(Circulation. 1996;94:1876-1885.)
© 1996 American Heart Association, Inc.
Articles |
Precapillary Servo Control of Blood Pressure and Postcapillary Adjustment of Flow to Tissue Metabolic Status
A New Paradigm for Local Perfusion Regulation
the Institut für Physiologie und Pathophysiologie, Johannes Gutenberg-Universität Mainz, Germany.
Correspondence to Priv Doz Dr med Karlfried Groebe, Institut für Physiologie und Pathophysiologie, Johannes Gutenberg-Universität Mainz, Duesbergweg 6, D-55099 Mainz, Germany. E-mail thews@mzdmza.zdv.uni-mainz.de.
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Abstract |
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Background There are several shortcomings in current understanding of how the microvasculature maintains tissue homeostasis. Presently unresolved issues include (1) integration of the potentially conflicting needs for capillary perfusion and hydrostatic pressure regulation, (2) an understanding of signal transmission pathways for conveying information about tissue energetic status from undersupplied tissue sites to the arterioles, (3) accounting for the experimentally observed interrelations between precapillary and postcapillary resistances, and (4) an explanation of how precise local adjustment of perfusion to metabolic demands is achieved.
Methods and Results A novel conceptualization of how local microvascular control mechanisms are coordinated is proposed, according to which blood flow is adjusted to the metabolic needs of the tissue by the venules. Arteriolar action is merely called on for controlling capillary pressure through myogenic response and shear stress–induced vasodilation. A mathematical model of this theory is introduced and evaluated using well-established experimental data from the literature on regulating mechanisms of microvessel diameters exclusively. The model results demonstrate the suggested mode of microvascular operation to be functional and efficient under conditions present in vivo. Moreover, the predicted vascular responses are large enough to cover the entire range observed in exercising skeletal muscle during adjustment of perfusion to higher performance levels.
Conclusions Precapillary pressure regulation combined with postcapillary adjustment of perfusion to tissue metabolic status is suitable to resolve the above shortcomings in our current understanding of microvascular control. With mathematical modeling based on experimental data, this mode of microvascular operation is shown to be functional and effective in controlling muscle microcirculation.
Key Words: regional blood flow • blood pressure • metabolism • microcirculation • physiology
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Introduction |
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In organs in which blood flow is adjusted primarily to meet their nutritional requirements, the demands that the local microvascular control system has to meet are threefold. First, perfusion needs to be sufficiently large and adequately distributed to maintain undisturbed function of all tissue cells, including the ones with least favorable supply conditions. Second, transmural pressure in exchange vessels must be kept small enough to prevent excessive fluid filtration and the formation of interstitial edema. Third, under conditions of high demand for oxygen or nutrients, overall organ blood flow should not exceed the amount actually necessary to avoid overloading the heart and/or underperfusing other organs. Obviously, there is a potential conflict between the first and the second and third requirements. As a further problem, all these tasks need to be accomplished in the presence of a wide range of tissue metabolic rates and, particularly under pathophysiological conditions, of substantial variations in local arterial blood pressures.
On the basis of numerous in vivo and in vitro studies, it is widely accepted now that (1) small arterioles are the sites at which the largest changes in perfusion resistance occur and hence represent the most powerful effectors in blood flow regulation1 ; (2) the myogenic response of arteriolar smooth muscle to changes in wall tension plays an important role in the control of capillary transmural pressure2 ; (3) important factors involved in perfusion control are concentration changes of vasoactive substances (the release or consumption of which depend on the level of organ cell activity or energetic status)1 3 ; and (4) there are mechanisms for amplification and propagation of local vasodilator signals like wall shear stress–dependent and conducted vasodilations.4 5 It is not clear, however, how these various mechanisms interact to result in an integrated and adequate response of the vascular system to changes in local metabolism or arterial blood pressure. In particular, there is no answer as to how the system reconciles the conflicting requirements for satisfying the nutritional demands and keeping capillary pressure and overall perfusion rate acceptably low. Moreover, understanding perfusion control through the release of vasoactive substances from nutritionally deprived microregions poses problems of its own (Fig l). The most critical sites in which tissue energetic status falls short earliest ("lethal corners") and that therefore are the first to generate a vasodilator signal are located hundreds of micrometers—in skeletal muscles, frequently >1 mm—away from the corresponding feeding arterioles that selectively need to dilate to restore adequate supply. To date, all suggestions on how this gap in the signal chain might be bridged6 7 8 9 fail to furnish satisfactory explanations (see Fig 1
).
