Inert gas clearance from tissue by co-currently and counter-currently arranged microvessels

Abstract

To elucidate the clearance of dissolved inert gas from tissues, we have developed numerical models of gas transport in a cylindrical block of tissue supplied by one or two capillaries. With two capillaries, attention is given to the effects of co-current and counter-current flow on tissue gas clearance. Clearance by counter-current flow is compared with clearance by a single capillary or by two co-currently arranged capillaries. Effects of the blood velocity, solubility, and diffusivity of the gas in the tissue are investigated using parameters with physiological values. It is found that under the conditions investigated, almost identical clearances are achieved by a single capillary as by a co-current pair when the total flow per tissue volume in each unit is the same (i.e., flow velocity in the single capillary is twice that in each co-current vessel). For both co-current and counter-current arrangements, approximate linear relations exist between the tissue gas clearance rate and tissue blood perfusion rate. However, the counter-current arrangement of capillaries results in less-efficient clearance of the inert gas from tissues. Furthermore, this difference in efficiency increases at higher blood flow rates. At a given blood flow, the simple conduction-capacitance model, which has been used to estimate tissue blood perfusion rate from inert gas clearance, underestimates gas clearance rates predicted by the numerical models for single vessel or for two vessels with co-current flow. This difference is accounted for in discussion, which also considers the choice of parameters and possible effects of microvascular architecture on the interpretation of tissue inert gas clearance.

in this paper, we develop a numerical model that examines the clearance of dissolved inert gases from tissue by its microvessels. Quite apart from their intrinsic interest, models of inert gas exchange between the blood and the tissues are central to three major problems of applied physiology: decompression sickness, anesthesia, and as the basis for one of the most widely used methods for measuring tissue blood flow (11, 13–18, 35). Because the problem cannot be solved analytically, assumptions have been made to achieve a simplified, asymptotic solution. The most widely used model of tissue-blood exchange of the inert gas is that introduced by Haldane and co-workers (3) to calculate the first decompression tables for naval divers ascending from depth. Haldane (13) reasoned that following a step change in arterial P_N2, the rate of equilibration in a given tissue depends on the blood flow (F), the tissue volume (V_T), and the solubility of nitrogen (N₂) in the tissue relative to its solubility in blood (λ). He also assumed that during a single-capillary transit, equilibration of N₂ between blood and tissue was complete. Although Haldane himself wrote no algebraic equations, Wagner (35) has summarized these ideas as the conductance-capacitance (C-C) model and expressed them as

$$-V_T·S_T·\frac{d{P}_T}{dt}=F{S}_b\left(P_v-P_a\right)=F{S}_b\left(P_T-P_a\right)$$ (1A)

where P_T, P_a, and P_v are partial pressures of N₂ in tissues, arterial, and venous blood, respectively; S_T is the solubility of N₂ in the tissues; and S_b is its solubility in the blood. Rearranging Eq. 1A with λ = S_T/S_b leads to

$$-\frac{d{P}_T}{dt}=\frac{F{S}_b}{V_T{S}_T}·\left(P_T-P_a\right)=\frac{F}{V_T\lambda }·\left(P_T-P_a\right)$$ (1B)

which on integration, becomes

$$P_T-P_a=\left[P_T\left(0\right)-P_a\right]·\exp \left(-F·t/\lambda V_T\right)$$ (1C)

where P_T(0) is P_T at t = 0. When P_a = 0

$$P_T=P_T=\left(0\right)e^{-\left(F/\lambda V_T\right)·t}.$$ (1D)

Appreciating that both the blood flow per unit volume and the tissue N₂ solubility can vary considerably in different parts of the body, Haldane (3, 13) calculated his tables using five parallel compartments to represent tissues with widely differing rates of equilibration. His schedules reduced both the morbidity and mortality of decompression sickness and were adopted almost immediately. Models of the tissues as a series of parallel compartments with different equilibration times have continued to be used for calculating decompression tables up to the present time. The rates of onset and recovery from an anesthetic have been modeled as a lung compartment in a series with the parallel tissue compartments (2, 11, 35). With the use of this approach, Wagner (35) was able to predict observed changes in arterial partial pressure of halothane and cyclopropane during induction and recovery from anaesthesia but failed to predict the anesthetic pressures in the mixed venous blood. More examples are given by Baker and Farmery (2) in their recent review.

A generalized form of Eq. 1D is the theoretical basis of one of the most widely used methods of determining tissue blood flow. The method, conceived by Kety (17), involves the injection of a solution containing a radioactive tracer into a tissue and recording the declining amount of tracer in this tissue depot as it is carried away from the tissue by blood flow using an external counter. When Kety introduced the technique, he used Na²² as the tracer, but since the clearance of Na⁺ ions becomes permeability limited as blood flow increases, most investigators over the past 50 years have used xenon Xe¹³³ as the isotope of choice (22, 27, 28, 33). Being minimally invasive, the method gains wide use in clinical studies (26, 32, 34). It has not been without its critics, however, and in several papers, inert gas clearance is reported to underestimate tissue blood flow when comparisons with other methods have been made (5, 19, 25). To account for these discrepancies, some investigators have extended the simple C-C model by incorporating a diffusion barrier at the blood-tissue interface but have retained the assumption of instantaneous mixing in the extravascular space (27, 30). Piiper et al. (30) have used their model to examine the effects of counter-current exchange of gas between small arteries that feed the tissue and adjacent veins that drain it. Doolette et al. (9, 10) have reported that relatively simple perfusion/diffusion models are able to fit both nitrous oxide and helium (He) clearance data from sheep skeletal muscle better than the classical perfusion-limited model.

