INTRODUCTION

TOP ABSTRACT INTRODUCTION METHODS RESULTS DISCUSSION APPENDIX REFERENCES

CORONARY FLOW CONTROL is the result of modulation of tone in resistance arteries. On the basis of measurements and theoretical predictions of pressure profiles, the resistance vasculature spans vessels with diameter up to 400 µm (3, 20, 43). Because of this substantial distribution of resistance, a coordinated action of control mechanisms at different domains in the arterial tree is required to adapt flow to metabolism. The control mechanisms proposed are 1) metabolic factors, which are believed to have an effect only on the small arterioles (2, 18, 19); 2) myogenic control, constricting the arteries and arterioles with rising pressure and vice versa (5, 6, 14, 15, 21-23, 30, 37); and 3) flow-induced dilation, dilating the vessels in response to an increase in shear stress, modulated by nitric oxide (NO) (18, 22, 24, 25, 40). These mechanisms have been shown to play a role in regulating tone both in isolated coronary arteries and arterioles and in vivo. Theories integrating these mechanisms are proposed in the literature (7, 8, 17, 28, 31); however, a quantitative analysis that bridges the knowledge from isolated vessel studies and whole organ studies is lacking. We developed a mathematical model to study the possible interactions of these control mechanisms during flow control.

In an earlier study, we (5) demonstrated that pressure-induced myogenic tone alone, taking into account the diameter dependence of myogenic strength, could predict well the course of coronary autoregulation curves. However, this model raises two concerns: 1) flow-dependent dilation will diminish autoregulation; and 2) tone (myogenic) in the smallest vessels is almost absent because of the low level of pressure. The current study has the hypothesis that flow-dependent dilation in upstream vessels is compensated by vasoconstriction of the metabolically controlled smaller vessels, resulting in 1) restoration of autoregulation properties to the levels found for the model with pressure-induced myogenic tone alone and 2) increase in tone in the smaller vessels, leading to the possibility of dilation by increasing metabolism and to a more equal distribution of tone over the resistance vessels.

	METHODS

TOP ABSTRACT INTRODUCTION METHODS RESULTS DISCUSSION APPENDIX REFERENCES

A theoretical model of flow control in the arterial tree was made based on a series arrangement of vessel units, each unit having either myogenic and flow-induced dilatory properties or metabolic properties. The parameters for these units are obtained from experimental data of Liao and Kuo (27) and Kuo et al. (25) on the myogenic response and flow-dependent dilation.

Compartmental Model of Arterial Tree

We assume that each compartment i (i = 1...9), represents N_i identical vessels in parallel. According to Poiseuille, the resistance of the first nine compartments (R_i) can then be described with

(1)

where i is the index referring to the resistance compartment number; _i is the viscosity of the perfusate and is taken to be dependent on the diameter of the vessel as determined in the rat mesentery vasculature by Pries et al. (35), assuming a systemic hematocrit of 45%

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L_i is the length of the vessel; N_i is the number of parallel vessels in the compartment; and d_i is the diameter of the vessel that is dependent on the hemodynamic parameters of flow and pressure in the vessel. Hence, the resistances of the first nine compartments vary according to diameter changes of their representative vessel units, which follow the control behavior measured in isolated vessel experiments.

The vessels in each compartment are characterized by their anatomic diameter, d_anat,i. This diameter is defined as the passive diameter at 100 mmHg. The choice for the anatomic diameters for each compartment was justified in a previous study (5), and these diameters are shown in Table 1.

The number of parallel vessels per compartment (N_i) follows from the condition that the total flow in each compartment (_iN_i) should be equal. Flow (_i) in each vessel unit was assumed to match the velocity (v_i)-diameter data of passive vessels measured by Stepp et al. (40) in the intact canine epicardial circulation
The actual passive diameters of the vessel units, d_i, follow from the passive pressure-diameter relation (5) d_i = d_anat,i{0.705 + 0.363 [P_i/(22.7 + P_i)]}, where P_i is the mean pressure in the vessel. On the basis of measurements of the pressure distribution in the dilated arterial tree (3), a gradual resistance distribution is assumed when inlet pressure equals 90 mmHg and outflow pressure of the ninth compartment equals 30 mmHg (5). These assumptions yield values for average pressure (P_i) in each vessel unit, which allowed us to calculate d_i and _i for the fully dilated tree at 90 mmHg, resulting in values for N_i, shown in Table 1.