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In the present article, a further suggestion is put forth, according to which the local regulation of perfusion resides in the venous system. The teleological reasons for regulation at this site are based on engineering principles that suggest a design that would (1) position a sensor for indicators of tissue energetic status released to the blood downstream of and close to the lethal corner to record the unabridged local signal with minimal delay; (2) place the effector next to the sensor and control flow by changing venous flow path conductance to enable tight spatial and temporal coupling of the response to the signal, to avoid distortions that might be caused by long-range signal transmission, and to minimize interferences with capillary pressure control; and (3) build a servomechanism into the arteriolar tree that keeps capillary pressure (almost) constant at various systemic pressures and adjusts it as needed to maintain the blood flow preset by the venular flow controller.
Thus, an engineer would solve the problem of integrating the conflicting requirements of pressure and perfusion controls by constructing two largely independent mechanisms: the first sets exchange vessel hydrostatic pressure as low as possible and as high as necessary to achieve the perfusion rate demanded by the second one.
Support for the idea that nature has applied the very same blueprint is provided by experimental studies that have shown that all the components constituting the control system outlined above actually do exist in the microcirculation. (1) Not only arterioles but also venules are reactive to vasoactive substances10 and thus may represent combined sensors of tissue energetic status and effectors in flow control. This also has been concluded from direct observations of venules during postcontraction hyperemia11 and is reflected in the decrease in postcapillary resistance with increasing perfusion.1 (2) Arterioles show myogenic response2 and (3) flow-induced dilation,4 which together may be viewed as an arteriolar servomechanism for control of capillary hydrostatic pressure during variations of blood flow rate or arterial pressure. (4) In response to small increases in venous pressure, precapillary resistance increases up to 10 times more sensitively than on corresponding changes in arterial pressure,1 demonstrating that arterioles may indeed be highly reactive to small changes in postcapillary resistance. Hence, the proposed mode of microvascular operation relies exclusively on long-known mechanisms. The novel feature about this is not a new property of microvessels or the like but rather a new interpretation of the changes in postcapillary resistance, which are regarded as the cause of rather than a terminal effect in the chain of processes taking place during adjustment of perfusion to metabolic needs. From a historical viewpoint, much respect is owed to the extensive thinking of Burton,12 who put forth the idea of a venoarterial feedback loop by which inflow into an organ may be reduced in response to congestion in its venous outflow path.
If all the above components of the microvascular controller are present, the following is going to happen on local vasodilator release from tissue cells: The draining venules will dilate by this lowering postcapillary resistance; consequently, capillary hydrostatic pressure will decrease slightly and blood flow will increase slightly. The small changes in pressure and flow exert synergistic effects on arterioles, which also will dilate, thus initiating increases in flow (and capillary hydrostatic pressure). Higher blood flow reduces the viscosity of venular blood, resulting in a further drop in postcapillary resistance that is fed back to the arteriolar system, and so forth. Hence, from the existence of venular reactivity to vasodilators and of pressure and shear stress dependence of arteriolar diameters, we may conclude that the outlined mechanism of microvascular control should be functioning to some extent, the practical relevance of which, however, needs to be evaluated.
Thus, the crucial question one must answer concerns the quantitative importance of this mechanism: Can the small active changes observed in postcapillary resistance by themselves enforce large arteriolar responses, even though they are only a small fraction of the changes in total resistance? Shifting focus to the arteriolar tree, one needs to assess the dependencies of arteriolar diameters on pressure and flow in their relation to the changes in postcapillary resistance; thus, an equivalent question arises: Are the known properties of arterioles suitable for constituting the proposed arteriolar servo system, ie, to quantitatively account for the characteristics that the latter is required to exhibit? Put more practically, Can the arteriolar tree, with its pressure- and flow-dependent mechanisms, transform small changes in postcapillary resistance into large changes in precapillary resistance? On the basis of the presently available data, it will be demonstrated here that the entire arteriolar system can be controlled by rather minute active changes in venular diameters.
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Methods |
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Model and Data
Calculations have been performed for the microvasculature in cat tenuissimus muscle because it represents one of the animal models for which the most complete morphometric and hemodynamic data sets are available. To study the relations between precapillary and postcapillary resistances in the absence of the modifying effects of heterogeneities, a symmetrically branching microvascular tree was considered that consisted of four generations of arterial and venous microvessels, each exhibiting average dimensions and blood flow rates with a capillary bed between them. Distortions introduced by network asymmetries will be explored in a future step.