Periodically, investigators have questioned the simplifying assumptions underlying Eqs. 1A–1D, i.e., that partial pressures of the inert gas equilibrate between blood and tissue during a single transit through the exchange vessels and that there are no gradients of partial pressure within the tissues. Hills (16) argued that much of the experimental data on diving could be accounted for by a diffusion theory, in which the diffusion coefficients of inert gases in tissues have values three orders of magnitude less than those in aqueous solutions. Stimulated by Hills' criticisms, Hennessy (14) combined diffusion and perfusion equations to study the desaturation of inert gas from a cylinder of tissue surrounding a single capillary. To obtain asymptotic solution of the partial differential equations (PDEs), Hennessy omitted axial diffusion in both tissue and capillary among other simplifications. With the use of this model to compare the predictions of the Haldane and Hills theories as the basis of the decompression schedules used by divers, Hennessy concluded that the Hills diffusion theory (15) required many fewer parameters to predict the schedules than did the Haldane model. The low diffusion coefficients required by the Hills model, however, have never been demonstrated experimentally, and experiments designed to reduce inert gas exchange by increasing tissue diffusion gradients have yielded negative results (28). Although Wienke (39) suggested that a better understanding of inert gas exchange might follow if the PDEs were solved numerically, it was several years later that Whiteley et al. (38) published a study of this kind. Here, a straight, single capillary surrounded by a cylinder of tissue was considered, and the authors reported large discrepancies between their numerical results and those calculated from the simple C-C model.

In the present study, inert gas clearance from tissue by co-current and counter-current capillaries has been investigated by solving PDEs using finite element analysis for gas transport by perfusion and diffusion. The aims of this investigation are to determine: 1) whether significant gradients of partial pressure develop in the tissues during inert gas clearance; 2) the relationship between the blood flow and inert gas clearance in tissues with co-current and counter-current microvessels; and 3) effects of different parameters, e.g., partition coefficient and diffusivity of the inert gas, on clearance of the dissolved gas from tissues.

THEORETICAL MODELS AND NUMERICAL METHODS

Transport of the inert gas in tissue is analyzed in a representative microcirculatory unit, similar to Krogh's model (20). Each unit consists of one or two straight capillaries of radius (r_b) and length (l) enclosed by a tissue cylinder of outer radius (r_T). Steady blood flow in capillaries is assumed. When only one capillary is considered, it is referred to as the single-capillary unit, which has been used in previous studies (14, 35, 38) (Fig. 1A). When two parallel capillaries are considered, it is referred to as the co-current system (Fig. 1B) or the counter-current system (Fig. 1C), depending on the blood flow direction in the two capillaries.

Fig. 1.

Schematic drawings of the cylindrical microcirculatory unit with a length of l and a radius of r_T (not to scale). Capillaries have identical radii r_b and constant velocity u. In the 2-capillary units, the distance between capillaries is d. A: a single-capillary perfusion unit. B: a 2-capillary co-current perfusion unit. C: a 2-capillary counter-current perfusion unit.

The inert gas concentration (C_b) in capillaries is governed by the convection-diffusion equation

$$\frac{\partial C_b}{\partial t}+\text{u}·\nabla C_b=D_b{\nabla }^2{C}_b$$ (2)

where D_b is the diffusivity of gas in blood plasma, u is blood velocity, and ▽ and ▽² are spatial gradient and divergence operators, respectively. The governing equation for gas concentration (C_T) in the tissue is

$$\frac{\partial C_T}{\partial t}=D_T{\nabla }^2{C}_T$$ (3)

where D_T is the diffusivity of gas in the tissue. The ratio of diffusivity (R_D) is defined as D_T/D_b. Dissolved gas in the tissue is cleared by diffusion through the capillary-tissue interface (r = r_b), where the flux on either side of the interface satisfies

$$D_b\frac{\partial C_b\left(r_{b,}z,t\right)}{\partial {\text{n}}^{+}}=D_T\frac{\partial C_T\left(r_{b,}z,t\right)}{\partial {\text{n}}^{-}},$$ (4)

where n is the normal direction of the capillary wall with “+” pointing from the center and “−” toward the center. Partial pressure of the gas is continuous across the interface

$$\frac{C_b\left(r_{b,}z,t\right)}{S_b}=\frac{C_T\left(r_{b,}z,t\right)}{S_T},$$ (5)

where S_b and S_T are the gas solubility in blood and in tissue, respectively. The solubility ratio, λ = S_T/S_b, is also known as the tissue-blood partition coefficient. At the outer boundary of the tissue cylinder, transport of the inert gas is assumed to be negligible. At the inlet of the capillary, the gas concentration is prescribed. At the outlet of the capillary, the diffusive loss of the gas is assumed to be negligible. Initially, the whole microcirculatory unit has a uniform gas pressure of P₀

$$C_b\left(r,z,0\right)=S_b{P}_0;C_T\left(r,z,0\right)=S_T{P}_0,\text{at}t=0$$ (6)

Although axis symmetry is observed in the single-capillary case, it does not hold in the two-capillary unit. In our study, three-dimensional finite element analysis is carried out. Considering the plane symmetry, one-half of the microcirculatory unit is decomposed into 50,000 hexahedral elements, which ensures that results are mesh independent. Equations 2–5 are solved numerically by the Galerkin finite element method (FEM; see appendix).

Parameters

Capillaries have the same radius r_b = 3 μm and length l = 500 μm. The outer radius of the microcirculatory unit, r_T = 50 μm, is a value typical for skeletal muscle (23). The vessels in the two-capillary units are separated by a distance of d = 24 μm, and their axes lie in the same plane as that of the tissue cylinder. Blood velocity has a typical value of 0.5 mm/s in capillaries but is allowed to vary between 0.01 and 2.0 mm/s to study its effects on gas clearance. Inert gas diffusivity in blood plasma is close to that in the aqueous condition but is higher than that in tissues. For a general inert gas, e.g., Xe, D_b = 1.0 × 10⁻⁵ cm²/s, whereas D_T = 0.5 × 10⁻⁵ cm²/s, given R_D = 0.5 (35). In the parametric study, D_T is allowed to vary between 1.0 × 10⁻⁷ and 2.0 × 10⁻⁵ cm²/s so that R_D varies between 0.01 and 2.0. The solubility of the inert gas in blood varies as a function of the temperature and hematocrit (7, 8), and in tissues, it increases with the fat content due to its lipophilic nature (24). The partition coefficient λ = 0.7 is used as the control (22), and it is allowed to vary between 0.5 and 5.0 in parametric study.