The resistance of the vessel compartments (R_i) at a perfusion pressure of 90 mmHg at full dilation can be calculated from the assumed pressure drop (P = 60/9 mmHg) and the flow distribution over the tree: R_i = P/(_iN_i). With Eq. 1, the values for the length of the vessel units (L_i) follow, which are also shown in Table 1. The constant resistance in the tenth compartment equals 15.4 × 10³ mmHg · ml¹ · s [= 30/(_iN_i)].

Control Mechanisms of Isolated Vessels

Experimental data defining control mechanisms. Liao and Kuo (27) and Kuo et al. (25) performed experiments in four different sizes of isolated porcine subepicardial microvessels with anatomic diameters of 65, 100, 165, and 255 µm. The vessels constricted to stepwise increase in luminal pressure (20, 40, 60, 80, 100, and 120 cmH₂O), and the strongest myogenic response was found in arterioles of 100 µm. The vessels also demonstrated flow-dependent dilation. The strongest response to shear occurred in arterioles of 165 µm. Furthermore, because NO mediates the flow-dependent response, they investigated the dose-response curve for nitroprusside in these vessels. For the four different sizes of microvessels, identical dose-dependent dilation for this NO donor was found.

Description of vessel unit in model. A set of equations was derived that describes the data of Liao and Kuo (27) and Kuo et al. (25) and allows for interpolation and extrapolation of responses within and outside the range of tone and diameters studied by these authors. We developed these equations on the basis of a biophysical model of vessel wall behavior as described earlier by VanBavel and Mulvany (42).

View larger version (17K): [in this window] [in a new window]   Fig. 1.   Diameter-tension relations for vessel wall behavior of a vessel unit with anatomic diameter of 255 µm. Shown are the normalized relations between 1) diameter (d*) and passive tension (T), 2) d* and sum of the passive and maximal tension (T + T), and 3) d* and tension relation as dictated by the law of Laplace (T) at mean pressure of 60 mmHg. Tone is defined as the ratio between active and maximal active tension.

View larger version (14K): [in this window] [in a new window]   Fig. 2.   Diagram of the model for a vessel unit, characterized by the anatomic diameter (danat,i). The inputs of the model are either pressure (Pi) and flow (i) or Pi and metabolism-dependent tone (Tonemeta). For a given initial diameter at a certain Pi, maximal active tension (T), passive tension (T), and wall tension (T) are determined. Tone multiplied by T gives T; adding T gives T. Comparing T with T results in an error () representing the imbalance in tensions in the vessel wall. When T is larger than T the diameter is increased; when T is smaller the diameter is decreased. With this new diameter, a new error is calculated. This iteration is continued until the error is <0.001. Tone is either determined by Tonemeta or by the product of TRFflow and Tonemyo, as is indicated by the switch. Tonemyo follows from the P-Tonemyo relation, which depends on danat,i. TRFflow is calculated in 3 steps. First, shear stress () is calculated:  = 32 · i · i/[ · (d · danat,i)3]. Second, the effective concentration of nitric oxide (NO) in the vessel ([NO]) follows from the shear stress-[NO] relation, which depends on the attenuation factor (A) and danat,i. Third, TRFflow follows from the [NO]-TRFflow relation. Tonemeta is only applied in the distal compartments in the network, and its value is determined by assuming a certain flow and having Tonemyo and TRFflow in the other proximal compartments intact.

Relations within vessel units. We first present equations describing T, T, and Tone_myo and estimate their parameters in the absence of flow-dependent influences (TRF_flow = 1). The relations for the flow-dependent tone reduction factor, which is considered to be an attenuation factor of pressure-induced myogenic tone (Tone = Tone_myo × TRF_flow), are then presented, and its parameters are estimated. The parameter estimation procedure is described in detail in the APPENDIX. Below we describe the basics for this approach and the results.

