Cardiovascular Research, 1976, 10, 13-19.
Mathematical formulation of post-occlusion hyperaemia and autoregulation of blood flow in the capillaron model HIROSHI MURAO
and
SIMON RODBARD
From the City of Hope Medical Center, Department of Cardiology, 1500 East Duarte Road, Duarte, California 91010, USA
A U T H O R S ’ S Y N O P S I S A mathematical formulation of the blood flow behaviour through a capillaron model is described, The formulation is applied to the analysis of post-occlusion (reactive) hyperaemia and autoregulation.
Theories which have been proposed to explain post-occlusion (reactive) hyperaemia, and autoregulation include : the myogenic response, local reflexes, cell separation, and metabolic and tissue pressure hypotheses (Johnson, 1964). Examination of the hydrodynamic properties of the capillaron-a tissue unit consisting of parenchyma, capillaries, and extracapillary fluid enclosed in a compliant capsule-has shown that many phenomena of the circulation including post-occlusion hyperaemia and autoregulation can be reproduced passively without involvement of arterioles or other active components (Rodbard, 1963; 1966a; 1966b). This report presents a mathematical formulation of the macroscopic behaviour of the flow through the capillaron model. No attempt has been made to make a mathematical analysis of the microscopic behaviour of the flow within the capillaron. Capillaron model The systemic capillaries permit ultrafiltration into the extracapillary space, and this ‘tissue fluid’ then filters across the downstream end of the capillary to return to the circulation. This arrangement implies a tissue unit in which the parenchymal cells, the tissue fluid, and soft-
walled permeable capillary and its contents are enclosed in a compliant capsule (Rodbard, 1966a). The conceptual arrangement of the capillaron model is depicted in Figure 1. Assumptions A certain relationship between the capillary transmural pressure and the vascular conductance (volume flow rate divided by longitudinal pressure difference) is assumed. The capillary is a non-distensible, thin-walled collapsible permeable passive tube. When the capillary transmural pressure (intracapillary minus extracapillary) is sufficiently negative (ie, when the intracapillary pressure is sufficiently lower than the extracapillary pressure), the capillary is collapsed and conductance through it approaches zero (negative region in Fig. 2). The capillary is fully open when the transmural pressure is sufficiently positive, (ie, when the intracapillary pressure is sufficiently higher than the extracapillary pressure). Since the capillary wall is non-distensible, the conductance of the vessel remains practically constant despite further elevation of transmural pressure (positive region in Fig. 2). Between these two regions, there is a region in which the capillary is partially
14 Murao and Rodbard VASCULAR
CONDUCTANCE vs. . TRANSMURAL PRESSURE VASCULAR CONDUCTANCE
I
!
I I 1
1
NEGATIVE
REGION
TRANSMURAL PRESSURE = P,.PE
TRANS~TIONAL I
POSITIVE REGION
I 1
REGION
I
F I G . 2 Vascular conductance vs transmural pressure. I n the negative region the extracapillary pressure is sufficiently higher than the intracapiflary pressure and the vessel is completely collapsed. In the positive region the intracapillary pressure is sufficiently higher than the extracupillary pressure and the vessel is completely open. I n this state the vascular conductance is constant and maximal. Between the negative region and the positive region exists a transition phase where the vascular conductance is dependent on the transmural pressure (PI-PE). It is assumed that the vascular conductance is linearly dependent upon the transmural pressure in the transitional region.
Conceptual model of the capillaron. The capillaron is a unit structure that encloses encapsulated tissue fluid and passive soft-walled permeable capillaries. The capillaries are represented by a single soft-walled collapsible tube which extends through a box (the extracapillary space) filled with extravascular fluid. Extravascular pressure (PE), is determined by the volume of extravascular fluid and the compliance ( C ) of the capsule. I n the model shown, compliance is determined by the volume/ pressure relationships of the PE standpipe. Fluid filters out of the upstream end of the capillary into the extravascular space at a rate Qz' Qz is determined by the upstream permeability (D1) and the difference between the upstream pressure (PA) and the extravascular pressure (PE). At the downstream end of the capillary, fluid from the extravascular space filters back into the capillary a ta rate Q3 Q3 is determined by the downstream permeability (Dz) and the difference between the extravascular pressure (PE) and the downstream pressure (Pv). Patency of the capillary is determined by the transmural pressure which is the difference between the intravascular pressure (PI) and the extravascular pressure (PE) FIG. I
1
open and the conductance increases with the transmural pressure. In this region we assume a linear relationship between the transmural pressure and the conductance which varies from zero to a maximum value (transitional region in Fig. 2). This assumption is supported by observations of the relationship between conductance and transmural pressure in models (Rodbard and Kuramoto, 1963; Fig. 3), and in the isolated, saline-filled lung(Rodbardand Kira, 1966; Fig.4). Intracapillary pressure PIis assumed to have a uniform value throughout its length. Transmural
1
-40
1
1
-20
5-5 (crnH,O)
,
I
0
'
I
20
I
I
I
40
F I G . 3 Flow (conductance) and transmural pressure in the model. (Redrawn and modified from Rodhard and Kuramoto, 1963.) Conversion: traditional to SI units: I cm H Z O z 0.098 kPa.
