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Department of Electronics, Computer Science and Systems, University of Bologna, I-40136 Bologna, Italy
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ABSTRACT |
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Top
Abstract
Introduction
Results
Discussion
Appendix
References
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The relationships among cerebral blood flow, cerebral blood volume, intracranial pressure (ICP), and the action of cerebrovascular regulatory mechanisms (autoregulation and CO2 reactivity) were investigated by means of a mathematical model. The model incorporates the cerebrospinal fluid (CSF) circulation, the intracranial pressure-volume relationship, and cerebral hemodynamics. The latter is based on the following main assumptions: the middle cerebral arteries behave passively following transmural pressure changes; the pial arterial circulation includes two segments (large and small pial arteries) subject to different autoregulation mechanisms; and the venous cerebrovascular bed behaves as a Starling resistor. A new aspect of the model exists in the description of CO2 reactivity in the pial arterial circulation and in the analysis of its nonlinear interaction with autoregulation. Simulation results, obtained at constant ICP using various combinations of mean arterial pressure and CO2 pressure, substantially support data on cerebral blood flow and velocity reported in the physiological literature concerning both the separate effects of CO2 and autoregulation and their nonlinear interaction. Simulations performed in dynamic conditions with varying ICP underline the existence of a significant correlation between ICP dynamics and cerebral hemodynamics in response to CO2 changes. This correlation may significantly increase in pathological subjects with poor intracranial compliance and reduced CSF outflow. In perspective, the model can be used to study ICP and blood velocity time patterns in neurosurgical patients in order to gain a deeper insight into the pathophysiological mechanisms leading to intracranial hypertension and secondary brain damage.
intracranial hemodynamics; cerebral blood flow; cerebral blood volume; cerebrovascular control mechanisms
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INTRODUCTION |
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Top
Abstract
Introduction
Results
Discussion
Appendix
References
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THE CEREBROVASCULAR BED in humans and animals is subjected to the action of sophisticated regulatory mechanisms that work to maintain an adequate cerebral blood flow (CBF) for functional and metabolic needs. Intracranial vessels respond promptly to changes in cerebral perfusion pressure (CPP; cerebral arterial pressure minus cerebral venous pressure); the response, called autoregulation, ensures that CBF remains approximately constant in the CPP range of 50-150 mmHg (38). Moreover, cerebral vasculature is particularly sensitive to chemical stimuli, especially O2 and CO2 tension in arterial blood. These mechanisms interact in complex nonlinear ways in response to perturbations affecting pressure and blood concentration levels simultaneously.
There are many factors that make the analysis of cerebrovascular regulation particularly complicated. First, consecutive vascular segments in the brain contribute differently to CBF control, a phenomenon often referred to as "segmental heterogeneity" (6). Some authors have observed that the large pial vessels play a major role in the regulatory response to moderate arterial pressure changes, whereas small pial arteries undergo massive vasodilation when CPP approaches the lower autoregulation limit (25, 29).
In addition, cerebral circulation takes place within a closed space (the skull and the craniospinal axis) with only a limited capacity to buffer blood volume changes; consequently, any modification in vessel caliber (either passive or induced by cerebrovascular regulatory mechanisms) may affect intracranial pressure (ICP) through changes in cerebral blood volume (CBV). Because ICP is the extravascular pressure of cerebral vessels and is almost equal to pressure in the large cerebral veins, its changes can have a significant impact on cerebral hemodynamics.
Because of the close relationship among the action of cerebrovascular control mechanisms, CBV, and ICP, CBF can be affected by several noncirculatory aspects of intracranial dynamics, especially craniospinal elasticity (defined by the intracranial pressure-volume relationship) and the circulation of cerebrospinal fluid (CSF).
The interaction among ICP, CBV, and cerebrovascular regulation must be taken into account in the treatment of patients with severe brain lesions. Changes in arterial CO2 pressure (PaCO2) and mean systemic arterial pressure (SAP) are frequently used in neurosurgical intensive care units to assess the status of cerebrovascular control mechanisms (8, 48). Moreover, manipulation of arterial CO2 concentration is routinely used as a therapeutic means in controlling ICP and adjusting CBF to metabolic requirements (34, 37). However, uncontrolled changes in CO2 or in arterial pressure can have a dramatic impact on ICP and CBF, leading to acute intracranial hypertension and cerebral ischemia (12).
The complexity of the aforementioned relationships makes it difficult to achieve a clear understanding of cerebrovascular control in simple qualitative terms. For this reason, various mathematical models of cerebral hemodynamics and ICP dynamics have been presented in past years (7, 21, 22, 31, 47, 49). We recently developed a mathematical model of the interaction among cerebral autoregulation, CBV changes, and ICP in humans (50, 51). The model was able to reproduce various phenomena concerning ICP, such as the genesis of plateau waves (51), the ICP response to arterial pressure perturbations (14), or the ICP changes induced by bolus injection within the craniospinal space (PVI tests) (52).