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View larger version (19K): [in this window] [in a new window]   Fig. 3.   A: normalized diameter-normalized tension relations (corresponds to Fig. 1). Symbols represent data calculated from pressure-diameter data for the 4 vessel groups measured by Liao and Kuo (27). The broken lines through the data of Liao and Kuo are the model fits that meet the equilibrium between the wall tension, the sum of normalized active (T = Tone × T) and normalized passive tension (T) on the one hand, and the Laplace tension at different mean pressures on the other hand. The accompanying sum of T and T, depending on danat,i, is shown as well and is calculated by Eqs. 3, 4, and A1. The diameter-wall tension relations for 3 anatomic diameters applied in the network, danat,i = 20, 123, and 400, are shown by the solid gray lines. B: transformed pressure-Tonemyo data of Liao and Kuo (27) obtained from the experimental pressure-diameter data, d*-T relation, and d*-T relations. The fits to the pressure-Tonemyo data (lines) are estimated by Eqs. 5 and A2-A4. With the pressure-Tonemyo relations of B together with d*-T and d*-T relations the fits through the data of Liao and Kuo in A were calculated.

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View larger version (18K): [in this window] [in a new window]   Fig. 4.   Relations for flow-dependent dilation. A: [NO]-TRFflow relations. Open symbols represent the transformed dose response curve for nitroprusside for the 4 vessel groups as obtained by Kuo et al. (25). Solid line shows the fit through this data (Eq. 6). Closed symbols are the expected [NO]-TRFflow data for the shear stress-induced dilation data for the 4 vessel groups as presented in Liao and Kuo (27). B: symbols represent the shear stress-induced dilation data expressed as shear stress-TRFflow data. Solid lines are the model fits. C: recalculated shear stress-[NO] data of Liao et al. (symbols) and fits through the data that are calculated with Eqs. 7 and A5-A7 (solid lines).

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Algorithm for Network Simulations

10

The data of Kuo et al. (25) show that the effect of shear stress on TRF_flow is saturated above 4 dyn/cm². The estimates for shear stresses in vivo are in general >4 dyn/cm²_, whereas flow-dependent dilation is believed to be still involved in control of blood flow. We considered the possibility that shear stress sensitivity is different in vivo from that in vitro and accordingly studied the effect of shifts in _50,i of Eq. 7. Therefore, we introduce an attenuation factor A by which ₅₀ is divided.

When metabolism-dependent tone was added to a number of the most distal compartments the procedure was as follows. The model solutions were obtained for pressure-induced myogenic tone alone, and the resulting flow was used as set point flow for the network. Flow-dependent dilation was made active, resulting in an increase in flow above the set point flow. Tone in the metabolic units was then increased to bring back flow to its set point level. The level of tone in the distal vessel units needed for this flow decrease is referred as metabolism-dependent tone (Tone_meta).

	RESULTS

TOP ABSTRACT INTRODUCTION METHODS RESULTS DISCUSSION APPENDIX REFERENCES

Simulations were performed for perfusion pressures ranging from 5 to 140 mmHg and outflow pressure of 0 mmHg. The simulated pressure-flow relations are shown in Fig. 5A. The simulated pressure-flow relation for maximal dilation is slightly curved as a result of the pressure dependence of resistances. When tone in the vessel units is determined by pressure-induced myogenic tone alone, autoregulation is revealed: flow rises less than proportionally with pressure. Flow-dependent dilation reduces the autoregulation characteristics of the pressure-flow line. The attenuation factor (A) that modifies the shear stress-[NO] relation is set to 0.046. 

