INTRODUCTION

Top Abstract Introduction Methods Results Discussion References

KNOWLEDGE OF THE relationship between structure and function of the normal pulmonary arterial tree is necessary for understanding normal pulmonary hemodynamics and the functional consequences of the vascular remodeling that accompanies pulmonary vascular diseases. It is also a step in the process of developing hypotheses regarding pulmonary arterial tree morphogenesis (49). As improved imaging techniques are applied to the pulmonary vasculature, the means for interpreting the morphometric data will become more important as well.

The complexity of the pulmonary arterial tree structure can make it difficult to recognize the functional significance of the quantifiable morphometric and biomechanical variables. One approach to clarifying relationships between structure and function has been to develop mathematical models that take advantage of data on the geometric and viscoelastic properties of the vessels and arterial tree to link structure and function with a much smaller number of parameters than the number of individual vessels comprising the tree. There are several examples of this approach (4, 6, 48, 51), with a progression of models with increasing ability to predict the complexity of the hemodynamic behavior of the vascular tree while minimizing the complexity of mathematical expression. The latter is valuable insofar as it allows for a more intuitive understanding of the quantitative contribution of different variables to system function than do more detailed representations. The present study is a further attempt to find efficient ways of expressing the influence of vascular geometry, vessel mechanics, and blood rheology on mean pressure-flow relationships within the intrapulmonary arterial tree.

One approach to this problem has been to apply Poiseuille's law for flow through fixed-diameter (16, 44) or distensible (20, 55) cylindrical tubes arranged in a symmetrical pattern with vessel dimensions based on morphometric measurements summarized according to a particular ordering scheme. Each order is characterized by a separate set of parameters, and then the equations for each order are concatenated (20, 33, 55). An alternative is to use a continuum model, which attempts to express the tree structure-function relationships with global parameters, minimizing the emphasis on particular ordering schemes (32, 48). In the present study we present some further attempts toward maximizing the efficiency of expression of the pulmonary arterial structure-function relationships, and we explore some model predictions. There is no expectation that the approach will provide model expressions that accommodate all the details of the pulmonary arterial tree function. Instead, the objective is to provide expressions that demonstrate aspects of the sensitivity of the hemodynamic function of the pulmonary arterial tree to tree geometry and vessel mechanics in a similar sense that Poiseuille's law is useful for that purpose for individual vessels or that the sheet flow model of Fung and Sobin (14, 15) is useful for the pulmonary capillary bed. As is the case for those models, the expressions may also serve as building blocks for more comprehensive model development in the future.

To help put parameter values used in model simulations in perspective, we also include some morphometric and hemodynamic measurements carried out on isolated dog lungs.

Glossary

a₂

Antilog of intercept of log L_j vs. log D_j, also defined by Eq. 2

Fractional change in vessel diameter per mmHg change in intravascular pressure

₁

Slope of log N_j vs. log D_j, also defined by Eq. 1

₂

Slope of log L_j vs. log D_j, also defined by Eq. 2

D

Vessel diameter

D_F

Diameter of vessels in which viscosity is midway between µ_p and µ_b

D_T

Terminal arteriolar diameter

D(x,P)

Arterial diameter-pressure relationship

D(0)

= a1/11

D(0,0)

Inlet diameter (i.e., at x = 0) at zero pressure

D_j

Diameter of vessels in order j

D₁

Diameter of parent vessel at a bifurcation

D₂

Diameter of the larger of two daughter vessels at a bifurcation

D₃

Diameter of the smaller of two daughter vessels at a bifurcation

D_in,j

Diameter at inlet of vessel of order j

D_out,j

Diameter at outlet of vessel of order j

G

= (1  2^₂/₁)/[₂D(0)^₂]

K

Defined by Eq. 31

L_j

Length of vessels in order j

µ

Blood viscosity

µ_b

Large vessel blood viscosity

µ_p

Plasma viscosity

N_j

Number of vessels in order j

P

Intravascular pressure

P_in,j

Intravascular pressure at inlet of order j

P_out,j

Intravascular pressure at outlet of order j

Q

Cumulative blood volume upstream from all locations at distance x from inlet

Q_j

Volume in order j

(0)

Total flow rate entering arterial tree

(x)

Flow rate in vessel at distance x from inlet to arterial tree

x

Distance from arterial inlet (x = 0) to a location within the arterial tree

z

Defined by Eq. 32

	METHODS

Top Abstract Introduction Methods Results Discussion References

Model

(1)

(2)

In the present investigation we take the continuum point of view (32, 48), from which vessel ordering is of minimal importance. For the continuum point of view expressed by Suwa et al. (48), any route the blood follows from pulmonary artery to a terminal arteriole may be visualized as being confined within a tapering tube through which the blood flow decreases with distance from the inlet to account for flow drawn off at the many bifurcations along the route. An alternative visualization leading to the same development might be to think of the vascular tree as branching continuously, such that N or j need not be thought of as having only integer values. Figure 1 is a diagrammatic representation of one such route through the tree depicted as a trunk with side branches pruned to emphasize the taper of the trunk. The taper is quantified by ₁, ₂, and a₂.

