Interactions of brain, blood, and CSF: a novel mathematical model of cerebral edema

Intracranial pressure (ICP) is the cornerstone of most treatment decisions in the neurointensive care unit. Monro and Kellie presented the hypothesis linking volume and pressure in the closed cranium that served as the basis for our understanding of raised ICP [1, 2]. The physiologist Hugh Davson described steady-state ICP in terms of the flow of cerebrospinal fluid, the resistance to its outflow from the cranium, and the pressure in the sagittal sinus where CSF is absorbed [3, 4]. This equation, which defines steady-state ICP under certain conditions laid the cornerstone upon which all further models of intracranial physiology were built. Davson’s simple, one-element model of CSF hydrodynamics accounts for the value of steady-state ICP when the primary pathology is a disturbance in the ability to reabsorb CSF but does not account for other pathologies that may lead to increased ICP. The pioneering work of Marmarou added the element of intracranial compliance in order to better describe the dynamic aspects of ICP and the pressure changes that occur with injection of different volumes into the closed cranium at varying rates [5]. Further investigators used these models to describe the importance of the rate of volume changes in the cranium on intracranial pressure dynamics [6, 7]. Others developed models to depict obstruction to the flow of CSF within the ventricular system and replicate various clinical aspects of obstructive hydrocephalus [8, 9]. Subsequently, Ursino and others added the important element of the cerebrovascular circulation to models of intracranial fluid dynamics to account for cerebral blood flow (CBF) and its key aspects such as cerebrovascular autoregulation [10‐17]. Recent studies have also modeled the ICP waveform and translated an electrical analogue model into a working physical model of the cerebral vascular circulation [18, 19]. Together these models of intracranial hydrodynamics built our current understanding of ICP, CBF, and their inter-relationships and served as the foundation for a vast body of literature describing important interactions between cerebral and systemic physiological parameters [20, 21].

While models of intracranial physiology to date have provided the basis of our understanding of ICP, it is important to point out that fundamentally they were based on two primary elements: the circulation of CSF and the cerebrovascular circulation. As such, they did not take into account a crucial aspect of intracranial physiology, swelling of the brain itself and the effect of raised ICP on the circulation of cerebral interstitial fluid (ISF). Cerebral swelling is a key aspect of many pathologies commonly treated in the neurointensive care unit, including traumatic brain injury (TBI) and stroke. The ability to model the effects of cerebral swelling on intracranial physiology is an important goal. In this work, we aimed to build on the previous classical models of intracranial fluid dynamics adding the additional component of “brain” to those models that have described the flow of “CSF” and “blood.” As such, we sought to develop a model of intracranial hydrodynamics that incorporates the flow of cerebral interstitial fluid. We sought to assess the ability of this model to predict clinically relevant changes in cerebral physiology that are known to occur with cerebral edema and raised ICP.

We developed a lumped parameter model to represent the interactions between brain tissue, CSF, and blood enclosed within the rigid cranium under both normal physiological conditions and under conditions of brain edema. As in the classical formulations of intracranial physiology, we describe the model using a mathematical formulation that is based on the physical relations in the system and included a representation of these relations using an electrical analog circuit. Figure 1 shows the intracranial anatomy and the corresponding electrical analog model. The model consists of four major compartments and flow conduits. Each compartment contains fluid within an elastic barrier that is represented as an elastic chamber with nonlinear pressure–volume relations. These nonlinear relations may take a form of either an exponent or a polynomial describing the chamber’s transmural pressure as a function of total volume enclosed within that chamber. The selection of exponential or polynomial functions were chosen to accord with accepted intracranial physiological relationships and are detailed below. Importantly, the enclosed volume in some chambers may include the volumes of other elastic chambers that are contained within it. From the outside-in these chambers are: (1) The subarachnoid space that is limited outwardly by the semi-elastic dura and the rigid skull and inwardly by the pia mater but encloses all intracranial volumes including the brain tissue, ISF, CSF, and the cerebrovascular circulation. (2) The brain compartment through which the cerebral interstitial fluid circulates and is limited outwardly by the pia. (3) The ventricles where CSF is formed and through which it flows, and (4) the cerebral vascular tree. The superior sagittal sinus, owing to the thick dural walls of the sinus, is a separate compartment within the cranium. It is the outflow for venous blood and the site of reabsorption for CSF. It maintains a pressure lower than the ICP under most physiological conditions [22‐27]. In addition, in the model the spinal thecal sac compartment is interconnected to the intracranial space through the link between the cranial and spinal subarachnoid spaces.