View larger version (20K): [in this window] [in a new window]   Fig. 5.   A: simulated pressure-flow lines. Flow is normalized to the flow in the fully dilated case (no tone) at 90 mmHg, being 1.94 × 103 ml/s. When tone in all compartments is only determined by pressure-induced myogenic tone (TRFflow = 1), autoregulation is fairly good (Myo). When in all compartments tone is determined by both pressure-induced myogenic tone and flow-dependent dilation [attenuation factor (A) = 0.046] autoregulation is diminished (Myo + Flow). B: effect of attenuation factor A on normalized flow at 90 mmHg. With increasing A the normalized flow increases rapidly; when A > 0.2, normalized flow is saturated at 0.74, indicating that the shear stress-induced dilation is saturated as well. To make sure that the flow-dependent mechanism is effective, A was chosen to be 0.046.

In Fig. 5B, the effect of the attenuation factor on normalized flow at perfusion pressure of 90 mmHg is demonstrated. When A = 1, the flow equals 74% of the flow at maximal dilation; when A < 0.2, the flow decreases rapidly until A approaches zero and the normalized flow converges to the flow in the case of pressure-induced myogenic tone alone. To make sure that the flow-dependent mechanism is effective, the attenuation factor was set for further analyses to 0.046.

The distributions of normalized diameter and tone at a perfusion pressure of 90 mmHg are depicted in Fig. 6. With pressure-induced myogenic activity alone, tone drops gradually from the larger resistance vessels to the smaller vessels. Note that normalized diameters of the largest and smallest vessels are close to one. Flow-dependent dilation changes the tone and diameter distribution over the compartments, but the resulting changes in pressure distribution introduce marginal changes in pressure-induced myogenic tone distribution (Fig. 6C). Flow-dependent dilation gives a plateau in the diameter-tone relation and gives an increase in diameter, most emphatically apparent in the larger resistance vessels (Fig. 6A).

View larger version (20K): [in this window] [in a new window]   Fig. 6.   Distributions of normalized diameter (A) and tone (B-D) as a function of the diameters of the vessel units in the compartments. B-D show tone, Tonemyo, and TRFflow, respectively, for a perfusion pressure (P) of 90 mmHg. Three cases are depicted: fully dilated (circles), all compartments having only pressure-induced myogenic tone (squares), and all compartments having both myogenic and flow-dependent properties (triangles). Inset, accompanying flows.

Figure 7 demonstrates the effect of balancing flow-dependent dilation with metabolic constriction on the distribution of diameter and tone. This effect is presented for a perfusion pressure of 90 mmHg and an attenuation factor of 0.046. Restricting metabolic compensation of flow to the most distal compartment requires a high tone (Tone_meta = 0.99). Distributing the required resistance increase equally over more than one compartment reduces the amount of tone per vessel unit (Fig. 7). Having four compartments under the influence of metabolism-dependent tone results in a more or less equal tone distribution over the compartments (Fig. 7A).

View larger version (21K): [in this window] [in a new window]   Fig. 7.   Distributions of tone for increasing number of compartments under influence of metabolism-dependent tone (Tonemeta) at perfusion pressure of 90 mmHg. A-D: tone, Tonemyo, TRFflow, and Tonemeta, respectively, as a function of the diameters of the vessel units. Distributions without influence of Tonemeta are also shown (circles). Inset, accompanying flows at P = 90 mmHg and the pressure-flow lines for the fully dilated case and with all compartments having pressure-induced myogenic tone alone. Tonemeta is set such that the total distal resistance under metabolic influence was increased to a value such that flow equals the value in the case of pressure-induced myogenic tone alone. Symbols refer to different numbers of distal compartments under metabolic influence. When Tonemeta is restricted to only the most distal compartment, Tonemeta equals almost unity, and thus wall tension equals almost our defined maximal wall tension. Distributing the required resistance increase over more than 1 distal compartment reduces the amount of tone per vessel unit.

In Fig. 8, the distributions of tone at perfusion pressures of 60, 90, and 120 mmHg are depicted. To bring flow back to the autoregulation curve obtained with pressure-induced myogenic tone alone, a lower level of metabolism-dependent tone is required at lower perfusion pressure. Most of the changes in tone of the larger vessels are due to pressure-induced myogenic tone variations. The changes in TRF_flow oppose the desired effect; however, the changes are small, which is to be expected because flow is kept in a narrow range.