View larger version (22K): [in this window] [in a new window]   Fig. 1.   Diagrammatic representation of a pulmonary arterial tree trunk with pruned side branches. Taper of trunk, expressed as diameter (D) as a function of distance (x) from inlet, is specified by 3 parameters: 1, 2, and a2. Aspect (diameter-to-length) ratio was chosen for diagrammatic purposes.

We begin with the examination of a route through a symmetrical, bifurcating tree. After Suwa et al. (48) and noting that, for a bifurcating tree, N_j = 2^(j  1), each vessel segment may be considered to be a frustum of a cone having diameter D_in,j = D_in,1/2^(j  1)/₁ at the inlet to the vessel segment and D_out,j = D_in,1/2^j/₁ at its exit with D_in,j = D_out,j  1. We define x as the cumulative length of a route from the arterial inlet to, but not including, the vessel having diameter D_in,j. If the route is visualized as consisting of discrete vessel segments, the cumulative length at the inlet would be x(D_in,1) = 0. At the end of order 1, x(D_out,1) = L₁ = a₂D_²in,1. At the end of order 2, x(D_out,2) = L₁ + L₂ = a₂D_²in,1(1 + 2^₂/₁), and at the end of order j

(3)

To make the transition to the continuum visualization of the tree, we replace D_out,j with D, and thus one can view

(4)

as the distance upstream from diameter D to the arterial inlet. Inverting Eq. 4, we obtain or simply

(5)

where D(0) = a1/11 and G = (1  22/1)/[a2D(0)2]. The concept of order or generation is thus replaced by establishing a relationship between the diameter D at any location along a route through the tree at distance x from the arterial inlet.

If the morphometric measurements were available for a given steady flow and terminal arterial pressure and under the assumption that the pressure-flow relationship is governed by Poiseuille's law, the vascular pressure (P) at any x could be calculated as follows

(6)

where

(7)

With use of Eq. 5

(8)

Thus

(9)

and

(10)

It will also be useful to keep track of the relationship between vascular pressure and cumulative vascular volume [Q(x)], the volume of the entire tree upstream from distance x into the tree (12, 31), which can be obtained as follows

(11)

With use of the morphometric relationship

(12)

the cumulative arterial tree volume (Q) as a function of length from the arterial inlet (x) is then given by

(13)

Thus the pressure at a distance x in a vessel having diameter D(x) at cumulative volume Q(x) can be determined.

Distensibility. In general, the morphometric measurements have been obtained under some arbitrary set of conditions, usually with no flow, i.e., with the pressure constant throughout the tree. Thus the effects of the relationship between vessel diameter and vascular pressure would have to be included to predict, e.g., P(x), for any other set of conditions of flow [(0)] and terminal arteriole pressure. In addition, an expression including the effects of vessel distensibility would be more generally useful for the purpose indicated in the introduction.

(14)

(15)

(16)

(17)

(18)

(19)

(20)

(21)

(22)

(23)

Fahraeus-Lindqvist effect. To this point, the blood viscosity (µ) has been a constant independent of vessel diameter. However, the blood viscosity is also dependent on the vessel geometry (Fahraeus-Lindqvist effect) (13, 18). For vessels larger than the terminal pulmonary arteriole diameter of ~20 µm, µ(D) can be approximated by Eq. 24

(24)

where µ_p is the plasma viscosity, µ_b is the blood viscosity in large vessels, and D_F is the vessel diameter in which the viscosity is (µ_b + µ_p)/2. Thus with use of D = D(x,P) in Eq. 24

(25)

and by a development analogous to that above

(26)

Equation 26 can be numerically integrated to calculate P(x) by solving the equivalent nonlinear ordinary differential Eq. 25, and Q(x) can then be calculated from Eq. 22.