pressure is expressed as PI-P,. Actually, PI and transmural pressure values change along the length of the vessel, but for the purpose of correlating the conductance through the entire
15 Post-occlusion hyperaemia and blood flow in the capillaron model 2.01Conductance
D, : upstream capillary permeability (ml-min-l. cm H,O-l) D2: downstream capillary permeability (ml min-l.cm H,O-') C: compliance of the extracapillary compartment (ml.cm H20-') (PAand Pv are applied to the pressure present at the immediate upstream and downstream ends of the capillaron unit, respectively. These are likely to be different from systemic pressures, and might better be called arteriolar and venular pressures, respectively.) Filtration through the upstream permeability: dQ Qz = 2 dt = Dl (PA-PE)
.
1
-%4 6 8 10 I2 14 Capillary transmural pressure (cm saline) (vascular minus airway)
16
18
Vascular transmural pressure and conductance of pulmonary vascular bed in isolated dog lung lobe. The left lower lobe was isolated and suspended in a bottle filled with saline. Its airway was filled with saline and the vascular bed was perjirsed with saline at an arteriovenous pressure difference of 3.3 cm saline. The zero reference level was at the hilum. Pressure in the bottle (pleural pressure) was - 2.5 cm saline. Starting from a level lower than airway pressure, arterial and venous pressures were raised simultaneously without changing arteriovenous pressure difference. The transmural pressure of the pulmonary vascular bed is intravascular pressure minus airway pressure. When transmural pressure was negative, vascular conductance was zero. As transmural pressure became positive, conductance increased gradually to a plateau. (Redrawnfrom Rodbard and Kira, 1966.) FIG. 4
length of the capillary with the transmural pressure change, the intracapillary pressure can be lumped into a mean value. The transmural pressure (PTM)in the whole system can therefore be expressed as PI - PE.
Derivation Some basic relationship in the capillaron system can be expressed in several equations (Fig. 1). PA: arterial (arteriolar or upstream) pressure (cm H20) P,: venous (venular or downstream) pressure (cm HZO) P,: extracapillary pressure (cm HzO) PI: intracapillary pressure (cm HzO) Q1:flow rate through the artery (ml.min-') Q,: filtration rate through the upstream permeability (ml.min-l) Q3:filtration ratL through the downstream permeability (ml.min-l)
Filtration through meability:
the downstream
per-
Relationships between the extracapillary tissue fluid volume and the extracapillary pressure :
(3) Combining the three equations above, we obtain:
+
= (Dip, DzPv)- PE (Di + Dz) Solving equation 4 for PE we obtain: PE = DlP.4 + DzPv pE (0) D1+ D2
(4)
+
where P, (0) is the value of PE when t=O. Equation 5 means that regardless of the value of parameters such as D1, Dz, PA, Pv, C, etc, PE has a characteristic that, starting from an initial value PE (0), P, exponentially approaches the steady-state value (D1PAf D,Pv)/(D, D2). It is necessary that D, be greater than D,. This is because the P, is the 'weighted' average of PA and Pv, weighted according to D, and D,, respectively. Accordingly, if D, is equal to or less than D,, then the net or mean transmural pressure, PTM,becomes equal to zero or negative In this state, the capillary is completely collapsed, conductance is zero, and this condition is not of physiological interest.