View larger version (17K): [in this window] [in a new window]   Fig. 8.   Autoregulation and distribution of tone with 4 distal compartments under metabolism-dependent tone. Results are for a perfusion pressure (Pp) of 60 (inverted triangles), 90 (triangles), and 120 (diamonds) mmHg. A-D, tone, Tonemyo, TRFflow, and Tonemeta, respectively. Inset, flows for the pressures simulated as well as the pressure-flow line for full dilation and with all compartments having pressure-induced myogenic tone alone.

Full dilation of the smaller vessels is simulated by reducing the tone of the metabolic vessels to zero. Such local dilation has been observed to result from adenosine infusion and metabolic dilation induced by NO synthesis inhibition of larger resistance vessels (18). The effect of metabolic dilation for these two conditions is demonstrated in Fig. 9. Distal dilation without inhibition of NO synthesis increases flow from 37% to 80% of maximal dilated values. A substantial part of the decrease in total resistance is due to the dilation of the proximal vessel units (14%). Tone in these vessels decreases (Fig. 9A), and it is clear that both the pressure-induced myogenic mechanism (Fig. 9B) and the flow-dependent mechanism (Fig. 9C) enhance the effect of distal dilation on reduction of overall coronary resistance.

View larger version (18K): [in this window] [in a new window]   Fig. 9.   Distal dilation (4 compartments) and its effect on the distribution of tone in the proximal vessel units. Results are for a perfusion pressure (P) of 90 mmHg. Circles, control conditions; squares, distal dilation; inverted triangles, distal dilation with NO synthesis inhibition (TRFflow = 1). A-D: tone, Tonemyo, TRFflow, and Tonemeta, respectively. Inset, flows at P = 90 mmHg as well as the pressure flow lines for full dilation and with all compartments having pressure-induced myogenic tone alone.

Inhibition of NO synthesis in our model is simulated by setting the TRF_flow of the proximal vessels to 1. Flow increases only from 37% to 47% of maximal dilated values. The behavior of the vessels in the proximal compartments depends on location in the circuit. As indicated in Fig. 9A, the vessels in the third compartment decrease diameter with increasing tone, whereas in the fifth compartment both diameter and tone decrease. The pressure-induced myogenic tone (Fig. 9B) in compartments 1-5 is always reduced by dilation of compartments 6-9. Thus constriction of the larger vessels by inhibition of NO synthesis is attenuated by the myogenic mechanism, indicating that the upstream constriction is not only compensated by the metabolic mechanism but the myogenic mechanism also contributes to this compensating effect.

	DISCUSSION

TOP ABSTRACT INTRODUCTION METHODS RESULTS DISCUSSION APPENDIX REFERENCES

Pressure-induced myogenic tone alone results in autoregulatory properties of the coronary model, independent of factors related to metabolic control, i.e., the flow is rather independent of arterial pressure. When the flow rate through this myogenically controlled tree matches flow demand, a stable perfusion system is obtained. However, flow-dependent dilation induces a flow rate higher than needed. Therefore, an increase in tone in the smaller downstream vessels should compensate for this. When vessels with diameters ranging from ~10 to ~40 µm are assumed to have a higher intrinsic tone than found in isolated vessels, this tone can then be modulated by metabolism and tone is more or less equally distributed over the arterial tree. Moreover, the level of tone in all compartments is such that a sufficient range of tone in these compartments is available to adjust flow to the large variations in tissue oxygen demand. To distinguish the responses in the smallest resistance vessels from those in the larger ones we introduced the notion of "metabolism-dependent tone" to indicate that this tone is modulated predominantly by metabolism.

Previous Model Studies

However, an essential difference is that their model is diameter based and ours is wall tension based, which seems to better fit a biophysical description of vessel wall mechanics. The effect of shear stress-dependent dilation on the adenosine-flow dose-response relation was investigated, a problem different from the one addressed here.