Cylindrical vessels. Zhuang et al. (55) utilized a distensible vessel model to express the influence of geometry and vessel distensibility on the pressure-flow relationship in each vessel order of the cat pulmonary arterial tree. To put the expressions derived in the present study in perspective, it will be useful to compare the continuum model predictions with those of a model similar to that of Zhuang et al. as a model representative of concatenated order models. That model will be referred to as the "ordered" model to distinguish it from the continuum model. For such comparisons, the ordered model can be written as follows

(27)

(28)

(29)

(30)

in,n  1

out,n  1

(31)

Experimental Methods

Experiments were performed using an isolated dog lung lobe preparation that has been described previously (3). Each of 32 dogs [19.7 ± 2.1 (SD) kg body wt] was anesthetized with pentobarbital sodium (30 mg/kg), heparinized (1,250 IU/kg), and exsanguinated via a carotid artery catheter. During the exsanguination procedure, 250 ml of saline solution containing 10% dextran (Rheomacrodex, 40,000 mol wt) were infused. Approximately 1 liter of the autologous blood [hematocrit (Hct) 37.7 ± 1.6% (SD)] was used to prime the perfusion system. After exsanguination the chest was opened and cannulas were placed in the left lower lobe artery, bronchus, and vein. The lobe, removed from the dog, was either vertically suspended by the bronchus or placed horizontally on its ventrolateral surface. The arterial cannula was attached to the temperature-controlled (36-37°C) perfusion system, which included a Masterflex roller pump that pumped the blood at a constant rate of 6.78 ± 1.82 (SD) ml/s from a reservoir into the lobar artery. The blood then drained from the lobar vein back into the reservoir. The height of the reservoir was adjusted to set the venous pressure at 3.6 ± 1.3 (SD) mmHg. The lobar arterial and venous pressures were referenced to approximately the level of the measured vessels. The pressure measured at the lobar arterial inlet was 10.4 ± 3.7 (SD) mmHg.

The lobar bronchus was attached to the ventilation system, and between measurements the lobe was ventilated using a piston respirator with a gas mixture containing ~15% O₂-5.6% CO₂-79.4% N₂. This resulted in average PO₂ of 105 ± 23 (SD) Torr, PCO₂ of 40 ± 4 (SD) Torr, and pH of 7.35 ± 0.05 (SD) in the blood in the recirculating perfusion system. The tidal volume was 75-100 ml at ~10 breaths/min, and the end-expiratory airway pressure was set at 3.3 ± 0.6 (SD) mmHg by using a water overflow valve.

The inflow tubing included a previously described (3) injection loop that allowed the introduction of a bolus of 4-5 ml of radiopaque contrast medium, 61% iopamidol (Isovue 300), into the lobe inflow tubing by activating a solenoid valve to redirect the inflow through the bolus-containing loop so that the bolus could be introduced without changing the pressure or flow. The contrast medium was maintained at the same temperature as the perfusion system.

The lung lobe was situated between the X-ray source and the image train of one of two X-ray imaging systems. One was a Nicolet NXR-200 X-ray imaging system with a 40-µm focal spot X-ray source and an X-ray-sensitive videocamera, as previously described (3). The other system included a Fein-Focus FXE-100.50 X-ray tube with a 3-µm focal spot, a North American Imaging AI-5830-HP 9/7/5 in. image intensifier, and a Sony (model AI-01-CCD) charge-coupled-device camera (9, 27). The lobes were situated horizontally and vertically, respectively, in the two systems. The video images were recorded as the bolus passed through the lobar vasculature with a videocassette recorder (model S-VHS). Before the injection of a bolus, the ventilator was stopped at end expiration.

