INTRODUCTION

A number of neurological diseases are associated with disorders of the cerebrospinal fluid (CSF) circulation. One of these pathologies is hydrocephalus, characterized by pathological enlargement of the ventricles of the brain due to excessive accumulation of CSF. One of the forms of hydrocephalus is normal-pressure hydrocephalus (NPH) in which intracranial pressure slightly differs from a normal value. This form of hydrocephalus, characterized by gait disorder, urinary incontinence, and dementia [1, 2], is difficult to diagnose, so it is difficult to make surgery plans. The most common treatment for NPH is bypass surgery, which removes excess CSF from the cranial cavity. However, this method of treatment can help one achieve a positive result only in 50–60% of cases [3]. Moreover, according to [3, 4], patients who are diagnosed and prescribed treatment at an early stage of NPH respond better to treatment.

Magnetic resonance imaging (MRI) is one of the main methods of in vivo study of the CSF system of humans. It is used for noninvasive quantitative and qualitative estimation of the dynamics of CSF circulation in both normal and pathological conditions. This research method can be used to visualize the CSF flow, quantify its velocity, and analyze the structural state of the brain substance [5]. Thus, MRI is currently the most effective method for diagnosing NPH and monitoring its development. Also of great importance is the observation of intracranial pressure, which can only be measured invasively.

However, the current state of medicine does not allow for a long-term and continuous reception of data on processes in the brain. Therefore, NPH has not been sufficiently studied: its origin is not entirely clear, and it is difficult to predict its course. As a result, various hypotheses have been put forward about the causes and pathophysiology of NPH. As these hypotheses have not yet led to a complete understanding of the causes and mechanisms of the formation and development of hydrocephalus, it is advisable to use mathematical modeling to study this pathological condition. Constructing an adequate mathematical model requires that the nature of the flow and interaction of CSF in the brain is investigated.[3]

Fig. 1
figure 1

CSF circulation diagram: (1) superior sagittal sinus, (2) vascular plexus, (3) third ventricle, (4) Sylvian aqueduct, (5) fourth ventricle, (6) central spinal canal, (7) right lateral ventricle, (8) subarachnoid space, and (9) arachnoid granulation.

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The extracellular fluids of the brain are blood, CSF, and intercellular fluid. CSF is a transparent liquid, whose density and viscosity are close to those of water. It is known that CSF is formed by filtration of capillary blood plasma through a vascular plexus epithelium in all four ventricles, which is a system of cavities located in the center of the brain. CSF flows from the ventricles into the subarachnoid space of the spinal cord and brain. The bulk of the CSF is drained into a system of venous sinuses of the dura mater (Fig. 1). Also, CSF is absorbed along blood vessels (a so-called glymphatic system) and along meningeal lymphatic vessels into an extracranial lymphatic system. It was shown in [6] that CSF seeps into the brain parenchyma and mixes with the intercellular fluid, thereby clearing the brain from the waste products of metabolites and cells. Moreover, CSF acts as a hydraulic shock absorber for the brain and ensures the delivery of nutrients.

Intracranial hemoliquorodynamics plays a major role in the healthy operation of the brain as disturbances in the dynamics of the fluid medium of the brain can cause multiple complications. It is believed that it is the violation of hemoliquorodynamics that is the cause of NPH. Therefore, obtaining data on pathological processes and providing better treatment can only be possible with understanding of the CSF hydrodynamics and the CSF relationship with blood flow and intracranial pressure.

1. MATHEMATICAL MODELING

Mathematical models describing intracranial dynamics in the normal condition and with a disease can be classified as follows: compartment models dependent only on time ((\(P{-}V\)) models (pressure — volume)) [7–9], models that consider the brain parenchyma as a rheologically complex deformable continuous medium [10–14], models using computational fluid dynamics (CFD) [15–18], and mechanical models that interpret the brain parenchyma as a porous material saturated with fluid and deforming in accordance with the interaction of tissue and pore fluids [19–22].