The outermost compartment in the model, the subarachnoid space, is represented as nonlinear elastic chamber in the electrical analog circuit by a capacitor with a nonlinear pressure–volume relationship in the form of an exponential function (Eq. 1):

\({ICP}_{SAS}\) is the subarachnoid pressure, \({V}_{tot}\) is the total cranial volume, which is the sum of the brain compartment volume, ventricular volume, subarachnoid space volume, and blood volume, and \({V}_{0}\) is the cranial unstressed volume. Coefficient values \({a}_{s}\), \({b}_{s}\), and \({c}_{s}\) as well as all the coefficients for the equations listed below are detailed in Appendix. As the capacity of the intracranial space to accommodate additional volume is limited, this equation accounts for the exponential-like rise in ICP when the total volume increases. Previous models have also included a similar equation to account for the exponential-like rise in pressure when intracranial volume reserve capacity reaches its limit [28, 29].

The brain compartment in turn encloses two additional compartments that are, for the most part, contained within it: the ventricles and the vascular bed. This compartment’s transmural pressure (\({ICP}_{{BR}_{TM}}\)) as a function of its volume is described by a polynomial (Eq. 2):where \({V}_{BR}\) is the total volume enclosed by the pia, including the volume in the ventricles and in the blood vessels. Coefficient values \({a}_{b}\), \({b}_{b}\), \({c}_{b}\) and \({d}_{b}\) are detailed in Appendix.

Ventricular pressure is defined as follows: external pressure is the \({ICP}_{BR}\) and the ventricular transmural pressure \(IC{P}_{{VEN}_{TM}}\) is defined as a function of its own volume as described by a polynomial (Eq. 3):where \({V}_{VEN}\) is the ventricular volume. Coefficient values \({a}_{v}\), \({b}_{v}\), \({c}_{v}\) and \({d}_{v}\) are detailed in Appendix.

The cerebrovascular circulation within the cranium is subdivided into three volume components: arterial and arteriolar, C_A, capillary, C_C and venous, C_V. The pressures in the three sub-compartments, arteriolar \({P}_{ARL}\), capillary \({P}_{C}\), and venous \({P}_{V}\) were described as a function of their blood volumes \({V}_{A}\), \({V}_{C}\), and \({V}_{V}\) respectively, by linear expressions (Eq. 4):

As noted above, the superior sagittal sinus is the only intracranial compartment that maintains a pressure lower than ICP and is the site for both venous outflow and CSF reabsorption.

The spinal thecal sac is connected to the intracranial space through fluid conduit between the intracranial and spinal subarachnoid spaces. Pressure–volume relations in the sac were also described by a polynomial (Eq. 5):where \({V}_{SPI}\) is the spinal thecal sac volume. Coefficient values \({a}_{t}\), \({b}_{t}\), and \({C}_{t}\) are detailed in Appendix.

Importantly, in the model transmural pressures of the ventricular and brain compartments are very small; however, they are required to maintain compartment volumes. For example, the ventricle is enclosed by the brain on its outer margin. The pressure in the ventricle (P_VEN) is slightly greater than the pressure in the brain at baseline physiological conditions, thereby maintaining a positive transmural pressure and ventricular volume. When this normal physiological condition is reversed by pathophysiological processes, collapse of the compartment occurs.

Production rates of CSF (Q_CSF) and ISF (Q_ISF) in the model are constant (Table 1), in line with experimental evidence that supports a constant rate of production independent of changes in ICP [30]. Initial baseline CBF (Q_CBF) is 910 ml/min, but varies in relation to changing volume and pressure in the other model compartments or changes in resistance within the cerebrovascular tree.

Fluid conduits connect between some of the different compartments. In the model, resistors represent viscous resistance to fluid flows within the brain and ventricular system and through the cerebrovascular tree. In the cerebrovascular circulation, R_A, R_C, and R_V and represent viscous resistance to flow in the circulation. The resistors in the vascular bed are affected by the transmural pressure of the vascular wall that is exposed to ICP_BR. Changes in ICP_BR affect the volume inside the vessel that in-turn alters the resistance. Radius changes affects resistance, according to Hagen-Poiseuille's law for a cylinder (Eq. 6):\(where V=\pi {r}^{2}L\)

\(\mu\) is viscosity, L segment length and r is the radius. \({R}_{0}\) is initial resistance of the compartment with compartment volume \(V={V}_{0}\) at baseline conditions.

The normally very high resistance to the passage of fluid from the cerebrovascular circulation into the brain parenchyma is depicted by R_BBB which has a very high value under normal physiological conditions. The model depicts pathophysiological disturbance that impair the normally nearly impenetrable blood–brain barrier as a decreasing R_BBB.