The vessel diameters were measured off-line from the videotaped images in one of two ways. Those obtained from the Nicolet system (n = 570) were analyzed as described previously using visual edge detection of background-subtracted images (3). For those obtained with the Fein-Focus system (n = 313), a region of interest (ROI) was placed over a portion of the image including the vessel, the diameter of which was to be measured. The videotape was automatically advanced frame by frame until the frame having the maximum ROI absorbance during passage of the contrast medium through the ROI was identified by the computer. An average background image was calculated by averaging image pixel intensities for 10 image frames (0.33 s) before injection of the contrast bolus. This background image was subtracted from the maximum absorbance image. Then the absorbance across the vessel cross section was measured, and the diameter was estimated as a fraction of the total image dimensions by using the cylindrical model-based algorithm described by Al-Tinawi et al. (1) and Clough et al. (10). The diameter-measuring algorithm included a modification, not previously reported, for automatically setting the absorbance line scans used to calculate the vessel diameter orthogonal to the vessel axis. This was accomplished by identifying within the rectangular ROI the lines of pixels that would ultimately need to be parallel to the axis of the vessel. Initially, these lines were oriented at an arbitrary angle relative to the vessel axis as the result of user placement of the ROI in approximately the correct orientation. The SD of all pixel intensities was calculated along each line. If the lines had, in fact, been oriented parallel to the vessel, the SD of pixel intensities along each line would be at a minimum. To find the box position that resulted in that minimum, the box was rotated ±20°, in 0.1° increments, and the SD of pixel intensities along each line was calculated at each angle. Finally, the box was oriented to the angle corresponding to the smallest sum of the SD of pixel intensities along the lines. To calibrate the imaging dimensions, a remotely controlled micrometer was used to move the lung lobe a known distance. The actual distance moved during the calibration displacement was divided by the displacement of the bifurcation on the image to provide a calibration factor for the vessels of that bifurcation.

Morphometry Methods

(32)

	RESULTS

Top Abstract Introduction Methods Results Discussion References

Morphometry

1

View larger version (15K): [in this window] [in a new window]   Fig. 2.   Frequency of z values for vessels in 4 size ranges; n, number of measured bifurcations in each size range.

Sources of Input Values for Simulations

Hemodynamic and rheological information used in the simulations, but not obtained from measurements made in the present study, were chosen with reference to values from previous studies based on the following information. In studies in which subpleural arteriole pressures were measured in arterioles with diameters between 20 and 50 µm in the dog lung, the estimated intrapulmonary arterial pressure drop has ranged between ~20% (5) and 50% (40) of the total arteriovenous pressure difference (21). On the basis of those measurements, for the lung lobes in the present study with a mean arteriovenous pressure difference of ~6.8 mmHg, the arterial tree pressure drop would be from ~1.4 to 3.4 mmHg. Because of the availability of these pressure data in vessels of ~35 µm diameter, the terminal arteriole diameter (D_T) in the model simulations was generally ~35 µm. The precapillary terminal pulmonary arteriole diameters, which have been measured, tend to be a little smaller than the average for which most of the pressure measurements are available and are listed in Table 1.

Left lower lobar arterial volume for a 20-kg dog under pressure and flow conditions of the present study was found to be ~10 ml by using the ether dilution method (11). Also a morphometric estimate of ~12 ml can be calculated from the dog right lung data of Gan and Yen (17) by taking into account the size of the dog on which the measurements were made and the relative sizes of the left lower lobe and right lung (41).

The value of  for the dog lung has been found to be ~0.02/mmHg with a range of ~0.01-0.04/mmHg (3).

In the present study, D(0,P = 10.4 mmHg) = 0.757 cm (i.e., the largest diameter from Fig. 2 and Table 2), which, for  = 0.02/mmHg, is also equivalent to D(0,0) = 0.627 cm.

The lower and upper bounds on D_F from the summary of blood viscosity vs. tube diameter data presented by Goldsmith et al. (18) are 0.0023 and 0.0050 cm, respectively.

Fitting the equation µ_b = µ_pexp(kHct) to the viscosity vs. Hct data for dog blood from Stone et al. (47), for which µ_p = 0.0119 Poise, yields k = 0.0267. Thus the Hct in the present study (37.7 ml cells per 100 ml large vessel blood volume) gives µ_b ~ 0.033 Poise. The Hct of 37.7% obtained in the present study compares with the normal mean large vessel Hct for the dog reported to be 39.7% by Rapaport et al. (42).

Model Simulations

View larger version (22K): [in this window] [in a new window]   Fig. 3.   Representations of continuum (dashed lines) and ordered (solid lines) model simulation results by using the following input parameters: 1 = 2.6, 2 = 1.1, a2 = 5, number of orders in ordered tree model = 21, DT = 0.0035 cm,  = 0.02/mmHg, flow = 6.8 ml/s, diameter-independent viscosity = 0.02 Poise, outflow pressure = 7.9 mmHg, D(0,0) = 0.627 cm, D1(0) = KD(0,0) = 0.6 cm. A: intravascular pressure vs. distance (x) into tree (only 1 line is seen, because the 2 model results superimpose). B: intravascular pressure vs. vessel diameter. Continuum model is a smooth curve. Ordered model is stair graph on B, E, and F. C: intravascular pressure vs. volume, where volume is cumulative volume into tree. Thus zero volume is at inlet to arterial tree, and graph ends at total arterial tree volume. D: volume vs. distance. E: volume vs. diameter. F: diameter vs. distance.