In the following, the structure of the model is outlined; for a more detailed description, see the "Appendix." Pressure drop per unit length of vessel, dP/dl (millimeters of mercury per millimeter) at blood flow rate, 
(nanoliters per second), and vessel diameter, d (micrometers), is governed by Hagen-Poiseuille's law:
 | (E1) |
(d, v) (centipoise), may be expressed as a function of vessel diameter, d, and mean blood velocity, v (millimeters per second). Arterioles exhibit a myogenic response to changes in wall tension and wall shear stress–induced vasodilation; thus, arteriolar diameter, da, may be represented as a function of transvascular pressure, P, and blood flow rate: da=da(da0, P, ), where da0 is the arteriolar reference diameter at zero pressure and flow for given levels of vascular tone and concentration of vasoactive substances. Capillaries are generally thought to behave passively, ie, their diameters are increased by rising pressure, and are independent of flow: dc=dc(dc0, P). The same is true for the venules: dv=dv(dv0, P); however, dv0 is taken to depend on vascular tone and vasodilator concentration. By substituting d and v in Equation 1 by da(da0, P, ) and 
, respectively, we obtain
 | (E2) |
, reference diameters, da0, and functions, da=da(da0, P, ), specific to the individual vascular generations, Equation 2 may be integrated numerically to find the pressure course, P(l), along the vessels, from which the corresponding diameters, d(l), may be computed. Similarly, the dependence of pressure and diameters on position along the vascular tree may be calculated for capillaries and venules. Input data for four arteriolar/venular generations have been selected from the literature to match the situation in cat tenuissimus muscle as closely as possible. Data are detailed in Figs 2 and 3 and in the "Determination of System Parameters" section in the "Appendix." Fig 2 displays the fits used for the vascular pressure-diameter relations. Fig 3 shows pressures and velocities in the various vascular generations as functions of mean diameter, calculated for standard conditions (resting perfusion rate, =rest, and mean pressure in the organ artery, Pa=105 mm Hg), which are overlaid on a plot of experimental data.13 Computed pressures and velocities are well within the range of measured data, suggesting that a reasonable choice of data specific to tenuissimus muscle has been met. Resulting venous outflow pressure is Pv=6.6 mm Hg.
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), 81.2 (
), 41.2 (
), 27.6 (
), and 18.3 (
)µm. Because measurements had been made in vitro, data were converted to in vivo conditions by application of the ratios of in vivo to in vitro myogenic indexes given in Reference 24. To the right of peak diameters, only the descending portions of the measured P-d relations were employed for the fits. Expression functions are given in the "Appendix" (first two factors of the right-hand side of Equation 9), in which the first factor represents passive vessel behavior (which was fitted simultaneously to experimental data
) and venules (
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) as functions of vessel diameter in resting cat tenuissimus muscle, measured by Fronek and Zweifach
According to the presently proposed view of perfusion control, the organism actively adjusts postcapillary resistance so that the resulting perfusion rate meets the actual requirements of tissue metabolism. To model this process, one needs data quantifying vasodilator release and venular responses. Owing to a lack of such data, the model needs to take an indirect approach: It uses a range of predefined perfusion rates, , rather than metabolic rates for assessing the changes occurring in the microvasculature during perfusion regulation. While for given , Pa, and a set of parameters describing intrinsic properties of the vascular system (see previous paragraph and the "Appendix") pressures in arterioles and capillaries are uniquely determined, the active adjustment of postcapillary resistance to vasodilator concentration is simulated in the following way: For each set of nonstandard conditions (
rest or Pa105 mm Hg), the model calculates the relative increase of venular unstressed diameters over corresponding diameters under standard conditions, which yields the above venous outflow pressure, Pv, of 6.6 mm Hg. (For more details, see the Appendix).
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Results |
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Fig 4
shows calculated transmural pressures and vessel diameters as functions of position along the microvasculature for a range of blood flows at Pa=105 mm Hg (Fig 4A) or arterial pressures at =rest (Fig 4B). Compared with standard conditions, a 95 mm Hg increase in Pa increases pressure at the capillary origin by 
6 mm Hg (Fig 4B), which demonstrates the potential of the system to autoregulate capillary pressure. On the other hand, an increase in perfusion pressure above rest (Fig 4A) reduces pressure drops in arterioles, and pressure at the capillary origin increases from 38 to 54 mm Hg at 8rest. This effect of flow-induced vasodilation (which is partly counteracted by myogenic response) occurs if the increase in diameter exceeds the fourth root of the increase in flow: (This is a consequence of Hagen-Poiseuille's law [Equation 1], which states that the pressure drop in a vessel is proportional to blood flow divided by vessel diameter to the fourth power. Thus, if because of flow-induced vasodilation vessel diameter increases more rapidly than the fourth root of blood flow, the denominator of this fraction grows faster than the numerator, and pressure drop decreases.) In working muscles, an increase in capillary perfusion pressure with blood flow rate is essential for achieving high performance levels at which capillary resistance may become limiting to perfusion. Owing to the small compliance of capillaries, their diameters and hence their flow resistances hardly change, and recruitment of capillaries that are unperfused at rest would account for no more than an approximate threefold increase in flow if pressure drop across the capillary bed were constant.