The resistances to flow of CSF and ISF are depicted as follow: R_VEN represents resistance to flow within the ventricles and out of the ventricular system into the subarachnoid space. R_BF represents resistance to the bulk flow of cerebral ISF within the brain’s extracellular space and out to the subarachnoid space where it merges with the circulating CSF. R_OUT represents the resistance to CSF absorption into the superior sagittal sinus. Lastly, R_FM represents the resistance to flow of CSF from the intracranial subarachnoid compartment to the spinal subarachnoid compartment. The resistors to the flow of CSF and ISF are affected by the volume of CSF or ISF in a compartment through which they flow. For example, an extracellular compartment with large fluid content will result in larger space through which cerebral ISF can flow leading to a lower resistance (low R_BF). In contrast, when the volume of the extracellular space is low as occurs in brain swelling, the resulting denser cellular space leads to greater resistance to cerebral ISF flow through the constricted ECF channels (high R_BF).

The solution of the model involves solving seven differential equations. Due to the nonlinear behavior of the parameters’ values and their dependency on pressure and volume values, the Euler method was employed to solve the model and reach a steady state solution for each pathological condition. The Euler method was used as follows:

Volume update for every capacitor:where n is capacitor index and \(i\) is discrete time.

Pressure update:where the function is one of those described above (exponential or polynomial)

Parameter values update:

Flow calculation:

Table 1 details initial baseline values and the model components’ characteristic pressure and volume relations. Most baseline values were designated based on normal values obtained from the literature, while others whose values may be uncertain due to the paucity of data in the literature were derived and estimated on the basis of normal physiological pressures and flows.

Initial assessment of the model demonstrated stable values for pressure, volume, and flow in all three model elements of brain, CSF, and blood. The hierarchy of pressures between the ventricles, cerebral parenchyma, and subarachnoid space on the one hand, and the cerebral venous compartment on the other, was maintained in the expected manner (Fig. 2). Importantly, simulations of rapid intracranial volume injections, demonstrated the exponential-like intracranial pressure–volume curve (Fig. 3A). Rising ICP_BR also resulted in changes to the ICP waveform that are typically seen when ICP is elevated, including a progressively increasing ICP pulse amplitude typical to states of high intracranial elastance (Fig. 3B). Changes in arteriolar resistance led to the expected changes in CBF and ICP_BR, simulating the decrease in ICP induced by hyperventilation and the increase in ICP induced by hypoventilation (Fig. 4). Hyperventilation resulted in a 20% decrease in CBF from baseline and an 18% decrease in ICP. Conversely, hypoventilation resulted in a 30% increase in CBF from baseline and a 15% increase in ICP.

As expected, the lumped parameter model demonstrated rising intracranial pressure under conditions of obstructive hydrocephalus, absorptive hydrocephalus, and cerebral edema (Fig. 5A–D). While ICP rises in each of these cases, the effect on the volumes in the different intracranial compartments varies between simulations of different pathophysiological conditions. When non-absorptive hydrocephalus (high R_OUT) or obstructive hydrocephalus (high R_VEN) is simulated, ventricular volume increases substantially to 140% and 152% of baseline values, respectively, as ICP reaches 20 mmHg. In contrast, when cerebral edema is simulated by an increased R_BF or decreased R_BBB, ventricular volume decreases markedly to 8% and 12% of baseline values, respectively, as ICP reaches 20 mmHg.

Table 2 details the changes in model output parameters that are induced by a change in the input values of R_OUT, R_VEN, R_BF, and R_BBB which lead to an ICP rise to a level of 30 mmHg. The changes in the volumes of the brain, ventricles, and subarachnoid space compartments are detailed in the Table. Importantly, in the model CBF decreases substantially with the rise in ICP. The results demonstrate that model output corresponds to accepted patterns associated with each of the distinct pathophysiological conditions. The model clearly indicates that simulating cerebral edema with an increase in the resistance to the flow of cerebral interstitial fluid (\({R}_{BF}\)) leads not only to raised ICP_BR (Fig. 5C), but also to an increase in total brain volume, and a marked decrease in ventricular and subarachnoid space volume (Table 2), accurately replicating the physiological changes known to occur with brain edema. We also modeled the effect of changes in the integrity of the blood–brain barrier on cerebral physiological parameters (Table 2). Not surprisingly, a breakdown of blood–brain barrier integrity represented by a decrease in \({R}_{BBB}\), resulted in an increase in the volume of the brain component in the model (Table 2) leading to a rise in ICP_BR (Fig. 5D) and a marked decrease in ventricular volume (Fig. 6D) that contrasts with the increasing ventricular volume that occurs with non-absorptive or obstructive hydrocephalus (Fig. 6A, B). ICP_BR increased dramatically as R_BBB decreased to low levels indicating an exponential-like effect of a rising permeability to fluid shifts across the BBB (Fig. 5D).