Compartment models make it possible to describe the physiology and pathophysiology of diseases, but they are often difficult to implement, which limits their application in a clinical setting. CFD modeling allows determining the pressure and velocity fields of the brain fluids for patient-specific geometry. However, the drawback of most of the existing models is that they do not describe an interaction among the circulatory network of the brain, parenchyma, and CSF. This disadvantage can be eliminated by refining the existing mechanical models based on the theory of poroelasticity. In this work, a model similar to that proposed in [21] is used.

1.1. Mathematical Model

A multifluid poroelastic filtration model is used. It is assumed that the brain parenchyma is modeled by a porous matrix, and there are liquid phases located at each point of the pore space and communicating with each other: arterial blood (values with subscript \(a\)), capillary blood (values with subscript \(c\)), venous blood (values with subscript \(v\)), and CSF (values with subscript \(e\)). Each liquid phase is characterized by its own pressure. The exchange of liquids between phases is shown in Fig. 2. This model takes into account the distribution and mutual influence of pressures in the phases under consideration, as well as the influence of these pressures on the displacement of the brain substance.

Fig. 2
figure 2

Blood and CSF transfer in the parenchyma of the brain: (1) arterial territory, (2) capillary territory, (3) venous territory, (4) CSF territory, and (5) parenchyma.

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Fig. 3
figure 3

MRI-based geometry of the parenchyma \(\Omega\).

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Mathematical modeling is performed in a two-dimensional approximation. The geometric model is a sagittal section of the brain [17] obtained using MRI (Fig. 3). The modeling of mechanical processes occurring in the brain using two-dimensional sections is widespread [10, 14, 17, 23]. The choice of a sagittal section is due to the fact that it is located in the symmetry plane, which is why possible differences between two- and three-dimensional deformations are minimized (this section remains generally flat in the case of three-dimensional deformations) and which is why it passes through the anatomical structures under study.

An inner boundary denoted as \(\Gamma_V\) represents a boundary between cerebral ventricles, and an outer boundary denoted as \(\Gamma_S\) corresponds to the boundary of the skull. According to clinical studies, the properties of white and gray matter differ. However, there are no experimental data to quantify these differences at the current time, so the brain parenchyma \(\Omega\) is modeled as a homogeneous material [21, 24, 25].

An equilibrium equation for the brain substance has the form [26]

$$ \mu\Delta \boldsymbol{u}+(\mu + \lambda)\nabla(\mathrm{div}\boldsymbol{u}) -(\alpha_a \nabla p_a+\alpha_c \nabla p_c +\alpha_e \nabla p_e+\alpha_v \nabla p_v)=0.$$
(1)

The law of conservation of mass and Darcy’s law for pore fluids are used to obtain the following equations

$$ -(k_a/\mu_a)\,\Delta p_a+|\dot{s}_{ac}|=0;$$
(2)
$$ -(k_c/\mu_c)\,\Delta p_c-|\dot{s}_{ac}|+|\dot{s}_{ce}|+|\dot{s}_{cv}|=0;$$
(3)
$$ -(k_e/\mu_e)\,\Delta p_e-|\dot{s}_{ce}|+|\dot{s}_{ev}|=0;$$
(4)
$$ -(k_v/\mu_v)\,\Delta p_v-|\dot{s}_{cv}|-|\dot{s}_{ev}|=0.$$
(5)

In Eqs. (1)–(5), \(\boldsymbol{u}\) denotes the displacement of the brain substance, \(\lambda\) and \(\mu\) are the elastic moduli, \(p_i\) is the pressure of the corresponding pore fluid, \(\alpha_i\) and \(k_i\) are the Biot and permeability coefficients, \(\mu_i\) is the viscosity of the pore fluids, and \(i = a, c, v, e\). It is assumed that the unidirectional fluid transfer from network \(x\) to network \(y\) is due to a hydrostatic pressure gradient:

$$ \dot{s}_{yx}=-\gamma_{yx}[p_x-p_y].$$
(6)

Here \(\gamma_{y x}\) denotes the coefficient setting the transfer of pore fluids between different territories. System (1)–(5) is supplemented with boundary conditions for the displacement and four pore pressures. On the boundary between the ventricles of the brain \(\Gamma_V\), the following conditions are set.