The aim of this study was to significantly improve and extend the previous model to include the CO2 reactivity of cerebral vessels, its nonlinear interaction with ICP and with cerebral autoregulation, and the description of the transcranial Doppler (TCD) velocity signal. The TCD technique, in fact, is routinely used today to assess cerebral hemodynamics due to its characteristics of noninvasiveness and continuous data acquisition (1).
There are two main justifications for developing this new model. We aimed to 1) demonstrate that several experimental results on cerebral autoregulation and chemical CBF regulation presented in past years can be summarized into a single theoretical setting, and 2) improve the interpretation of clinical maneuvers concerning cerebrovascular control and ICP dynamics.
The present preliminary study presents the model, its justification on the basis of current physiological knowledge, and its validation through experimental results taken from physiological literature. A second, related study (27) presents a sensitivity analysis of model parameters and validation of the model on the basis of clinical data from real patients with severe head trauma.
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QUALITATIVE MODEL DESCRIPTION |
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The model is qualitatively presented, with attention focused on its new aspects. A more thorough analysis can be found in previous reports (50, 51). The complete set of mathematical equations is given in the APPENDIX.
Intracranial Hemodynamics and CSF Dynamics
The main biomechanical law at the basis of intracranial dynamics is the constancy of the overall volume contained in the craniospinal space. This principle, known as the Monro-Kellie doctrine, implies that any volume variation in an intracranial compartment causes a compression or dislocation of the remaining volumes, accompanied by an alteration in ICP. This phenomenon is mathematically reproduced by means of intracranial compliance (Cic), which represents the capacity of the craniospinal system to store a volume load. We assumed that this quantity is inversely proportional to intracranial pressure (Pic) through a constant parameter called the elastance coefficient (kE) (5, 31)
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The electric analog in Fig. 1 represents the four intracranial compartments considered in the model and their mutual relationships, together with the extracranial venous drainage pathway. These are briefly described in the following sections.
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Pial arteries and arterioles. Pial arteries and arterioles comprise the sections of the cerebrovascular bed directly under the control of regulatory mechanisms. We make a distinction between two consecutive segments, the large pial arteries (proximal segment) and the medium and small pial arteries (distal segment). In the following equation, subscript j = 1 indicates a quantity or a parameter that belongs to the proximal arteries, whereas subscript j = 2 refers to the distal arterioles. Each segment is represented by a transverse hydraulic capacity (Cj), at which all blood volume changes are concentrated, and a longitudinal hydraulic conductance (Gj), which reproduces the pressure drop upstream and downstream of the capacity. The distinction between two segments is justified by the heterogeneity of cerebrovascular regulation (6).
Conductance and blood volume changes are calculated by simulating each segment as the parallel arrangement of several vessels with equal radius. Blood volumes are proportional to the second power of the inner radius, whereas conductances are proportional to the fourth power of the inner radius, according to the Hagen-Poiseuille law (33). This representation does not aim at reproducing the actual anatomic configuration of the cerebrovascular bed; rather, it aims only to describe how a certain variation of vessel caliber, induced by cerebral autoregulation and/or CO2 reactivity, can influence vascular resistance and blood volume in a given segment. A value for the inner radius (rj) is computed starting from the equilibrium of forces acting on the wall of the vessel, supposedly cylindrical (Laplace law)
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Cerebral venous circulation. The circulatory pathway from cerebral capillaries down to the dural sinuses also consists of a series arrangement of venous capacity and upstream and downstream longitudinal conductances. Because autoregulation plays a negligible role in cerebral veins (4), we adopted a simpler description for venous mechanics than for the pial arterial-arteriolar vasculature. The venous compliance (Cvi) is inversely dependent on the local transmural pressure, implying a monoexponential pressure-volume relationship. The terminal portion of the intracranial venous bed (bridge veins and lateral lacunae or lakes, represented by the conductance Gvs in the model) passively collapses or narrows during intracranial hypertension, with a mechanism similar to that of a Starling resistor (4, 33). Because of this mechanism, cerebral venous pressure (Pv) is always slightly higher than ICP, and upstream large cerebral veins and venules remain open during intracranial hypertension. Consequently, conductance of the proximal venous circulation (Gpv) is constant. The remaining part of the vascular bed, from dural sinuses down to the heart, is summarized by the hydraulic conductance (Gve) and compliance (Cve) of the extracranial venous pathways.
CSF compartment. We have assumed that the processes of both CSF production at the cerebral capillaries and CSF outflow at the dural sinuses are passive and unidirectional. Hence, they are proportional to the corresponding transmural pressure value and fall to zero when this becomes negative.
All the parameters that describe the venous and CSF compartments have been given the same basal values used in previous studies, for which thorough discussions of these quantities can be found (50, 51).Middle cerebral artery blood flow velocity.
Additional equations in the model have been included to estimate an
approximate value for blood flow velocity in the middle cerebral artery
(MCA) and to compare it with experimental data as recorded with the TCD
technique. On the basis of studies conducted on healthy subjects and
neurosurgical patients (13, 36), we assumed that the behavior of the
MCA is essentially passive in nature; the transmural pressure (SAP 
ICP) is a monoexponential function of the inner radius (Fig.
