=Paper=
{{Paper
|id=Vol-2783/paper15
|storemode=property
|title=The Numerical Assessment of Cerebral Blood Flow in Immature Brain of Preterm Infants (short paper)
|pdfUrl=https://ceur-ws.org/Vol-2783/paper15.pdf
|volume=Vol-2783
|authors=Irina Sidorenko,Varvara Turova,Andrey Kovtanyuk,Renée Lampe
|dblpUrl=https://dblp.org/rec/conf/birthday/SidorenkoTKL20
}}
==The Numerical Assessment of Cerebral Blood Flow in Immature Brain of Preterm Infants (short paper)==
https://ceur-ws.org/Vol-2783/paper15.pdf
The Numerical Assessment of Cerebral Blood
Flow in Immature Brain of Preterm Infants ? ??
Irina Sidorenko1[0000−0003−4158−9416] , Varvara Turova2[0000−0002−6613−3311] ,
Andrey Kovtanyuk2[0000−0002−3286−110X] , and
Renée Lampe2[0000−0002−1016−0249]
1
Mathematical Faculty, Chair of Mathematical Modelling,
Technical University of Munich, Garching, Germany
sidorenk@ma.tum.de
2
School of Medicine, Klinikum rechts der Isar, Orthopedic Department,
Research Unit for Pediatric Neuroorthopedics and Cerebral Palsy
of the Buhl-Strohmaier Foundation,
Technical University of Munich, Munich, Germany
turova@ma.tum.de, andrei.kovtaniuk@tum.de, renee.lampe@tum.de
Abstract. Intracerebral hemorrhage is the most dangerous complica-
tion in the development of premature infants. It is strongly connected
with disturbances in cerebral blood flow (CBF ) and fragility of small
blood vessels in germinal matrix (GM ), which is a highly vascularized
layer of the premature brain. Permanent control of CBF value and its
reaction on changes in mean arterial pressure (M AP ), arterial carbon
dioxide partial pressure (pCO2 ), and oxygen partial pressure (pO2 ) are
of great importance in the clinical treatment of preterm newborns. The
mathematical model for the calculation of CBF in immature brain, ear-
lier proposed by Nikolai Botkin and his colleagues, included the depen-
dence of the number of cerebral vessels, their diameter and length on
the gestational age, volume of GM and brain weight. Furthermore, the
vascular response of CBF to the change of M AP and pCO2 was in-
corporated into the model by increasing or decreasing the diameter of
blood vessels (i.e. vasodilation or vasoconstriction). The objective of the
present study is modeling of pO2 effect on CBF in the immature brain
of preterm infants by accounting the phenomenological (experimental)
dependence of CBF on pO2 changes. Numerically calculated CBF reac-
tivity to changes of M AP , pCO2 and pO2 demonstrates similar values
as those observed in experimental studies. The developed mathemati-
cal model for CBF calculation can be a useful tool in both theoretical
research of blood circulation and clinical nursing of preterm infants.
Keywords: Cerebral blood flow · Preterm infant · Immature brain ·
Germinal matrix · Intracerebral hemorrhage · Mean arterial pressure ·
Carbon dioxide partial pressure · Oxygen partial pressure.
?
Supported by the Klaus Tschira Foundation, Würth Foundation, and Buhl-
Strohmaier Foundation
??
Copyright ©2020 for this paper by its authors. Use permitted under Creative Com-
mons License Attribution 4.0 International (CC BY 4.0).