+
16 Murao and Rodbard
Application of the basic formulae to reactive hyperaemia For any value of PE (0), PE approaches an equilibrium value PEas sufficient time elapses: PE pE = (DIPA D7,PV)/(Dl + D7,)a We can take P, as the control value of P, before occlusion of the arterial inflow. The mean intracapillary pressure before arterial occlusion is : (PA + pV)/2* Immediately after arterial occlusion, the intracapillary pressure PI becomes equal to Pv, and according to equation 5, PE exponentially approaches the new steady-state value Pv (which can be obtained by setting PA= Pv in equation 5, or setting Q2=0 in equation 1,and solving equations 2 and 3 simultaneously. On release of the arterial occlusion, PI returns almost instantaneously to (PA+ Pv)/2 while PE begins to return exponentially to the control value (D1PA DPPV)/(D1 D2) (Fig. 5). The transmural pressure change can consequently be depicted as in the top row in Fig. 6. If the transmural pressure remains within the transitional region in Fig. 2 during the entire
Transitional reqion
Poslhve reqlon
+
+
+
Flow I
I
FIG. 6 Flow in arterial occlusion. The horizontal axes show time. The top row shows the transmural pressure as derived from Fig. 5. The second row shows conductance. The third row shows the pressure difference between the artery and Gein. The bottom row shows the j o w which is deriued as the product of conductance and pressure difference. In the transitional region, the post-occlusion overshoot of the transmural pressure is rejected in the postocclusion overshoot of conductance. Correspondingly, the flow shows the post-occlusion hyperaemio. In the positive region, there is no overshoot of conductance and no postocclusion hyperaemia.
process, the vascular conductance curve will show a pattern similar to the transmural pressure curve. As the flow through the capillary vessel is expressed as the product of the conductance and the arterial-venous pressure difference, the capillary flow will show the pattern in the bottom row of the left column in Fig. 6. Thus, with the release of the arterial occlusion, the flow rises above the pre-occlusion level (post-occlusion hyperaemia) and then returns toward the preocclusion level. From Fig. 5 it is predicted that I the longer the period of occlusion, the higher the OCCLUSION level of the peak of post-occlusion transmural pressure, hence the higher the level of the peak of FIG. 5 Intracapillary, extracapillary, and transpost-occlusion hyperaemia. Fig. 7 shows the mural pressures in arterial occlusion: the solid line result obtained in an actual capillaron model, shows the change of the intracapillary pressure PI. which is consistent with the present theoretical The broken line shows the change of the extra- analysis (Rodbard, 1966a).l capillary pressure PE.The vertical distance between The diference in the shapes of flow peaks in theoretically obtained PI and PErepresents the magnitude of the transmural flow patterns and that in experimental observations (Fig. 7) can be pressure PTM. Before arterial occlusion, PI is attributed to simultaneous changes of the calibre of the capillary and the transmural pressure. Consideration of the time lag between the (PA+ Pv)/2, and P E is &. Upon arterial occlusion, PI of transmural pressure change and the capillary calibre change will make drops to Pv and PE exponentially approaches Pv. the ascending limb of the flow curve curvilinear over a short span of time instead of a n instantaneous vertical rise and the flow pattern thus Upon releasing the occlusion, PI returns to (PA+ obtained will be much closer to that obtained in actual tissue or in the Pv)/2 and PE returns exponentially to ward PE. capillaron model.
-
17 Post-occlusion hyperaemia and blood flow in the capillaron model Occlurion
(I)
P I G . 7 Post-occlusion hyperaemia in a model of the capillaron. The arterial inflow was occludedfor I , 2, 6, and 20 s, as indicated. The eflects during occlusion and the post-occlusion hyperaemias are shown. (From Rodbard, 1966a.)
The rate of change of P, is determincd by the time constant of the system C/(D1+Dz); the larger this value, the slower the rate of change. Consequently, the larger the compliance of the extravascular compartment and the smaller the permeabilities, the slower the change of PE. The slower the change of PE, the smaller the level of post-occlusion hyperaemia for a given duration of occlusion. If the transmural pressure changes beyond the transitional region in Fig. 2, the flow pattern will * -TIME 0 differ from that described above, since the linear relationship between transmural pressure and F I G . 8 A sudden rise in the arterial pressure (PA) conductance will then no longer hold. At such from PA1 to PA2.The intracapillary pressure (PI) high intracapillary pressures the transmural also rises suddenly. The extracapillary pressure (PE) rises exponentially to a new steady-state value. The pressure levels before and after the occlusion will transmuralpressure (PTM),which is PI - PE,increases remain the positive region, and no post-occlusion suddenly and then decreases gradimlly. hyperaemia will be observed (Fig. 6, right column).
Application of the basic formulae to autoregulation Suppose the control value of PE is P,= (D1PA1+DzP,)/(Dl Dz). Elevation of the arterial pressure from PA1 to PA2is followed by an exponential rise in PEtoward its new steadystate value (DIPAD+ D,P,)/(D, + DJ according to equation 5 (Fig. 8). On the other hand, the mean intracapillary pressure rises instantaneously from (PA1 P,)/2 to (PA2 P,)/2. As a result, the PI- PE difference will rise instantaneously and then decline to a new level. If the transmural pressure is within the transitional region, the flow will correspondingly show a pattern as in the bottom row of the left column in Fig. 9, ie, first it rises suddenly and then declines to approximately the control value (autoregulation).