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According to the present hypothesis, precapillary resistance is matched to postcapillary resistance through arteriolar responses to alterations in pressures and flows that are induced by adjustment of postcapillary resistance to the actual metabolic needs of the tissue. However, in addition to the active, metabolite-mediated control mechanism, postcapillary resistance is affected by at least two more passive mechanisms that are pressure-dependent dilation and shear-dependent blood viscosity and that consequently should also influence regulation of precapillary resistance and flow. Moreover, total postcapillary pressure drop is a fourth determinant of venous return. Because it has been a concern whether the adaptations taking place in the venous system during pressure or perfusion changes are merely passive, the relative contributions of these four factors toward postcapillary adjustment of flow are compared in Fig 5. Plots are in the form of, for example, actual postcapillary pressure drop divided by postcapillary pressure drop under control conditions. Fig 5A shows that for increasing , postcapillary perfusion pressure rises by up to 35% to 40%. Because equals perfusion pressure times conductance, it follows that conductance at =8rest must be roughly six times that at rest. Shear-dependent viscosity and passive dilation can increase conductance up to 
1.6-fold and 1.1-fold, respectively. Consequently, in cat tenuissimus muscle, postcapillary conductance would be 6÷(1.6x1.1)3.4 times smaller than necessary to accommodate perfusion rates of =8rest if there were no active diameter control in venules (which includes shear stress–induced venular dilation14 ). This factor is too large to be explained by errors in the underlying data. For further details, see the legend of Fig 5.
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Discussion |
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By means of model calculations using currently available measured relations between microvascular diameters and luminal pressures and flows, it has been demonstrated that the changes in conductance of the arterial tree experimentally observed during muscular work at increasing performance may be caused exclusively by arteriolar responses to the corresponding (and much smaller) active reductions in postcapillary resistance. The entire vasculature can, in principle, be controlled by effects of vasoactive metabolites on venular diameters; neither direct vasodilator action on precapillary vessels nor propagation of any dilator signals along the vascular tree from venules to arterioles is required. (Alternatively, one could state that it is the bloodstream itself that carries the signal to the arterioles in the form of a slightly increased flow and reduced transmural pressure, where it is amplified by shear stress–induced dilation and myogenic response. For conducted vasodilation, see also the next paragraph.) If confirming this theory experimentally were possible, the problems of signal transmission between lethal corners and feeding arterioles (see Fig 1
) and of coordination of pressure and perfusion controls would be resolved.
The above statement is not meant to imply that there in fact are no influences on arteriolar diameters besides transmural pressure and luminal flow. Rather, this article is intended to propose a plausible and realistic concept of a stable feedback loop for metabolic perfusion regulation, the individual components of which may, however, be modified in their actions by a variety of factors. In this way, the operation of the controller could be adapted to extraordinary conditions, its regulatory range could be shifted, etc, but the feedback loop as such should remain intact and become modulated only in its function. An example of modulating factors is central control of the microvasculature through the autonomic nervous system and humoral factors such as epinephrine and angiotensin that one would expect to change the "set points" of the local mechanisms (which may be represented by unstressed diameters at zero flow, the functions describing P-d or -d relations, or dose-response curves of vasodilator action on vessel diameters). Moreover, there is conducted vasodilation in arterioles5 (which has most likely contributed to the experimental -d relations underlying the present simulations4 and therefore should, at least in part, be implicitly considered in the model), and even direct effects of vasoactive metabolites on arterioles may be present (which again may serve to establish new set points for the proposed mechanism to operate at greatly altered supply-demand ratios and which should play a role, at least under most critical supply conditions in which metabolite concentrations in all the tissue are expected to be high).