The four-compartment model we describe simulates the important interactions of brain, blood, and CSF within the closed cranium and accurately replicates important and well-accepted aspects of intracranial physiology. The model replicates the pressure–volume curve in response to rapid volume injection into the intracranial space (Fig. 3) and the responses to hyper and hypoventilation (Fig. 4). Importantly, the incorporation of a novel compartment to the model, the brain, through which flows cerebral interstitial fluid, adds an essential component that has not been previously included in models of intracranial hydrodynamics. This is in effect a modification of Davson’s original formulation:where the superscript, ^D, denotes the parameters referred to by Davson: ICP^D a lumped ICP for a single intracranial compartment, Q^D_CSF representing flow of all CSF within the cranium, which in the context of the current model corresponds to flow of CSF in the subarachnoid space or the sum of ISF and CSF flow, and R^D_OUT representing the resistance to reabsorption of CSF into the superior sagittal sinus. Rather than treating the intracranial space as a single lumped compartment with a unitary pressure, the model we describe stresses the small but important differences between pressures in each of the different cerebral compartments, ventricles, brain, and subarachnoid space. In our model, steady-state pressure in the brain parenchyma and subarachnoid space, respectively, may be described as follows:

Substituting Eq. 9 in Eq. 8 yields Eq. 10.

This modification of the Davson equation considers both CSF and ISF production and flow as essential elements in determining steady-state cerebral intraparenchymal pressure. This additional element is vital because it allows modelling of pathologies in which cerebral edema is the primary component leading to raised ICP and an increased resistance to bulk flow of ISF. Our results clearly demonstrate the rise in ICP caused by an increase in R_BF (Fig. 5) and the consequent changes in volumes of the other intracranial compartments that occur with brain swelling (Table 2). Pathologies in which cerebral edema predominates, either as the primary insult or the consequence of secondary brain injury occur frequently in neurocritical care and continue to present difficult therapeutic challenges. The current model offers the ability to separate two distinct pathways leading to ICP elevation, those caused by obstruction to CSF flow (\({R}_{VEN}\), \({R}_{OUT}\)) and those caused by swelling of the brain itself which leads to an increased resistance to cerebral interstitial fluid flow (\({R}_{BF}\)). The model also depicts the changes caused by a breakdown of the normally nearly impenetrable blood–brain barrier (\({R}_{BBB}\)) (Figs. 5D, 6D). The ability to model these important pathophysiological conditions may provide potential benefit by affording an improved categorization of the underlying pathophysiology that leads to raised ICP. Crucially, in addition to demonstrating raised ICP, the model replicates well known clinical manifestations that occur with brain swelling such as a marked decrease in volume of the ventricles and the subarachnoid space (analogous to cisternal and sulcal effacement) which most previous models of intracranial hydrodynamics have not addressed (Table 2). Opening of the blood–brain barrier is another well-recognized pathological process known to lead to severe cerebral edema and raised ICP [31‐33]. The current model is successful in emulating the multifaceted effects of increased BBB permeability, indicating its potential utility in modelling the blood–brain barrier disruption that occurs with brain injury. Importantly, the model simulates the often-complex multivariate interactions between key cerebral physiological parameters. By simulating a primary disturbance in one of four key resistances (\({R}_{VEN}\), \({R}_{OUT}\), \({R}_{BF}\), and \({R}_{BBB}\)) the model is able to demonstrate the ensuing changes in the volumes, pressures, and resistances in each of the intracranial compartments. The model also depicts the changes in intracranial compliance that occur with rising ICP and with a rapid volume load to the intracranial space (Fig. 3A). Similar to previous models of ICP pressure dynamics it replicates the changes in the ICP waveform that result from raised ICP and decreased intracranial compliance (Fig. 3B) [11, 34]. The ability to study interrelationships is an important advantage of lumped parameter modelling. Although this is a preliminary study to establish the theoretical underpinnings of modelling intracranial pressure–volume dynamics in relation to the resistance to cerebral ISF flow, our hope is that future translational studies will be able to build on this work and use large animal experimental models or data from brain injured patients in order to assess the model’s ability to help predict thresholds of intracranial volume-reserve capacity in cerebral edema.