In results, distribution of the inert gas is expressed as the partial pressure normalized by P₀. The total inert gas clearance is shown as the fraction of inert gas that has been cleared. The absolute value of P₀ does not affect results.

RESULTS

Spatial Distribution of the Inert Gas

The changes in partial pressure of the inert gas in the blood along the capillary axis are shown in Fig. 2. Results for a single-capillary unit are shown in Fig. 2A, and those for co-current and counter-current are shown in Fig. 2, B and C, respectively. Figure 2A shows the distribution of the gas along the capillary at three different times after the start of gas washout. From zero time onward, blood enters the capillary with P/P₀ = 0. As blood flows down the vessel, P/P₀ rises, as gas diffuses in from the surrounding tissue. At t = 4.5 s, gas partial pressure in the blood appears to have equilibrated to the initial P₀ value after it has traveled ∼40% of the length of the capillary. As time progresses, e.g., at t = 31.5 s and 135 s, the inert gas pressure in the blood drops further, as the inert gas content in the tissue decreases (shown later in Fig. 3). For the co-currently arranged capillaries, P/P₀ in both vessels are identical, and its distribution in one capillary is shown in Fig. 2B. Here, the initial shape of the profile, i.e., at t = 4.5 s, is similar to that in the single-capillary unit. Note that the total blood flow in the co-current and counter-current systems (Fig. 2, B and C) is twice that in a single-capillary unit (Fig. 2A). As time increases, more rapid decreases in the blood P/P₀ values are observed compared with Fig. 2A. For the counter-current unit, P/P₀ distribution along the two capillaries is antisymmetric to the mid-point, z/l = 0.5. In Fig. 2C, inert gas pressure in one vessel only is shown. As in a single-capillary unit and the co-current system, there is an initial, sharp rise in P/P₀ over the first 20% of the vessel length. Unlike in the other two units, where the blood gas pressure increases monotonically along the direction of the flow, P/P₀ falls near the exit of the capillary as gas diffuses back to the surrounding tissue, where the gas content has been lowered by the partner vessel of the counter-current system. This cross-diffusion between the two counter-current capillaries slows down clearance of the dissolved gas from tissue. By comparing Fig. 2, B and C, it can be seen that by t = 135 s, the blood inert gas pressure in the counter-current system is significantly higher than that in the co-current system. The effects of counter-currently arranged capillaries on clearance of the inert gas from tissue are shown later in more details.

Fig. 2.

Axial distribution of the inert gas partial pressure in a capillary at 3 different times (t) following the start of clearance: t = 4.5 s (solid lines), 31.5 s (dashed lines), and 135 s (dotted lines). In the figure, λ = 0.7, R_D = 0.5, and all capillaries have the same flow rate, u = 0.5 mm/s; therefore, the tissue perfusion rate in the single-capillary unit is only 1/2 of the 2 capillary units. A: a single-capillary unit. B: a 2-capillary co-current unit. C: a 2-capillary counter-current unit. z, axial position in the cylindrical coordinate system.

Fig. 3.

Partial pressure contours of inert gas in tissue at 3 different time points in a single-capillary unit and co-current and counter-current units. In the figure, λ = 0.7, R_D = 0.5, and all capillaries have the same flow rate, u = 0.5 mm/s; therefore, the tissue perfusion rate in the single-capillary unit is only 1/2 of the 2 capillary units. All pressure contours refer to the same legend.

The corresponding changes in tissue gas partial pressure are shown in Figs. 3. Here, we observe similar axial gradients of the inert gas pressure as those in capillaries. Close examination of results at early times, e.g., at t = 4.5 s, shows that there are substantial radial gradients of gas pressure in the tissue near capillary entrances. These gradients are much less conspicuous around the entrances of the co-current vessels, and all diminish quickly with the distance, e.g., all radial gradients are small beyond z/l = 0.2. In the counter-current system, as expected, significant radial gradients exist at both ends of the unit. The co-current system, with twice the blood flow of the single-capillary unit, shows a much quicker decrease in the tissue inert gas content than the latter. The counter-current system, by contrast, retains the inert gas in the middle section, e.g., between 0.3 < z/l < 0.7, of the unit. This can be seen most clearly at t = 135 s, where the tissue gas content in the counter-current system is noticeably higher than that in the co-current system. Not obvious in Fig. 3 is a step change in the radial pressure gradient at the capillary-tissue interface. This is a consequence of different values in gas diffusivity and solubility.

Clearance of Dissolved Inert Gas from Tissue

Clearance of the inert gas from the microcirculatory unit is defined as (∫_v C₀dv − ∫_v Cdv)/∫_v C₀dv, where V represents the total volume of the unit. Figure 4 shows the clearance of dissolved gas with time predicted by the C-C model and by the numerical models for different configurations of the capillaries and the directions of flow. Figure 4A compares the predictions of the C-C model with those of the numerical model for a single-capillary unit. The predictions of both models approximate closely to a single exponential decay, but clearance declines more rapidly with the numerical model. If the exponential constant of the C-C model is increased by a factor of 2ln2 (for which the rationale is given in discussion), its predictions approximate closely to those of the numerical model. A comparison between clearance by one vessel and two vessels with co-current flow is shown in Fig. 4B. Here, the flow velocity in the single vessel is the same as that in both vessels with co-current flow so that the flow per unit volume of tissue (F/V_T) of the single vessel system is only one-half of that for co-current flow. When flows in the single-capillary and co-current units are adjusted so that F/V_T is the same, i.e., at the same blood perfusion rate per unit volume of tissue, the clearance curves show close agreement with each other. Figure 4C shows the difference between clearance rates by the counter-current and co-current systems. The counter-current arrangement of capillaries slows down the clearance process by trapping inert gas in the central region of the tissue (see Fig. 2C). When Fig. 4, B anc C, are compared, the rate of clearance from the counter-current unit is more comparable with that by a single capillary (with one-half of the blood flow) than that seen in the co-current unit with the same F/V_T.

