Coupled Autoregulation Models T.David1, S, Alzaidi1, R. Chatelin1 and H.Farr 1 1

Centre for Bioengineering, University of Canterbury, Christchurch, New Zealand

Abstract— Cerebral tissue requires a constancy of both oxygen and nutrients (notably glucose). During periods of pressure variation, which occur normally as well as in cases of hypo- and hyper-tension, the body's cerebral autoregulation mechanism cause the arterioles to vasoconstrict/dilate, thus maintaining a relatively constant cerebral blood flow. A nondimensional representation of autoregulation coupled with an asymmetric binary tree algorithm simulating the cerebrovasculature has been developed. Results are presented for an autoregulation algorithm of the cerebro-vasculature downstream of the efferent arteries, in this case the middle cerebral artery. These results indicate that due to the low pressures found in the arteriolar structure the myogenic mechanism based on the increased open probability due to pressure of stretched activated ion channels does not provide enough variation in the vascular resistance to support constancy of blood flow to the cerebral tissue under variable perfusion pressure. Results show that under variations in pressure the metabolic mechanism provides sufficient variation peripheral resistance to accommodate constancy of cerebral blood perfusion. Keywords— autoregulation, blood flow.

has focused on the myogenic mechanism where the systemic blood pressure exerts an influence on vascular smooth muscle. Fundamental studies were carried out by Harder [6, 7]. Less research has been done for the metabolic condition where variations in pH and CO2 are considered to be the prime movers for dilation/contraction of the small arterioles deep in the vascular tree. The most comprehensive study of blood flow control has been that of [8], however even in this case several important constants remain unknown making the model somewhat constrained. The present model uses previous work by GonzalezFerrandez and Ermentrout [9] to closely couple a myogenic model and a ne metabolic model of cerebral autoregulation with a representation of a vascular tree taken from the work of Steele et al [10]. In order to allow all parts of the tree to vary in its arterial radius (and hence resistance) the myogenic and metabolic models are non-dimensionalised enabling a single algorithm to be utilised. The work shows that in order to replicate the well-known variation of peripheral resistance with pressure the metabolic model is by far the dominant mechanism, so much so that the myogenic mechanism seems to hardly play a role.

I. INTRODUCTION Cerebral tissue requires a constancy of both oxygen and nutrients (notably glucose). During periods of pressure variation, which occur throughout the normal day as well as in cases of pathological hypo- and hyper-tension, the body's cerebral autoregulation mechanism cause the arterioles to vasoconstrict/dilate in response to changes in cerebral perfusion pressure over a certain range, thus maintaining a relatively constant cerebral blood flow. These effects are of particular importance when investigating how blood is redistributed not only via the circle of Willis but throughout the cerebral tissue. It should be noted that there have been a number of cerebral autoregulation models proposed [1, 2], including models incorporated with a circle of Willis [3-5]. However, the combination of an autoregulation model with a fully populated arterial tree able to regulate dynamically remains a relatively unexplored field. Considerable research effort has been expended in furthering the understanding of how an artery alters its radius and the associated chemical pathways. Particular interest

II. METHODS The creation of vascular tree is based on the binary algorithm of Steele et al [10]. The parent vessel bifurcates in to two daughter arteries according to a phenomenological law described in [10]. The present model utilizes the early work of Kamiya and Togawa [11]which allows an evaluation of the angle between parent and daughter arteries. Such that

cos θ1 =

rp4 + rd41 − rd42 2rp2 rd21

; cos θ 2 =

Figure 1 shows the basic construction.

rp4 + rd42 − rd41 2rp2 rd22

; (1)

F (τ w ) is a strain energy function varying with wall shear stress

τ w , m∞

the equilibrium open channel prob-

2+

Figure 1 In order to map the binary tree into a 3D space we randomise the normal vector of the plane defined by the two daughter artery vectors. This randomization allows for “forcing” the arteries to grow in particular directions as used by Schreiner et al [ref Schreiner]. The tree is terminated when the arterial size reaches a lower limit and we consider the end terminals to be connected to a network of capillaries, each of which is modeled by Krogh cylinder. For our present work the binary tree can be made from up to 1,000,000 segments with approximately 500,000 terminal arteries. Passing “up” and “down” the tree structure determines the peripheral resistance of the vascular bed by assuming Poiseuille flow through the arterioles. The myogenic model uses the work of FerrandezGonzalez and Ermentrout [9] to associate systemic blood pressure with the opening of calcium activated ion channels. This produces variations in cytosolic Ca2+ allowing MLCK (myosin light chain kinase) activation and subsequent smooth muscle contraction. We have introduced extra equations to model the consequencial production of eNOS from both Ca2+ variations and wall strear stress values . We have non-dimensionalised the resulting equations so as to be able to implement this myogenic response over all arteries in the vascular tree. The membrane potential is given by N dv (2) = −∑ gi ( v − vi ) dt i The gi ’s are ion channel conductances and vi Nernst

C

potentials. Ca2+ is given by

dCa 2+ = − ρ m∞ ( v − 1) − kCa 2+ + γ F (τ w ) dt

(

)

(3)

ability for Ca , and k a reverse reaction rate. A similar equation is written for eNOS. Although selective ion channels in the smooth muscle cell of the cerebro-vasculature respond to variations in both CO2 and pH our initial mode for the metabolic response uses the simple assumption that the set of arterioles feeding the capillary bed are in close proximity to the venous return. Excess carbon dioxide in the venules is diffused to the arterioles and induces a relaxation/contraction of the arteriolar radius thus allowing increased/decreased blood flow to convect away carbon dioxide and hence maintain the correct CO2 concentration. This model was also used in the 1D models of Alastruey et al. [12].