+
+
+
As in Fig. 10, lowering of the arterial pressure from an initial value PAl to PA3, lowers PI instantaneously, while PE changes exponentially from the control value (D1PA1+ D2P,)/(Dl D2) toward the new steady-state value (D1PA3+ DzPv)/(Dl D2). PI - PE drops instantaneously and then increases slowly to a new value. If the transmural pressure is within the transitional region, the flow also shows the initial fall and then the gradual increase (Fig. 11, left column, bottom row). In either case, if the transmural pressure is high and in the positive region, the flow does not show autoregulation (Fig. 10 and Fig. 11, right column, bottom row), The basic mathematical formulae can also be applied to the analysis and elucidation of a wide variety of other phenomena observed in the capillaron system including basal ‘tone’, closing
+
+
18 Murao and Rodbard
Transitional reqion
Positive reqion
Transitional reqion Transmural I pressure
Transmural pressure
Conductance I
Conductance
Positive reqion I
I
~~~~~
Pressure dlfference
Pressure difference
Flow I
% -
F I G . 9 Autoregulation of flow accompanying a sudden rise in the arterial pressure. If the transmural pressure is within the transitional region, the sudden rise in the transmural pressure followed by the gradual decrease is reflected in a similar pattern of change in flo w (autoregulution). If the transmural pressure is within the positive region, the flow does not show autoregulation.
1
Flow I
I
F I G . I I Autoregulation of flow accompanying a sudden fall in the arterial pressure. If the transmural pressure is within the transitional region, the sudden fall in the transmural pressure followed by the gradual increase is reflected in a similar pattern of change in flow (autoregulation). If the transmural pressure is within the positive region, the flow does not show nutoregirlation.
la does not give a precise reproduction of either an actual capillaron model or an actual capillary segment in tissue. For example, in the real capillary, permeability is not lumped but is distributed continuously along the length of the capillary vesse1.l Also the actual relationship between the vascular conductance and the transmural pressure is curvilinear rather than rectilinear. Such a precise analysis will result in nonlinear equations which require a computer for solution. I n spite of the drawback of siniplification cited above, the present analysis provides F I G . 1 0 A sudden in the arterial pressure (PA) a good approximation to the situation in the ,from PA^ to PAa. The intracapillory pressure (PI) also tissues, and a means for analysis and prediction falls suddenly. The extracapillory pressure (PE)falls of the flow behaviour through the capillary esponentiully to a new steady-state uiilue. The vessels. This also can provide an excellent basis transmural pressitre (PTbt)decrenses srtddcril.v, then for the formulation of more detailed analyses in:renser grotiitaily. using a computer. D. Rodbard provided a critical review of the manuscript.
pressure, opening pressure, etc (Rodbard, 1966a). Since several assumptions have been made to facilitate the mathematical analysis, the formu-
In both the animal and model experiments it has been observed that the autoregulatory flow change is slower when the arterial pressure is lowered than when it is raised. This asymmetry in the autoregulatory flow pattern can be demonstrated also in mathematical analysis if the continuous distribution o f capillary permeability is taken into consideration.
19 Post-occlusion hyperaemia and blood flow in the capillaron model
References Johnson, P. C.(1964). Review of previous studies and current theories of autoregulation. American Heart Association Monograph No. 8, Autoregulation of Blood Flow,pp. 1-9. Rodbard, S. (1963). Autoregulation in encapsulated, passive, soft-walled vessels. American Heart Journal, 65, 648-655. Rodbard, S. (1966a). Evidence that vascular conductance is regulated at the capillary. Angiology, 17, 549-573.
Rodbard, S. (1966b). A hydrodynamic mechanism for auto regulation of flow. Cardiologia, 48, 532-535. Rodbard, S., and Kira, S. (1966). Mechanical forces and pulmonary vascular conductance. American Heart Journal, 72,498-508. Rodbard, S., and Kuramoto, K. (1963). Transmural pressure and vascular resistance in soft-walled vessels. American Heart Journal 66, 786-791.
ANNOUNCEMENT An international symposium on Cardiac Receptors will be held at the Department of Cardiovascular Studies, University of Leeds, Leeds LS2 9JT, from 14-17 September 1976. It is spon5ored by Cardiovascular Commission of International Union of Physiological Sciences. Topics for discussion include histology and electrophysiology of vagal cardiac receptors, reflex responses of atrial and ventricular receptors, efferent and central nervous mechanisms ; reflex responses following bleeding, infusion, and infarction; electrophysiology and reflex responses of sympathetic cardiac afferents. Further details and preliminary programme can be obtained from Dr C . Kidd, Department of Cardiovascular Studies, University of Leeds, Leeds LS2 9JT.
