one segment, characterized by its values of hydraulic resistance (Rpa) and compliance (Cpa). As described in Cerebrovascular Regulation Mechanisms, both parameters are actively regulated by cerebrovascular control mechanisms.
Venous intracranial circulation. The intracranial venous vascular bed is described by the series arrangement of two segments. The first extends from the small postcapillary venules down to the large cerebral veins and includes the venous resistance (Rpv) and the intracranial venous capacity (Cvi). Because the impact of cerebrovascular mechanisms on the venous vasculature is negligible and large veins do not collapse even at elevated ICP (3), venous resistance has been maintained constant throughout the simulations. By contrast, venous capacity is inversely proportional to the local transmural pressure level, which implies a monoexponential pressure-volume relationship for the veins. We are aware that the exact mathematical expression for venous compliance may be affected by the head injury. However, changes in intracranial venous transmural pressure are always modest in our model (because of the Starling resistor mechanism described below); as a consequence, the exact expression for venous compliance has only a minor impact on the results.
The last segment in Fig. 1 represents the terminal intracranial veins (lateral lakes and bridge veins). During intracranial hypertension these vessels collapse or narrow at their entrance into the dural sinuses, with a mechanism similar to that of a Starling resistor (50). Because of this mechanism, pressure in the large cerebral veins (Pv) remains a little higher than ICP even during elevated intracranial hypertension (provided ICP < SAP).CSF circulation.
The circulation of CSF is described as a passive process. CSF is
produced at the cerebral capillaries because of a positive transmural
pressure gradient (Pc
Pic, where
Pc is capillary pressure and Pic is
intracranial pressure) and is reabsorbed at the dural sinuses because
of a negative transmural pressure gradient (Pvs
Pic, where Pvs is venous sinus pressure). The
resistances to CSF formation and CSF outflow are
Rf and Ro, respectively. However, both processes are unidirectional, so we have assumed that
resistances rise to infinity when the corresponding transmural pressure
is reversed.
Intracranial compliance.
Because the overall intracranial volume must remain constant, any
change in the content of one of the previous compartments (i.e., pial
artery volume at Cpa, venous volume at Cvi, and
CSF volume) must be accompanied by an opposite change in the remaining intracranial volumes with a concomitant variation in ICP. This phenomenon is described through the intracranial storage capacity (Cic). According to Marmarou et al. (28) and
Avezaat et al. (4), an expression for the craniospinal
capacity can be computed by assuming that the intracranial
pressure-volume relationship is approximately monoexponential in type.
We then have
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Cerebrovascular Regulation Mechanisms
Cerebrovascular regulation mechanisms work by modifying Rpa and Cpa (and hence blood volume) in the pial arterial-arteriolar vasculature. However, changes in these two parameters are not independent but are related through biomechanical and geometrical laws.Two distinct control mechanisms are considered in this work, i.e.,
autoregulation and CO2 reactivity (see Fig.
2, top). The role of arterial
oxygen is not included because O2 pressure was maintained constant throughout the present trials.
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As shown in the upper branch of Fig. 2, we assumed that autoregulation
is activated by changes in CBF. Its action on the arterial-arteriolar pial vessels includes a static gain (Gaut) and first-order
low-pass dynamics with the time constant
aut.
The lower branch in Fig. 2, top, represents CO2
reactivity. This includes a static gain as well
(GCO2) and first-order dynamics with the time
constant
CO2. However, we used the logarithm
of PaCO2 as input to the controller. The latter choice
is justified because the response of pial vessels to CO2 is
correlated quite linearly with pH changes in the perivascular space.
pH, in turn, depends on the logarithm of CO2 concentration
via the Henderson-Hasselbalch equation. The minus sign in the upper
branch signifies that an increase in CBF causes vasoconstriction, with
a consequent decrease in pial vessel compliance and an increase in
resistance. The dynamics of each mechanism are summarized through the
low-pass frequency response shown in Fig. 2, bottom.