3). The parameters of this relationship
have been set to reproduce the pressure-radius curves concerning human
intracranial basal arteries (20). Finally, the MCA velocity
(VMCA) is
calculated as the ratio of blood flow to vessel cross-sectional
area. To this end, we assumed that about one-third of
total CBF passes through each MCA. Based on these assumptions, the
basal value of
VMCA in the model
is as high as 60 cm/s, which agrees with the values reported by Aaslid
et al. (1) (normal range 33-90 cm/s, mean 62 cm/s). Furthermore,
during maneuvers that affect SAP or
CO2 pressure, the
VMCA changes are in agreement with those measured with the TCD technique (see
RESULTS). This aspect has been more
deeply analyzed in a second, related paper (27).
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Cerebrovascular Regulation Mechanisms
Cerebrovascular regulation mechanisms act on smooth muscle tension of large pial arteries and arterioles. To reproduce their action, we assumed that the amplitude of the "active tension-inner radius" relationship (Tmax,j) depends on an activation factor (Mj) that represents the degree of smooth muscle contraction in the given segment. Hence
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The block diagram in Fig. 4 shows how
feedback mechanisms modulate the activation factor. The upper branch
refers to cerebrovascular autoregulation mechanisms, whereas the lower
branch refers to CO2
reactivity. The contributions of autoregulation and
CO2 reactivity to smooth muscle
tension in the jth segment are
described by two state variables,
xaut,j
and
xCO2,j, respectively. The dynamics of each is reproduced by means of a first-order low-pass filter, characterized by a gain factor
(Gaut,j or
GCO2,j)
and a time constant
(
aut,j
or
CO2,j).
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As shown in Fig. 4, we have assumed that autoregulation acts on large pial arteries and small arterioles by means of two distinct feedback mechanisms: the mechanism on large arteries depends on changes in CPP, reflecting a myogenic or neurogenic mechanism; and the mechanism on small arteries is triggered by alterations in blood flow to cerebral tissue, according to a metabolic or endothelium-dependent response. This distinction aims at reproducing the heterogeneity of autoregulation observed by Kontos et al. (25) and MacKenzie et al. (29) in cats.
In contrast, CO2 is assumed to affect smooth muscle tension, in both large and small pial arteries, through a similar mechanism. Because the prevalent hypothesis ascribes the vasoactive effect of CO2 to pH changes in the perivascular space (24, 34), we have assumed that model response depends on the logarithm of CO2 changes rather than on PaCO2 by itself. Hence, the following input quantity is used in the block diagram in Fig. 4
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Furthermore, we have included two kinds of nonlinear interaction between autoregulation and CO2 reactivity. First, on the basis of several experimental studies (8, 18), we have assumed that the strength of the CO2 mechanism is not independent on CBF but, rather, is attenuated during severe brain ischemia. Ischemia, in fact, causes tissue acidosis, which has a buffer effect on the influence of CO2 on perivascular pH. Accordingly, the CO2 reactivity in the model decreases exponentially when CBF falls below normal (see inset in Fig. 4).
Second, Fig. 4 shows that autoregulation and CO2 reactivity are not simply summed to obtain the activation factor (Mj); rather, they interact through a sigmoidal relationship, which accounts for the existence of lower and upper limits for cerebrovascular reactivity.
A value characterizing the parameters in the sigmoidal relationship has been set so that smooth muscle tension falls to zero during maximal vasodilation and can rise to twice the basal value during maximum vasoconstriction.
The autoregulation time constants were set at the same values used in
previous studies (50, 51). In particular, we assumed that the
pressure-dependent mechanisms acting on large pial arteries exhibit
rapid dynamics (aut,1 = 10 s),
whereas the flow-dependent response of pial arterioles is slightly
slower (aut,2 = 20 s).
The choice of time constants for
CO2 reactivity deserves further
comment. Kontos et al. (24) found that arteriolar dilation in response
to a pH decrease takes place rapidly, with a time constant of <10 s.
Schneider et al. (43), during experiments on feline pial arteries with
the use of a microelectrode, observed that pH and vascular diameter
varied with a delay of ~30 and 45 s, respectively, after a
CO2 change; Severinghaus and
Lassen (45) reported a time constant of 20 s for
CO2 reactivity in humans, whereas
Auer (3) in his study on cats measured a delay between 7 and 48 s. It
is worth noting that, in the last three cases, the measured quantity
was the end-tidal CO2 pressure;
the observed delays therefore also included the transit time for blood
to pass from the lung to the brain. Because in our model the input
quantity is the arterial CO2
pressure, the time constant includes only the time necessary for
PaCO2 to alter perivascular pH and that required for the vessel to react to a pH change. Hence, we have chosen
a value of
CO2,j = 20 s for both large and small pial arteries. In the analysis of
neurosurgical patients reported in our related paper (27), we have used
a greater value for the parameter
CO2,j,
because the monitored quantity is end-tidal rather than arterial
CO2 pressure.