Fig. 4.

Clearance of inert gas in different systems. In the figure, λ = 0.7, R_D = 0.5, and u = 0.5 mm/s in all capillaries. A: clearance curves for the single-capillary unit based on numerical model and the conductance-capacitance (C-C) model. The dashed curve shows the predictions of the C-C model after it has been adjusted by the factor 2ln2. B: clearance curves for the single-capillary unit and the 2-capillary co-current unit. Solid curves are with the same flow velocity in all vessels; therefore, the single-capillary unit only has 1/2 of the flow rate (F/2) of the 2-capillary co-current unit. The dotted curve close to the co-current result represents doubling of the velocity in the single vessel so that both systems have equal total flow rate, F. Dashed curves are predictions of the C-C model after adjustment by the factor 2ln2. C: clearance curves for co-current and counter-current units. Note that counter-current flow clears the gas more slowly than the co-current flow. Half-time of clearance is defined as the time when 1/2 of the dissolved gas is cleared.

As already noted, all clearance curves approximate to a single exponential decay function, as predicted by the simple C-C model (Eq. 1C) over, at least, the first one-half of tissue clearance. When describing the effects of different factors on the inert gas clearance, we assume that the exponential relation is a reasonable one and use the half-time of clearance, τ, i.e., the time for 50% clearance of the initial amount of gas, as a measure of the clearance curve. Therefore, 1/τ can be regarded as a measure of the clearance rate and is proportional to F/V_T (Eq. 1).

Effect of Different Parameters

Blood flow velocity.

Our calculations assume plug flow in capillaries and that all microvessels have equal radii so that the volume flow rate is equal to the product of the blood flow velocity (u) and the cross-sectional area of the capillary, u·πr_b². The tissue perfusion rate, F/V_T, is proportional to u and the number of capillaries in the unit.

The effects of varying blood flow velocity on τ (the one-half time) are compared between co-current and counter-current systems in Fig. 5. As expected, there is an approximate inverse relation between the velocity and the half-time for both co-current and counter-current systems, with greater values of τ for counter-current flow at most perfusion velocities (Fig. 5A). This is shown more clearly in Fig. 5B, where the same data are used to plot 1/τ against perfusion velocity. Figure 5B also reveals that clearance is slower under counter-current than under co-current conditions over a wide range of perfusion velocities. Furthermore, the simple C-C model overestimates the blood velocity necessary to achieve the clearance rates predicted by the numerical model for co-current flow.

Fig. 5.

Effects of the perfusion rate on the gas clearance in co-current and counter-current units. In the figure, R_D = 0.5, and λ = 0.7. A: half-time of clearance decreases with the perfusion velocity in both co-current (□) and counter-current (■) systems. B: reciprocal of the half-time is approximately linearly related to perfusion velocity in both co-current and counter-current models. Symbols indicate the data for co-current (□) and counter-current (■) systems. The points follow linear trends that back extrapolate to small, positive intercepts. The solid line passing through the origin gives the prediction of the C-C model without adjustment; the dashed line shows the predictions of the C-C model after adjustment by the factor 2ln2 (see discussion). 1/τ, clearance rates.

At very low flow rates, counter-current flow appears to be more effective than co-current flow in clearing the tissues. This phenomenon is linked to the presence of the positive intercept in Fig. 5B, which arises from diffusional loss of gas through the capillary entrance regions. Closer inspection of Fig. 5A reveals that τ has the same value for both systems when u = 0.15 mm/s.

Partition coefficient.

Whereas the solubility of inert gases in blood can be measured directly and is constant in a particular individual, solubility varies considerably from tissue to tissue so that the partition coefficient (λ) may be quite different in one region of the body from its value in another. For example, λ may be 0.7 in muscle and 4.0 in fat. The effects of changing values of λ upon 1/τ are shown in Fig. 6. For both co-current and counter-current systems, 1/τ declines approximately inversely with increasing λ. Such an inverse relation is consistent with Eq. 1C, which is shown in Fig. 6. At low values of λ, the C-C model predicts values for 1/τ, similar to those for counter-current flow but below those for co-current flow. At higher values of λ, the C-C model predicts lower values for 1/τ for both co- and counter-current systems. When the exponential constant of the C-C model is multiplied by 2ln2 (“corrected model”), however, it predicts relations that are consistent with the numerical model for co-current flow over a wide range of λ. The counter-current system results in slower gas clearance than the co-current system for all λ values, particularly when λ is small. This is due to the trapping, or recirculation, of the gas between two capillaries in the counter-current system. As λ increases, the disparity between the curves diminishes, and it becomes negligible when λ ≥ 4. Under conditions of high λ, tissue-storage capacity of the inert gas reduces interactions between adjacent vessels. Results in Fig. 6 were for u = 0.5 mm/s. At higher perfusion velocity, co-current and counter-current results converge at higher values of λ.

Fig. 6.

Effects of partition coefficient λ on the inert gas clearance. In the figure, R_D = 0.5, and u = 0.5 mm/s in all capillaries. Reciprocals of the half-time are calculated for both co-current (□) and counter-current (■) systems. The solid curve represents the prediction of the C-C model and the dashed curve, its predictions after adjustment by the factor 2ln2.

Diffusivity ratio.