Finally, the two mechanisms do not superimpose linearly on pial vessels, but their interaction is characterized by significant nonlinearities (17, 56). To account for experimental and clinical evidences, we introduced two main nonlinearities. First, the strength of CO2 reactivity in the model is not independent of the level of CBF but decreases significantly during severe ischemia (see Fig. 2, corrective factor ACO2). In fact, severe ischemia is associated with tissue acidosis, which, in turn, buffers the effect of CO2 changes on perivascular pH. A second nonlinearity considers that the overall regulatory action is not the sum of the two mechanism actions but is instead passed through a sigmoidal static relationship with upper and lower saturation levels. The sigmoid accounts for the existence of maximal limits for the vascular response, i.e., total vasodilatation and maximal vasoconstriction.
Values were given to all model parameters under normal conditions to
reproduce the intracranial hydro- and hemodynamics of a healthy
subject. A list of all model parameters and their values under basal
conditions can be found in Table 1.
Details on the main parameter values can be found in previous papers
(50, 53). A deeper comment on normal and pathological
values for the six parameters individually estimated is provided in the
DISCUSSION.
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Figure 3 shows how the model can
reproduce the regulatory mechanism actions in steady-state conditions
with the use of basal values for the autoregulation and CO2
reactivity gains. Figure 3A shows the CBF percent changes
vs. CPP during normocapnia, while Fig. 3B reports CBF
percent changes vs. PaCO2 during normotension. In both
cases, we can observe a satisfactory agreement between model results
and physiological data.
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MATERIALS AND METHODS |
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The model was used to reproduce the time pattern of ICP and VMCA in 13 neurosurgical patients during maneuvers that alter PaCO2 and mean SAP. We examined several tracings, obtained at different hours after head injury, in all patients. A total of 44 distinct tracings was thus simulated, each with a duration ranging between 12.5 and 93 min.
This study was approved by the local Ethics Committee, and informed consent of each subject's next of kin was duly obtained.
Patients
The main characteristics of the patients are summarized in Table 2. All patients had severe head injuries (Glasgow coma scale < 8, mean age 30.5) typified by closed multifocal contusions or diffuse axonal injuries within the first few days after head trauma. After the removal of large epidural or subdural hematomas, where present, patients were put into intensive care and positioned supine with the head elevated 30° above the horizontal plane. They were sedated (midazolam and fentanyl) and paralyzed (vecuronium) at the time of the measurement. SAP (radial catheter), ICP (ventricular catheter), and end-tidal CO2 pressure (PETCO2) were monitored continuously. Jugular venous saturation in O2 (SvO2) was continuously monitored via a fiber-optic catheter (Opticath, 5.5 Fr; Abbott Laboratory) inserted retrogradely up to the jugular bulb of the dominant jugular vein. Moderate hyperventilation was induced to obtain a PETCO2 between 20 and 35 mmHg. According to the routine management protocol of our establishment, if CPP was <80 mmHg and ICP was >20 mmHg and/or jugular SvO2 was
55%, norepinephrine was continuously infused after optimization
of blood volume. In such cases, CPP was stabilized between 80 and 100 mmHg to normalize ICP and jugular SvO2. When
CPP was compromised between 65 and 80 mmHg but with ICP <20 mmHg and
jugular SvO2 >55%, no therapeutic action was
initiated. Second-range therapies included bolus mannitol infusion
and/or hyperventilation under jugular SvO2 control.
None of our patients had received mannitol in the 6 h preceding
the study. Before the study, when a steady-state hemodynamic condition was achieved, all patients underwent TCD measurement (Angiodyn DMS,
2-MHz probe) of both MCAs (1) to eliminate gross
interhemispheric differences in the basal state. The probe was then
secured in a special helmet in front of the temporal windows to
continuously record VMCA. The spectral outline
of MCA Doppler time recording, SAP, ICP, and
PETCO2 signals were sampled at 200 Hz
(digital-to-analog converter, National Instruments, Houston, TX; and a
personal computer) and stored for off-line analysis. The signals were
then numerically low-pass filtered (cut-off frequency 0.1 Hz), and a
sample recorded every 4 s was stored again for comparison with
model results.
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We performed CO2 challenges at different levels of SAP. The studies were conducted as follows. After 30 min in a steady basal condition, patients were gradually hyperventilated by increasing respiratory frequency at constant tidal volume until VMCA no longer fell. During hyperventilation, jugular SvO2 was not allowed to drop below 50%. A short period of hypoventilation was then allowed by decreasing the respiratory rate. Hypoventilation was allowed up to a PETCO2 of ~40 mmHg or less if the ICP increased above 35 mmHg.