Values for the gain factors of regulation mechanisms have been set to reproduce results obtained in animals when experiments are executed at constant ICP (see RESULTS).
A list of all model parameters and their values at basal conditions can be found in Table 1.
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RESULTS |
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Top
Abstract
Introduction
Results
Discussion
Appendix
References
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Model simulations have been performed to analyze
1) cerebral autoregulation,
2)
CO2 reactivity,
3) the interactions between these
two mechanisms, and 4) the effect of
acute changes in CO2 on ICP. With
regard to the first three topics, only results obtained under
steady-state conditions are presented and compared with experimental
data. Moreover, to simulate experiments performed with the dura opened
or at constant ICP, during these trials we used an extremely low value
for the intracranial elastance coefficient. Hence, ICP changes were
negligible throughout the simulations (ICP 
9.5 mmHg). In contrast,
the last trials concern dynamic conditions, with ICP free to change as
a consequence of blood volume and CSF volume alterations.
Cerebral Autoregulation
Figure 5 shows the percent changes in inner radii in large and small pial arteries, percent change in CBF, and percent change in VMCA evaluated under steady-state conditions at different values of mean SAP. A value was applied to the parameters that describes the autoregulation gain in both segments (i.e., Gaut,j in Fig. 4 and Eqs. 23A and 24A in APPENDIX) to reproduce the inner radius percent changes measured by Kontos et al. (25) and MacKenzie et al. (29) on animals; the correspondence is quite good. The results show that the importance of each pial segment depends on the arterial pressure level. During moderate SAP alterations, the changes in vessel caliber are more pronounced in the large pial arteries than in the arterioles. When arterial pressure is further decreased and approaches the autoregulation lower limit, dilation of small pial arteries becomes progressively greater, causing an enlargement of the cerebrovascular volume, whereas the response of the larger vessels is rapidly exhausted. At very low or very high values of mean SAP, the inner radii behave in a completely passive way. The simulated pattern of CBF and VMCA versus mean SAP also agrees with that of experimental data (19, 29, 35). Autoregulation is able to maintain a fairly constant CBF in the range of 50-150 mmHg (38); VMCA does not decrease because of the passive variation in cross-sectional area (see Eq. 21A in APPENDIX). This result agrees with Nelson et al. (35), who found that velocity is always higher than CBF. Beyond the lower limit of autoregulation, CBF and VMCA start to decrease, despite the evident vasodilation of the small pial arteries.
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CO2 Reactivity
Figure 6 shows the dependence of inner radii, CBF, and VMCA on PaCO2 under steady-state conditions. The value indicating normocapnia has been fixed at PaCO2 = 40 mmHg. Also in this case, a value for the gain factors describing CO2 reactivity (i.e., GCO2,j in Fig. 4 and Eq. 25A in APPENDIX) was set to reproduce the inner radius percent changes observed during experiments on animals. The comparison between simulation results and the corresponding experimental and clinical data proves to be suitable (18, 26, 30, 40, 41, 44, 53). Also in this condition, as during autoregulation, the amount of vessel variation is more pronounced in the small pial arteries than in the large pial arteries. This finding agrees with the observation made by Wei et al. (53), who demonstrated that the CO2 reactivity in the arterioles of cats is dependent on vessel size. Moreover, the vasoconstrictive effect of hypocapnia reduces CBF to one-half its normal value. During hypercapnia, vasodilation can even double CBF. VMCA percent changes closely resemble those in CBF.
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Interaction Between Autoregulation and CO2 Reactivity
Figure 7 (bottom) shows the percent changes in small pial artery radius induced by arterial hypotension during normocapnia and hypercapnia. Figure 7 (top) displays the difference between vessel diameters during normocapnia and hypercapnia, which is considered an index of the CO2 reactivity at different arterial pressure levels. As the model (Fig. 7, left) clearly shows, CO2 reactivity initially increases with lowering mean SAP; afterward, when vasodilation in the small pial vessels becomes appreciable, CO2 reactivity rapidly decreases. These findings qualitatively concur with the data reported by Wei et al. (53) (Fig. 7, right). The behavior shown in Fig. 7 can be attributed to the presence of nonlinear sigmoidal interaction between the two mechanisms that reflects the existence of upper and lower saturation levels for smooth muscle tension (see block diagram in Fig. 4).
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Figure 8 describes the dependence of CBF on mean SAP (autoregulation curve) under steady-state conditions at various PaCO2 levels. Obviously, an increase in PaCO2 causes an increase in CBF at a given arterial pressure value. At the same time, the lower autoregulation limit shifts toward higher pressure; in fact, cerebral vessels are nearly maximally dilated because of hypercapnia and therefore are unable to dilate further in response to arterial hypotension. These findings are confirmed by several authors who studied the effects of induced hypotension at various PaCO2 levels (16, 17, 37, 39).