Granger (11) used a three-compartment model. However, each compartment was characterized by a single control mechanism, which is different from the present model, in which the strength of myogenic tone and flow-dependent dilation vary gradually in a realistic way over the different compartments. Ursino et al. (41) developed a theoretical network model based on wall tension with five branched elements to analyze the functional role of dynamic vasomotion in blood flow control, but they did not study interactions between control mechanisms.

Evaluation of Model Structure and Parameter Choice

In the vascular wall of veins and venules smooth muscle cells are observed, as well as diameter changes in response to sympathetic activity and certain agonists. However, the contribution to resistance of this part of the circulation under physiological conditions is small compared with the contribution to resistance of the vessels studied here (resistance vessels ranging from ~20 to 500 µm). Therefore, the assumption of a constant resistance contribution of capillaries and veins during flow control under physiological conditions seems quite realistic.

Important to the model is the law of Laplace. We used the law of Laplace for thin-walled vessels, whereas a more general formula is T = 0.5[P₁d  P₂(d + 2t)] (34), where P₁ and P₂ are the pressure inside and outside the vessel, respectively, d is the inner diameter, and t is the wall thickness. The deviation between the thin and thick wall equations increases with the wall thickness-to-diameter ratio but is diminished when the transmural pressure (P₁  P₂) increases. This deviation is not the same for all vessel diameters, and for a single vessel it varies with tone. All these effects are neglected in the model. The application of the thin wall formula simplifies the mathematics of the model because it results in a linear tension-diameter relation for a given pressure. The simplification seems justified in the context of other uncertainties, especially with respect to the diameter-T_max, pressure-Tone_myo, and shear-TRF_flow relations in the model.

The relations chosen to determine T_max and Tone_myo are not uniquely defined by the data of Liao and Kuo (27). For example, applying a different diameter-T_max relation, where T_maxtop,i is twice the original value and d_m is 90% of the original value, results in less steep pressure-Tone_myo,i relations. However, the resulting autoregulation curve for myogenic tone alone is hardly different from the original one and, furthermore, the number of compartments under influence of metabolic control required to have an approximately equal distribution of tone is still four. The only difference is that the value of tone is smaller under these circumstances than when the original diameter-T_max and pressure-Tone_myo relations are used.

Flow-dependent dilation is mediated by shear stress-induced production of NO. We have modeled this in three steps to reconcile the observations that the shear stress-TRF_flow relation is dependent on anatomic diameter but the [NO]-TRF_flow relation is not. Both relations were measured by Kuo et al. (25). Our model fits their relations by having components providing the diameter-dependent relation between shear stress and [NO] and the relation between [NO] and TRF_flow. The data of Kuo et al. (25) suggest that for high shear stresses, [NO] saturates such that vasodilatation is submaximal. The level of [NO] would therefore depend on vessel size. The reason for submaximal vasodilatation is unknown.

The attenuation factor (A) in the model modifies the relation between shear stress and [NO] and by definition equals unity for the experiments of Kuo et al. (25). This factor deviates from unity to compensate for scavengers and other factors that may affect the shear stress-TRF_flow relations. This factor only affects the effective working range of the [NO]-shear stress relation and not the level of [NO] at infinitely high shear stress. As shown in Fig. 5B, a large A results in a large flow, and thus, in our concept, a large counteracting metabolic resistance is needed. However, because the shear stresses in the vessel units of the network are all >7 dyn/cm², the flow-dependent mechanism would hardly be effective: possible changes in shear stress do not result in changes in [NO] and thus do not result in changes in TRF_flow. When A is too low, the counteracting metabolic resistance is low and metabolic control in the sense of metabolic dilatation is limited. With, for instance, A = 0.01, it is sufficient to set only the last compartment under the influence of metabolic control. Tone_meta would then be only 0.47.