1. Stresses are assumed to be continuous:

$$ 2 \mu \varepsilon(\boldsymbol{u}) \cdot \boldsymbol{n}+\lambda \epsilon(\boldsymbol{u}) \boldsymbol{n} =\sum_{i=a, c, e, v}(\alpha_i-1) p_i\boldsymbol{n}$$
(7)

(\(\varepsilon(\boldsymbol{u})\) denotes the strain tensor, \(\epsilon(\boldsymbol{u})= \mathrm{tr}\,\varepsilon(\boldsymbol{u}) = \varepsilon(\boldsymbol{u})_{ii} = \mathrm{div}\boldsymbol{u}\), and \(\boldsymbol{n}\) is the outer unit normal vector).

2. No flow for arterial and venous networks:

$$ \nabla p_a \boldsymbol{n}= \nabla p_v \boldsymbol{n} = 0.$$
(8)

3. CSF is secreted at a constant rate \(Q_{p}\) in the ventricles of the brain. Under the condition of conservation of the fluid mass, the following parameters are accounted for in the ventricular system: the volume of the CSF produced by the vascular plexuses, the volume of CSF that seeps through the wall of the ventricles, the outflow of the cerebrospinal fluid through the Sylvian aqueduct.

$$ Q_{p}=\frac{\pi d^4}{128 \mu L}\,(p_e\big|_{\Gamma_V}-p_e\big|_{\Gamma_S}) -\oint\limits_{\Gamma_V}\Big(-\frac{k_e}{\mu_e}\,\nabla p_e\Big) \cdot \boldsymbol{n}\,d S$$
(9)

(\(d\) and \(L\) denotes the diameter and length of the Sylvian aqueduct).

4. The formation of CSF from blood reduces pressure in the capillary network:

$$ \varkappa_{cv} \nabla p_c \boldsymbol{n}=Q_{p}.$$
(10)

Here \(\varkappa_{cv}\) is the resistance of the flow penetrating from the capillary network into the ventricles through the vascular plexus.

The following assumptions are made on the skull boundary \(\Gamma_S\).

1. As this study deals with an adult person’s brain, the skull is considered to be rigid. Thus, the displacements of the skull boundary are equal to zero:

$$ \boldsymbol{u} = 0.$$
(11)

2. There is no capillary flow on the skull boundary:

$$ \nabla p_c \boldsymbol{n} = 0.$$
(12)

3. The values of arterial and venous pressures are set:

$$ p_a=p_{art},\qquad p_v=p_{ven}.$$
(13)

4. CSF absorption into the venous network increases the pressure:

$$ p_e=p_v+\mu_e R Q_{0}$$
(14)

(\(R\) is the resistance caused by the presence of arachnoid granulations, \(Q_{0}\) is the CSF outflow into the venous network, and \(\mu_e\) is the CSF viscosity).

According to [27, 28], the quantities characterizing the pore fluids and the porous matrix have the following values: \(E =584\) Pa [29], \(\nu = 0.35\), \(k_{a, c, v} = 10^{-10}\) m2, \(k_e = 1.4\cdot10^{-14}\) m2, \(\mu_{a, c, v} = 2.67\cdot10^{-3}\) (N\(\,\cdot\,\)s)/m2, \(\mu_e = 8.9\cdot10^{-4}\) (N\(\,\cdot\,\)s)/m2, \(p_{art} = 8000\) Pa, \(p_{v e n} = 650\) Pa, \(Q_p = Q_0 = 5.8\cdot10^{-9}\) m3/s, \(d = 0.004\) m, \(L = 0.07\) m, \(R = 8.5\cdot10^{13}\) m-3, \(\varkappa_{cv} = 6\cdot10^{-4}\) (m\(^5\,\cdot\,\)s)/kg [22, 28], \(\alpha_{a, e, c, v} = 0.99\).