The intravascular pressure is presented as a function of diameter for the two models in Fig. 3B. Again, the stair graph is the ordered model. In this case, the risers, which appear to be vertical, are actually slightly pitched, reflecting the pressure drop through each vessel and the vessel distensibility. The treads are flat, because there is no pressure difference between the outlet of one vessel and the inlet to the next. The continuum model produces a smooth curve. The curvature of the pressure vs. diameter graphs is determined by ₁, ₂, and a₂. The graphs of the pressure vs. distance x into the tree are indistinguishable between the two models (Fig. 3A). On the other hand, the volume (Q) accumulated through all parallel pathways from the inlet to distance x (Fig. 3D) or the volume upstream from a given diameter (Fig. 3E) or a given pressure (Fig. 3C) is larger for the continuum than for the ordered model. These differences again reflect the fact that a frustum and a cylinder of equal length cannot have equal volume and equal resistance and the continuous vs. stepwise progression of the changes in total cross-sectional area of all the pathways with distance x into the tree. Having made these comparisons, subsequent simulations are from the continuum models.

In the model development we distinguished between models that predict the pressure-flow relationship and the longitudinal pressure and volume profile for the vascular tree, given the availability of morphometric parameters obtained under one specific set of conditions (e.g., Eqs. 10 and 13), and a model that can make predictions over a range of conditions (e.g., Eqs. 19 and 23 or 27 and 30). We will refer to the former as fixed-diameter models, because no accommodation for vessel distensibility is included, since the vessel morphometry is assumed to be available for a specific set of relevant pressures and flow. The latter are referred to as distensible-vessel models, because they predict the results under any set of pressure and flow conditions within the range for which Eq. 14 is a useful representation of the diameter-pressure relationship. This distinction can also serve to provide a sense of the sensitivity of the morphometric parameters to the conditions under which they are measured. To obtain morphometric data from the continuum model, ordering was imposed by calculating a length L_j from

(33)

and then "measuring" the diameters at x = L_j. N_j was set equal to 2^j  1.

Two sets of simulated morphometric data are shown in Fig. 4 as plots of log N_j vs. log D_j or log L_j vs. log D_j. One set is for the vessels at P = 0. The other is that obtained from the distensible-vessel model during the simulated flow (6.8 ml/s) condition. Linear regression of the latter using Eqs. 1 and 2 provided the input morphometric parameters for the fixed-diameter model. In this set of simulations, ₁, ₂, a₁, and a₂ input into the distensible-vessel model was 2.560, 1.150, 0.303, and 5.000, respectively. The pressures that resulted from flow (6.8 ml/s) through the model distended the vessels such that when the morphometry was carried out "with flow," ₁, ₂, a₁, and a₂ were 2.534, 1.139, 0.741, and 3.987, respectively. The largest differences are in a₁ and a₂, because the vessel diameters increase with pressure ( = 0.02/mmHg). Figure 5 shows the longitudinal pressure profiles obtained for the two sets of parameters used as input to the distensible-vessel and fixed-diameter models, respectively. The differences in pressure between the distensible-vessel and fixed-diameter models reflect the small differences in the shape of a vessel in the two models. These pressure differences are small enough to suggest that the simpler fixed-diameter model would be useful, in the sense indicated in the introduction, given morphometric measurements made while the lungs are being perfused.

View larger version (13K): [in this window] [in a new window]   Fig. 4.   Simulated number (Nj) vs. diameter (Dj, in cm) and length (Lj, in cm) vs. Dj from distensible-vessel model. , Diameters at undistended or zero pressure values; , diameters with flow through model. Model 1, 2, a1, and a2, values input into distensible-vessel model (i.e., no-flow values) were 2.56, 1.15, 0.303, and 5.0, respectively. Values obtained by fitting straight lines through  were 2.534, 1.139, 0.741, and 3.987, respectively.

View larger version (15K): [in this window] [in a new window]   Fig. 5.   Pressure vs. distance (x) from arterial inlet for distensible-vessel (solid line) with 1, 2, a1, and a2 set at 2.56, 1.15, 0.303, and 5.0, respectively, and fixed diameter (dashed line) models with input values of 2.534, 1.139, 0.741, and 3.987, respectively, obtained by linear regression of morphometric data from distensible-vessel model simulation data on Fig. 4. Small differences reflect the fact that vessels are not exactly the same shape in the 2 mode