1. Introduction
Understanding the mathematics behind the cerebral autoregulation process is of primary importance in the mathematical modeling of cerebral blood circulation and regulation. Impaired cerebral blood flow autoregulation is one of the crucial factors that can cause cerebral hemorrhage in preterm infants. An impaired cerebral autoregulation mechanism is unable to maintain constant cerebral blood flow despite the changes in systemic arterial pressure. As a result, an increase in cerebral blood flow caused by an acute systemic arterial pressure increase can lead to bleeding in the germinal matrix of a preterm newborn [
1], which is a specific region in the immature brain between the thalamus and caudate nucleus, with high vascularity and a fragile capillary network [
2]. Among a variety of cerebral blood flow models (see, e.g., [
3,
4,
5,
6,
7,
8,
9,
10,
11]), one can highlight lumped parameter models based on the analogy to electric circuits [
4,
5,
6,
7,
8,
9,
10]. In [
9,
10], the influence of the germinal matrix on blood flow is taken into account by modeling the capillary level as two parallel connected lumped objects describing the germinal matrix and the remaining part of the brain. In [
4], a cerebral blood flow model is presented in the form of nonlinear ordinary differential equations, which can be seen as a starting point to the automatic control theory oriented cerebral autoregulation modeling, because of its simplicity and, at the same time, ability to reproduce various clinical results [
4].
The first steps towards considering the cerebral blood flow autoregulation modeling problem as a feedback control problem were undertaken in [
7,
8]. In [
7], maintenance of the cerebral autoregulatory function was studied as an optimal conflict control problem. In [
8], a mathematical model of cerebral autoregulation was proposed in the form of a heuristic feedback controller, verified using the techniques of the viability theory [
12].
In the current work, we continue to develop the automatic control theory-based autoregulation considerations in a systemic way. Based on the cerebral blood flow dynamical model of [
4], the nonlinear control theory tools are applied to construct the feedback control laws that describe the mathematics behind the cerebral autoregulation mechanisms. The main results of this paper originate from the suggested idea to interpret the cerebral blood flow autoregulation modeling challenge as an automatic control problem. This is the first time, at least to our knowledge, that the cerebral blood flow model introduced in [
4] is studied as a dynamical system with control input and its controllability properties are analyzed. Because of the model’s intrinsic nonlinearity, such a well-known and effective nonlinear control tool as integrator backstepping combined with barrier Lyapunov functions is used to construct the control laws that recover the cerebral autoregulation performance of healthy humans.
The remaining part of the paper is organized as follows. In
Section 2, the cerebral hemodynamics model equations presented in [
4] are revisited and written in the form of a nonlinear dynamical system with control input. The cerebral blood flow autoregulation modeling problem is formulated as a nonlinear output tracking control problem. In
Section 3, we show the differential flatness property of the cerebral hemodynamics in question, which is used further in
Section 4 to construct the state feedback control laws that model the cerebral autoregulation performance. Numerical simulation results of the suggested cerebral blood flow autoregulation scheme are given in
Section 4. Finally,
Section 5 concludes with some remarks.