With account for Young’s modulus \(E\) and Poisson’s ratio \(\nu\), the elastic moduli \(\lambda\) and \(\mu\) calculated using the known expressions

$$\lambda = \frac{\nu E}{(1 + \nu)(1 - 2 \nu)},\qquad \mu = \frac{E}{2(1 + \nu)}.$$

1.2. Solution Method

Equations (1)–(5) the boundary conditions (7)–(14) are solved using the finite element method in an open-source software [30]. A weak formation of the problem is obtained by multiplying Eqs. (1)–(5) by the test functions \(\bar{\boldsymbol{u}}\), \(\bar{p}_a\), \(\bar{p}_v\), \(\bar{p}_c\), \(\bar{p}_e \in H^1(\Omega)\), respectively, and integrating them with respect to domain \(\Omega\) with account for the boundary conditions

$$\bar{\boldsymbol{u}}\big|_{\Gamma_S} = 0, \quad\bar{p}_a\big|_{\Gamma_S} = 0, \quad\bar{p}_v\big|_{\Gamma_S} = 0, \quad\bar{p}_e|_{\Gamma_S} = 0.$$

The Gauss formula and the boundary conditions (7)–(14) are applied to formulate the problem as follows: we should determine functions \(\boldsymbol{u}\), \(p_a\), \(p_v\), \(p_c\), and \(p_e \in H^1(\Omega)\), that satisfy the boundary conditions (7)–(14) and the integral relations

$$\int\limits_{\Omega}\Big(\Big( -\lambda \mathrm{div}\boldsymbol{u} + \sum_{i=a, c, e, v} \alpha_i p_i\Big)\mathrm{div}\bar{\boldsymbol{u}} - 2 \mu \varepsilon(\boldsymbol{u}) : \varepsilon(\bar{\boldsymbol{u}})\Big)\,d\Omega -\int\limits_{\Gamma_V} \Big(\sum_{i=a, c, e, v} p_i \boldsymbol{n} \cdot \bar{\boldsymbol{u}}\Big)\,ds = 0,$$
$$\int\limits_{\Omega} \Big( \frac{k_a}{\mu_a}\,\nabla p_a \cdot \nabla \bar{p}_a + |\dot{s}_{ac}| \bar{p}_a\Big)\,d\Omega = 0,$$
$$ \int\limits_{\Omega} \Big( \frac{k_v}{\mu_v}\,\nabla p_v \cdot \nabla \bar{p}_v +( - |\dot{s}_{cv}| - |\dot{s}_{ev}|)\bar{p}_v\Big)\,d\Omega = 0,$$
(15)
$$\int\limits_{\Omega} \Big( \frac{k_c}{\mu_c}\,\nabla p_c \cdot \nabla \bar{p}_c +( - |\dot{s}_{ac}| + |\dot{s}_{ce}| + |\dot{s}_{cv}|) \bar{p}_c\Big)\,d\Omega - \int\limits_{\Gamma_V}\frac{k_c}{\mu_c}\, \frac{Q_p}{\varkappa_{cv}}\,\bar{p}_c\,ds = 0,$$
$$\qquad\int\limits_{\Omega} \Big(\frac{k_e}{\mu_e}\, \nabla p_e \cdot \nabla \bar{p}_e +( - |\dot{s}_{ce}| + |\dot{s}_{ev}|) \bar{p}_c\Big)\,d\Omega -{}$$
$${}-\int\limits_{\Gamma_V} \Big(\frac{Q_p}{4 \pi (r_1 + u)^2 } + \frac{\pi d^4}{4 \pi (r_1 + u)^2 \cdot 128 L \mu_e }\, (p_{v e n} + \mu_e R Q_0 - p_e )\Big) \bar{p}_e \,ds = 0\qquad$$

for the arbitrary test functions \(\bar{\boldsymbol{u}}\), \(\bar{p}_a\), \(\bar{p}_v\), \(\bar{p}_c\), and \(\bar{p}_e\), which satisfy the boundary conditions (15). In the last equation of system (15), the denominator contains the factor \(4 \pi (r_1 + u)^2\), which corresponds to the spherical approximation of the ventricle surface for the second (integral) term in Eq. (9).