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Figure 9 describes the effect of arterial hypotension on the relationship between CBF and PaCO2. Model results are compared with those obtained by Harper and Glass (18) during experiments on dogs. It can be seen that arterial hypotension progressively reduces CO2 reactivity. The most considerable effect of hypotension is visible during hypercapnia, when the arterial bed is already dilated to the maximum. On the other hand, experimental data suggest that the vasoconstrictory effect of hypocapnia is also reduced during severe hypotension (18). The model can reproduce this behavior, attributing it to a reduction in the CO2 reactivity during cerebral ischemia (see block diagram in Fig. 4 and Eqs. 25A and 26A in APPENDIX).
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Effects of Hypocapnia and Hypercapnia on ICP
The purpose of the following simulations, performed with ICP free to change, is to analyze the possible effects of CO2 variations on ICP in patients with head injuries. This matter has been analyzed in depth in a second, related paper (27). We shall consider a hypothetical patient with moderate increase in CSF outflow resistance, hence with moderate intracranial hypertension. Three different cases, characterized by a progressive reduction in craniospinal elasticity, are presented.Figure 10 shows the ICP response to an acute rise in PaCO2. Hypercapnia causes a vasodilation with an increase in CBV, leading to a sudden ICP rise; subsequently, however, ICP returns monotonically toward baseline due to CSF reabsorption. The final ICP value remains slightly high with respect to the initial situation because, in the model, the increase in CBF causes a parallel increase in CSF production rate. As intracranial compliance diminishes, the capacity to buffer CBV variations decreases, and ICP shows a dramatic transient rise. In the extreme case, the ICP pattern resembles that of the single A wave or plateau wave observed by Lundberg (28).
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In Fig. 11 the ICP reaction to induced hypocapnia is plotted. The conditions are similar to those in Fig. 10; in this case, however, CO2 change produces a clear drop in ICP that is partially recovered by the process of CSF production. The final ICP level is lower than the initial level due to a decrease in CSF production rate. Also in this case, the lower the craniospinal storage capacity, the greater the ICP alteration.
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DISCUSSION |
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Top
Abstract
Introduction
Results
Discussion
Appendix
References
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The present work improves our previous model significantly, including the analysis of CO2 cerebrovascular reactivity and the interaction among CO2, autoregulation, and ICP. The model aims at representing an acceptable compromise between physiological accuracy and clinical usefulness. In fact, even though the emphasis of this study is on the modeling of physiological experiments, an understanding of the pathophysiology of intracranial dynamics is an essential prerequisite in the treatment of severe brain diseases.
An important methodological aspect of this study is that many different experimental results, concerning various phenomena of intracranial dynamics often considered separately in the literature, can be summarized into a single theoretical setting. The main aspects elucidated in this work are discussed separately below.
Cerebrovascular Segmental Heterogeneity
A critical choice in the model concerns the subdivision of the cerebrovascular bed into several consecutive segments, each characterized by its own parameters and a specific functional response to pressure and/or CO2 changes. Our segmentation largely concurs with that proposed by Kontos and colleagues (25, 53); these authors observed a vessel-size dependence in the response to blood pressure and CO2 changes between small and large pial arteries. This different behavior of the pial vascular bed, confirmed in other studies (29, 32), is shown by our model to be a mere consequence of the Laplace law and of the different values of intravascular pressure at consecutive levels of the pial vasculature.Moreover, in this study, as in a previous one (51), we assumed that autoregulation of the large pial artery is dominated by pressure-dependent mechanisms, whereas the response of small pial arteries mainly depends on blood flow changes. This hypothesis permitted us to reproduce not only the active response of the vessels quite well but also all CBF and VMCA changes at many levels of arterial pressure and CO2. Hence, we are confident that this subdivision into two regulated segments actually represents the best compromise between accuracy and simplicity and allows cerebrovascular reactivity to be reproduced with a limited number of parameters. Moreover, the good correlation between model and experimental CBF data confirms the minor role that the venous cerebrovascular bed plays in CBF regulation.
Cerebrovascular Response to CO2
The genesis of CO2 reactivity has been discussed in many papers over recent decades (24, 34, 43, 45, 46). Some authors hypothesized that the effect of CO2 on cerebral vessels is mediated by a neurogenic mechanism, probably involving the brain stem (9, 12, 46). However, most experimental results suggest that CO2 reactivity is mediated principally by local factors, mainly pH changes in the perivascular space (24, 34, 43). Accordingly, we assumed a logarithmic relationship between smooth muscle tension and CO2 pressure to emphasize that vessel response depends on perivascular pH rather than on CO2 pressure per se. This relationship allowed us to reproduce various experimental and clinical results concerning vessel caliber percent changes, CBF, and VMCA variations during hypercapnia and hypocapnia (18, 26, 30, 35, 40, 41, 44, 53).It is well known that CO2 reactivity tests are frequently conducted on neurosurgical patients to assess cerebral vessel vasomotor capacity (8, 12, 48). Usually CO2 reactivity is quantified by an index equal to CBF percent changes per mmHg of PaCO2 change. We assessed the sensitivity of CBF to CO2 variations in the model by computing the central slope of the curve shown in Fig. 6. The value obtained is ~2.5% CBF change/mmHg PaCO2 change. This value quite closely concurs with those obtained during physiological experiments on animals [2.2%/mmHg by Jones et al. (23) in rats; 2.1%/mmHg by Göbel et al. (15) in rats; and 2.87%/mmHg by D'Alecy et al. (9) in dogs]. With regard to human subjects, Tenjin et al. (48) suggested the value 2.0%/mmHg as a threshold to discriminate between patients with preserved and impaired CO2 reactivity.