As with λ, the diffusion coefficient of an inert gas in the blood can be regarded as a constant for a particular individual, but in the tissues, D_T may vary, leading to variations in diffusivity ratio (R_D) in different regions of the body. Although most measurements of gaseous diffusion coefficients in tissues suggest that R_D is unlikely to be <0.1, it has been argued that tissue-gas diffusivity has been overestimated (15, 16). So far in this paper, we have considered R_D to have a value of 0.5. Here, we use the model to allow R_D to vary between 0.01 and 2.0 by varying D_T from 10⁻⁷ cm²/s to 2 × 10⁻⁵ cm²/s, whereas D_b is held constant at 1.0 × 10⁻⁵ cm²/s. Figure 7 shows the effects of R_D on 1/τ in the co-current and counter-current systems, with R_D plotted on a logarithmic scale to cover more of the range investigated. It is seen that the inert gas clearance rate (as indicated by 1/τ) increases almost monotonically with R_D in the co-current system. In the counter-current system, however, the effect is much more complicated. At R_D ≈0.2, the counter-current system is least effective in gas clearance. As R_D decreases from 0.2, the low value of the gas diffusivity in the tissue reduces the diffusive shunting between the two vessels. As a result, the clearance rate by the counter-current system rises slightly. On the other hand, as R_D increases from 0.2, the higher gas diffusivity in tissue promotes more rapid transport of the gas. Although the diffusive trapping is also enhanced, it results in an overall faster clearance of the dissolved gas from the tissue. At high values, e.g., R_D > 2.0, the clearance rates by co-current and counter-current arrangements become similar. Under such conditions, the retardation effect of cross-diffusion becomes less significant compared with the high diffusive gas transport. In Fig. 7, differences in 1/τ of 0.005 (s⁻¹) may seem to be very small, but their importance can be evaluated by using them to estimate the F/V_T, which would be calculated from Eq. 1C. With R_D = 0.5, λ = 0.7, and co-current flow in two capillaries, 1/τ has a value of 0.021 (s⁻¹) in Fig. 7, and this is equivalent to a blood flow of just over 60 ml/(min 100 ml tissue). When flow in the two capillaries is counter-current, and R_D = 0.5, and λ = 0.7, 1/τ has a value of 0.014 (s⁻¹), which is equivalent to blood flow of just less than 40 ml/(min 100 ml tissue).

Fig. 7.

Effects of the diffusivity ratio R_D on the inert gas clearance. In the figure, λ = 0.7, and u = 0.5 mm/s in all capillaries. Reciprocals of the half-time are calculated for both co-current (□) and counter-current (■) systems. Note the logarithmic scale for R_D.

Vascularity of tissues.

So far, we have considered clearance by varying rates of perfusion through either one or two vessels from a cylindrical block of tissue of constant dimensions. In this section, we consider variations in the diameter of the tissue cylinder on clearance by a single vessel and variations of the distance separating the capillaries in a two-vessel system. In this way, we hope to gain insight into clearance from tissues of different vascularity, where vascularity is defined as the fractional volume of the blood vessels in the tissue.

Figure 8A shows the relations between 1/τ and blood velocity through three cylindrical blocks of tissue, each containing a single vessel, all having the same length (500 μm) and vessel radii (3 μm), but each cylinder of tissue with a radius r_T, which differed from the others. The values of r_T are 15 μm, 30 μm, and 50 μm. One would expect from Eq. 1 that the slope of the relation between 1/τ and perfusion velocity would be inversely proportional to the volume of the tissue. Inspection of Fig. 8A reveals that this is so. At any given perfusion velocity, 1/τ for the different vessels increases with the reciprocal of the tissue cylinder radius squared (which is directly proportional to tissue volume).

Fig. 8.

Effects of tissue dimensions and distances between capillaries upon inert gas clearance. In the figure, λ = 0.7, and R_D = 0.5. A: relations between 1/τ and perfusion velocity in 3 single capillary units surrounded by cylindrical blocks of tissue with radii of 15 μm, 30 μm, and 50 μm. B: relations between 1/τ and perfusion velocity in 4 counter-current units where the vessels are separated by different distances.

Another potentially important variable is the distance, d, between adjacent vessels. We have examined the effects of varying d on clearance by both co-current and counter-current flow from a cylinder of tissue with a 50-μm radius. The effects were negligible when flow was co-current (results not shown). This might be expected since the clearance of inert gas from a 50-μm radius tissue cylinder by a single capillary is very nearly the same as that by two vessels in which flow is co-current, provided that volume flow through the single vessel is equal to that through two vessels (i.e., velocity in the single vessel is twice that in the two co-current vessels). When there is counter-current flow in the vessels, increasing d increases the rate of clearance from the system, particularly at high flows. Figure 8B shows the effects of increasing d upon the relations between 1/τ and flow velocity. As the vessels become increasingly separated, clearance increases as the influence of counter-current exchange is diminished.

DISCUSSION

Results reported here allow us to answer three major questions about the clearance of dissolved inert gas from tissue. 1) We have shown that significant gradients of gas pressure develop within the tissue during inert gas clearance. These gradients exist in both the radial and axial directions of the cylindrical tissue unit. The step change in the gradient of gas pressure at the capillary-tissue interface could be relevant to the aetiology of decompression sickness as a site of bubble formation (6). 2) We have confirmed that the clearance rate of inert gas from tissue is approximately linearly related to the microvascular flow rate for both co-currently and counter-currently arranged capillaries. The rate at which the inert gas is cleared by the circulation is, however, considerably less in a tissue with counter-currently arranged microvessels. 3) Higher partition coefficient and lower diffusivity of inert gas in tissue slow down the clearance process, but the effects become more complicated for the gas diffusivity when counter-current capillaries are involved.

Also, we have found that the predicted rates of clearance by the numerical models for a single capillary or a co-current capillary pair are more rapid than the predictions of the simple C-C model when the same values are used for F/V_T. Whiteley et al. (38) reported a similar finding for inert gas uptake but did not discuss its possible cause. It occurs because the C-C model assumes that gas partial pressure throughout the tissue is uniform (a mean tissue gas pressure) and that this mean partial pressure is equilibrated with the blood leaving the tissue (Eqs. 1A–1D). Our model results (Figs. 2 and 3), however, show a different story. For a single capillary and a co-current capillary pair, gas clearance in tissue starts from the inlet end of the capillary and moves gradually toward the exit end. At t = 31.5 s, for example, the mean partial pressure of the gas in the tissue is well below its original value P₀, but near the capillary outlet, partial gas pressure in the exiting blood is still very close to P₀. The assumption used in the C-C model, therefore, results in an underestimation of tissue gas clearance by blood flow.

