INTRODUCTION

TOP ABSTRACT INTRODUCTION MODEL VASCULAR NETWORK DISCUSSION APPENDIX A APPENDIX B REFERENCES

PULMONARY CAPILLARY PERFUSION and alveolar ventilation are adequately matched for efficient gas exchange over a wide range of cardiac output from rest to heavy exercise. Normally, this matching is achieved in a largely passive manner, despite the fact that the heterogeneous and asymmetric vascular geometry (10) results in a wide distribution of local flows (16, 17, 22). The pulmonary arteries are also quite distensible, as required to provide the appropriate impedance for the right ventricle output. Even though the pulmonary arterial wall structure varies considerably from the main pulmonary artery to the precapillary terminal arteries (8, 44), the distensibility, defined as the fractional change in vessel diameter per unit change in pressure, is essentially constant and independent of vessel diameter and vessel wall composition (2, 8, 28). The same is true for the veins (1). We have observed that, in model arterial (diverging flow) or venous (converging flow) treelike structures having one common outflow or inflow pressure, respectively, common distensibility results in the fraction of the total flow passing through each vessel segment of the heterogeneous asymmetric tree being constant, regardless of the total flow or the pressure at the inlet(s). This is true, despite the fact that in such a tree the distending pressures and, therefore, the diameters of individual vessels of identical unstressed diameter may diverge substantially when the total flow or inflow pressure is changed. Depending on the functional relation between pressure and diameter, given two identical vessels located in different parts of the tree, the diameter of one may increase while the diameter of the other decreases in response to a given change in total flow; yet the ratio of flows passing through the two vessels will remain the same. Thus the flow distribution, normalized to total flow, will be the same as if the vessel walls were rigid. This perhaps counterintuitive observation led us to the conclusion that the vessel diameter-independent distensibility of the pulmonary blood vessels may be an adaptation that helps fix the pulmonary flow distribution in the face of the large variations in total pulmonary blood flow (30). This conclusion was met with some skepticism, at least in part because of the stipulation of a common terminal outlet pressure (for an arterial tree) or inlet pressure (for a venous tree). Thus, because the capillary inlet pressure distribution is not known, it is not clear to what extent this stipulation might affect the degree to which the idealized model might reflect the behavior of the real system. In the present study, we extend the theoretical analysis to the more general case of an entire vascular network diverging from a single inlet and then converging to a single outlet, with some additional observations on multiple inlet-outlet networks.

Glossary

	Distensibility parameter defined by D/D0 = f(P) = 1 + P (see Fig. 7)
	Parameter in a general vessel segment resistance per unit length formula
	Pressure as a function of length, P = (L)
µ	Individual vessel blood viscosity; viscosity may be different in each vessel but is assumed to be constant within a vessel as flow or diameter is changed
b	A vector in a linear system of equations: Ax = b
b and c	Distensibility parameters defined by D/D0 = f(P) = b + (1  b)ecP
fi	Reference flow in vessel segment i in a vascular network (see Fig. 1); if the vessels were rigid, with diameters fixed at geometry given at zero pressure, fi would be flow in vessel segment i when total network inlet flow is 1 and outlet pressure is 0
i and j	Vessel segment (flow through or resistance of a vessel segment) or a node (pressure or nonlinearly transformed pressure at the inlet or outlet of a vessel segment)
f(P)	Vessel diameter-distensibility relation
pi	Reference pressure at node i in a vascular network (see Fig. 1); if vessels were rigid, having diameters fixed at geometry given at zero pressure, pi would be pressure at node i when total network inlet flow is 1 and outlet pressure is 0
nj	Occasionally used to emphasize that subscripting refers to node number, rather than vessel segment number
r0	Resistance in a vessel segment if vessel diameter is fixed at geometry given at zero pressure
ri	r0 for vessel segment i in a vascular network
x	Solution to a linear system of equations: Ax = b
y	Solution to a linear system of equations: Ay = Fb
A	Matrix containing conservation of flow and vessel segment pressure drop using the usual Ohm's law (reference calculation) or the nonlinearly transformed pressure Ohm's law (see Eq. 6)
C	Parameter in a general vessel segment resistance per unit length formula
D	Vessel diameter, which changes with pressure (P) according the model D(P) = D0f(P)
D0	Vessel diameter at zero pressure: D(0) = D0
F	Flow in a single vessel segment or total network flow
Fi	Flow in distensible vessel segment i
FR and FL	Flow in right and left daughter vessels, respectively, at a bifurcation
L	Length variable
P	Transmural and vascular pressures, which, for simplicity, are taken as equal
Pi	Pressure at node i
Pin	Pressure at the inlet of a vessel segment
Pout	Pressure at the outlet of a vessel segment
P	Upstream-downstream pressure drop across a vessel segment: P = Pin  Pout
(P)	Nonlinear transformation of P, e.g., for Poiseuille flow and f(P) = 1 + P, .  enables use of an equivalent Ohm's law for distensible vessels where upstream-downstream pressure drop for a single vessel [(Pin)  (Pout)] is directly proportional to flow through the vessel segment; constant of proportionality is resistance if vessel diameter is fixed at zero pressure
nj	Nonlinear transformed pressure: nj = (Pnj) at node nj
	Nonlinear transformed upstream-downstream pressure drop for distensible vessels:  = (Pin)  (Pout)