Interaction Between Autoregulation and CO2 Response
An important aspect of CBF control, documented in many physiological and clinical studies, is the presence of a significant nonlinear interaction between CO2 reactivity and autoregulation. Simulating the nature of this interaction is not only a problem of physiological importance but also is one that may have a great clinical impact. First, it is generally accepted that CO2 reactivity is an important prognostic index significantly correlated with clinical outcome. Nevertheless, knowledge of this index alone may be misleading if not integrated with information on arterial pressure, CBF, and autoregulation status. Second, hyperventilation is an effective means routinely used to control ICP in neurosurgical patients. However, the effectiveness of this maneuver hinges on mean SAP and autoregulation, a fact that should be kept in mind by clinicians when planning patient treatment (12). Finally, anyone trying to identify the parameters characterizing CBF control (e.g., the gains and time constants of the mechanisms) needs to consider their complex interactions and nonlinear "inhibitory-facilitatory" summation. The latter problem is addressed in the second, related paper (27).Several experimental results show that arterial hypotension can critically interfere with the cerebrovascular response to CO2 changes. Harper and Glass (18), in a classic study, demonstrated that reducing arterial pressure to about one-third that of normal completely eliminates the vasodilatory response to hypercapnia and seriously weakens the vasoconstrictory response to hypocapnia. The latter result was further confirmed by others (2, 3).
A thorough analysis of the interaction between autoregulation and hypercapnia on pial vessel response can be found in the study by Wei et al. (53). According to their findings (see Fig. 7), the interaction between the two mechanisms is significantly nonlinear, i.e., the superimposition of the effects does not hold. In particular, as blood pressure decreases starting from a basal level, the responsiveness to CO2 becomes initially greater, suggesting a "facilitatory" overlapping of the two mechanisms. However, when arterial pressure approaches the autoregulation lower limit, CO2 responsiveness progressively disappears, indicating an "inhibitory" overlapping of autoregulation and hypercapnia.
It is well known that hypercapnia and hypocapnia significantly modulate the autoregulation curve. Hypercapnia shifts the autoregulation plateau to higher CBF levels, and hypocapnia reduces it to a lower CBF. Moreover, data collected by several authors concur, showing that the lower autoregulation limit is fairly shifted by CO2 changes (16, 17, 37, 39). More controversial results can be found for the effect of CO2 on the upper limit of autoregulation (11, 16, 38).
We were able to qualitatively reproduce most of the results mentioned above (Figs. 7-9) by supposing the presence of only two significant nonlinearities in the interaction between autoregulation and CO2 response. Both nonlinearities are confirmed by physiological considerations.
A first nonlinearity in the model derives from a sigmoidal static relationship and accounts for the natural bounds of smooth muscle tension (see last block in Fig. 4) and permits reproduction of most of the nonlinearities observed during combined stimuli. However, the presence of this sigmoidal relationship for smooth muscle tension did not allow us to explain the lack of vasoconstrictory response to hypocapnia observed during severe arterial hypotension (2, 3, 18). It has been claimed that the failure of cerebral vessels to constrict to hypocapnia during hypotension indicates that maintenance of adequate tissue O2 supply prevails over the maintenance of adequate CO2 and acid-base levels (18). To account for this phenomenon, we assumed that cerebrovascular CO2 reactivity depends on CBF through a sharply nonlinear relationship (see inset in Fig. 4); when CBF falls below an alarm threshold (indicating the initiation of ischemia), CO2 reactivity is suddenly withdrawn. As a consequence of this assumption, cerebrovascular response to hypocapnia progressively disappears when mean SAP decreases below the autoregulation lower limit (SAP < 50 mmHg) in accordance with data by Artru and Colley (2). The existence of an "ischemic threshold" for CBF, below which CO2 reactivity drops to zero, is also confirmed by clinical observations (8).