200 I. Sidorenko et al.
1 Introduction
According to the World Health Organization statistics, more than 10% of infants
are born preterm, i.e. before 37 completed weeks of gestation, and the preterm
birth rate is increasing worldwide [3]. Intracranial hemorrhage (ICH) is the
major complication of the preterm birth that occurs in 20% to 25% of neonates
born before the 30th week of gestation (W G) and/or with body weight less than
1500 grams at birth [31]. It often leads to lifelong impairment, such as cerebral
palsy, and may cause permanent disorder of the postural and musculoskeletal
system, learning disabilities, behavioral problems, speech disorders, perception
deficits and seizure disorders. ICH typically originates in the germinal matrix
(GM ) [1], which is a highly vascularized area of the developing brain. Volumet-
ric analysis of the germinal matrix provided using 3D MR measurements [20]
has shown that the GM reaches its maximum size at 23 W G and practically
disappears by 34 W G. A highly fragile microvessel network of GM is vulnerable
to destruction, which may occur due to the spontaneous fluctuations of cerebral
blood flow (CBF ) caused by the impaired autoregulation [35]. Due to this, the
continuous monitoring of CBF is important issue in clinical nursing of preterm
infants. During the last decades, several techniques, such as near-infrared spec-
troscopy [12], Xenon-133 clearance measurements [19], transcranial Doppler
ultrasonography [27], MRI based arterial spin labeling [11], and diffusion cor-
relation spectroscopy [12] became available for measurements of the entire brain
CBF . However, none of these techniques are currently used in neonatal clinical
practice for the regular monitoring of CBF . The numerical assessment of CBF
using standard clinical records [4, 5, 21, 30] may become a promising approach
for clinical applications.
The previous mathematical models [21, 29, 30] evaluated CBF from the brain
weight (BW ) estimated from the gestational age (GA), mean arterial pressure
(M AP ), and carbon dioxide partial pressure (pCO2 ) available from the standard
clinical records. The arterial oxygen partial pressure (pO2 ) is another important
clinically measured parameter that affects CBF . There is evidence [22, 25] that
the brain reacts to pO2 changes by inverse CBF changes, which is described
by negative pO2 reactivity. Moderate changes in arterial pO2 do not influence
CBF noticeably. Acute hypoxia causes an increase in CBF via vasodilation of
cerebral arteries and arterioles [14, 26]. This elevation appears to be a threshold
phenomenon [22]: CBF does not change until pO2 falls below 50 mmHg [10, 18],
but beneath this limit CBF increases substantially [22, 24]. A strong decrease
of pO2 can increase CBF up to 400%, comparing to normoxia condition [7, 18].
While CBF is greatly increased by a reduction in pO2 , the elevation of pO2
level typically causes less pronounced, but regular reduction in CBF [34]. The
study [26] has shown significant decrease in CBF velocity in 15 of the 17
premature infants with hyperoxia during the first few days of life.
The purpose of the present work is accounting for the effect of pO2 on CBF
in the immature brain of preterm infants. The enhancement of the mathematical
model for CBF calculation [21, 30] is performed by altering the vessels’ diameter
during hypoxia or hyperoxia as a response to changes in arterial pO2 [14, 26].
The numerical assessment of cerebral blood flow in immature brain 201
2 Methods
2.1 Modeling of CBF in the Immature Brain with a Germinal
Matrix
A mathematical model for the calculation of CBF [21, 30] is based on a hier-
archical cerebrovascular model for the adult brain [29] in which the cerebral
vascular system is described by 19 levels according to the morphological charac-
teristics of the vessels. Levels from 1 to 9 correspond to arteries and arterioles,
level 10 accounts for capillaries, and levels from 11 to 19 simulate venules and
veins. To adjust the model to the immature brain, the number of vessels mj as
well as their length lj and diameter dj are scaled down on each level j according
to the brain weight (BW ) of infant as follows:
−1
mj = Mj · 1200/BW − (1200/BW − 1) · |j − 10|/9 , (1)
−1
lj = Lj · 1 + 0.1 · (1200/BW − 1) · |j − 10|/9 , (2)
−1
dj = Dj · 1 + 0.1 · (1200/BW − 1) · |j − 10|/9 . (3)
Here, Mj , Lj and Dj are the number, the length, and the diameter of vessels
in level j of the adult brain, respectively, and the value of 1200 g corresponds
to the approximate weight of the adult brain [29]. The coefficient 0.1 is used to
scale the vessel length and diameter to the experimental measurements [2, 32].
