INTRADURAL VASCULAR SYSTEM
Elastance Co-efficient of the Intradural Vascular System in
Human Beings: A Modelling Study
Dr Bhupal Singh1 , Prof K.B Sahay2 and Prof. Sneh Anand3
1Department of Electrical Engineering, AjayKumarGargEngineeringCollege, PO Adhyatmic Nagar, Ghaziabad 201009 UP
2, 3Indian Institute of Technology, New Delhi 110016
, 2 ,
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Abstract—In head injuries, hydrocephalus and other neurological conditions, quantification of the fluid induced cerebrospinal fluid (CSF) pressure changes has significant clinical implications. There is prevailing assertion amongst the neuroscientists that the CSF pressure-volume relationship is primarily determined by the elasto-mechanical properties of the brain tissues such as blood vessels and dura. In the present study a model characterizing the relative elastance coefficient (EC) of the intradural vascular system has been proposed and analyzed. The present model proposes that absolute value of EC of the intradural vascular system (IDVS) is inversely dependent upon theintradural blood volume. Hence it supports fact that there is increase in the elastance coefficient of the craniospinal system during hydrocephalus. Employing reported empirical relationship between hydraulic resistance of the venous bed, its blood volume and Hagen -Poiseuille’s equation, it is possible to arrive at result that EC is inversely proportional to the intradural blood volume (IDBV). The negative value of EC within the auto regulatory limits explains the negative slope in the response of the intracranial pressure (ICP) with mean arterial blood pressure (MAP)variation as observed by some researchers. This model also predicts the discontinuity in the EC versusmean arterial pressure (MAP) curve at the upper and the lower auto regulatory limits which support the auto regulatory break down at these limits.
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Keywords: Autoregulatory Reserve Factor, Cerebral Autoregulation, Intradural Blood Flow, Elastance Coefficient and Intradural Blood Volume.
I.INTRODUCTION
IN the case of head injuries, hydrocephalus and few other neuro-pathological conditions, cerebrospinal fluid (CSF) pressure - volume changes have significant clinical implications.
Neuroscientists assert that the CSF pressure- volume relationship is determined primarily by the elastomechanicalproperties of the brain tissues (Lofgren at al.1973). Concerted efforts have also been made in the past to study and quantify the elastomechanical properties of the brain issues. Several ways for determining the elastomechanical properties of the brain tissues have been proposed in the past. Amongst the earliest one is direct catheter coupled manometeric measurement of CSF pressure. Later Marmarou et al. (1971) introduced the concept of compliance to characterize the CSF pressure-volume relationship and found that it is inversely dependent on the CSF pressure. In an attempt for better characterization of brain tissue elastomechanical properties Avezaat et al. (1976) introduced a new parameter, which have been defined as the elastance at the given CSF pressure and is found inversely proportional to the pressure volume index defined by the Marmarou et al. (1971).
Elastance is defined as inverse of compliance, which is the measure of distendibility of a system. The elastance coefficient (EC) is a measure of the toughness of the brain tissues complex, which is primarily determined, by the toughness of the intradural blood vessels (Pronoy et.al.1983).
Various types of tests such as balloon inflation, bolus injection of the mock CSF, pressure displacement and constant rate infusion have been employed to induce change in the CSF pressure in man and animals for assessing elasto-mechanical behavior of the brain tissues. These tests have inherent disadvantages of being time consuming with the allied risks of infection and brain herniation.
In the past minimally invasive method of estimation of EC of the craniospinal tissues has also been proposed by the Avezaat et al. (1971), which involves estimation of pulsatile change in the intracranial pressure (ICP) and cerebral blood volume change per cardiac cycle. A technique for noninvasive estimation of elastance of the intracranial system tissues, proposed by Alexender et al. (1999) is based on the measurement of the blood flow velocity change during the body tilt test. But little work has been done with regard to the factors affecting the EC of the IDVS and its possible non-invasive estimation. Bearing this purpose in mind a model EC of the IDVS has been proposed and analyzed employing the reported data.