View larger version (20K): [in this window] [in a new window]   Fig. 1.   Vascular network consisting of 22 vessel segments. Each vessel is assigned a unique vessel segment number, i, ranging from 1 to 22. Filled circles denote the 18 locations or nodes (n1-n18) where pressures are determined. Vessel 1 is inlet vessel.

	MODEL VASCULAR NETWORK

TOP ABSTRACT INTRODUCTION MODEL VASCULAR NETWORK DISCUSSION APPENDIX A APPENDIX B REFERENCES

The first model vascular networks we consider are those with a single arterial inlet and a single venous outlet. To simplify the notation, intravascular pressure at a point in the network will be taken to be the same as transmural pressure at the point, which we will refer to jointly as the pressure (P). We employ the standard development for relating pressure to flow within distensible vessels (5, 9, 13, 28, 30, 36, 53).

First, consider a single distensible vessel subjected to constant, nonpulsatile flow. For ease of exposition, assume Poiseuille flow within the vessel (13). If a vessel segment is modeled as a distensible right circular cylinder and entrance effects are ignored, the local frictional pressure drop per unit length from inlet to outlet of a single vessel is represented as

(1)

where P denotes pressure, L is vessel length, µ is blood viscosity, F is vascular flow, and D is vessel diameter (13). The model vessels share a common diameter-pressure relation given by

(2)

We refer to Eq. 2 as the vessel distensibility relation. For example, f(P), which has been discussed in the pulmonary circulation literature, includes f(P) = 1 + P and f(P) = b + (1  b)e^cP (1, 2, 13, 53).

Throughout the analysis, we assume that 1) f(P) is sufficiently smooth, so when D = D₀f(P) is used in Eq. 1, the differential equation has a unique solution, 2) f(0) = 1 and f(P) > 0 for all P, and 3) distensibility is constant throughout the vascular network; i.e., f(P) in Eq. 2 applies to each vessel within the vascular network. No additional restrictions are placed on f(P). Under the above assumptions (13, 30, 36) we can conclude the following.

Lemma 1. If (P) denotes an antiderivative of the fourth power of f(P), then

(3)

where P_in and P_out are the inlet and outlet pressures of the vessel, respectively, and r₀ is the vascular resistance at the zero pressure diameter (D₀).

Remark 2. In lemma 1, individual vessel blood viscosity is part of r₀, rather than part of . Thus µ appears as a multiplicative constant separated from . This separation allows us to view µ as possibly being different from vessel to vessel, and µ affects only the vessel's r₀.

Remark 3. As pointed out elsewhere (30), lemma 1 is not the most general result possible, because any resistance per unit length formula that allows the separation of D₀ and f(P) could be employed, giving rise to an appropriate . An example of a separable local resistance per unit length relation that describes how P changes with L would be as follows: dP/dL = CµDF, where F is flow in a particular vessel segment, D is vessel segment diameter,  > 0 is fixed, and C and µ > 0 are constant within an individual vessel but might change from vessel to vessel; however, they do not change with diameter or flow. With a distensibility relation, f(P), such that
has a unique solution: P = (L); then, as shown elsewhere (30)

(4)

or

(5)

where P_in and P_out are the inlet and outlet pressures of the vessel segment, respectively, and  and r₀ are modified from their Poiseuille flow-derived formula.

(6)

(7)

Remark 4. Because antiderivatives differ by at most an additive constant, we may select  such that (P_n1) = 0 at pressure P_n1. This is equivalent to selecting the nonlinear outlet pressure to be the zero baseline, simplifying notation and computations without affecting the generality of the results. We do not require that P_n1 = 0.

(8)

(9)

(10)

Theorem 5. Suppose that, in an arbitrary single inlet-single outlet vascular network, 1) every vessel segment has the same distensibility relation, D/D₀ = f(P) (each vessel segment, however, may have a different D₀), 2) µ, although it may be different in each vessel segment, remains constant within a vessel segment as flow or diameter changes, and 3) up to a multiplicative constant, every vessel segment has the same separable local resistance per unit length relation. Then, for each vessel segment i in the vascular network, there exists a unique constant f_i, independent of F, such that

(11)

relating the flow in the vessel (F_i) to the nonzero total inlet flow (F).