Even though currently no direct proof is available that the proposed conceptualization of microvascular control plays an important role, experimental evidence in support of it exists and is outlined here (manuscript in preparation): (1) The present paradigm postulates and relies on largely independent mechanisms for pressure and flow controls. From a number of experimental studies (most explicitly from Davis,15 who also emphasized the role of venous resistance), such mechanisms may be inferred to exist. (2) Direct observations of vascular diameters before and after electrical stimulation of rat spinotrapezius muscles reveal an active and possibly even a leading role of venules during flow adjustment to muscular work.11 (3) In hamster cremaster muscle in which a step increase in oxygen consumption rate was induced,8 the relations between the subsequent increases in diameters of first-order arterioles, red cell velocities, and wall shear stresses suggest that arteriolar dilation was caused by faster blood flow rather than a direct metabolic vasodilator effect and that flow control was exerted at locations other than the observation site. (4)The present theory of perfusion control predicts that for flow to be autoregulated at decreasing perfusion pressure, postcapillary resistance must decrease, at least slightly. From experimental studies in which perfusion pressure was decreased, it indeed appears that the absence of flow autoregulation correlates with rising postcapillary resistance (eg, see Reference 1 for references). Moreover, changes in postcapillary resistance calculated from the results of the above-cited experiments in bat wing15 are similar to those predicted by the present model. (5) From their analysis of vascular resistance in arteriolar networks of cat sartorius muscle, Popel et al16 concluded ". . . that postarteriolar resistances may play a very important role in distribution of flow in the microvascular network."
In this context, an important question is, How can the relevance of the proposed theory be tested experimentally? In a first and indirect approach, one could systematically compare simultaneously observed time courses of arteriolar and venular responses to a change in muscular performance (cf, Reference 11). However, it is known that in perfusion control in muscle, at least two distinct processes with a slow and a fast time course are involved17 ; the latter may be due to direct nervous or ionic (K+) action on arterioles. Thus, one would have to ensure that stimuli last long enough to evoke a vascular response to changes in muscle metabolism and compare the times at which peak dilation is reached, not the onsets of vasodilation. Moreover, the time courses for the return to resting tone after cessation of muscle stimulation may reveal relevant information. Second, in systematic studies of the responses of local precapillary and postcapillary resistances (and their mutual relations) to changes in perfusion rate or pressure (cf, Reference 15), these responses may be checked for consistency with the proposed mechanism for perfusion control. For a more direct third approach, one could try to lower postcapillary resistance without affecting arterioles directly. This might be done by injecting small amounts of a nondiffusible vasodilator into a small venule. In this type of experiment, a diffusible vasodilator also might be used if an arteriolar supply path remote from the venules can be identified and observed for its responses. Alternatively, one could simulate lowered postcapillary resistance by applying suction to a pipette inserted into a venule and observe the reactions of the feeding arterioles. To avoid venular collapse during such an experiment, it may be necessary to increase transvascular pressure by placing the body of the animal in a pressurized chamber.15
Because of a lack of data pertaining to the microvascular system of one specific muscle, data from different sources had to be used (eg, from bat wing arterioles for myogenic response or from rat cremaster arterioles for flow-dependent dilation). Furthermore, quantitative information on the recruitment of terminal arterioles or collecting venules with increasing muscle perfusion is not available. The same holds true for the effects of further mechanisms that (in addition to active metabolite-induced dilation, passive distensibility, and shear rate–dependent viscosity) may affect postcapillary resistance, like active venular responses to pressure or shear stress.18 These deficits, along with the simplified treatment of the vascular tree (see below), make quantitative interpretation of the results difficult. Moreover, concentrations of the various vasoactive substances in the venular blood under different experimental conditions and (joint) venular dose-response curves to these substances are unknown. Therefore, it was not possible to simulate the adjustment of perfusion to metabolic needs explicitly; rather, the responses of the microvasculature to a range of predetermined perfusion rates were modeled. Specific quantitative information about vasodilator release, perfused network geometry, and intrinsic properties of the vasculature in one and the same muscle are needed to obtain a more quantitative description of microvascular control mechanisms.
In the model calculations, a symmetrically branching vascular tree was considered, and structural and functional heterogeneities were neglected. In the framework of the proposed theory, diameter and flow heterogeneities should exclusively be a consequence of heterogeneities in flow path length and connectivity or in demand (which could be heterogeneous volume-related demand and/or heterogeneous magnitudes of the tissue regions supplied or drained by the vessels of a given generation). For a rigorous test of such heterogeneity effects, a direct description of vasodilator release and the impact on venules (instead of the currently used indirect approach; see above) needs to be implemented in the model. However, some very preliminary calculations show that the model behaves reasonably if heterogeneous flows, eg, in microvascular generations 3 and 4, are assumed. More precisely, arteriolar diameters in the branches with higher flows are greater than in those with lower flows, so pressure drops are much less variable than perfusion rates. Moreover, diameters and pressure drops in generations 1 and 2 depend on total flow only and are the same, regardless of whether homogeneous or heterogeneous flows in generations 3 and 4 are assumed.