Ventilation was then adjusted to achieve the same
PETCO2 obtained at the basal state. In a
second session we kept PETCO2 constant and
modulated norepinephrine perfusion to generate variations in SAP. When
a new steady level of SAP was achieved, a second CO2
challenge was performed. Quantitative details on the maneuvers (duration, PETCO2, and mean SAP ranges)
can be found in Table 3. Changes in
PETCO2 always took a few minutes to be
completed. Hence, the rate of PETCO2
variation per unit time ranged between 2 and 5 mmHg/min. Arterial
pressure changes exhibited wider dynamics, ranging approximately
between ±4 mmHg/min for the slowest maneuvers and ±36 mmHg/min for
the fastest ones.
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Model Identification
In the model, the time pattern of ICP and VMCA following SAP and PaCO2 changes depends on the specific value ascribed to the parameters. Of course, the parameter values reported in Table 1 were assigned with reference to a hypothetical healthy individual and, hence, are not representative of such pathological conditions as those occurring in severe brain damage. Moreover, parameter values may vary widely from one patient to another.As explained in classic identification textbooks (5),
individual values to the parameters can be assigned a posteriori through a best fitting procedure. This consists of minimizing a
suitable criterion function of the difference between in vivo data (in
our case, ICP and VMCA) and the corresponding
model outputs, assuming that the model is stimulated by the same input
perturbations used in the real trials. The general process of
model identification is schematized in Fig.
4.
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Unfortunately, the model contains too many parameters for them all to
be individually estimated. Hence, when performing the best fitting
procedure, we decided to estimate only the parameters that have the
greater physiological and clinical meaning, i.e., those that summarize
intracranial elasticity, CSF circulation, and cerebrovascular
regulation. Hence, the parameters chosen are the CSF outflow resistance
(Ro), the intracranial elastance coefficient (kE), the gains of cerebral autoregulation
(Gaut) and CO2 reactivity (GCO2-), the time constant of the
CO2 response (
CO2-), and the position of the sigmoidal regulation curve. To this end, we
individually estimated a "set point" for the CO2
regulation (PaCO2 n) (see Eq. A15 in APPENDIX A). Changing the latter parameter is equivalent to shifting the normal working point along the sigmoidal regulation curve, that is, modifying the position of the upper and
lower autoregulation limits. The latter choice is justified by the
observation that the position of upper and lower limits for
cerebrovascular regulation is affected by the patient's metabolic need
and by adaptation to gas level in the blood (32).
It is worth noting that the CSF production resistance
(Rf) was not included in the previous list
because, according to our earlier experience (52), this
parameter is inversely correlated with CSF outflow resistance; hence,
it cannot be estimated independently. Moreover, the time constant of
autoregulation (
aut) was not individually estimated
because it is significantly smaller than the other time constants in
the model and, hence, has only a minimal impact on the fitting process.
Automatic estimation of the aforementioned parameters was achieved by minimizing a least-square criterion function. To assess the accuracy of parameter estimates, we also evaluated the coefficient of variation (the percent standard deviation of the estimates) by using classic statistical techniques (5). A mathematical description of the criterion function and the equations for the assessment of the coefficient of variation are reported in APPENDIX B.
When the minimization procedure was performed, the initial values of model ICP and VMCA were assigned on the basis of clinical data. In particular, the scaling factor kV in Eq. 1 was computed separately in each trial to ensure that model VMCA at the beginning of the simulation is the same as the first velocity data point.
The entire software program for mathematical computations was written using the scientific programming language Fortran 77 (AbSoft Fortran 77 for Windows) running on a Pentium-based personal computer. A user-friendly interface was also written in Visual Basic (Microsoft Visual Basic 5.0); the latter program permits handling of the parameter estimation procedure, visualization of parameter values, and graphic plotting in a simple and straightforward way.