The brain weight of the preterm infant (BW ) is computed from the gestational
age in weeks (W G) according to the regression formula [16]:
BW (W G) = 255.25 − 35.44 · W G + 1.52 · W G2 − 0.01 · W G3 . (4)
Such a model keeps the number of main arteries and veins constant across age
and increases the number of arterioles, capillaries and venules according to the
amount of the brain tissue, which grows with age. In contrast to the vessel’s
number, the length and diameter rise with age for large vessels, but remain the
same for capillaries.
Anatomic analysis of blood vessels in a germinal matrix [1, 2, 32] have shown,
that their morphological and histological characteristics are close to other brain
capillaries. Therefore, the presence of GM is modeled as an additional parallel
circuit on the capillary level (j = 10). The number of non-GM capillaries in the
larger circuit is given by:
mB = M10 /(1200/BW ) · (1 − GMvf ). (5)
The number of GM capillaries in the smaller circuit is given by:
mGM = M10 /(1200/BW ) · GMvf · 1.5. (6)
Here, M10 is the number of capillaries on the 10th level of the adult vascular
network [29], GMvf is the volume fraction of the germinal matrix relative to the
202 I. Sidorenko et al.
total brain volume [20], and the factor 1.5 describes a vascular density correction
factor [2] for the GM . The values of capillary length and diameter are taken
from the literature as follows: lGM = 40 µm, dGM = 6.7 µm for the GM [2, 32],
and lB = 60 µm and dB = 5.6 µm for the rest of the brain [29].
The total CBF is calculated from the Kirchhoff’s law as follows:
19
X −1
CBF = (M AP − Pic ) · RESjlevel . (7)
j=1
Here, Pic is the intracranial pressure taken for preterm infants as 5 mmHg [13]
and RESjlevel is the total vascular resistance of the level j. All levels except for
the capillary one consist of mj parallel connected vessels with the individual
resistance RESj . Thus, the total resistance can be calculated as follows:
RESjlevel = RESj /mj . (8)
The total resistance of the capillary level j = 10, consisting of two parallel
circuits (GM and non-GM capillaries), can be calculated as:
−1
level
RES10 = (RESGM /mGM )−1 + (RESB /mB )−1 . (9)
The resistances RESj , RESB , and RESGM are calculated using a micropolar
fluid model [15, 28]. Thus, the conservation of angular momentum results in new
equations describing the rotation of fluid particles on the micro-scale [17, 33].
2.2 Accounting for Arterial pO2
According to the experimental data, the effect of the pO2 on CBF is different for
three different states: hypoxia with pO2 < 50 mmHg, normoxia with 50 mmHg
≤ pO2 ≤ 70 mmHg, and hyperoxia with pO2 > 70 mmHg. Experimental study
[25] has shown that the direct effect of pO2 on CBF in normoxia state was about
10 times less than the effect of pCO2 , suggesting that the influence of pO2 on
CBF in this case is negligible. Therefore, in the mathematical model presented
here, the effect of pO2 in normoxia is neglected. The effect of pO2 in hyperoxia
or hypoxia is modeled by the decrease or increase of the vessels’ diameter. An
important point is that pO2 and pCO2 influence blood circulation independently
of each other and have an additive effect on the vessel diameter [14, 26]. In the
mathematical model, first the influence of pCO2 on the reference diameter of
the vessel is accounted for, as described in [30], and then the effect of pO2 is
added. The myogenic response to the M AP changes is included afterwards as it
is specified in [30].
The vasodilation during hypoxia and vasoconstriction during hyperoxia is
calculated as a linear increase or decrease of the vessel diameter as:
dpO2 = dpCO2 + pv ∗ dpCO2 . (10)
The numerical assessment of cerebral blood flow in immature brain 203
Here, dpCO2 is the diameter of the vessel after accounting for the pCO2 effect
as described in [30]. The coefficient pv depends on vessel type and is equal to
the relative change of the vessel’s diameter measured in animal experiments [14,
26]. Neither dilations nor constrictions due to pO2 alterations were detected for
capillaries. During hypoxia, a more obvious vasodilation in the veins than in the
arteries was observed. In mild hypoxia, with 40 mmHg ≤ pO2 < 50 mmHg, an
increase from the reference diameter size was 6% for arteries and 9% for veins.