Figure 2 Figure 2 shows the proximity of arteriole and venule connected by the capillary bed. Equation (4) represents a conservation equation where the rate of change of carbon dioxide is balanced by the production (due to metabolism) and that convected away by the blood flow. dCO2,tissue (4) = CMRO2 + CBF ( CO2,artery − CO2 ,tissue ) dt where CO2,tissue is the tissue concentration of carbon dioxide, CO2;artery is the arterial concentration of carbon dioxide (assumed to be 0.49ml/ml) CMRO2 is the cerebral metabolic rate of oxygen consumption (assumed to be a constant 0.035ml/g/min for all parts of the brain), which due to the stoichiometry of the aerobic metabolism in the brain tissue, is equal to the cerebral metabolic rate of carbon dioxide production. CBF is the flow rate in the terminal artery and thus entering the capillary bed. We use a reverting differential equation for the radius of the arteriole given by

dr = G ( CO2, sp − CO2 ) dt

(5)

here

CO2,sp is the solution to the homogeneous equation

for CO2,tissue (equation (4)). CMRO2, the metabolic rate of oxygen consumption is determined by a space-time partial differential equation given by

⎛ ∂ 2C 1 ∂C ⎞ ∂C = D⎜ 2 + ⎟ − Ψ (C ) ∂t r ∂r ⎠ ⎝ ∂r

(6)

modelling convection of blood flow through the capillary and oxygen diffusion into the tissue of the Krogh cylinder. Ψ ( C ) is the depletion rate of oxygen in the cerebral tissue. Boundary conditions at the capillary/tissue surface are formed by solving the equation of Blum [ref Blum]. This provides a value of CMRO2 which is a function of the cerebral blood flow in the capillary. Equations 2-5 are solved using a standard Runge-Kutta adaptive time-step algorithm whilst equation 6, due to it’s non-linearity uses an implicit Crank-Nicholson scheme. The binary tree mapping is also non-linear in determining the daughter angles and is solved at each bifurcation by a Newton-Raphson algorithm with an analytical Jacobian.

Figure 3 For comparison, the no-autoregulation case and the ideal autoregulation response are also shown. Our analysis of the pressure distribution in the binary vascular tree shows that the smaller arterioles of this arterial tree contribute the vast majority of the total resistance of the tree, as expected.

III. RESULTS Figure 3 shows a representation of a binary tree mapped to 3D space. The number of terminal segments is small compared to those used for the Figures 4 and 5. Figure 4 demonstrates the relationship between systemic pressure plotted and normalized cerebral blood flowrate for the myogenic model. This is a representation of the blood flow required to perfuse the brain tissue supported by the middle cerebral artery. The plot shows that although the myogenic autoregulation model does provide some change from the no autoregulation case, it does not successfully emulate the ideal observed autoregulatory response.

Figure 4 These smaller arterioles have very low pressures, and according to the myogenic autoregulation model only very small changes occur at these low pressures, indeed these changes could well be dilatory! The percentage change of the smaller arterioles (in the range of 10 and 100 microns) that is required to reach the ideal autoregulation profile is of the order of 30%. This change cannot be induced using the current myogenic autoregulation model.

tree at the arteriolar level. Results show that the metabolic mechanism seems to be the dominant mechanism for cerebral autoregulation.

ACKNOWLEDGMENT This work was partially supported by the neurological Foundation of New Zealand and a Summer Scholarship from the University of Canterbury.

REFERENCES Figure 5

1.

The metabolic algorithm described above is now substituted for the myogenic model where the look-up table contains values of radius and pressure evaluated as before for large times so as to reach asymptotically stable values. For each pressure value ranging from zero to 200 mmHg and for each pressure value in the tree the radii of segments, where these radii lie between 10 and 100 microns, are changed in an iterative manner (using the look-up table previously evaluated from equations 4 and 5 until a converged value of the total peripheral resistance is found. Segments of the tree with radii outside of the range given above are unchanged. This algorithm has been used to yield the following results. Figure 5 plots physiological perfusion pressure vs the total normalised flow rate in the tree. This flow rate is normal−1

ised to that known value ( 2.85mLs ) required to perfuse the volume of the brain supported by the middle cerebral artery at "normal" physiological conditions (systemic pressure of 100 mmHg). As can be seen the metabolic autoregulation model provides the correct variation in peripheral resistance to ensure constancy of flow rate to the cerebral tissue without myogenic support.

IV. CONCLUSIONS By developing a non-dimensional representation of the arterial wall model of Gonzalez-Ferrandez and Ermentrout and the asymmetric binary tree algorithm of (Oufsen3) results are presented for an autoregulation algorithm of the cerebro-vasculature downstream of the efferent arteries, in this case the middle cerebral artery. These results indicate that due to the low pressures found in the arteriolar structure the myogenic mechanism does not provide enough variation in the vascular resistance to support constancy of blood flow to the cerebral tissue under variable perfusion pressure. A metabolic model has been developed under the assumption of close proximity between venules and the vascular

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