Mathematical formulation of post-occlusion hyperaemia and autoregulation of blood flow in the capillaron model HIROSHI MURAO
and
SIMON RODBARD
From the City of Hope Medical Center, Department of Cardiology, 1500 East Duarte Road, Duarte, California 91010, USA
A U T H O R S ’ S Y N O P S I S A mathematical formulation of the blood flow behaviour through a capillaron model is described, The formulation is applied to the analysis of post-occlusion (reactive) hyperaemia and autoregulation.
Theories which have been proposed to explain post-occlusion (reactive) hyperaemia, and autoregulation include : the myogenic response, local reflexes, cell separation, and metabolic and tissue pressure hypotheses (Johnson, 1964). Examination of the hydrodynamic properties of the capillaron-a tissue unit consisting of parenchyma, capillaries, and extracapillary fluid enclosed in a compliant capsule-has shown that many phenomena of the circulation including post-occlusion hyperaemia and autoregulation can be reproduced passively without involvement of arterioles or other active components (Rodbard, 1963; 1966a; 1966b). This report presents a mathematical formulation of the macroscopic behaviour of the flow through the capillaron model. No attempt has been made to make a mathematical analysis of the microscopic behaviour of the flow within the capillaron. Capillaron model The systemic capillaries permit ultrafiltration into the extracapillary space, and this ‘tissue fluid’ then filters across the downstream end of the capillary to return to the circulation. This arrangement implies a tissue unit in which the parenchymal cells, the tissue fluid, and soft-
walled permeable capillary and its contents are enclosed in a compliant capsule (Rodbard, 1966a). The conceptual arrangement of the capillaron model is depicted in Figure 1. Assumptions A certain relationship between the capillary transmural pressure and the vascular conductance (volume flow rate divided by longitudinal pressure difference) is assumed. The capillary is a non-distensible, thin-walled collapsible permeable passive tube. When the capillary transmural pressure (intracapillary minus extracapillary) is sufficiently negative (ie, when the intracapillary pressure is sufficiently lower than the extracapillary pressure), the capillary is collapsed and conductance through it approaches zero (negative region in Fig. 2). The capillary is fully open when the transmural pressure is sufficiently positive, (ie, when the intracapillary pressure is sufficiently higher than the extracapillary pressure). Since the capillary wall is non-distensible, the conductance of the vessel remains practically constant despite further elevation of transmural pressure (positive region in Fig. 2). Between these two regions, there is a region in which the capillary is partially
14 Murao and Rodbard VASCULAR
CONDUCTANCE vs. . TRANSMURAL PRESSURE VASCULAR CONDUCTANCE
I
!
I I 1
1
NEGATIVE
REGION
TRANSMURAL PRESSURE = P,.PE
TRANS~TIONAL I
POSITIVE REGION
I 1
REGION
I
F I G . 2 Vascular conductance vs transmural pressure. I n the negative region the extracapillary pressure is sufficiently higher than the intracapiflary pressure and the vessel is completely collapsed. In the positive region the intracapillary pressure is sufficiently higher than the extracupillary pressure and the vessel is completely open. I n this state the vascular conductance is constant and maximal. Between the negative region and the positive region exists a transition phase where the vascular conductance is dependent on the transmural pressure (PI-PE). It is assumed that the vascular conductance is linearly dependent upon the transmural pressure in the transitional region.
Conceptual model of the capillaron. The capillaron is a unit structure that encloses encapsulated tissue fluid and passive soft-walled permeable capillaries. The capillaries are represented by a single soft-walled collapsible tube which extends through a box (the extracapillary space) filled with extravascular fluid. Extravascular pressure (PE), is determined by the volume of extravascular fluid and the compliance ( C ) of the capsule. I n the model shown, compliance is determined by the volume/ pressure relationships of the PE standpipe. Fluid filters out of the upstream end of the capillary into the extravascular space at a rate Qz' Qz is determined by the upstream permeability (D1) and the difference between the upstream pressure (PA) and the extravascular pressure (PE). At the downstream end of the capillary, fluid from the extravascular space filters back into the capillary a ta rate Q3 Q3 is determined by the downstream permeability (Dz) and the difference between the extravascular pressure (PE) and the downstream pressure (Pv). Patency of the capillary is determined by the transmural pressure which is the difference between the intravascular pressure (PI) and the extravascular pressure (PE) FIG. I
1
open and the conductance increases with the transmural pressure. In this region we assume a linear relationship between the transmural pressure and the conductance which varies from zero to a maximum value (transitional region in Fig. 2). This assumption is supported by observations of the relationship between conductance and transmural pressure in models (Rodbard and Kuramoto, 1963; Fig. 3), and in the isolated, saline-filled lung(Rodbardand Kira, 1966; Fig.4). Intracapillary pressure PIis assumed to have a uniform value throughout its length. Transmural
1
-40
1
1
-20
5-5 (crnH,O)
,
I
0
'
I
20
I
I
I
40
F I G . 3 Flow (conductance) and transmural pressure in the model. (Redrawn and modified from Rodhard and Kuramoto, 1963.) Conversion: traditional to SI units: I cm H Z O z 0.098 kPa.