Furthermore, in severe hypoxia, with pO2 < 40 mmHg, the diameter increased
by 20% for arteries and by 34% for veins. While hypoxia caused considerable
increases in blood flow, hyperoxia produced only a moderate decrease. During
hyperoxia, both arteries and veins constricted slightly and similarly. In mild
hyperoxia, with 70 mmHg < pO2 ≤ 80 mmHg, a decrease from the reference
diameter size was 5% for arteries and 6% for veins, while in severe hyperoxia,
with pO2 > 80 mmHg, the decrease was 7% both for arteries and for veins.
3 Results and Discussion
The number of vessels on different levels of the hierarchical cerebrovascular model
versus gestational age in weeks is shown in Fig. 1. While the number of large
arteries (j = 1) and veins (j = 19) stays constant, the number of smaller vessels
(2 ≤ j ≤ 18) increases with gestational age. On capillary level (j = 10), the
number of non-GM capillaries also grows with gestational age, whilst the number
of GM capillaries rapidly decreases and becomes zero after 33 W G (see Fig. 1b).
This is in agreement with the observation that the number of ICH cases rapidly
decreases after 34 W G [1].
Fig. 1. Number of vessels for different gestational ages on different levels of the hier-
archical cerebral vascular model (a) and on the capillary level (b).
204 I. Sidorenko et al.
The initial vessel diameter slowly increases with gestational age (see Fig. 2a).
The CBF reactivity on changes in pO2 , pCO2 , and M AP is regulated by the
vascular activity, i.e. vasoconstriction and vasodilation, which is modeled as the
alteration of the vessel’s diameter. The effect of pO2 changes on diameters of
the largest arteries and veins is demonstrated in Fig. 2b. The most considerable
alteration of the vessel’s diameter is observed for veins during hypoxia. The
diameter of the largest veins increases from 1.53 mm to 2.1 mm at age 23 W G
and from 2.37 mm to 3.2 mm at age 33 W G.
Fig. 2. Initial diameter of vessels for different gestational ages on different levels of the
hierarchical cerebral vascular model (a) and at different values of pO2 (b).
The dependence of CBF on changes in pO2 , pCO2 , and M AP is demon-
strated in Fig. 3. The CBF stays constant for pO2 values between 50 mmHg
and 70 mmHg, as it have been described in [10, 18]. In hypoxia condition, CBF
demonstrates a considerable increase with a threshold phenomenon at pO2 = 50
mmHg described in [18, 22]. At pO2 = 50 mmHg, CBF starts to increase with
decreasing pO2 , reaching a twofold elevation of CBF value with respect to nor-
moxia level (see Fig. 3a) at pO2 = 30 mmHg, as it has been observed in [8, 18].
In hyperoxia condition, the increase of pO2 from 70 mmHg to 80 mmHg causes
the 20% decrease in CBF (see Fig. 3a). Such a reactivity is in good agreement
with experimental studies [6, 8, 23], where CBF reactivity of 10 – 30% per 1 kPa
(7.5 mmHg) increase of arterial pO2 was measured in preterm infants.
Simultaneous dependence of CBF on M AP , pCO2 , and pO2 is shown on
Figure 3b. Whilst the CBF reactivity on pCO2 changes has linear behavior,
the CBF reactivity on M AP demonstrates a plateau which corresponds to the
cerebral autoregulation observed in experiments [1, 9]. The calculated values of
CBF and its reactivity to changes in main medical parameters are in good agree-
ment with experimental measurements presented in the literature [1, 6, 8–10, 18,
22, 23]. Thus, the mathematical model developed provides a realistic description
The numerical assessment of cerebral blood flow in immature brain 205
of physiological processes and can be proposed as a useful tool both for the
theoretical research of cerebral circulation and the clinical nursing of preterm
infants.
Fig. 3. Dependence of CBF on pO2 (a), pCO2 and M AP (b). Both plots correspond
to the 23 W G. The red and green points in both plots correspond to the same values
of pO2 , pCO2 and M AP .
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