Corollary 6. Under the same suppositions as for theorem 5, if F_R denotes the flow in one daughter vessel at a bifurcation and F_L denotes the flow in the other daughter vessel, then F_R/F_L is the same for every nonzero total inlet flow (F) through the vascular network.

View larger version (11K): [in this window] [in a new window]   Fig. 2.   An autoregulatory-like distensibility relation. Vertical axis is distensibility D/D0 = f(P); horizontal axis is pressure (P). Peak distension occurs at P ~ 3.6.

View larger version (13K): [in this window] [in a new window]   Fig. 3.   Antiderivative  for 4th power of f(P) in Fig. 2. Vertical axis, abstraction of the concept of pressure in distensible vessels [(P)]; horizontal axis, actual pressure (P).

View larger version (25K): [in this window] [in a new window]   Fig. 4.   Pressure-flow relation for individual vessels for vascular network in Fig. 1. Horizontal axis, total vascular network flow (F). Individual vessel flow is a constant fraction of total vascular network flow. Vertical axis, pressure (P) at nodes.

View larger version (24K): [in this window] [in a new window]   Fig. 5.   Individual pressure drop vs. flow for each vessel in network in Fig. 1. Horizontal axis, individual vessel flow; vertical axis, pressure difference between nodes at each end of vessel segment. Numbers (1-22) indicate vessel segments corresponding to plot.

View larger version (41K): [in this window] [in a new window]   Fig. 6.   Hemodynamic calculations for network topology in Fig. 1 using distensibility relation D/D0 = 1 + P. Each column represents a different choice of :  increases with decreasing vessel size (left),  is the same for all vessels (middle), and  decreases with decreasing vessel size (right). From top to bottom: pressure at 18 nodes as a function of total network flow, vessel segment flow as a function of total network flow, fraction of total flow in a vessel segment as a function of total network flow, and pressure drop across a vessel segment as a function of vessel segment flow. Negative values indicate retrograde flow.

	DISCUSSION

TOP ABSTRACT INTRODUCTION MODEL VASCULAR NETWORK DISCUSSION APPENDIX A APPENDIX B REFERENCES

The primary observation of this study is that if a heterogeneous asymmetric vascular network (having the stated properties) consists of blood vessels, each of which has the same distensibility relation, despite potentially wide variations in pressures within vessels that have a common diameter at a given pressure, the flow distribution within the network will be unaffected by changes in total network flow and the accompanying redistribution of pressures. A similar observation was made previously for diverging (arterial) and converging (venous) trees, with the restriction that there was a common outlet or inlet pressure, respectively. Because there is little available experimental information on capillary pressure distributions (24), it is unknown how damaging this restriction might be to the relevance of the theorem to any real vascular system. Extension to the single inlet-single outlet network helps address this question to the extent that an arterial (or venous) tree can now be considered a part of a network for which there is somewhere downstream (or upstream) a common pressure and in which distributed arterial outlet or venous inlet pressure would be the normal condition for a heterogeneous asymmetric network.

As indicated in the introduction, this study was motivated by an attempt to understand the significance of observations indicating that the distensibility of the pulmonary arteries (Fig. 7) and veins (1) is virtually independent of vessel size. The network model does not completely resolve the question, because the specific assumption invoked is that the arteries, capillaries, and veins have the same distensibility relation. Over the physiological range of pulmonary pressures, that assumption appears to be reasonable for the pulmonary arteries and veins (1, 2, 26). Whether the same can be said for the capillaries is not so clear; in part, it depends on whether the capillaries are viewed as cylinders (21, 47), which distend with a uniform increase in diameter, or as a punctuated sheet, wherein distension is only orthogonal to the alveolar surface (13, 14, 47), or somewhere between these extremes. For a cylindrical capillary, the geometric component of the resistance would involve the fifth power of the diameter, as in the cylindrical arteries and veins, whereas the sheet resistance involves the fourth power of the dimension orthogonal to the alveolar surface (13). On the other hand, the available data suggest that the capillary distensibility is at least within the same order of magnitude as the arteries and veins, with values of capillary distensibility (defined as the fractional change in the vessel dimension orthogonal to the alveolar surface per unit change in pressure), obtainable from the literature, ranging from ~0.023/mmHg for the dog lung (40) to ~0.07/mmHg for cat and dog lungs (13, 15).

View larger version (19K): [in this window] [in a new window]   Fig. 7.   Pulmonary arterial distensibilities () from 6 species obtained from 26 studies of vessels of various internal diameters (D), where  is defined by D(P) = D(0) + P/D(0). Values were reported as such in cited reference or estimated from data provided in the reference. Transmural pressure (P) range was generally 0-30 mmHg. Dashed line, representing  = 0.02/mmHg, appears to reflect central tendency of data reasonably well.