In the early stage of gas clearance, gas partial pressure in the exiting blood remains at P₀. The amount of gas that is cleared is FtP₀S_b. When the half-time, τ, is reached before a significant drop of the gas partial pressure occurs near the capillary outlet, we have

$$F_{\tau }{P}_0{S}_b=\frac{1}{2}{P}_0{V}_T{S}_T,$$

which gives τ = $\frac{1}{2}$ λ V_T/F.

From Eq. 1D, the half-time of the C-C model, τ_c-c, is

$${\tau }_{c-c}-ln2\left(\lambda V_T/F\right);$$

hence, τ = τ_c-c/2ln2. If the C-C model is “corrected” by multiplying the exponential constant in Eq. 1C by 2ln2, its predictions become consistent with those of the numerical models for clearance by a single vessel and by two vessels with co-current flow, as seen in Figs. 4–6.

Counter-Current and Co-Current Exchange

Our results show large differences between co-current and counter-current flow upon the clearance rate at the capillary level. The difference is not unexpected, but whereas others have considered counter-current exchange of inert gas between arterioles and venules entering and leaving a capillary bed (30), counter-current exchange of inert gas at a capillary level does not appear to have been considered in this context. This is surprising, because the effects of counter-current flow between capillaries have been analyzed for steady-state transport of oxygen (O₂) (1, 12), plasma proteins (36), and heat (37, 40). Inspection of Figs. 2 and 3 provides clear insight into why clearance is slower with counter-current flow. In Fig. 3, it is seen that whereas gas is cleared progressively from the entrance end by co-current and single-vessel perfusion, with counter-current, flow clearance proceeds from both ends of the unit with dissolved gas apparently trapped in the central region. It is as if diffusion between counter-current flowing vessels sets up a nonproductive recirculation of gas molecules, which delays their clearance from the tissue. The size of this recirculation region depends on the blood flow, the partial pressure differences between the inflow and outflow vessels, the tissue diffusion coefficient of the gas, and its partition coefficient. The relations between the trapped gas and these factors are not straightforward and will be discussed in terms of their effects on the clearance rates.

Blood Flow and Inert Gas Clearance Rate from Tissues

Our models predict an almost linear relation between tissue clearance rates of inert gas, as estimated by 1/τ and blood flow for both co-current and counter-current systems (Fig. 5B). The slope of the relation for counter-current flow is well below that for co-current flow. The approximate linear relation between 1/τ and perfusion rate for counter-current flow indicates a constant fraction of the clearance is trapped by counter-current exchange. The difference between the slopes of the relations in Fig. 5B indicates that for these conditions, it amounts to 30–40% of the co-current clearance. Apart from the differences in slope between clearance rates by co-current and counter-current systems, both lines extrapolate back to a positive value of 1/τ when the perfusion rate is zero, suggesting that clearance continues in the absence of blood flow. This reflects the diffusion of gas through the entrance ports of the capillaries in the model where P/P₀ remains at zero. Although diffusional losses through the entrance ports of the capillaries may be considered as an artifact of the boundary conditions of the model, analogous phenomena arise in living tissues, e.g., diffusion from poorly perfused to well-perfused tissues (29) and diffusion from the borders of pools of labeled inert gas, introduced into tissues for the measurement of blood flow.

Over the range of low perfusion velocities, e.g., u < 0.5 mm/s, equivalent to F/V_T < 44 ml/(min 100 ml tissue) in our models, the clearance rate deviates considerably from a simple exponential decay, and 1/τ is not a good indicator of the rate of clearance. It must be emphasized that the linearity of the relation between 1/τ and perfusion rate is only approximate. The nonlinearity of the counter-current data is the more conspicuous, and an upward curvature is visible for the data shown in Fig. 5B. It was noted earlier that whereas a simple exponential decay function is a reasonable approximation for the first 50% or more of clearance by both counter-current and co-current perfusion, data from counter-current systems have been shown to deviate in the later stages. So far, these discrepancies have not been pursued in detail.

Partition Coefficient and Diffusivity

The inverse relation between the clearance rate (1/τ) and partition coefficient (λ), shown in Fig. 6, is to be expected from Eqs. 1A–1D. The higher rates of clearance by co-current as opposed to counter-current flow might also be anticipated from the relation between clearance and perfusion rates, but the diminution of this difference as λ increases merits comment. Increasing λ expands the capacity of the tissues to hold gas molecules, and the product λV_T is the volume of distribution of the dissolved gas in the tissue. During transient changes in blood partial pressure in adjacent capillaries, this increased capacity decreases the volume of tissue with which the blood can equilibrate in single transit. As λ increases to four and above, cross-diffusion between counter-current partner vessels is negligible during the first one-half of tissue clearance, and when it does occur in the later stages, the mean gradients of pressure within the tissues are too small for intercapillary diffusion to be comparable with the convective efflux. Because the product λV_T represents the volume of distribution of the dissolved gas in the tissue, the effects of high λ upon clearance rate could be viewed as equivalent to those of tissues with low F/V_T, i.e., a wider spacing between the microvessels.

Changes in relative diffusivity (R_D = D_T/D_b) have more complicated effects. Remembering that here, R_D has been varied by changing D_T, while holding D_b constant, the abscissae in Fig. 7 may be interpreted as a logarithmic scale of D_T. The general trend is a falling clearance rate with a reduction in D_T for both co-current and counter-current flow, with higher clearance rates for co-current flow over a wide range of values for R_D but with the differences diminishing at both high and low values of R_D. When D_T is very high, both radial and axial gradients of P/P₀ in the tissues are reduced quickly, and clearance rates become the same for co-current and counter-current perfusion. At very low values of D_T, each vessel clears dissolved gas only from the immediate surrounding tissue, and the differences in P/P₀ in blood leaving and entering the unit are reduced considerably. The latter greatly diminishes the gradients of P/P₀ in the tissue, which drive cross-diffusion, ultimately reducing it to zero when clearances by co-current and counter-current flow become identical. When R_D is in the range of 0.1–1.0, intercapillary diffusion through the tissues is maximal, while longitudinal gradients are able to persist, amplifying differences in clearance rates between co-current and counter-current flow.