As a final caveat, the model describes steady-state responses of the vasculature to different levels of demand for perfusion; transients occurring during the adjustments to changes in performance or local perturbations are not accounted for. Considering transients not only would be a much more complex mathematical problem but also would require a number of time constants for arteriolar responses to pressure and shear stress, vasodilator release after a change in performance, venular reactions to altered vasodilator concentrations and pressure, etc, to be known, which are not readily available. Moreover, for modeling the time courses of these adjustments, an explicit description of venular responses to performance changes is needed (see above). Even though a lot of work on refining the model is needed and a large number of additional data will have to be acquired, mathematical simulation of the transients during flow adjustment appears to be a rewarding goal. With such a model, one could check the underlying mechanisms of blood flow regulation for stable operation and could compare the time courses of the various adaptive responses to experimental observations, which probably will provide strong evidence in favor of or against the proposed theory of perfusion control.
All of the above shortcomings relate to the suitability of the model to give a comprehensive description of the processes taking place during local adjustment of perfusion to metabolic demand. However, this is not what the model was designed for. Rather, the question raised and answered by the model was, Does a combination of venular diameter responses to vasoactive metabolites in the draining blood on the one hand and arteriolar diameter control by pressure- and shear stress–dependent mechanisms exclusively on the other hand possess the potential to account for the changes in vascular conductance observed during altered perfusion rates or arterial pressures? To answer this question, heterogeneities, transients, and the exact nature of venular interactions with tissue metabolism are of no relevance. Relevant are a reasonably precise description of the dependence of arteriolar diameters on pressure and flow rate and a realistic range for the magnitudes of active venular diameter changes necessary to bring about the observed variations in precapillary resistance; both requirements are met. Beyond its original purpose, the model demonstrates for the first time how known mechanisms of microvascular control may interact to form integrated responses of the vascular system and gives a clue as to how precapillary and postcapillary resistance may be coordinated.
In conclusion, the present article shows that blood flow autoregulation by metabolite-mediated active changes in venular diameters must be present to some degree. Moreover, combined with control of capillary pressure by myogenic response and shear stress–induced dilation in arterioles, this mechanism may be powerful enough to play a dominant role in the adjustment of perfusion to tissue metabolic needs, which would resolve a number of problems in our current understanding of local microvascular control: (1) The proposed theory is based on a well-defined and unambiguous closed feedback loop for metabolic perfusion regulation that enables (2) strictly local perfusion control and (3) tight temporal coupling between demand and microvascular response. (4) Pressure and perfusion controls are largely uncoupled, allowing easy integration of their potentially conflicting requirements. (5) Blood flow by itself is suggested to transport vasodilator information from downstream of the lethal corners to the arterioles, which solves the problem of long-range signal transmission.
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Acknowledgments |
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I wish to acknowledge the helpful critiques and comments from many colleagues in developing the ideas underlying the conceptualization of local perfusion control proposed in this article. In particular, I am indebted to Prof Dr G. Thews, Prof Dr C. Honig,
and Prof Dr U. Pohl for their invaluable contributions.
Derivation of Model Equations
Pressure Gradients in Single Unbranched Vessels
Pressure drop per unit length of vessel, dP/dl, at blood flow rate and vessel diameter d is governed by Hagen-Poiseuille's law (Equation 1). In this equation, blood apparent viscosity, (d, v), is a function of vessel diameter and mean blood velocity, v. Its dependence on d (µm) was determined in vitro19 :
 | (E3) |
p is the plasma viscosity (p=1.2 cP at 37°C), Hd is the discharge hematocrit (Hd=0.4 in the cat20 ), and dm is the minimum diameter the red cell can attain. For known red cell volume, V, and resting diameter, dr (V=57 fL, dr=6 µm in the cat20 ), red blood cell surface, A, and dm may be obtained according to Reference 21 from
 | (E4) |
 | (E5) |
As shown by Lipowsky22 under in vivo conditions, there is an additional dependence of the viscosity on mean blood flow velocity, v (mm/s)—more precisely, on the reduced velocity 
=
 | (E6) |
This relation was implemented by multiplying d(d) from Equation 3 by a dimensionless factor that specifies the velocity dependence of . For each diameter d, this factor takes on a value of 1 at a reference velocity, vref, at which v(ref) (from Equation 6) becomes equal to d(d) (from Equation 3). Moreover, a lower limit for the apparent blood viscosity, (d, v), is given by the plasma viscosity, p. Hence, one obtains
 | (E7) |
(d and v from Equations 3 and 6) for each vessel diameter d. To eliminate mean blood velocity, v, from Hagen-Poiseuille's law (Equation 1), v may be substituted by 4000/(
d2), and one obtains
 | (E8) |
(from Equation 7) are functions of d and exclusively.