pressure is expressed as PI-P,. Actually, PI and transmural pressure values change along the length of the vessel, but for the purpose of correlating the conductance through the entire
15 Post-occlusion hyperaemia and blood flow in the capillaron model 2.01Conductance
D, : upstream capillary permeability (ml-min-l. cm H,O-l) D2: downstream capillary permeability (ml min-l.cm H,O-') C: compliance of the extracapillary compartment (ml.cm H20-') (PAand Pv are applied to the pressure present at the immediate upstream and downstream ends of the capillaron unit, respectively. These are likely to be different from systemic pressures, and might better be called arteriolar and venular pressures, respectively.) Filtration through the upstream permeability: dQ Qz = 2 dt = Dl (PA-PE)
.
1
-%4 6 8 10 I2 14 Capillary transmural pressure (cm saline) (vascular minus airway)
16
18
Vascular transmural pressure and conductance of pulmonary vascular bed in isolated dog lung lobe. The left lower lobe was isolated and suspended in a bottle filled with saline. Its airway was filled with saline and the vascular bed was perjirsed with saline at an arteriovenous pressure difference of 3.3 cm saline. The zero reference level was at the hilum. Pressure in the bottle (pleural pressure) was - 2.5 cm saline. Starting from a level lower than airway pressure, arterial and venous pressures were raised simultaneously without changing arteriovenous pressure difference. The transmural pressure of the pulmonary vascular bed is intravascular pressure minus airway pressure. When transmural pressure was negative, vascular conductance was zero. As transmural pressure became positive, conductance increased gradually to a plateau. (Redrawnfrom Rodbard and Kira, 1966.) FIG. 4
length of the capillary with the transmural pressure change, the intracapillary pressure can be lumped into a mean value. The transmural pressure (PTM)in the whole system can therefore be expressed as PI - PE.
Derivation Some basic relationship in the capillaron system can be expressed in several equations (Fig. 1). PA: arterial (arteriolar or upstream) pressure (cm H20) P,: venous (venular or downstream) pressure (cm HZO) P,: extracapillary pressure (cm HzO) PI: intracapillary pressure (cm HzO) Q1:flow rate through the artery (ml.min-') Q,: filtration rate through the upstream permeability (ml.min-l) Q3:filtration ratL through the downstream permeability (ml.min-l)
Filtration through meability:
the downstream
per-
Relationships between the extracapillary tissue fluid volume and the extracapillary pressure :
(3) Combining the three equations above, we obtain:
+
= (Dip, DzPv)- PE (Di + Dz) Solving equation 4 for PE we obtain: PE = DlP.4 + DzPv pE (0) D1+ D2
(4)
+
where P, (0) is the value of PE when t=O. Equation 5 means that regardless of the value of parameters such as D1, Dz, PA, Pv, C, etc, PE has a characteristic that, starting from an initial value PE (0), P, exponentially approaches the steady-state value (D1PAf D,Pv)/(D, D2). It is necessary that D, be greater than D,. This is because the P, is the 'weighted' average of PA and Pv, weighted according to D, and D,, respectively. Accordingly, if D, is equal to or less than D,, then the net or mean transmural pressure, PTM,becomes equal to zero or negative In this state, the capillary is completely collapsed, conductance is zero, and this condition is not of physiological interest.
+
16 Murao and Rodbard
Application of the basic formulae to reactive hyperaemia For any value of PE (0), PE approaches an equilibrium value PEas sufficient time elapses: PE pE = (DIPA D7,PV)/(Dl + D7,)a We can take P, as the control value of P, before occlusion of the arterial inflow. The mean intracapillary pressure before arterial occlusion is : (PA + pV)/2* Immediately after arterial occlusion, the intracapillary pressure PI becomes equal to Pv, and according to equation 5, PE exponentially approaches the new steady-state value Pv (which can be obtained by setting PA= Pv in equation 5, or setting Q2=0 in equation 1,and solving equations 2 and 3 simultaneously. On release of the arterial occlusion, PI returns almost instantaneously to (PA+ Pv)/2 while PE begins to return exponentially to the control value (D1PA DPPV)/(D1 D2) (Fig. 5). The transmural pressure change can consequently be depicted as in the top row in Fig. 6. If the transmural pressure remains within the transitional region in Fig. 2 during the entire
Transitional reqion
Poslhve reqlon
+
+
+
Flow I
I
FIG. 6 Flow in arterial occlusion. The horizontal axes show time. The top row shows the transmural pressure as derived from Fig. 5. The second row shows conductance. The third row shows the pressure difference between the artery and Gein. The bottom row shows the j o w which is deriued as the product of conductance and pressure difference. In the transitional region, the post-occlusion overshoot of the transmural pressure is rejected in the postocclusion overshoot of conductance. Correspondingly, the flow shows the post-occlusion hyperaemio. In the positive region, there is no overshoot of conductance and no postocclusion hyperaemia.