General Comments

Discrepancies between the predictions of the simple C-C theory and experimental observations have been shown most clearly when clearance has been used to estimate blood flow. Xe clearance curves are described by not one but at least two exponentials with a fast component dominating the first 90% of clearance (33). Two or more exponential terms are necessary to describe the entire clearance curve from our models, and the first 90% are described by a single exponential with a half-time that approximates closely to the half-time of the fast component derived from experimental data. When the C-C model has been used to interpret clearance data, the fast component of Xe clearance has been shown to correlate well with tissue blood flow. Some investigators have found that estimates of F/V_T calculated using the C-C model from the half-times of the fast component agree with direct measurements of flow (33). Others, however, using the same methods of analysis, have reported that estimates of flow based on Xe clearance average only 50% of the flow that has been measured either directly in isolated, perfused skeletal muscle or by using the microsphere technique in intact muscle. In the latter studies, however, direct proportionality between clearance rates and flow was maintained over a wide range of flows (5). This combination of underestimation of the absolute values of flow but maintenance of proportionality between flow and clearance rates would be consistent with our model's predictions for counter-current flow in the microcirculation. The effect would be amplified if counter-current exchange also occurred between the small arterioles and venules feeding the muscle capillaries, and this would seem possible given the architecture of the muscle microcirculation (30).

Our numerical models do not provide an immediate quantitative solution to this discrepancy between experimental observations and calculations of flow, based on the C-C model's interpretation of tissue clearance. Figure 5B shows that a given clearance rate (indicated by 1/τ) is predicted to require a higher value of F/V_T (indicated by perfusion velocity) by the uncorrected C-C model than by our numerical model for co-current flow and a lower velocity than the counter-current flow model. Thus if the values of 1/τ, predicted by the numerical model for co-current flow, were interpreted using the C-C model, then, the resulting values for F/V_T would be overestimates and not underestimates. For high blood flows (u > 0.5 mm/s), clearance predicted numerically for counter-current flow, however, would be underestimated by the C-C model. Whereas this agrees qualitatively with the observed discrepancies, the maximum underestimation (based on Fig. 5B) would only be 15%. This is considerably less than 50% difference between direct measurements of flow and estimates using the C-C theory to interpret clearance measurements in skeletal muscle (5). The rates of clearance predicted by our numerical models appear to be greater than those measured in experiments. Considering only those factors that have been discussed so far, there are at least three possible explanations for the high clearance rates of the numerical models. These are: 1) λ has been underestimated; 2) D_T has been overestimated; and 3) the simplified geometry of our models does not represent the arrangement of microvessels in tissues.

First, Fig. 6 shows that a relatively small increase in λ from 0.7 to 1.0 would be accompanied by a substantial fall in 1/τ for both co-current and counter-current models. For co-current flow, 1/τ would be reduced from 0.021 to 0.014 s⁻¹, and for counter-current flow, 1/τ would fall from 0.014 s⁻¹ to 0.010 s⁻¹. If diffusional losses at the entrances of the vessels were eliminated, both values of 1/τ would be reduced by a further 0.003 s⁻¹, bringing the value 1/τ for counter-current flow to less than one-half of its value for the C-C model if this retained a value for λ of 0.7 .

A second possible explanation would be that D_T (or R_D) has been overestimated. Since D_T is not considered in the C-C model, it is reasonable to question its value. Figure 7, however, suggests that to reduce 1/τ to values well below those predicted by the C-C model, D_T would have to be lowered by 10- to 100-fold. Particularly striking is the near constancy of 1/τ for counter-current flow, as R_D falls from 0.5 (the value used in most calculations) to 0.05. To have effects on 1/τ comparable with those that would result from an increase in λ, from 0.7 to 1.0, D_T would have fallen below 1.0 × 10⁻⁷ cm²/s, with R_D < 0.01. This is not impossible and cannot be refuted by precise measurements of D through tissues for Xe, which as Xe¹³³, has been used most often for clearance measurements of blood flow. In view of the very much higher measurements of D_T for O₂, hydrogen, N₂, and He (21), it does not seem very likely.

The third factor to be considered is the arrangement of the vessels in the tissue. The cylindrical block of tissue, with one or two straight vessels running axially through it, is an obvious simplification. Apart from one or two specialized microcirculations, adjacent capillaries in most tissues are not parallel, and the distance between them varies considerably. Both co-current and counter-current flow are often observed in adjacent microvessels in the same area of tissue in vivo (4). Where the distance between adjacent vessels within a tissue varies, but distances between a given vessel and its near neighbors vary less (or continuously, as vessels slowly converge or diverge), clearance by each vessel might approximate to that in a parallel array, and the time course of clearance from the tissue as a whole should reflect a summation of each of its component units. Figure 8A shows that the rate of equilibration varies inversely with square of radius of surrounding tissue, suggesting that the summation of clearances from a population of units might be skewed toward those with largest values of F/V_T, which might represent a small fraction of the total flow through the tissue. As soon as one moves from the considering tissue clearance by one or two exchange vessels to clearance from an entire tissue by its microvascular bed, one has to understand how to integrate clearances from a heterogeneous population of exchange units, i.e., the different blood flow and vessel surface area of each vessel and the tissue volume that it clears. This conclusion favors a stochastic approach to the analysis of clearance data, as proposed by Zierler (41). Models such as those considered in this paper can provide information about the components of the frequency distributions and how the distributions may be skewed by factors such as counter-current flow and so ultimately, lead to more complete interpretations.