2. RESULTS AND DISCUSSION

This work describes dependences between the displacement of the ventricular wall and capillary pressure on \(\gamma_{ac}\), \(\gamma_{cv}\), \(\gamma_{ce}\), and \(\gamma_{ev}\) for the geometry obtained using MRI scans of a real patient’s brain. To obtain these dependences, each parameter \(\gamma_{ac}\), \(\gamma_{cv}\), \(\gamma_{ce}\), \(\gamma_{ev}\) independently acquires values from a 15-element set covering a range of physiologically acceptable parameter values: 0.0001, 0.000 68, 0.0149, 0.1489, 0.8872, 3.8154, 13.1054, 38.2225, 98.5754, 231.567, 507.698, 1063.08, 2181.07, 4531.31, 10 000. Thus, \(15^4\) variants have been calculated. One of them was calculated for approximately 15 s. The results of these calculations are given below in interpolated form.

2.1. Displacement of the Ventricular Wall

The value of the ventricle displacement is barely affected by \(\gamma_{ev} = 10^{-4} {-} 10^3\) D/(N\(\,\cdot\,\)s) (D is Darcy). In the case of larger values of \(\gamma_{ev}\), the value of the displacement begins to become larger, increasing at \(\gamma_{ev} = 10^4\) D/(N\(\,\cdot\,\)s) by 3.5 mm. With an increase in \(\gamma_{cv}\) and \(\gamma_{ce}\), the displacement of the ventricular walls decreases. In the case of small \(\gamma_{ce}\), larger \(\gamma_{ac}\) correspond to larger displacements of the ventricular wall. For \(\gamma_{ce} = 10^{2} {-} 10^4\) D/(N\(\,\cdot\,\)s), the nature of the dependence of the displacement of the ventricular wall on \(\gamma_{ac}\) is reversed.

In the domain of small values of \(\gamma_{ce}\), the ventricle displacements are sufficiently large, which does not correspond to the physiological norm. Small displacements of the ventricles are observed only in the domain of large values of \(\gamma_{ce}\) and at values \(\gamma_{ac} > 10^2\) D/(N\(\,\cdot\,\)s). In this domain, there is a displacement of the ventricular wall, corresponding to the physiological norm: \(|\boldsymbol{u}| \le 2\) mm [31].

2.2. Pressure Distribution on the Ventricular Wall

The capillary pressure is barely affected by \(\gamma_{ev}\). As \(\gamma_{cv}\) and \(\gamma_{ce}\) increase, the capillary pressure on the ventricular wall decreases. Larger values of \(\gamma_{ac}\) correspond to larger values of the capillary pressure.

The values of arterial, venous, and CSF pressures at the ventricle boundary remain approximately constant throughout the parameter range (\(p_a = 60.15\) mm Hg, \(p_v =4.88\) mm Hg, and \(p_e =8.18\) mm Hg) and correspond to the physiological norm [32]. The spread in the pressure values does not exceed 0.01 mm Hg.

Based on the computational results, a domain of parameters \(\gamma_{ac}\), \(\gamma_{cv}\), \(\gamma_{ce}\), and \(\gamma_{ev}\) is constructed, corresponding to the physiological norm for the values of the displacement of the ventricular wall and capillary pressure.

2.3. Displacement of the Ventricular Wall and the Behavior of Capillary Pressure in the Cases of Cylindrical and Clinical Geometry

In [33], for the simple geometry of the brain parenchyma, considered in cylindrical approximation, the nature of the displacement of the ventricular walls and the behavior of the capillary pressure at various values of \(\gamma_{ac}\), \(\gamma_{cv}\), \(\gamma_{ce}\), and \(\gamma_{ev}\) are analyzed. Such a simple geometric model allows one to study the behavior of pore pressures and the deformation of the brain substance using ordinary differential equations, thereby having an almost 20-fold increase in the rate of computations. Therefore, the goal is to compare the values of the displacements of the ventricular wall and the capillary pressure obtained for the simplified cylindrical and clinical geometries. As shown by the comparison results, the dependences of the displacement of the ventricular wall and the capillary pressure on \(\gamma_{ac}\), \(\gamma_{cv}\), \(\gamma_{ce}\), and \(\gamma_{ev}\) are in good qualitative agreement. Figure 4 shows the projections of the tolerance range of \(\gamma_{ac}\), \(\gamma_{cv}\), \(\gamma_{ce}\), and \(\gamma_{ev}\) for these two cases onto a three-dimensional space of \(\gamma_{ac}\), \(\gamma_{cv}\), and \(\gamma_{ce}\), which are curved quadrangular tubes.