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RESULTS |
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The parameter values estimated in 44 tracings on 13 patients are
reported in Table 4,
together with the coefficient of variation of the estimates. To
facilitate the analysis, all parameters in Table 4 (with the exception
of PaCO2 n) were normalized to the values in Table 1. The accuracy of the best fitting is summarized in
two other columns of Table 4, in which the SD of the residuals between
the predicted and the actual measurements (
ICP and
VMCA) are presented. A column scatter graph
showing these values in all 44 tracings is presented in Fig.
5. Figure 5 reveals that, in most cases,
the SD of the residuals is of the same order as measurement accuracy
(~1-2 mmHg for ICP, 4-5 cm/s for Doppler velocity),
indicating that the model's reproduction of clinical tracings is
satisfactory. As for ICP, only four tracings exhibit residuals >3
mmHg, denoting insufficient fitting, but three of these tracings refer
to the same patient (patient Der). Velocity exhibits nine
tracings in which residuals are >6 cm/s, suggesting poor fitting.
However, six of these are concerned with only two patients
(patients All and Gra). These patients had high
basal values of velocity (>100 cm/s), which might suggest the
existence of either vasospasm or hyperemia and could explain the poor
velocity fitting.
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By looking at Table 4, one can observe that the estimated values of the
parameter PaCO2 n are significantly
lower than normal. This result may reflect adaptation to the low level
of PETCO2 set before the maneuvers. To
check this aspect, Fig. 6 shows the correlation between the basal PETCO2 and
the parameter PaCO2 n for all 44 tracings. The correlation is high, apart from a few diverging points.
However, three of these points refer to tracings (tracings
Der4, Pra5, and Tho5) in which
PCO2 was maintained almost constant (see Table
3); hence, estimation of CO2 reactivity parameters is
scarcely significant. If these three points are excluded, correlation is as high as r2 = 0.86.
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Examples of the results of the best fitting procedure, obtained in two
different patients, are shown in Figs. 7
and 8. The time pattern of the input quantities (i.e., mean SAP and
PaCO2) are shown in Figs. 7 and 8,
top, while a comparison between model outputs (ICP and
VMCA) and the corresponding clinical data is shown in Figs. 7 and 8, bottom.
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Figure 7 shows four different tracings obtained on the same day from patient Bab. The first two tracings contain only a CO2 challenge, while arterial pressure was maintained constant. The third and fourth tracings comprise both SAP and PaCO2 maneuvers. It is clear from the first two tracings that a mild increase in CO2 pressure, performed by starting from a baseline hypocapnic level (~20 mmHg), gave rise to a velocity increase and a significant ICP rise. Finally, even a modest decrease in PaCO2 was able to interrupt the uncontrolled increase in ICP, leading to rapid restoration of the baseline levels. The model is able to explain this chain of events with reliable values of its parameters (Table 4), ascribing the ICP time pattern to changes in cerebral blood volume.
The third and fourth tracings in Fig. 7 differ from the previous tracings because they comprise a hypertensive maneuver as well, and the CO2 challenge occurs at a higher level of CPP. In the third tracing, a rapid increase in mean SAP causes a change in VMCA, so the patient is classified as having weak autoregulation (normalized Gaut = 0.65, see Table 4). An index commonly used to estimate autoregulation in the clinical literature is the static autoregulation index (sARI) (30, 48), defined as the percentage changes in cerebrovascular resistance (CVR) divided by the percent changes in CPP. Values of sARI in normal subjects are >0.85 (see DISCUSSION). However, if this index is computed from the two arterial pressure maneuvers in the third clinical tracings of Fig. 5, one obtains the values sARI = 0.47 and sARI = 0.39, respectively, in accordance with model estimation of weak autoregulation.
By contrast, the increase of mean SAP in the fourth tracing in Fig. 7 does not evoke a clear velocity change, whereas it causes a moderate decrease in ICP, ascribed to active blood volume reduction. Hence, in this case the estimation procedure provides a high value for the normalized Gaut (1.50, see Table 4). Accordingly, sARI computed from the clinical tracing is as high as 0.89, which belongs within the range of normality.