process, the vascular conductance curve will show a pattern similar to the transmural pressure curve. As the flow through the capillary vessel is expressed as the product of the conductance and the arterial-venous pressure difference, the capillary flow will show the pattern in the bottom row of the left column in Fig. 6. Thus, with the release of the arterial occlusion, the flow rises above the pre-occlusion level (post-occlusion hyperaemia) and then returns toward the preocclusion level. From Fig. 5 it is predicted that I the longer the period of occlusion, the higher the OCCLUSION level of the peak of post-occlusion transmural pressure, hence the higher the level of the peak of FIG. 5 Intracapillary, extracapillary, and transpost-occlusion hyperaemia. Fig. 7 shows the mural pressures in arterial occlusion: the solid line result obtained in an actual capillaron model, shows the change of the intracapillary pressure PI. which is consistent with the present theoretical The broken line shows the change of the extra- analysis (Rodbard, 1966a).l capillary pressure PE.The vertical distance between The diference in the shapes of flow peaks in theoretically obtained PI and PErepresents the magnitude of the transmural flow patterns and that in experimental observations (Fig. 7) can be pressure PTM. Before arterial occlusion, PI is attributed to simultaneous changes of the calibre of the capillary and the transmural pressure. Consideration of the time lag between the (PA+ Pv)/2, and P E is &. Upon arterial occlusion, PI of transmural pressure change and the capillary calibre change will make drops to Pv and PE exponentially approaches Pv. the ascending limb of the flow curve curvilinear over a short span of time instead of a n instantaneous vertical rise and the flow pattern thus Upon releasing the occlusion, PI returns to (PA+ obtained will be much closer to that obtained in actual tissue or in the Pv)/2 and PE returns exponentially to ward PE. capillaron model.
-
17 Post-occlusion hyperaemia and blood flow in the capillaron model Occlurion
(I)
P I G . 7 Post-occlusion hyperaemia in a model of the capillaron. The arterial inflow was occludedfor I , 2, 6, and 20 s, as indicated. The eflects during occlusion and the post-occlusion hyperaemias are shown. (From Rodbard, 1966a.)
The rate of change of P, is determincd by the time constant of the system C/(D1+Dz); the larger this value, the slower the rate of change. Consequently, the larger the compliance of the extravascular compartment and the smaller the permeabilities, the slower the change of PE. The slower the change of PE, the smaller the level of post-occlusion hyperaemia for a given duration of occlusion. If the transmural pressure changes beyond the transitional region in Fig. 2, the flow pattern will * -TIME 0 differ from that described above, since the linear relationship between transmural pressure and F I G . 8 A sudden rise in the arterial pressure (PA) conductance will then no longer hold. At such from PA1 to PA2.The intracapillary pressure (PI) high intracapillary pressures the transmural also rises suddenly. The extracapillary pressure (PE) rises exponentially to a new steady-state value. The pressure levels before and after the occlusion will transmuralpressure (PTM),which is PI - PE,increases remain the positive region, and no post-occlusion suddenly and then decreases gradimlly. hyperaemia will be observed (Fig. 6, right column).
Application of the basic formulae to autoregulation Suppose the control value of PE is P,= (D1PA1+DzP,)/(Dl Dz). Elevation of the arterial pressure from PA1 to PA2is followed by an exponential rise in PEtoward its new steadystate value (DIPAD+ D,P,)/(D, + DJ according to equation 5 (Fig. 8). On the other hand, the mean intracapillary pressure rises instantaneously from (PA1 P,)/2 to (PA2 P,)/2. As a result, the PI- PE difference will rise instantaneously and then decline to a new level. If the transmural pressure is within the transitional region, the flow will correspondingly show a pattern as in the bottom row of the left column in Fig. 9, ie, first it rises suddenly and then declines to approximately the control value (autoregulation).