II.MATHEMATICAL DESCRIPTION OF THE MODEL
This analysis deals with a vascular system contained within the craniospinal dural sac. This vascular bed is termed as intradural vascular system (IDVS). The blood flow rate through the vascular bed and blood volume within the IDVS is named as intradural blood flow (IDBF) rate and intradural blood volume (IDBV) respectively. The quantities involved in the model such as IDBF rate, IDBV, perfusion pressure and transmural pressure are the steady state values, time averaged
AKGEC JOURNAL OF TECHNOLOGY, Vol.2, No.1
over a cardiac cycle.Blood that carries oxygen and nutrients to the central nervous system (CNS) and its supporting tissues
is supplied through carotid, vertebral and spinal arterial system. This blood after circulating through arteries, arteriole, capillary, and venous bed and finally drains out of CNS through the jugular and emissary veins. The blood volume leaving IDVS per second has been termed as IDBF rate.
The steady state IDBF rate q (ml/s) is offered hydraulic conductance G (ml/s/mmHg) by the IDVS. Therefore applying the Ohm’s law at steady state intradural vascular blood perfusion pressure, p is given below
or
(1)
The flow rate of blood to the CNS has been observed to be relatively constant in spite of the changes in the intradural perfusion pressure; this behavior has been termed as autoregulation. The autoregulation involves changes in the calibers of the arterial blood vessels, in response to the changes in the transmural pressure of arterial blood vessels and its blood flow rate, through the various auto regulatory mechanisms such as humoral, myogenic and neurogenic mechanisms (G. Mchedlishvili 1980).
We define a factor called ARF to quantify autoregulatory behavior of IDVS as given below
(2)
Now from equation (1) and (2) it follows
(3)
where
R is named as blood flow regulation factor; smaller the positive value better the blood flow regulation is considered.
Let the IDVS be comprised of n number of blood vessels. The blood vessels have been assumed to be cylindrical in their shape. Let blood flow rate qi though the ith blood vessel offers conductance Gi then by the Ohms law perfusion pressure pi can be related as follows.
or
(4)
But conductance Gi of the ith blood vessel can be related to blood volume vi contained within the vessel as follows
(5)
Assuming that coefficient of viscosity of blood iand length li of the ith blood vessel remains constant then from equation (5) becomes
(6)
Now applying volume balance within the IDVS, IDBV v at any instant must be equal to the sum of the blood volume contained within the individual blood vessels of the IDVS. i.e.
(7) or
(8)
But from the equations (6) and (8)
(9)
Dividing both side of the equation (9) by v we obtained
(10)
Now defining ratios
and
where
kGi is some factor.
Then equation (10) becomes
(11)
where, K is
Now simplification of equations (2) and (10) and (11) gives
(12)
Figure 1 Shows 2D diagram of the ith blood vessel. It shows the internal diameter 2riand length of the blood vessel li. Also indicates the perfusion pressure pi (mmHg) that causes blood flow qi (ml/s) through the blood vessel and transient accumulation rate of the blood volume vi (ml) within the blood vessel.
Since blood volumes of the IDVS can be related to the EC of the IDVS employing a concept similar to EC as proposed by the Avezaat et al (1976).
Figure 2 (a) Experimental IDBF rate versus MAP data on normal human subjects has been taken as reported by Ursino et al (1997). Data shown with continuous line is polynomial fit of the reported data (shown with stars).
Figure 2(b) ARF versus MAP has been obtained by employing equation (11).
(13)
Where pt= Transmural pressure of the IDVS system, which depends upon the arterial blood pressure and ICP.
Equations (12) and (13) give
INTRADURAL VASCULAR SYSTEM
(14)
Since IDVS comprises of collapsible venous bed, hence venous blood pressure is affected by the ICP. Thus an increase in ICP causes decrease in both perfusion pressure and transmural pressure of the IDVS. On the other handan increase in both the arterial pressure causes increase in the transmural pressure and the perfusion pressure of the IDVS. It is therefore % change in the perfusion pressure and the transmural pressure of the IDVS are assumed to be related through a ratio kr as given below.