Fig. 4
figure 4

Domains of the physiological norm in the space of parameters \(\gamma_{ac}\), \(\gamma_{cv}\), and \(\gamma_{ce}\) for \(\gamma_{ev} = 10^2\) (a–c) and the tolerance range of the capillary pressure (1) and of the displacement of the ventricular wall (2) (d): (a) for the case of simple geometry [33], (b) for the case of clinical geometry [17], (c) for the cases of simple and clinical geometries.

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Figures 4a and 4b show the tolerance ranges of \(\gamma_{ac}\), \(\gamma_{cv}\), and \(\gamma_{ce}\), constructed for cylindrical and clinical geometries, respectively. As shown by the results of comparing these areas in Fig. 4c, the domain corresponding to the cylindrical geometry is nested within the domain corresponding to the clinical geometry. Two pairs of opposite sides of this quadrangle correspond to constraints on the displacement of the ventricular wall and on the capillary pressure on its wall (see Fig. 4d). It can be seen that the pressure constraints mark practically the same region of the space of parameters \(\gamma_{ac}\), \(\gamma_{cv}\), and \(\gamma_{ce}\) for both geometries. The sections of the boundaries of the tolerance ranges of parameters \(\gamma_{ac}\), \(\gamma_{cv}\), and \(\gamma_{ce}\) for different geometries are approximately parallel to each other. With an increase in \(\gamma_{ev}\), the relative position of the three-dimensional projections remains the same, but they are synchronously shifted to the domain of large values of \(\gamma_{ce}\). This arrangement of the tolerance ranges of \(\gamma_{ac}\), \(\gamma_{cv}\), \(\gamma_{ce}\), and \(\gamma_{ev}\) allow one you to select physiologically acceptable values of \(\gamma_{ac}\), \(\gamma_{cv}\), \(\gamma_{ce}\), and \(\gamma_{ev}\) based on a model with cylindrical geometry, which ensures the physiological validity of these values of \(\gamma_{ac}\), \(\gamma_{cv}\), \(\gamma_{ce}\),[3] and \(\gamma_{ev}\) for a clinical case.

CONCLUSIONS

This work describes the modeling of normal-pressure hydrocephalus in which the intracranial pressure slightly differs from a normal value. The investigation is carried out on the basis of data on the brain geometry obtained with the help of MRI of a real patient’s brain. The mathematical basis of the study is a stationary model of multifluid poroelastic filtration. The parenchyma of the brain is modeled using a porous matrix, and communicating liquid phases exist at each point of the pore space: arterial blood, capillary blood, venous blood, and cerebrospinal fluid (CSF). Each liquid phase has its own pressure, and the distribution and mutual influence of these phases are taken into account. The effect of these pressures on the displacement of the brain substance is also taken into account. The equations of poroelastic filtration are supplemented with boundary conditions recorded on the basis of medical data. The resulting boundary-value problem is solved numerically using the finite element method. The displacement of the ventricular wall of the brain and the pressure thereon are considered. The behavior of these quantities is studied for various values of the model parameters in hydrocephalus. For the geometry of the brain with different levels of detail, it is shown how accurately the simplified geometry of the brain allows one to estimate the critical values of pressures and displacements in the case of more complex geometries. In this case, the values of the critical pressure on the ventricular wall coincide with a high degree of accuracy in the case of simplified and clinical geometry. The displacements of the ventricular wall in the case of simplified geometry make it possible to determine the upper bound of the displacement with high accuracy for the clinical case. This mutual arrangement of the tolerance ranges of the parameters allows one to select their physiologically acceptable values based on a model with simplified geometry, while ensuring the permissibility of these parameters for the clinical case at the same time.

This work was financially supported by the Russian Science Foundation (Grant No. 19-11-00069).