An important aspect arising from the tracings in Fig. 7 is that the
percent changes in TCD velocity per mmHg of CO2 change (
VMCA%/
PaCO2, which is
the index commonly used in the clinical literature to estimate
CO2 reactivity) depend strongly on the level of CPP. Hence,
this empirical index is not representative of the "true"
CO2 reactivity. By contrast, the gain
GCO2 in the model is quite independent of the
CPP level. In fact, if the empirical index is computed in the first
tracing, one obtains the value 1.84%/mmHg, while CPP decreases from 57 to 37 mmHg, i.e., below the autoregulation range. In the second tracing
in Fig. 7, the same index is as low as +0.28%/mmHg, suggesting
exhausted cerebrovascular regulation reserve, while CPP decreased
during the maneuver to 34 mmHg. However, in the third tracing, the
index is as high as +3.97%/mmHg. In fact, mean SAP was increased
before the maneuver, and CPP always remained >66 mmHg, i.e., the
overall maneuver was performed within the autoregulation range.
Finally, in the fourth tracing in Fig. 7, we computed a value of
VMCA%/
PaCO2 = 2.94%/mmHg, while CPP decreased from 84 to 61 mmHg. We can thus
conclude that, when the CO2 challenge is performed at a
higher CPP level (in the range of 60-80 mmHg), the
VMCA change per mmHg of PaCO2
change is significantly higher compared with the case of low CPP
(35-60 mmHg). Hence, the compensatory response to CO2
is depressed by hypotension. It is worth noting that the model can
explain this behavior, ascribing it to a nonlinear interaction between
CO2 reactivity and autoregulation (Fig. 2).
In some patients we performed several simultaneous CO2 and
SAP challenges in a single tracing, and we monitored the consequent ICP
and VMCA changes over a long time period
(40-90 min). Two tracings of this kind, monitored in patient
Dug, are shown in Fig. 8. It is
worth noting the capacity of the model to reproduce the consequences of
several consecutive maneuvers at different SAP and
PaCO2 levels using just a single set of parameters.
The dependence of the CO2 reactivity index
VMCA%/
PaCO2 on CPP is evident by comparing the results of the two consecutive CO2
challenges in the first tracing in Fig. 8. In the first maneuver,
performed at a CPP always >72 mmHg (i.e., within the autoregulation
range), we computed a
VMCA%/
PaCO2 = 4.11%/mmHg, which is quite normal. By contrast, the second
CO2 maneuver provides an index
VMCA%/
PaCO2 as
low as 2.13%/mmHg, while CPP decreased from 57.4 to 50 mmHg, approaching the autoregulation lower limit. Accordingly, the model estimates that
GCO2- is
quite normal in this patient (normalized value 0.93) and ascribes the
apparent reduction in CO2 reactivity of the second maneuver
to the attainment of maximal vasodilation at low CPP. In tracing
Dug2, the CO2 maneuver is performed at normal CPP
(always >74 mmHg), the index
VMCA%/
PaCO2 = 4.55%/mmHg, and estimated CO2 reactivity in the model is
rather normal.
Finally, the sARI computed during the arterial pressure maneuvers in the two tracings of Fig. 8 is 0.42 and 0.69, respectively, denoting a weak autoregulation. Accordingly, the model provides a Gaut lower than normal (normalized values close to 0.6).
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DISCUSSION |
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The present work was designed with two main purposes: to find a reliable quantitative description of the main hemodynamic changes occurring in head-injured patients during SAP and CO2 pressure challenges, and to verify the possibility of characterizing intracranial dynamics in each patient through a limited number of parameters. Before the significance and limitations of the obtained results are examined, it is important to provide a critical analysis of both the measurement technique employed and the quality of clinical data available.
A first important problem concerns the use of the TCD technique to achieve dynamic information on CBF percent changes. This choice is acceptable because previous studies in head-injured patients demonstrated the existence of a good proportionality, with a slope close to 1, between percent variations in TCD velocity and CBF estimated with the Fick principle. However, in patients with elevated VMCA >100 cm/s, the slope was not strictly close to 1, and TCD variations underestimated CBF variations (47). This result is probably due to the fact that hyperemia or vasospasm is associated with modifications in velocity profile and/or interhemispheric heterogeneity of vascular reactivity. This problem may have caused some of the high differences (>6-7 cm/s) between model predictions and clinical data observed in a few patients (especially patients Gra and All) who had elevated blood flow velocity.
