The Numerical Assessment of Cerebral Blood Flow in Immature Brain of Preterm Infants ? ??

Abstract

Intracerebral hemorrhage is the most dangerous complication in the development of premature infants. It is strongly connected with disturbances in cerebral blood ow (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, earlier proposed by Nikolai Botkin and his colleagues, included the dependence 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 incorporated 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 e ect on CBF in the immature brain of preterm infants by accounting the phenomenological (experimental) dependence of CBF on pO2 changes. Numerically calculated CBF reactivity to changes of M AP , pCO2 and pO2 demonstrates similar values as those observed in experimental studies. The developed mathematical 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 ow
• Preterm infant
• Immature brain
• Germinal matrix
• Intracerebral hemorrhage
• Mean arterial pressure
• Carbon dioxide partial pressure
• Oxygen partial pressure

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 de cits and seizure disorders. ICH typically originates in the germinal matrix (GM ) [ 1 ], which is a highly vascularized area of the developing brain. Volumetric 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 uctuations of cerebral blood ow (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 spectroscopy [ 12 ], Xenon-133 clearance measurements [ 19 ], transcranial Doppler ultrasonography [ 27 ], MRI based arterial spin labeling [ 11 ], and di usion correlation 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 a ects 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 in uence 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 signi cant decrease in CBF velocity in 15 of the 17 premature infants with hyperoxia during the rst few days of life.

The purpose of the present work is accounting for the e ect 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 ].

Methods

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 hierarchical cerebrovascular model for the adult brain [ 29 ] in which the cerebral vascular system is described by 19 levels according to the morphological characteristics 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: mj = Mj 1200=BW (1200=BW 1) jj 10j=9 1; lj = Lj dj = Dj 1 + 0:1 (1200=BW 1 + 0:1 (1200=BW 1) jj 1) jj 10j=9 10j=9 1; 1: 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 coe cient 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: The number of GM capillaries in the smaller circuit is given by: mB = M10=(1200=BW ) (1

GMvf ): mGM = M10=(1200=BW ) GMvf 1:5: 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 (1) (2) (3) (5) (6) 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 m, dGM = 6:7 m for the GM [ 2, 32 ], and lB = 60 m and dB = 5:6 m for the rest of the brain [ 29 ].

The total CBF is calculated from the Kirchho 's law as follows: CBF = (M AP

Pic) Here, Pic is the intracranial pressure taken for preterm infants as 5 mmHg [ 13 ] and RESlevel is the total vascular resistance of the level j. All levels except for j 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 : The total resistance of the capillary level j = 10, consisting of two parallel circuits (GM and non-GM capillaries), can be calculated as:

RES1le0vel = (RESGM =mGM ) 1 + (RESB=mB) 1 1 : The resistances RESj , RESB, and RESGM are calculated using a micropolar uid model [ 15, 28 ]. Thus, the conservation of angular momentum results in new equations describing the rotation of uid particles on the micro-scale [ 17, 33 ]. (7) (8) (9) 2.2

Accounting for Arterial pO2 According to the experimental data, the e ect of the pO2 on CBF is di erent for three di erent 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 e ect of pO2 on CBF in normoxia state was about 10 times less than the e ect of pCO2, suggesting that the in uence of pO2 on CBF in this case is negligible. Therefore, in the mathematical model presented here, the e ect of pO2 in normoxia is neglected. The e ect 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 in uence blood circulation independently of each other and have an additive e ect on the vessel diameter [ 14, 26 ]. In the mathematical model, rst the in uence of pCO2 on the reference diameter of the vessel is accounted for, as described in [ 30 ], and then the e ect of pO2 is added. The myogenic response to the M AP changes is included afterwards as it is speci ed 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) Here, dpCO2 is the diameter of the vessel after accounting for the pCO2 e ect as described in [ 30 ]. The coe cient 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 ow, 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 di erent 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 ].

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 e ect 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.

The dependence of CBF on changes in pO2, pCO2, and M AP is demonstrated 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 normoxia 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 agreement with experimental measurements presented in the literature [1, 6, 8{10, 18, 22, 23]. Thus, the mathematical model developed provides a realistic description 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.

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