Vascular Diameters
Within the framework of the present paper, is the independent variable, and d depends on transmural pressure, (in arterioles), and vascular tone and vasodilator concentration (in arterioles and venules). (For details on this approach, see the "Model and Data" section and below.) More specifically, the following diameter dependencies were used.
1. Arterioles exhibit a myogenic response to changes in wall tension and wall shear stress–induced vasodilation; thus, arteriolar diameter, da, may be represented as a function of transvascular pressure, P, and blood flow rate, da=da(da0, P, ), where da0 is an arteriolar reference diameter (eg, that at zero pressure and flow for given levels of sympathetic tone and concentration of vasoactive substances) that identifies the intrinsic properties of the respective vessel. To obtain analytical expressions for these dependencies, expression functions of the form
 | (E9) |
, the power [(P-P0)/
P] is defined only for P
P0. For P<P0, [(P0-P)/P] was used instead.) The first factor in this product represents passive vessel behavior (which is a hyperbola increasing from diameter d0 at zero pressure to d
for infinitely high pressure at a rate given by a); the second describes the development of active tone with pressure (represented by a dimensionless sigmoid curve centered at P0 and falling between P=- and P= from c to b, where the steepness and the shape of this decline are given by P and ); the third accounts for the effects of wall shear stress–induced dilation. This factor specifies the ratio of vessel diameter in the presence of flow to the diameter at no flow as a function of actual flow divided by resting flow. As the latter expression approaches 0 for 
0, which is unrealistic, this factor has been bounded from below by 1 (ie, vessel diameter 90% that at resting flow23 ). While the third factor was derived by nondimensionalizing the d- relation ln()=2.87 ln(1/2d)-5.47 given in Reference 4, the first two factors were fitted to active and passive P-d relations in arterioles of five different branching orders measured in vitro by Davis24 and converted to in vivo conditions by applying the ratios of in vivo to in vitro myogenic indexes given in Reference 24 (cf, Fig 2A). The family of curves in Equation 9 is parameterized by a reference diameter da0, eg, the diameter at zero pressure and flow, da0=da(0, 0). P-d relations (ie, the first two factors in Equation 9) for arterioles of sizes different from those displayed in Fig 2A are obtained by linear interpolation.
2. Capillaries are generally thought to behave passively; ie, their diameter, dc, is increased by increasing pressure and is independent of flow, dc=dc(dc0, P), with unstressed diameter, dc0. Strain-stress relations of capillaries were measured by Swayne et al,25 and these data were fitted by a nonlinear least-squares algorithm. As a result, in a capillary of unstressed diameter, dc0 (micrometers), its diameter, d (micrometers), at a given pressure, P (millimeters of mercury), is the solution of
 | (E10) |
3. Venules also exhibit largely passive responses to pressure changes; thus, their diameter, dv, may be expressed similarly as capillary diameter, dv=dv(dv0, P), with unstressed diameter, dv0. From the venular strain-stress relations,25 dc may be calculated by solving
 | (E11) |
Pressures and Diameters Along the Vascular Tree
By substituting d in Equation 8 with da(da0, P, ), dc(dc0, P), or dv(dv0, P) and rearranging, one obtains differential relations between intravascular pressure, P, and longitudinal coordinate, l, in arterioles, capillaries, and venules, respectively:
 | (E2B) |
 | (E12) |
 | (E13) |
, da, dc, and dv are defined in Equations 7 and 9 through 11, respectively. Equations 2, 12, and 13 are first-order ordinary differential equations with separated variables describing the pressure drops along individual arterioles, capillaries, or venules that may be solved by straightforward integration. For a given set of vessel lengths, li, vascular blood flow rates, i, and reference diameters, da0i, dc0, and dv0i, in arteriolar and venular generations i (i=1...4) and capillaries (i=5), one starts with the mean arterial pressure at the origin of arteriolar generation A1 and integrates Equation 2 numerically between l=0 and l=l1. This procedure is reiterated accordingly for consecutive vascular generations A2, A3, etc, by use of the pressure at the end of the preceding generation as the initial pressure in the next one. Thus, one finds the pressure course, P(l), along the vascular tree from which the corresponding diameters, d(l), may be computed according to Equation 9, 10, or 11.