There may be several reasons why the in vitro vessels behave differently in response to shear stress compared with the in vivo vessels. An important reason is obviously the fact that in vitro the red blood cells are missing that are metabolizing NO at a high rate. To study this would require a dedicated model in itself, to compensate for all the factors involved in controlling it, including the endothelial glycocalyx (4). Moreover, we left out other mechanisms that influence local vascular tone and concentrated on the interaction of two mechanisms well defined in in vitro studies. Hence, it is assumed that the attenuation factor, scaling the data from isolated single vessels to the whole heart, takes all these unknown factors into account.

Application of Model to Interpretation of Distributed Response in Coronary Circulation

Distributed vascular diameter response during metabolic regulation. Several elegant experiments (18, 40) have attempted to study the distributed response of flow-dependent dilation and metabolic influence over the vessels of different diameters in the coronary tree. However, the pressure-induced myogenic tone is always present as an additional factor of tone generation, and it is not always clear how this is affected by the interventions.

View larger version (19K): [in this window] [in a new window]   Fig. 10.   Changes in diameter induced by adenosine (A) and inhibition of NO synthesis (B). Open symbols reflect experimental data obtained from the epicardial microcirculation of an open-chest dog and are redrawn from Jones et al. (18). Filled symbols are the model results. Metabolic dilation caused by adenosine is simulated by reducing metabolism-dependent tone of the 4 distal compartments to zero (A; in Fig. 9, circles vs. squares). Inhibition of NO by NG-nitro-L-arginine methyl ester is simulated by setting TRFflow of the proximal compartments to 1 and reducing metabolism-dependent tone of the 4 distal compartments to 0 (B; Fig. 9, circles vs. inverted triangles). Model predictions of the percent diameter change after administration of adenosine and after administration of the NO synthesis inhibitor are in fair agreement with the experimentally observed values.

Distributed shear stress and velocity response during metabolic regulation. In a study of Stepp et al. (40) microvascular diameters and microsphere velocities were measured. Measurements in arterioles (30-160 µm) and small arteries (160-450 µm) were obtained under basal conditions and after administration of adenosine. Wall shear stress was calculated using the formula _wall = 8v/d, where _wall is the wall shear stress,  is blood viscosity, d is vascular diameter, and v is the mean velocity in the vessel cross-sectional area, which was assumed to equal the microsphere velocity. The data of these experiments are presented in Fig. 11, A and B.

View larger version (25K): [in this window] [in a new window]   Fig. 11.   Distribution of velocity (A) and shear stress (B). Open symbols represent measured and calculated data of Stepp et al. (40). Filled symbols are model simulations. Administration of adenosine was simulated by full dilation of the 4 distal compartments (Fig. 9, circles vs. squares). C: percent changes in shear stress (diamonds) and percent changes in velocity (circles). Open symbols are the average changes as provided by Stepp et al. (40). Filled symbols are the simulated changes. Our model could not predict the much smaller increase in shear stress than in velocity in small arteries as reported by Stepp et al. (40).

Predictions of flow and flow reserve by model. The model predicts a flow reserve a little over a factor of 2, whereas in humans and dogs this can easily be a factor of 4, although in goats it is less; a factor of ~3. The only conclusion can be that tone in our reference network is too low to provide sufficient vasoconstriction to allow the flow reserve to be a factor of 4. It should be noted, however, that we found an agreement between experimental and predicted diameter changes, which suggests that the distribution in tone is quite realistic. Therefore, both tone in the proximal vessels as well as tone in the distal metabolic vessels are somewhat too low. Tone in the proximal vessels was based on in vitro studies, and the conclusion must then be that in these studies the level of intrinsic tone was too low as well. Such differences between in vivo and in vitro observations may well be possible by changing vascular smooth muscle preconstrictor factors between dissection and mounting in the pressure myograph. For example, endothelin concentrations (26) might have been altered. The required increase in tone does not need to be very large. The law of Poiseuille dictates an inverse fourth-power relation between resistance and diameter. Hence, a 10% reduction in diameter of all segments would result in an increase of 40% of resistance and almost double the flow reserve in the model. Hence, the usefulness of the model is not in predicting quantitatively changes in flow but in the distribution of responses of control mechanisms in terms of vessel diameters and vascular tone.