Interaction Between CBF Control and ICP
The existence of a close relationship among the action of CBF control mechanisms, cerebral blood volume changes, ICP, and intracranial dynamics is a peculiar characteristic of cerebrovascular regulation that is not shared by other vascular beds. This has been stressed by several authors in past years, beginning with the pioneering works by Lundberg (28), up to the fundamental studies that appeared in the mid-1980s by Rosner and Becker (see e.g., Ref. 42). Using a previous version of the present model (51), we were able to support most of Rosner's ideas; the model predicted in pathological conditions the generation of self-sustained ICP oscillations very similar to plateau waves. This is a result of a potential positive feedback among autoregulation, CBV changes, and ICP, called by those authors "vasodilatory cascade."The new simulation results presented in Figs. 10 and 11 confirm the existence of a similar vasodilatory cascade in response to maneuvers that alter CO2 concentration in blood. Two main aspects of these simulations deserve consideration. First, the impact of CO2 changes on ICP crucially depends on the status of intracranial compensatory mechanisms (CSF outflow and intracranial elastance). The same maneuver, with almost no effect on ICP in physiological conditions, can potentially induce dramatic ICP changes in pathological states characterized by reduced CSF outflow and poor intracranial capacity. In the worst conditions, even a mild hypercapnia can induce a disproportionate transient ICP rise, similar to that of a single plateau wave, with the risk of acute intracranial hypertension and cerebral ischemia (Fig. 10).
Furthermore, the results of Fig. 11 indicate how hyperventilation can be used to control ICP in patients at risk of intracranial hypertension. It should be noted that, in our model, the effect of CO2 changes on ICP depends on two concurrent mechanisms that usually operate with different time constants: 1) active CBV changes, which cause rapid alterations in ICP, generally within a minute of the maneuver; and 2) alteration in CSF production rate, which takes place with a longer time constant (several minutes) and contributes to set the final steady-state ICP level. In the model, CSF production rate depends on capillary pressure and is therefore proportional to CBF.
An important new aspect of this model concerns the relationship between CBF and VMCA. In this study we assumed that the MCA behaves passively, i.e., its inner radius is a monotonic function of transmural pressure; neither autoregulation nor CO2 significantly affects the caliber of the MCA in the model. The consequence of such an assumption is that, during moderate variations in arterial pressure, velocity does not decrease, unlike CBF, due to variations in cross-sectional area. This behavior is supported by the experimental results by Nelson et al. (35) in the rabbit. In contrast, during CO2 alterations, the velocity percent changes are of the same order as CBF changes, because transmural pressure and cross-sectional area in the large arteries remain almost unaffected. This model behavior agrees with data reported previously (30, 44). It is possible, however, that a different behavior for the MCA (inner radius being dependent on PaCO2 or on an active myogenic response) may have significant consequences on model performance.
In perspective, the possibility of simulating the time pattern of ICP and TCD velocity in response to maneuvers affecting CO2 and/or mean SAP could be used to estimate the main parameters of intracranial dynamics from measurements taken in neurosurgical intensive care units. On the basis of knowledge of these parameters in the individual patient, a greater understanding of pathophysiological mechanisms can be achieved and a more well-founded treatment for severe brain damage developed. This model application on real velocity and ICP tracings is the subject of a second, related study (27).
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APPENDIX |
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Top
Abstract
Introduction
Results
Discussion
Appendix
References
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Quantitative Model Description
Cerebral hemodynamics and CSF dynamics. The fundamental equations describing cerebral hemodynamics are written by imposing volume preservation equations at all nodes of Fig. 1.
Volume balance at the proximal arterial segment (large pial arteries) is
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(2A) |
|
(3A) |
|
(4A) |
|
|
(5A) |
|
|
|
(6A) |
|
(7A) |
Biomechanics of arterial-arteriolar cerebrovascular bed. In Eq. 8A, subscript j = 1 denotes a quantity belonging to the segment of large pial arteries, whereas subscript j = 2 refers to pial arterioles.
The equilibrium of tangential forces in a vessel wall is provided by the Laplace law
|
(8A) |
|
(9A) |
|
(10A) |
e is elastic stress,
r0 is the inner
radius in the condition of "unstressed wall," and
K
,
0, and
coll are constant parameters.
In particular, coll has been
introduced to permit the vessel to sustain a moderate negative tension
before collapsing.
Smooth muscle tension depends on inner radius through a campanular
relationship, which reflects the different overlapping of actin and
myosin filaments. Active tension is therefore maximal at a particular
optimal value of inner radius and then progressively decreases. We have
|
(11A) |
v) has a linear dependency
on the time derivative of wall strain. Thus
|
(12A) |
|
(13A) |
j denotes wall
viscosity.
An expression for wall thickness, to be used in Eq. 8A, has been computed by assuming that vessel wall
volume remains constant. By neglecting longitudinal strain, we have
|
(14A) |
|
(15A) |
|
(16A) |
|
(17A) |
Biomechanics of cerebral venous circulation. Because the intracranial venous circulation behaves in a passive way (4), we assumed that venous compliance is inversely proportional to transmural pressure. Hence, we have
|
(18A) |
|
(19A) |
Blood flow velocity in middle cerebral artery.
The inner radius of the MCA
(rMCA) is
computed as a function of transmural pressure, assuming a
monoexponential pressure-radius relationship (20). The
rMCA is therefore
a logarithmic function of transmural pressure,
Pa Pic
|
(20A) |
|
(21A) |
Cerebrovascular regulation mechanisms. Cerebrovascular control mechanisms affect smooth muscle tension in pial arteries by modifying the amplitude of the active tension relationship (i.e., Tmax in Eq. 11A). We have
|
(22A) |
1 < M < 0 indicates
vasodilation, whereas 0 < M < 1 represents vasoconstriction.