(15)
Equations (14) and (15) give
(16)
But, from the equation (3) and (16) it follows that
(17)
Figure. 3(a) Ursino et al (1997) have also simulated the IDABV variation with MAP as originally reported by the Tomita et al (1988). To obtain IDBV, venous blood volume (54ml) has been added to the reported arterial blood volume. Data with the continuous line shows the polynomial fit of the reported data (shown with stars).
Figure 3(b) RelativeEC versus MAP has been obtained by employing equation (19).
AKGEC JOURNAL OF TECHNOLOGY, Vol.2, No.1
Let ECn be the elastance of the IDVS at the basal condition then from the equation (17) ECn can be related as follows
(18)
where, ARFn and vn are the basal value of ARF and IDBV at the normal MAP.
Therefore from the equations (17) and (18) it follows
(19)
With an additional condition the variation in ratioK /Kris inappreciable.
Equation (19), clearly indicates that EC of the IDVS besides, being inversely dependent on the IDBV also affected by the cerebral autoregulation. In the following section behavior of the EC of the IDVS will be analyzed employing the reported experimental data.
III. RESULTS AND DISCUSSION
Human ARF and IDBV or cerebral blood volume (CBV) data with variation in MAP were generated by the polynomials that had been fitted in the least square sense to reported data (Ursino et al 1997) employing MATLAB 6.1 graphical user interface. CBV or IDBV were obtained by adding constant venous blood volume to the arterial blood volume. According to Tomita et al. (1988) venous blood volume normal human subject constitutes 80% (54ml) of the total CBV (67.5 ml Cantos et al. 1978).
A MATLAB 6.1 program which generates and plots: IDBF or cerebral blood flow (CBF) rate (Fig. 2a) and CBV (Fig. 3(a)) employing the polynomial fits; hydraulic conductance of the IDVS using equation (2); ARF (Fig. 2b) employing equation (2); and relative elastance coefficient (Fig. 3b) of the IDVS defined by equation (11) with variation in MAP. Relative EC is defined as the absolute value of EC divided by the basal value of EC of the IDVS.
Figure 2 (a) and Fig. 3(a) show respective plots of reported CBF rate and CBV data (starred) and their polynomial fits (continuous line) together, which indicates good polynomial fit hence it has been assumed that ARF, CBV and ARF generated using these polynomials are error free. It is evident from Fig. 2(b) that ARF has least value (-80) corresponding to MAP value of 110 mmHg.
The ARF curve crosses zero line earlier in case of vasoconstriction that of vasodilation hence supports physiological observation that vasoconstriction is more potent than vasodilation. The vasoconstriction operates in the smaller range (110-150 mmHg) compare to the vasodilation (50-110
mmHg). Within autoregulatory limits (where ARF is negative) between 50 mmHg–150 mmHgof MAP, CBF is controlled actively through the arterial blood vessels caliber changes. Negative value of ARF implies that a change in CBF has an opposite effect on the hydraulic conductance of the IDVS as evident from the equation (2).
Hence smaller (more negative) the value of ARF faster and better the blood flow control is. But outside that auto regulatory limits ARF assumes positive values, which implies that conductance change will produces proportional changes in the CBF rate.
Relative EC (Fig 3(b)) of IDVS is constant (mean relative EC=15 and standard deviation=20) and negative within autoregulatory limits. Its valuefrom negative to positive exhibiting jump type of non linearity at the upper and lower autoregulatory limits.
Such behavior indicates major change in the elastic properties of the vascular bed and CBF dynamics. At upper and lower autoregulatory limits EC respectively jumps from –96.5 to 89.6 and –282.8 to 531.0, which than approaches to respectively high and low pressure values1.4 and 15.0. It is evident from Fig 3(b) that within that autoregulatory limits EC have no significant correlation with MAP (as EC is constant).
Avezaat et al. (1976) found that Elastance Coefficient of intracranial system of the Man has no correlation with MAP. Pronoy et al.(1983) have observed that cerebral vascular bed is major determinant of EC of the craniospinal system hence this supports above result.