A second problem is that we measured end-tidal CO2 during the clinical trials, whereas arterial CO2 is the true stimulus for cerebrovascular regulation. We are aware that in some head-injured patients the gradient between end-tidal and arterial PCO2 can be increased as a consequence of neurogenic pulmonary edema, which increases ventilation perfusion inhomogeneity. However, none of the patients analyzed exhibited symptoms of pulmonary edema or had pulmonary disease.
A final problem is that SAP was modified with norepinephrine. In the present simulations we assumed that infusion of norepinephrine does not modify the diameter of the insonated vessel or change cerebrovascular reactivity. This assertion is speculative, but it is supported by data from Olesen (33). This author observed that intracarotid application of adrenalin or norepinephrine does not affect regional cerebral blood flow in humans, and these drugs do not alter the diameter of large intracranial arteries, as shown by angiography. This result can explain why similar VMCA variations were observed during autoregulatory tests realized by means of pharmacological infusion of norepinephrine and by means of nonpharmacological methods (44, 48).
In most tracings examined, the model was able to show rather good reproduction of the observed ICP and VMCA time patterns, with a standard error of the residuals of the same order as measurement errors. Moreover, the capacity of the present model to reproduce ICP and velocity tracings is equivalent to that of the previous, more complex model (25) but with much shorter computation time. However, it is important to emphasize that reproduction of real tracings was not achieved a priori, based on knowledge of a patient's condition, but only a posteriori through a best fitting procedure. In other words, the present simplified model needs to be used together with a fitting procedure to assign some parameter values.
Among the estimated parameters, two are of particular interest because they contribute to intracranial hypertension and clinical status in patients with severe head injury; i.e., intracranial elastance and CSF outflow resistance. Clinicians are strongly interested in a quantitative knowledge of these parameters because their alterations require therapeutic intervention before the occurrence of secondary brain damage. However, the methods presently available to estimate these parameters often provide contradictory results (24, 27) and are not without criticism (15, 16). As a consequence, no gold standard method to derive these parameters is yet at hand.
Furthermore, knowledge of the autoregulation and CO2 reactivity gains could also be important in clinical practice to choose the correct management and to distinguish between the patients who need sustained elevated CPP and those who need hyperventilation (9, 39). Estimation of the strength of regulation mechanisms, however, as obtained by classic indexes (such as CBF percent change or resistance changes per mmHg), is misleading because, as in the example of Figs. 7 and 8, this may depend on CPP, ICP changes, and the working point along the regulation characteristic.
In the present study, parameters have been estimated by using a classic identification schema (Fig. 4), i.e., by minimizing the square of residuals between clinical data and model results. A crucial problem is whether the obtained parameter values have a reliable meaning or are merely the result of curve fitting. This problem will be critically examined below for each estimated parameter.
CSF Outflow Resistance
The values of the CSF outflow resistance (Ro) reported in the clinical literature differ significantly among authors, depending on the specific technique used to derive this parameter. In general, the bolus injection technique provides values of Ro three or four times smaller than the values obtained with the steady-state methods (42, 45). Basically, values of Ro evaluated with steady-state techniques in subjects with normal CSF circulation range between 4.4 and 15.6 mmHg min/ml (normalized values 0.5-1.78, with reference to our basal value reported in Table 1) (12, 29). These values are supported by those measured by Cutler et al. (10) based on isotope clearance in 11 patients (Ro value of 9.7 ± 1.0).Values of Ro, measured in patients with
severe head injury with stationary withdrawal (24)
show a significant increase compared with normal values, ranging
between 21 and 85 mmHg · ml
1 · min
(normalized range 2.4-9.7). By contrast, lower values of Ro were obtained by Marmarou et al.
(27) in the severe head injury with the bolus
technique (3.15-36.62
mmHg · ml
1 · min; normalized range
0.4-4.17).
The values of Ro estimated in the present study show a good agreement with those measured by Kosteljanetz (24). In 12 of 13 patients, Ro appears significantly increased compared with the basal value (normalized range 2.0-9.0). This result supports the idea that impairment in CSF outflow may represent a major contribution to intracranial hypertension in severe head injury.