Previous models of intracranial hydrodynamics and cerebrovascular circulation established the underlying principles needed to depict the interactions between the pressure, volume, and flow of CSF and CBF [35]. However, a substantial gap in these models remained in that none described the important changes that occur when the brain itself swells and the flow of cerebral interstitial fluid is impaired by that swelling. This lacunae in the models of intracranial hydrodynamics is of crucial importance because many of the pathologies dealt with by neurosurgeons and neurocritical care specialists involve severe cerebral edema. Without a means to accurately represent the changes that occur when the brain swells a substantial gap exists in the ability of models of intracranial physiology to describe the interactions between cerebral physiological parameters in a clinically meaningful way. The physiological importance of the flow of cerebral interstitial fluid has received renewed attention in recent years [36‐38]. When cerebral intracellular swelling occurs the normal flow of brain ISF may be impaired [39, 40] with important consequences for ICP. Intracellular swelling leads to a substantial reduction in the volume of the cerebral extracellular space [41‐43], leaving smaller constricted channels through which the cerebral ISF must flow. Under these conditions, the flow of cerebral ISF is impaired by cellular swelling. The model depicts this pathophysiological condition as an increased resistance to bulk flow of ISF through the brain parenchyma (increased R_BF). An increasing R_BF value represents the heightened resistance to the efflux of the water that is a by-product of cerebral metabolism out of the brain parenchyma by way of its normal passage from the extracellular space into the subarachnoid space. The effects of the resulting increased resistance to cerebral ISF flow are apparent in the model where a marked rise in ICP results from an increase in \({R}_{BF}\). The resulting rise in ICP may, in turn, further affect other cerebral physiological parameters leading to a cascade effect with deleterious consequences. An advantage of the lumped parameter model is that it allows the simulation of these interactions.

The model we describe also needs to be examined in light of recent controversies regarding cerebral ISF production and flow. In recent years, the development of the glymphatic and intramural peri-arterial drainage (IPAD) theories has challenged traditional conceptions of ISF flow within the brain [44‐48]. Strongly held views are held by both sides of the on-going debate regarding the experimental evidence and whether it supports the glymphatic theory or more traditional theories of ISF production and circulation. We sought to describe a model that describes the importance of ISF circulation as a key component in determining ICP, especially under conditions of brain edema. We aimed to ensure that the model is compatible with both traditional and glymphatic theories of cerebral ISF circulation. The model only assumes that ISF is produced at a constant rate (0.00083 ml/sec) as noted in Table 1. This comprises just under 20% of the summed total of CSF plus ISF production, which is in line with ratio of production of metabolic water produced by cellular metabolism in relation to the rate of CSF production. Whether this fluid is produced by mechanisms proposed by the glymphatic theory, the IPAD theory, or by traditional theories of ISF production and flow does not bear directly on the model’s results. The model postulates that cerebral ISF flows from the extracellular space of the brain parenchyma into the subarachnoid space. An important aspect of the model is that it obeys the principles of hydrodynamics in that fluid flows from a higher pressure into a compartment of lower pressure. The subarachnoid space which envelopes the brain is postulated to have a slightly lower pressure than both the brain parenchyma which it surrounds and the ventricles which empty directly into it. The slightly lower pressure of the intracranial subarachnoid space results from the fact that it itself is the site of CSF exit out of the cranium by way of reabsorption into the superior sagittal sinus. Importantly, when cerebral swelling or hydrocephalus lead to a smaller subarachnoid space, resistance to flow within this compartment increases and CSF reabsorption is impaired. While it is difficult to definitively establish whether small pressure differences between the intracranial compartments exist, it is not unreasonable to postulate that they may occur. The small number of clinical studies that have measured ICP concomitantly in the ventricles and brain parenchyma have found differences in these simultaneous measurements [49‐53]. A meta-analysis of these studies identified that 70% of concomitant measurements in the same patient could vary by as much as ± 6 mmHg and an overall mean pressure difference of 1.6 mmHg exists [54]. Admittedly, it remains difficult to ascertain to what degree these differences are due to actual differences in pressure between compartments and to what degree they may be due to measurement error. Further meticulous experimental and clinical studies will be required to establish whether the small differences in transmural pressures proposed in the model can be demonstrated and quantified by experimental data.

The four-compartment model of intracranial physiology incorporates a depiction of cerebral interstitial fluid flow and resistance that allows modelling of brain edema. The model accounts for the complex interactions that occur between different cerebral physiological parameters and accurately depicts the changes in pressure, volume, and resistances to flow in the different intracranial compartments under specific pathophysiological conditions. It may serve as a platform for improved modelling of the cerebral edema and blood–brain barrier disruption that occur following brain injury.