Negative value of EC within the auto regulatory range of MAP explains the negative slope observed in steady state ICP versus MAP simulation results reported by the Czonsyka et al. (2000). That simulated response shows positive slope out side the autoregulatory limits.
Negative value of EC in the equation (13) impliesincrease in transmural pressure due to increase in MAP causes decrease in the in CBV, triggers by autoregulatory vasoconstriction and there by providing more space for brain tissues hence decrease in the ICP. MAP increase is balanced by the increase in the transmural pressure of the vascular bed hence vascular bed acts as buffer between the arterial pressure and ICP.
Out side the autoregulatory limits EC of the IDVS is positive. Thus increase in MAP causes increase in the transmural pressure hence from the equation (13) there must be increase in CBV which in turn compresses brain tissues resulting increase in ICP.
Chopp et al (1983) develop a sterling resistor model of the IDVS to be precise the cerebral venous bed. Employed an
empirical relationship of the sterling resistor to the venous bed and related the resistance of the venous bed and blood volume change within the venous bed. That relationship has been employed by them to predict the ICP volume curve, which was found close to the experimental results. Starting from that empirical relationship and applying Pouiseille’s equation to the intradural venous be it can be shown that so derived EC is 2/(blood volume of the venous bed). Hence this analysis also supports the fact that EC of the IDVS is inversely proportional to the IDBV.
The present model explains the implications of EC on the steady state ICP versus MAP response reasonably well. The inverse dependence of elastance of EC of IDVS on cerebral blood volume can be further supported by the observation that in the case of hydrocephalus,EC of the intracranial system increases. As a matter of fact in the hydrocephalic patient venous blood volume is decreased due collapse of venous bed under the raised ICP. Hence increase in EC of both the IDVS and the intracranial system is expected.
The limiting factors of the present model are the assumptions that values of K and Kr do not change appreciably with MAP variation. Thus under the above assumptions relative EC of the IDVS could be estimated non invasively as there are the techniques to estimate CBF and CBV non invasively are reported (Gupta et al. 1997). This model suggests that ARF, hence autoregulation have a more significant effect on the EC of the IDVS as blood flow approaches in the neighborhood the upper and lower autoregulatory limits then that the effect of the ARF on EC of the IDVS in the basal conditions.
IV. CONCLUSION
In the present study relative EC of IDVS involving non-invasively assessable parameters such as blood flow and blood volume has been presented and analyzed. EC of IDVS reflects in the EC of the intracranial system as observed by researchers. Hence EC of IDVS has been used to explain the steady state ICP versus MAP behavior observed in the literature. In the basal condition EC of the IDVS is not affected by autoregulation and its effect is prominent at the upper and lower autoregulatory limits. EC of the IDVS is inversely dependent upon the IDBV.
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/ Dr Bhupal Singh is currently Professor in the Department of Electrical and Electronics Engineering, Ajay Kumar Garg Engineering College Ghaziabad. He is PhD (Biomedical Instrumentation and Control) from IIT Delhi, ME (Control & Instrumentation, Electrical Engineering) from MNNIT Allahabad and BTech. (Electrical Engineering) from IET Lucknow. His areas of interests include: Digital Signal Processing, Intelligent Instrumentation, Nonlinear Control Systems and BiomedicalInstrumentation.
/ Dr Sneh Anand is Professor IIT Delhi. She is Ph.D (Biomedical Engineering) IITD, MTech (Control System) IITD and BE Electrical Engineering form PunjabEngineeringCollege. Her areas of interests include Biomedical Instrumentation, Rehabilitation Engineering, Biomedical Transducers and Sensors, Biomechanics Technology in Reproduction Research, Controlled Drug Delivery System.
Dr K.B. Sahay is former Professor IIT Delhi. He is PhD (Biomedical Engineering) University of Monash, Australiaand B. Tech (Mechanical Engineering) IIT Kharagpur. His areas of interest include Morph. mechanics, Brain Mechanics and Population problems.
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