Intracranial Elastance Coefficient
Normal values for kE, computed by means of bolus tests, lie in the range 0.076-0.108 ml
1
(normalized range 1.0-1.4) (43). Values >0.128
ml
1 (normalized value 1.66) are generally considered
abnormal and indicative of a decreased intracranial storage capacity.
Values in patients with severe head injury were measured by
Kosteljanetz (24). kE ranged
between 0.08 and 0.76 ml
1 (normalized range
1.05-10.0). Values of kE during bolus tests were estimated by Ursino et al. (55) in head-injured
patients by using a simple model of intracranial dynamics. The values
(normalized at the basal value shown in Table 1) extended between 0.63 and 3.22.
Normalized values of kE estimated in this study range between 0.7 and 2.9, i.e., we observed both patients with normal elastance and patients with a significant increase in kE, suggesting a reduction in the intracranial storage capacity. On average, the elastance coefficient is increased in head injury compared with normality. This result agrees with the observation by Kosteljanetz (24) and our previous observation with the bolus technique (55).
Autoregulation Gain
In normal subjects, sARI, which is defined as the percentage change in CVR per percent change in CPP (i.e., sARI =
CVR%/
CPP%), ranges between 0.85 and 0.95 (30, 44,
48). In our model, similar values of sARI can be obtained during
normocapnia by changing Gaut (normalized) between 1.0 and
5.0. This can be assumed to be a normal autoregulation range in our model.
Autoregulation is known to be significantly impaired by many cerebral diseases, including head injury. Results from Bouma and Muizelaar (6) and Matta et al. (30) suggest that patients with head injury can be discriminated in three classes: normal autoregulation (sARI > 0.85), moderately impaired autoregulation (0.85 > sARI > 0.5), and severely impaired autoregulation (0.5 > sARI). According to our simulations, the values of Gaut parameters (normalized) that discriminate between these three classes are Gaut > 1.0 (normal), 0.5 < Gaut < 1 (moderately impaired), and Gaut < 0.5 (strongly impaired).
The values of Gaut estimated in this study range between 0.2 (impaired autoregulation) and 1.5 (normal autoregulation) in accordance with the literature.
CO2 Reactivity
An index for CO2 reactivity frequently used in the clinical literature is the percent CBF change (or percentage velocity change) per PaCO2 change (in mmHg). Normal values (22, 34) are approximately as great as 4-4.5 CBF%/mmHg with an SD as great as ±1 CBF%/mmHg. This range of normal CO2 reactivity is covered in the model when GCO2 is varied between 1.0 and 4.0.CO2 reactivity may decrease below 2 CBF%/mmHg in some patients with severe head injury, although in other patients CO2 reactivity is preserved despite the trauma. Data in Tenjin et al. (46) suggest that the value 2 CBF%/mmHg may be considered as a threshold to discriminate between patients with preserved and impaired CO2 reactivity. According to our simulations, this threshold corresponds to a value of normalized GCO2 approximately as low as 0.66.
In this work we estimated values of normalized GCO2 ranging between 0.21 and 2.32 (average 0.85). This range agrees with values reported in the clinical literature and confirms the existence of patients with preserved CO2 reactivity as well as patients with almost absent CO2 response.
The parameter
CO2, estimated in the present
trials, ranges between normal and threefold normal values (i.e., 2 min). These high values might depend on the impairment of some feedback
mechanisms in head-injured patients or may reflect a high time delay
between changes in PETCO2 and changes in
pH at cerebral arterioles.
Finally, the set point PaCO2 n is significantly lower than normal in most patients. We think that these low values should not be ascribed to the traumatic pathology but, rather, that they represent adaptation to the low PCO2 level set in the patients before the maneuvers, probably due to a reset of CSF pH to normal values after prolonged hyperventilation (41). This assumption is strongly supported by the high correlation between PaCO2 n and PETCO2 shown in Fig. 6.