GRANTS

W. Wang thanks the Royal Academy of Engineering for a Global Research Fellowship, which supports his secondment at Harvard University. The project was supported in part by Medical Research Council/Engineering and Physical Sciences Research Council (EPSRC) Discipline Bridging Initiative Grant G0502256-77947 to W. Wang and EPSRC Grant EP/J011959/1 to Y. Lu.

DISCLOSURES

No conflicts of interest, financial or otherwise, are declared by the authors.

AUTHOR CONTRIBUTIONS

Author contributions: C.C.M. and W.W. conception and design of research; Y.L. performed experiments; Y.L., C.C.M., and W.W. analyzed data; C.C.M. and W.W. interpreted results of experiments; Y.L. prepared figures; Y.L. drafted manuscript; Y.L., C.C.M., and W.W. edited and revised manuscript; C.C.M. and W.W. approved final version of manuscript.

Glossary

A

Advection matrix in finite element formulation

b (subscript)

Blood

C_b

Inert gas concentration in blood

C-C model

Conduction-capacitance model

C₀

Initial concentration of inert gas in blood

C_T

Inert gas concentration in tissue

d

The distance between capillaries in a two-capillary unit

D_b

Inert gas diffusivity in blood

D_T

Inert gas diffusivity in tissue

e (superscript)

Arbitrary element in finite element formulation

F

Blood flow

K

Diffusion matrix in finite element formulation

M

Mass matrix in finite element formulation

n

The number of elements

n

Normal direction of capillary wall

N₂

Nitrogen

P

Inert gas partial pressure

P_a

Partial pressure of given inert gas in the arterial end of a capillary

P_N2

Nitrogen partial pressure

P₀

Initial partial pressure

P_T

Partial pressure of given inert gas in the tissue space

P_v

Partial pressure of given inert gas in the venous end of a capillary

r

Radial position in the cylindrical coordinate system

R_D

Ratio of inert gas diffusivity in tissue to that in blood

S_b

Solubility of given inert gas in the capillary blood

S_T

Solubility of given inert gas in the tissue space

t

Time

T (subscript)

Tissue

u

Blood perfusion velocity in a capillary

V

Volume of both tissue and capillaries

V_T

Tissue volume

z

Axial position in the cylindrical coordinate system

Γ

Boundary associated with element(s)

λ

Inert gas tissue:blood partition coefficient

τ

Inert gas clearance half-time

φ

Interpolation function in finite element formulation

Ω

Space occupied by element(s)

▽

Gradient operator

▽²

Divergence operator

APPENDIX

The Galerkin FEM is used to formulate these governing equations, where both weighting function (φ) and interpolation function assume the identical form. Equation of C_b in the capillary over a typical finite element (Ω_b^e) reads

$${\displaystyle {\int }_{{\Omega }_b^e}\left(\frac{\partial C_b}{\partial t}+\text{u}\nabla C_b-D_b{\nabla }^2{C}_b\right)\varphi d\Omega =0}$$ (A1)

After the operation of integration-by-part, it becomes

$${\displaystyle {\int }_{{\Omega }_b^e}\left(\varphi \frac{\partial C_b}{\partial t}+\varphi \text{u}\nabla C_b+D_b\nabla C_b·\nabla \varphi \right)d\Omega =\underset{{\Gamma }_b^e}{\oint }{D}_b\varphi \nabla C_b·\text{n}d\Gamma }$$ (A2)

Accordingly, for the gas concentration in a finite element of tissue (Ω_T^e)

$${\displaystyle {\int }_{{\Omega }_b^e}\left(\varphi \frac{\partial C_T}{\partial t}+D_T\nabla C_T·\nabla \varphi \right)d\Omega =\underset{{\Gamma }_b^e}{\oint }{D}_b\varphi \nabla C_b·\text{n}d\Gamma }$$ (A3)

Considering all elements, final equations are expressed in the following matrix form as

$$M_b{\dot{C}}_b+A_b{C}_b+K_b{C}_b=F_b$$ (A4)

$$M_T{\dot{C}}_T+K_T{C}_T=F_T$$ (A5)

where the dot denotes temporal derivation. M corresponds to the assembled mass matrix, A for the advection matrix, and K for the diffusion matrix. These coefficient matrices are defined as

$$\begin{array}{c}{M}_b={\displaystyle \sum _{n_b^e}{\displaystyle \underset{{\Omega }_b^e}{\int }\varphi {\varphi }^{T}d\Omega ,}}\\ M_T={\displaystyle \sum _{n_T^e}{\displaystyle \underset{{\Omega }_T^e}{\int }\varphi {\varphi }^{T}d\Omega ,}}\\ K_b={\displaystyle \sum _{n_b^e}{\displaystyle \underset{{\Omega }_b^e}{\int }{D}_b\nabla \varphi ·\nabla {\varphi }^{T}d\Omega ,}}\\ K_T={\displaystyle \sum _{n_T^e}{\displaystyle \underset{{\Omega }_T^e}{\int }{D}_T\nabla \varphi ·\nabla {\varphi }^{T}d\Omega ,}}\\ F_b={\displaystyle \sum \underset{{\Gamma }_b^e}{\oint }{D}_b\varphi \nabla C_b·\text{n}d\Gamma ,}\\ F_T={\displaystyle \sum \underset{{\Gamma }_T^e}{\oint }{D}_T\varphi \nabla C_T·\text{n}d\Gamma ,}\\ A_b={\displaystyle \sum _{n_b^e}{\displaystyle \underset{{\Omega }_b^e}{\int }\varphi \text{u}·\nabla {\varphi }^{T}d\Omega }}\end{array}$$ (A6)

During the assembly process, most boundary integrals (F_b and F_T) on the right side cancel each other at the interelement boundary. At the capillary-tissue interface, the boundary integral also vanishes with the flux boundary condition (Eq. A3). More details about how to implement it numerically can be found in Reddy and Gartling (31). For the temporal gradient, a fully implicit scheme is used to ensure numerical stability. The main computer code used in the numerical study can be found in the supplement document of the paper.