2. Problem Formalization
In this paper, we consider the cerebral hemodynamics model introduced in [
4]. For convenience’s sake, let us first summarize the model equations. The model accounts for the hemodynamics of the arterial–arteriolar cerebrovascular bed and large cerebral veins, cerebrospinal fluid circulation and intracranial pressure dynamics; see
Figure 1. The arterial–arteriolar cerebrovascular bed is modeled as a windkessel and is described by means of the hydraulic compliance (storage capacity)
and the hydraulic resistance
variables [
4]. The arterial–arteriolar blood volume
is calculated as below
where
and
stand for the systemic arterial and intracranial pressure variables, respectively. The difference
represents transmural pressure in the arterioles. The rate
q of blood flow through the arterial–arteriolar cerebrovascular bed that enters the skull is written as
where
denotes the capillary pressure variable, and the difference
is the perfusion pressure of the arterioles. Here, by the Hagen–Poiseuille law, the arterial–arteriolar resistance
is inversely proportional to the second power of the blood volume
with a coefficient
[
4], i.e.,
For the large intracranial veins, autoregulatory mechanisms and venous elasticity are neglected. The transmural pressure in the cerebral veins and the venous blood volume are supposed to remain constant. Then, the rate
of the blood flow through the venous cerebrovascular bed is described using a hydraulic resistance
as
with the venous hydraulic compliance being ignored and
treated as a constant [
4]. Here, in (
4), the difference
represents the perfusion pressure of the cerebral veins. Similarly, cerebrospinal fluid production at the cerebral capillaries and reabsorption at the dural sinuses are modeled as static processes. The cerebrospinal fluid production and reabsorption rates
and
are characterized by constant hydraulic resistances
and
, respectively, as below
where
is the venous sinus pressure, which is taken as a constant [
4]. The quantities
q,
and
given by the Formulae (
2), (
4) and (
5) are linked through the algebraic equation
The intracranial pressure
dynamics by Monro–Kellie doctrine are written as follows [
4]:
where
is the intracranial compliance (craniospinal storage capacity), and
stands for the constant rate of possible mock cerebrospinal fluid injection in surgery. In (
7), the intracranial storage capacity
is inversely proportional to the intracranial pressure
with a craniospinal compartment elastance coefficient
, i.e.,
Moreover, one should note that the following condition
on the pressure values is required to hold for all the above considerations to be valid [
4].
Finally, within the cerebral hemodynamics model in question, cerebral blood flow autoregulation is supposed to function only at the level of arterioles and is described in terms of the arterial–arteriolar compliance
. Vasodilation or vasoconstriction of the arterioles is modeled through positive or negative values of the compliance rate
, respectively. In [
4], the following heuristic cerebral autoregulation model is suggested:
where
is a sigmoidal function with saturation,
is a time constant, and
denotes a basal value of the arterial–arteriolar blood flow rate
q required for tissue metabolism.
Notice that the cerebral autoregulation model (
9) is intuitively clear but the authors do not provide any mathematical proof or strict mathematical considerations of its validity in [
4]. In this paper, as is done in [
7], the arterial–arteriolar compliance rate
is considered as a control input. Then, the control purpose is to force the nonzero values of the difference
to zero using nonlinear control theory tools and, thus, provide rigorous mathematical insights into the cerebral blood flow autoregulation mechanism of a healthy human.
In the current work, we adopt the arterial–arteriolar blood volume
and the intracranial pressure
as system state variables, instead of considering the
and
dynamics, as is done in [
4]. This choice of the state variables is more suitable for a thorough analysis of the cerebral hemodynamics in question and control design since
is a crucial quantity defining the behavior of the overall system. In view of (
1), the blood volume
dynamics are governed by the formula
and are determined by the cerebral autoregulation mechanism
, the intracranial pressure dynamics
and the systemic arterial pressure alterations rate
.
After rearranging the terms in the differential-algebraic Equations (
7) and (
10) and taking into account the relations (
5) and (
8), one can avoid algebraic loops and obtain the following dynamical system:
which describes time behavior of the cerebral blood volume
and the intracranial pressure
. Using the Formulae (
1)–(
6), the capillary pressure
and the arterial–arteriolar compliance
quantities in the right-hand side of the system (
11) can be represented as functions of the system state variables
and
in the following way:
Notice that the functions in the right-hand side of the system (
11) depend on the systemic arterial pressure
time behavior. In this paper, we suppose that the arterial blood pressure dynamics are in a steady state, i.e.,
, and the arterial pressure
has a constant value, which is possibly different from the basal one. Then, the choice of the arterial–arteriolar compliance rate
as a control input
u results in the cerebral hemodynamics model
with the arterial-,arteriolar blood flow rate
q being considered as a system output function and in view of (
2), (
3) and (
12), written as
Thus, in summary, the cerebrovascular autoregulation modeling problem in question can be formulated as a constrained (e.g., asymptotic) output regulation control problem for the nonlinear dynamical system (
13), i.e., we find a feedback control law
such that
for all reasonable initial values
,
of the system state variables. In addition, for a proper range of the systemic arterial pressure values
, the quantities
and
are required to remain positive during transients and within reasonable bounds
3. Differential Flatness of Cerebral Hemodynamics
First, let us check the controllability properties of the nonlinear dynamical system (
13) by showing that it is differentially flat [
13]. Recall that a dynamical system of the form (
13) is differentially flat if and only if there exists a scalar function
of the system state variables
,
and, in a general case, of the control input
u with its time derivatives
,
, ⋯,
for some finite natural number
such that
,
and
u can be represented as functions of
and its time derivatives of up to some finite order [
13]. Then, it is well known that differentially flat systems possess good controllability properties (see, e.g., [
13,
14]). To find such a function
, called the flat output, one can exploit the linearity of the functions on the right-hand side of the system (
13) with respect to the control variable
u. The coefficients of
u in (
13) form the vector field
As a flat output candidate for the system (
13), one can take a first integral of the vector field (
17) (see, e.g., [
15]). To find the first integrals of
given by (
17), consider an auxiliary system of ordinary differential equations written in the symmetric form
Integration of (
18) results in the following function:
which has constant values on solutions of the auxiliary system (
18), with its Lie derivative
[
15] along the vector field
being identically zero. Then, the first- and second-order time derivatives of (
19) along solutions of the dynamical system (
13) can be written, respectively, as
where
and
are corresponding functions of their arguments.