Cox (8) was apparently the first to point out that the mechanical properties of the pulmonary arterial vessel walls are essentially independent of vessel diameter and the composition of the individual vessel walls defined by relative amounts of connective tissue and smooth muscle. This is reiterated by the compilation of data in Fig. 7, updated from the findings reported by al-Tinawi et al. (2), wherein the vessel diameter independence of the distensibility coefficient  is reflected by the fact that the data from several studies obtained using various methods can be correlated by a virtually constant value of  over several orders of magnitude in D₀ from the main pulmonary artery to terminal arterioles and represent many more orders of magnitude in individual vessel segment resistances. Despite the diameter independence, there is, in fact, variability in the individual values within a given diameter range, even between studies on the same species. The reasons for this are not clear but may reflect sensitivity to some aspect(s) of study conditions that has not been systematically identified. Thus it seems probable that the variability within a given diameter range in Fig. 7 is greater than would be expected within any particular individual lung. However, the objective of the analysis is not to provide an argument that the distensibility is constant. Rather, it points out that limits on the distribution of individual vessel distensibilities would be a logical result of evolutionary pressure to maintain gas exchange efficiency (i.e., the ventilation-perfusion distribution) over a wide range of cardiac output.

Determination of the impact of the various obvious differences between the model and the real system (e.g., pulsatile flow and gravity effects) will probably require numerical simulations beyond the scope of present study. Thus, even having generalized the model to encompass an entire network, it remains idealized. This allows for the analytic approach to understanding the model behavior, and we believe that the observations provide a reference point for understanding the implications of vascular network design in a sense similar to other idealizations, including "Poiseuille's law," "sheet flow," the "fifth power law" (13, 53), "Murray's law" (32), and others (31, 36, 43, 49).

It may also be useful to reiterate that the theorem presented here is not dependent on the assumption of Poiseuille flow. Rather, in the deviation of Eq. 9, the existence of a  and the reference r₀ is what is needed, where the reference r₀ might be thought of as the resistance that would exist if the vessel diameters were fixed at their zero pressure values. This is accomplished if, up to a multiplicative constant, each vessel in the vascular network has the same local normalized resistance per unit length expression. Although the results allow for blood viscosity being different in each vessel, the restriction of constant viscosity within a vessel may be viewed as more limiting. However, changes in viscosity within any single vessel segment due to physiologically reasonable flow or diameter changes are small (29).

Although our primary goal was to examine the potential for fixed flow partitioning within a heterogeneous asymmetric vascular network, the methodology can be employed to determine flows within multiple input-multiple outlet vascular networks where each vessel experiences the same distensibility relation (see APPENDIX B). It is further clear that reference flow distribution calculations apply to the distensible vessel case under any of the following conditions: 1) a multiple inlet-single outlet vascular network, where the inlet flows may increase or decrease, but inlet flows are delivered in a fixed ratio, 2) a single inlet-multiple outlet vascular network, where all outlet pressures are fixed at the same value, and 3) a multiple inlet-multiple outlet vascular network, where the inlet flows may increase or decrease but the inlet flows are delivered in a fixed ratio and all outlet pressures are fixed at the same value. In each case, the individual vessel segment flows throughout the network would follow the constant partitioning results described above.

The observation that distensibility of the pulmonary arteries and veins is diameter independent over several orders of magnitude in vessel diameter may reflect a design feature that takes advantage of the observations described above. A structure with vessels sharing a common distensibility should result in a stabilizing effect on the impact of changing cardiac output on the pulmonary capillary flow distribution without requiring an elaborate controlling mechanism (28). When the distensibility is not constant throughout the network, in particular, when it is diameter dependent, the fraction of total flow within any one branch of the network may diverge from the initial flow distribution, and even reversal of flow in some segments is possible (Fig. 6). Some observations of the effects of changing cardiac output on the pulmonary flow distribution (4, 23, 46) appear to be generally consistent with a nearly constant flow distribution.

These observations may have implications for the function of diseased lungs that are somewhat analogous to the effect of the distribution of airway mechanics on the breathing frequency dependence of the distribution of ventilation. That is, in diseased lungs having an abnormally broad distribution of time constants among respiratory units, the increase in breathing frequency generally accompanying an increase in total ventilation results in a redistribution of the fraction of the total ventilation received by a given respiratory unit (41). Likewise, increasing cardiac output in a lung with a disease extended distribution in individual vessel distensibilities and would tend to result in a redistribution of blood flow. Little information is available regarding any changes in the longitudinal or parallel distributions of distensibilities of vessels that might occur as the result of pulmonary vascular remodeling in pulmonary diseases.