The activation factor is the result of two contributions,
xaut and
xCO2,
which account for the effect of autoregulation and
CO2 reactivity, respectively. Both
are state variables, the dynamics of which have been assumed to be of
the first order (see below).
CEREBRAL AUTOREGULATION.
In accordance with physiological literature, we assumed that
autoregulation acts on large pial arteries and pial arterioles through
two distinct feedback mechanisms. Smooth muscle tension in the proximal
segment depends on perfusion pressure changes. Hence, the following
differential equation holds for the variable xaut,1
|
(23A) |
aut,1 are the mechanism gain
and time constant and Pa n Pv n is normal
CPP. According to Eq. 23A, any decrease in CPP causes a vasodilation in large
pial arteries, whereas a perfusion pressure increase causes
vasoconstriction.
In the small pial arteries (second segment in Fig. 1), we assumed that
autoregulation is triggered by changes in CBF. Accordingly, the
quantity xaut,2
is governed by the following differential equation
|
(24A) |
aut,2 are gain and time
constants, and the parameter qn
represents the value of CBF that satisfies the metabolic need (and so
does not activate the autoregulatory action). Equation 24A implies that a decrease in CBF below
qn causes vasodilation in small
pial arteries, whereas the opposite effect is produced by an increase
in CBF.
CO2 REACTIVITY.
Because CO2 acts mainly through
changes in the pH of the perivascular space, we assumed that the state
variables
xCO2,j (j = 1, 2) depend on the
logarithm of CO2 pressure in
arterial blood. Moreover, the nature of
CO2 reactivity does not differ substantially in large and small pial arteries. The following differential equations can thus be written
|
|
(25A) |
CO2
denote the mechanism gain and time constant and the subscript n is used
to represent the PaCO2 value that does
not affect the basal smooth muscle tension. According to
Eq. 25A, hypercapnia causes
vasodilation in both pial segments, whereas hypocapnia causes
vasoconstriction. Finally, in accordance with the corrective factor
ACO2,
we assumed that the strength of the
CO2 regulation mechanism remains
quite constant when CBF is normal or moderately decreased but rapidly
drops to zero during severe ischemia. We used the following
dependence of
ACO2
on CBF
![<IT>A</IT><SUB>CO<SUB>2</SUB></SUB> = <FR><NU>1</NU><DE>1 + exp [−<IT>K</IT><SUB>CO<SUB>2</SUB></SUB>(q − q<SUB>n</SUB>)/q<SUB>n</SUB> − <IT>b</IT><SUB>CO<SUB>2</SUB></SUB>]</DE></FR> <IT>j</IT>=1, 2](/content/vol274/issue5/fulltext/H1715/img035.gif)
|
(26A) |
![<IT>M<SUB>j</SUB></IT> = <FR><NU><IT>M</IT><SUB>min,<IT>j</IT></SUB> + <IT>M</IT><SUB>max,<IT>j</IT></SUB> exp [(<IT>x</IT><SUB>aut,<IT>j</IT></SUB> + <IT>x</IT><SUB>CO<SUB>2,</SUB><IT>j</IT></SUB>)/<IT>K</IT><SUB><IT>M</IT>,<IT>j</IT></SUB>]</NU><DE>1 + exp [(<IT>x</IT><SUB>aut,<IT>j</IT></SUB> + <IT>x</IT><SUB>CO<SUB>2,</SUB><IT>j</IT></SUB>)/<IT>K</IT><SUB><IT>M</IT>,<IT>j</IT></SUB>]</DE></FR>](/content/vol274/issue5/fulltext/H1715/img036.gif)
|
|
(27A) |
Mmin,j)/4.
Because of this assumption, the gain of each mechanism acting
individually is equal to Gaut and
GCO2 in
Eqs. 23A-25A.
| |
FOOTNOTES |
|---|
Address for reprint requests: M. Ursino, Dipartimento di Elettronica, Informatica e Sistemistica, viale Risorgimento 2, I-40136 Bologna, Italy.
Received 31 July 1997; accepted in final form 28 January 1998.
| |
REFERENCES |
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Top
Abstract
Introduction
Results
Discussion
Appendix
References
|
|---|
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denotes basal value of inner
radius at equilibrium; denotes equilibrium level following
vasodilation induced by complete smooth muscle relaxation, assuming
constant intravascular and extravascular pressures. Note greater
vasodilatory capacity of small pial arteries compared with large pial
arteries.
PaCO2 = log10
(PaCO2/PaCO2n),
where
PaCO2n
represents PaCO2 value that does not
affect basal smooth muscle tension. In contrast, input quantity for
autoregulation is different in the two segments: large pial arteries
actively respond to changes in cerebral perfusion pressure (
) for
VMCA.
4
mmHg