The previous analysis confirms that parameter estimates belong within the range reported in the clinical literature for patients with severe head injury. This observation suggests the possibility that the present method may be of clinical value to assess pathological alterations in intracranial dynamics. An important problem, however, concerns the accuracy by which parameters can be estimated from the clinical tracings and the uniqueness of the obtained solutions. Contradictory conclusions can be drawn on the basis of this major point. In favor of the accuracy of the estimates, we can observe that, in several cases, parameter estimation achieved in the same patient using tracings recorded a few hours apart furnished very similar results (compare tracings All4 vs. All5, Bab1 vs. Bab2, Dbr1 vs. Dbr2, Dug1 vs. Dug2, Hel1 vs. Hel2 and Hel3, and Tho1 vs. Tho2). The repeatability of parameter estimates is especially satisfactory when the clinical maneuvers comprise both CO2 and SAP challenges performed at different times in the same tracing (as in Fig. 8). In a few other cases, however, parameters exhibited large fluctuations when passing from one trial to the next. This intrinsic variability might depend on an effective change in a patient's intracranial status but might also be caused by insufficient accuracy in parameter estimates. At present we have not enough elements to discriminate between these two possibilities, but a few arguments can be developed.
First, we can observe that, in most cases, the variability in the
estimates of the CSF outflow resistance within the same patient is
small enough to permit discrimination among patients with normal CSF
outflow (normalized Ro
1-2), moderate
elevation in Ro (~2.5-5), or large
elevation in Ro (~6-9). Similarly, values of kE maintain certain coherence within the same
patient. In most cases, estimation of kE remains
rather constant or decreases during the treatment, suggesting an
improvement in intracranial elasticity. The remarkable case of
patient Rvm will be examined below.
By contrast, the autoregulation gain, as well as the gain and time constant of CO2 reactivity, exhibits a wide dispersion, which is evident not only from one patient to the next but also within the same patient. We think that the reason for this variability may be the significant correlation existing among these parameters, which often precludes the possibility of their accurate unequivocal estimation. To attenuate this problem, it would be important to use tracings in which SAP and CO2 challenges are performed separately (such as tracings in Fig. 8); multiple information, in fact, may provide more accurate estimation of correlated parameters.
A final important issue is that the estimated parameter values should have some relationship with the patient's clinical status to effectively support the clinical practice. Naturally, clinical validation of the model is well beyond the limits of the present work and should be the subjects of future, more clinically oriented studies. However, patient Rvm represents an exemplary case. This patient developed a secondary compressive chronic subdural hematoma during the recovery. Surgical evacuation was performed at the 16th day after trauma (i.e., between tracings Rvm2 and Rvm3). It is interesting to observe that the estimation procedure predicts an increase in Ro and a strong increase in kE (indicating reduced intracranial elasticity) at days 15 and 16 before the surgical procedure. However, these parameters returned rather close to normal after surgical evacuation (Table 4).
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APPENDIX A |
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All model equations are presented in APPENDIX A.
Parameters used in the estimation algorithm are as follows:
Ro in Eq. A3,
kE in Eq. A4, Gaut in
Eq. A14, and GCO2,
CO2, and
PaCO2 n in Eq. A15.
Intracranial Hemodynamics and Hydrodynamics
Model equations for intracranial dynamics are written by imposing the mass preservation at all nodes in Fig. 1.Application of mass preservation at the intracranial storage capacity,
Cic, leads to the following differential equation
|
(A1) |
|
(A2) |
|
(A3) |
Intracranial storage capacity is inversely proportional to ICP, which
implies a monoexponential pressure-volume relationship. Hence
|
(A4) |
Mass preservation at node pa gives us the following
differential equation
|
(A5) |
An expression for capillary pressure can be obtained by writing mass
preservation at node c. We have
|
(A6) |
Furthermore, by imposing mass preservation at the cerebral veins
(node vi), a further differential equation is obtained
|
(A7) |
|
(A8) |
To complete the model, we need expressions for blood volume changes in the pial arterial-arteriolar and venous vascular beds (i.e., dVvi/dt and dVpa/dt) to be used in Eqs. A1, A5, and A7.
Because cerebral veins behave passively, their blood volume variations
can only be ascribed to changes in transmural pressure. Hence
|
(A9) |
|
(A9′) |
By contrast, blood volume changes in the arterial-arteriolar pial
vascular bed can be ascribed to both transmural pressure changes and
the action of cerebrovascular control mechanisms. In particular, as
shown in the block diagram of Fig. 2, control mechanisms dynamically
modify the compliance Cpa. Hence
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