Let
,
and
. One can show that the Jacobian matrix of the mapping
defined by the relations (
19) and (
20) is nonsingular at a point
,
if and only if the following condition holds:
Hence, the map
is a diffeomorphism defined for all values
,
such that the inequality (
22) is satisfied. The relationships (
19) and (
20) qualify as a change of coordinates in the state space of the system (
13), with its inversion being written as
. Thus, the system state variables
,
are expressed as functions of the flat output
given by (
19) and its time derivative
defined as (
20). Finally, from (
21), we deduce that the control variable
u can be represented for the nonzero values of
as below
where
Note that one can check that the inequality
at a point
,
is equivalent to the condition (
22). This fact is an outcome of a more general theory of nonlinear dynamical systems presented in [
15]. As a consequence of the above considerations, we conclude that the system (
13) is differentially flat.
4. Nonlinear Output Regulation Control Design
To guarantee the cerebral blood flow regulation (
15), one can first try to find the constant reference values
and
such that the condition
holds. Then, a reasonable control strategy would be to force the differences
and
to zero as
in a controllable way to meet the constraints (
16) by the choice of a state feedback
.
In this paper, the reference value
of the intracranial pressure variable
let us select to nullify the
rate given by (
20) under the basal value
of the arterial-arteriolar blood flow rate, i.e.,
One can easily check that the right-hand side of the expression (
25) under the basal values of model parameters given in [
4] and revised for convenience’s sake in
Table 1 coincides with a basal value of the intracranial pressure
in a healthy human [
16]. Note that, in case some of the model parameter values in (
25) differ from the basal ones of a healthy human, still, one could use, for instance, the constant rate of mock cerebrospinal fluid injection
to obtain a reference value
of the intracranial pressure that stays within medically reasonable bounds [
16].
Then, from the relations (
14) and
, it is deduced that
Note that, for the reference blood volume value (
26) to be correctly defined, the following conditions on the model parameters and the intracranial pressure reference value have to be satisfied:
It is worthwhile to indicate that the validity of the relationships (
27) is inherently related to the model parameters’ consistency and can be easily verified for the parameter values given in
Table 1 within the systemic arterial pressure
autoregulatory lower and upper limits
mmHg and
mmHg, respectively [
4].
In what follows, we exploit the differential flatness property of the cerebral hemodynamics model (
13) conceived in the above section. In the new coordinates
defined by the Formulae (
19) and (
20), the dynamical system (
13) takes the form
with the functions on its right-hand side being introduced in (
24). Further, in view of (
19) for the
variable, we take the reference value
, where
and
are given by (
25) and (
26), respectively. Moreover, by combining the relations (
20) and (
25), one obtains
as the reference value of the
variable.
To achieve the regulation
and
as
, a straightforward control selection would be the state feedback linearization-based design
which results in the following regulation error dynamics:
Then, for any positive gain coefficients
and
, the equilibrium point
,
of the system (
30) is (globally) asymptotically stable.