Determination of System Parameters
Computations were made on the basis of the following data: Pressure in organ artery Pa=105 mm Hg.29 Blood viscosity in arterioles and venules as a function of wall shear rate was taken from Lipowsky's fit to experimental data in cat mesentery,22 and viscosity in capillaries was calculated according to19 (discharge hematocrit Hd=0.420 ). For generations i (i=1...5) in cat tenuissimus muscle, flow path lengths, li, were estimated from References 26 through 28 (3, 6, 1, 1, and 1 mm, respectively), average vascular blood flow rates during rest, i, were obtained from velocity-diameter data27 29 according to the method of Lipowsky and Zweifach30 (Qi=22.5, 12.4, 1.28, 0.121, 0.0110 nL/s); and total number of perfused vessels in generation i, ni, was derived from the parent/daughter flow ratios (ni=1, 1.81, 17.5, 186, and 2039). Under conditions different from rest, i may be found by dividing total flow through the network by the number of vessels, ni, in the respective generation. It should be noted, however, that in resting tenuissimus muscle, no more than approximately one third of the capillaries are perfused31 and that additional capillaries are recruited with increasing performance. For calculations at higher blood flows, perfused capillary fractions of 75% (=2rest) or 100% (4rest) were used.32
A difficulty in this study was determining appropriate estimates for the reference diameters da0i, dc0, and dv0i (which are needed to solve the differential equations) because there is no direct way to relate the vascular generations in cat tenuissimus muscle to those in the vascular systems in which the data specifying vessel characteristics had been collected. For the arteriolar reference diameters, this problem was solved by guessing an initial set of radii da0i, i=1...4, and then systematically varying their values until, in the solution for the resting case, the arteriolar P-d and v-d relations from Reference 29 were reproduced reasonably well (see Fig 3). Similarly, average diameters of capillaries and venules at resting perfusion, dc and dvi, i=1...4, were obtained from References 27 and 29. Average capillary diameter was 5.3 µm,27 and resting venular diameters were taken to be 1.5 times the corresponding arteriolar diameters for V1 and V227 29 or calculated from a least-squares fit to transverse arteriolar and venular diameter data27 for V3 and V4. Then the differential equations (Equation 12 for capillaries and Equation 13 for V1 to V4) were successively solved with these diameters replacing the functions dc(dc0, P) and dv(dv0, P) (which may be performed analytically). From the resulting longitudinal pressure distribution, mean pressures in the individual generations, 
c and vi, i=1...4, were computed, and capillary and venular reference diameters were calculated by solving the nonlinear equations dc=dc(dc0, c) and dvi=dv(dv0i,vi) for dc0 and dv0i, respectively. As documented in Fig 3, the capillary and venular P-d and v-d relations ensuing from this approach also are well matched with the data from Fronek and Zweifach.29 The resulting pressure at the venous end of the vascular system, PV, is 6.6 mm Hg, and reference diameters (unstressed diameters at zero flow) and the corresponding mean diameters at resting perfusion in arteriolar generations A1 to A4, capillaries C, and venular generations V4 to V1 are as given in the Table.
Model Application
After the basic descriptors of the system were determined by solving the differential equations (Equations 2, 12, and 13) for the resting case at Pa=105 mm Hg, the proposed theory of perfusion control was studied in situations with altered perfusion rates or arterial blood pressures. While in this view, pressures in arterioles and capillaries are uniquely determined for given values of and Pa, postcapillary resistance becomes actively adjusted so that the resulting perfusion rate meets the actual requirements of tissue metabolism. In this process, the organism keeps central venous pressure, PV, largely constant. Because venous responses to vasodilators are not modeled explicitly and is an input variable to the model (cf, "Model and Data" section), the constancy of PV is the decisive condition that the simulated perfusion control needs to meet. As mentioned, simulation of the adjustment of postcapillary resistance is performed by varying venular reference diameters—more precisely, by multiplying the diameters dv0i, i=1...4, obtained for standard conditions (=rest and Pa=105 mm Hg) by a common "dilatation factor," df. Thus, for nonstandard conditions, the model calculates the value of the multiplier df for which the venous outflow pressure becomes equal to PV=6.6 mm Hg.
Received October 2, 1995; revision received April 17, 1996; accepted April 24, 1996.
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