+
+
+
As in Fig. 10, lowering of the arterial pressure from an initial value PAl to PA3, lowers PI instantaneously, while PE changes exponentially from the control value (D1PA1+ D2P,)/(Dl D2) toward the new steady-state value (D1PA3+ DzPv)/(Dl D2). PI - PE drops instantaneously and then increases slowly to a new value. If the transmural pressure is within the transitional region, the flow also shows the initial fall and then the gradual increase (Fig. 11, left column, bottom row). In either case, if the transmural pressure is high and in the positive region, the flow does not show autoregulation (Fig. 10 and Fig. 11, right column, bottom row), The basic mathematical formulae can also be applied to the analysis and elucidation of a wide variety of other phenomena observed in the capillaron system including basal ‘tone’, closing
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18 Murao and Rodbard
Transitional reqion
Positive reqion
Transitional reqion Transmural I pressure
Transmural pressure
Conductance I
Conductance
Positive reqion I
I
~~~~~
Pressure dlfference
Pressure difference
Flow I
% -
F I G . 9 Autoregulation of flow accompanying a sudden rise in the arterial pressure. If the transmural pressure is within the transitional region, the sudden rise in the transmural pressure followed by the gradual decrease is reflected in a similar pattern of change in flo w (autoregulution). If the transmural pressure is within the positive region, the flow does not show autoregulation.
1
Flow I
I
F I G . I I Autoregulation of flow accompanying a sudden fall in the arterial pressure. If the transmural pressure is within the transitional region, the sudden fall in the transmural pressure followed by the gradual increase is reflected in a similar pattern of change in flow (autoregulation). If the transmural pressure is within the positive region, the flow does not show nutoregirlation.
la does not give a precise reproduction of either an actual capillaron model or an actual capillary segment in tissue. For example, in the real capillary, permeability is not lumped but is distributed continuously along the length of the capillary vesse1.l Also the actual relationship between the vascular conductance and the transmural pressure is curvilinear rather than rectilinear. Such a precise analysis will result in nonlinear equations which require a computer for solution. I n spite of the drawback of siniplification cited above, the present analysis provides F I G . 1 0 A sudden in the arterial pressure (PA) a good approximation to the situation in the ,from PA^ to PAa. The intracapillory pressure (PI) also tissues, and a means for analysis and prediction falls suddenly. The extracapillory pressure (PE)falls of the flow behaviour through the capillary esponentiully to a new steady-state uiilue. The vessels. This also can provide an excellent basis transmural pressitre (PTbt)decrenses srtddcril.v, then for the formulation of more detailed analyses in:renser grotiitaily. using a computer. D. Rodbard provided a critical review of the manuscript.
pressure, opening pressure, etc (Rodbard, 1966a). Since several assumptions have been made to facilitate the mathematical analysis, the formu-
In both the animal and model experiments it has been observed that the autoregulatory flow change is slower when the arterial pressure is lowered than when it is raised. This asymmetry in the autoregulatory flow pattern can be demonstrated also in mathematical analysis if the continuous distribution o f capillary permeability is taken into consideration.
19 Post-occlusion hyperaemia and blood flow in the capillaron model
References Johnson, P. C.(1964). Review of previous studies and current theories of autoregulation. American Heart Association Monograph No. 8, Autoregulation of Blood Flow,pp. 1-9. Rodbard, S. (1963). Autoregulation in encapsulated, passive, soft-walled vessels. American Heart Journal, 65, 648-655. Rodbard, S. (1966a). Evidence that vascular conductance is regulated at the capillary. Angiology, 17, 549-573.
Rodbard, S. (1966b). A hydrodynamic mechanism for auto regulation of flow. Cardiologia, 48, 532-535. Rodbard, S., and Kira, S. (1966). Mechanical forces and pulmonary vascular conductance. American Heart Journal, 72,498-508. Rodbard, S., and Kuramoto, K. (1963). Transmural pressure and vascular resistance in soft-walled vessels. American Heart Journal 66, 786-791.
ANNOUNCEMENT An international symposium on Cardiac Receptors will be held at the Department of Cardiovascular Studies, University of Leeds, Leeds LS2 9JT, from 14-17 September 1976. It is spon5ored by Cardiovascular Commission of International Union of Physiological Sciences. Topics for discussion include histology and electrophysiology of vagal cardiac receptors, reflex responses of atrial and ventricular receptors, efferent and central nervous mechanisms ; reflex responses following bleeding, infusion, and infarction; electrophysiology and reflex responses of sympathetic cardiac afferents. Further details and preliminary programme can be obtained from Dr C . Kidd, Department of Cardiovascular Studies, University of Leeds, Leeds LS2 9JT.