Notice that the control law (
29) and, hence, the resultant closed-loop dynamics (
30) are defined whenever the control coefficient
in (
28) is not zero. It can be shown that the inequality
holds for the reference values
,
of the arterial–arteriolar blood volume and intracranial pressure variables defined as (
25) and (
26), respectively, under the autoregulatory range of the systemic arterial pressure values
mmHg and model parameter basal values given in
Table 1. Hence, due to the continuity property of the function
, the condition
is satisfied at least in some neighborhood of the point
,
of the system (
28) state space.
It is well known that the control law (
29) cannot explicitly guarantee that the nontrivial trajectories
,
of the closed-loop dynamics (
30) (and, hence, the variables
and
during the cerebral autoregulation transients) remain bounded within the prescribed bounds for all
or do not approach and become stuck in the set
(
).
To avoid the control singularity, i.e., obtaining
at some
, let us redesign the control law (
29) to provide the conditions
with proper positive bounds
,
consistent with the constraints (
16). Notice that, since
, which, by virtue of the relations (
24), is equivalent to the inequality
, there always exist positive constants
,
such that, from (
31), it can be deduced that
for all
. Then, since the change of coordinates
given by (
19) and (
20) defines a diffeomorphism whenever
, the inequalities (
31) can be rewritten in the variables
,
as
with some relevant bounds
and
.
To provide the regulation
,
as
and satisfy the conditions (
32) on the transients simultaneously, we suggest to redesign the control law (
29) by using the integrator backstepping approach [
17] based on barrier Lyapunov functions; see, e.g., [
18].
To this end, consider first a (barrier) function
where
,
,
is a positive design constant. Introduce the error variable
. Here,
is a continuously differentiable function to be defined later, which accounts for the desired reference behavior of the
variable.
The time derivative of
along solutions of the system (
28) is written as
The choice
, where
is a positive gain coefficient, results in
Hence, is negative definite if .
Then, as a Lyapunov function candidate for the whole system, employ the barrier function
where
,
is some positive design constant. Let the values
and
,
be such that
. Note that the function
is positive definite in the domain
and grows unbounded
as
or/and
.
The time derivative of
along solutions of the system (
28) is calculated as below
The control selection
where
is a positive gain coefficient, yields
By completing the squares, one obtains
Hence, the time derivative
is negative definite in the domain
. Moreover, for any positive gain coefficients
,
in the control law (
33), by taking the ratio of the positive design parameters
,
within the Lyapunov function
small enough, one can obtain the values of
, which are arbitrary close to
.
Thus, the equilibrium point
,
of the system (
28) under the control (
33) is asymptotically stable with the domain of attraction
, which is positively invariant (see, e.g., [
19]).
Notice that the difference between the integrator backstepping control law (
33) and the basic feedback linearization-based design (
29) under
and
is in the presence of the extra term
in (
33), which resulted in the desired boundedness with the required bounds property of the transients.
Finally, in summary, in view of the relationships
and (
33), one obtains the following cerebral blood flow autoregulation mathematics:
The numerical simulation results of the cerebral blood flow autoregulation design (
34) performance under the model parameter values indicated in
Table 1 are shown in
Figure 2,
Figure 3,
Figure 4,
Figure 5,
Figure 6 and
Figure 7. First, the arterial pressure steady state value
mmHg, which is deviated from the basal quantity
mmHg of a healthy adult [
4], was considered.
Figure 2,
Figure 3,
Figure 4 and
Figure 5 demonstrate the autoregulation response to high arterial pressure for various control gain coefficients
,
and the selected bounds
,
based on initial values of the system state variables.
Figure 6 and
Figure 7 illustrate the sensitivity of the arterial–arteriolar blood flow rate
q autoregulation response to changes in systemic arterial pressure steady state values. All arterial pressure changes started from the basal steady state value
mmHg.
Notice that the simulation results show the good performance and flexibility of the cerebral blood flow autoregulation scheme (
34). By adjusting parameters
,
and
,
of the control law (
33), one can provide medically reasonable transients within required regulation times and bounds. Even if one control parameter set fails to yield satisfactory autoregulation responses for the whole range of arterial blood pressure values
mmHg, as shown in
Figure 6, one can readily adjust, e.g., the gain coefficient
, as demonstrated in
Figure 7, to obtain reasonable cerebral blood flow autoregulation time behavior.