J. theor. Biol. (1975) SO, 353~361
Deter~&Wm
of Pdsatile Flow Rata by Indicabdilution Method!!! KENNETH H. NORWICH
Department of Physiology ond Imtitute of Biomedical Engineering, University of Toronto, Toronto, Ontorio, iU5S 1AS, Cona& (Received 7 February 1974, and in revtied fom
6 June 1974)
A new method is proposed for measuring no@steady flow rates when this flow is pulsatile in nature. This method invol-&s the use of indicators and does not require direct access to the vessel carrying the fluid. No knowledge of the associated mathematics is required for its application. The investigator infuses indicator into the vessel leading into a capillary or similar labyrinth at a time-varying rate such that theI indicator concentration at the outflow from the labyrinth remains constant in time. When this
condition at the outflow has heen achieved, t&e puhatile flow rate at the inflow is given simply as the ratio of the varying infusion rate to the
constant outfiow indicator concentration. 1. rntrodnl!tion Almost invariably, mathematical formulae for calculating flow-related parameters using indicators can be associated ‘with analogous formulae for calculating metabolism-related parameters using isotopic tracers. For example, the “direct Fick” method for calculat$ng steady flow rates has as its analogue the constant tracer infusion method for determining steady state appearance rate.7 The Stewart-Hamilton formula for computing steady flow rate by applying an indicator injection and measuring the area under the indicator-dilution curve has as its analogue the formula for computing steady state appearance rate by applying a tracer injection and measuring the area under the specific activity-time curve (Shipley & Clark, 1972). when appearance rate is not constant, this rate may be calculated using a variation of the constant infusion method (Norwich, 1973). If tracer is infused at a varying rate in such a way that plasma spccik activity is held constant, then the nonsteady rate of appearance of tracee (unlabelled metabolite) is t The rute of appemmce of a metabolite is the rate (mass/time) at which the substance entcaa its distribution space due to a process of se&t@ production from chunkal prscurson, etc. 333
354
K.
H.
NORWICH
given by the ratio of the varying infusion rate to the constant specific activity. Such experiments are rather difficult to execute because specific activity is usually measured in all samples after the experiment has been terminated, denying the hapless investigator the knowledge of specific activity which he is trying to control during the course of the experiment. The nonsteady flowrelated analogue of this procedure consists of infusing indicator at a timevarying rate at the fluid inlet to a labyrinth of a system in cyclical steady state (to be defined precisely below) in such a way that the indicator concentration at the outlet remains constant-in the absence of recirculation of indicator. The nonsteady flow rate is then obtained as the ratio of the time-varying indicator infusion rate to the constant indicator concentration at the inlet. This flow-related analogue is, however, more readily applicable than its metabolic counterpart because certain indicators, such as thermal indicators or optical indicators with fibre optic detectors, can be detected nearly instantly-i.e. effectively zero time elapses between a change in indicator concentration in the fluid stream and its detection in the laboratory. Problems do arise with the recirculation of indicator and these will be discussed below. Little mathematical labour is required to deduce this formula for a rather general system.
2. Mathematical
Demonstration
Consider a capillary labyrinth with a single inlet and a single outlet (Fig. 1). Flow must be in a cyclical steady state-i.e. a series of identical pulses-and the various conduits and capillaries may be elastic or distensible. We shall assume that the indicator is injected into the inflow conduit in such a way that it rapidly distributes itself uniformly over a cross section of the conduit. (Such a condition is unlikely to be met experimentally and this eventuality will be discussed below.) The fluid density is taken to be uniform throughout, and both indicator mass and fluid volume are conserved within the system. Conservation of indicator mass is expressed by the equation of convective diffusion (Levich, 1962; Perl, 1962; Norwich, 1971; Slattery, 1972).
where the indicator is infused into a very small volume, and the flow of the i&sate is very small in comparison with the total fluid flow; where C(r, t) is
INDICATOR
METHODS
FOR
PULSATILE
FLOWS
355
FW.1. Aschanatiocdviewisgivenofacapillarylabyrinthwitha~inflowanda single outflow channel. The horizontal arrows dcaig@te the diroztion “f fluid flow; the verticd arrow represents the infusion of indicator. An origin of co+rdmatcs is taken at some arhitrary point in space. The vectors rh (just downstmam from the site of infusion) and rmt designate reapa$ivcly the pc&k.ms of the inflow and out&v or&es.
the indicator concentration at a point designatedby position vector r at time t; v(r, t) is the velocity of the fluid point at (r, t); D is a constant coefficient of diffusion; &(t) and C,&) are the indicator concentrations (mass/volume) at the inlet and outlet respectively; Q,,(t) and Q,,&) are the periodic functions describing the intiow and outflow rates (volume/time) (not necessarily equal if the capillary bed is elastic; 6(r - r,,,) and 6(r - r,,) are delta functions in three dimensions; and h(t) and R,,(t) are the net rates of appearanceat the inlet and disappearancefrom the outlet respectively (massftime).R&) is the variable rate of infusion of indicator by the experimenter. From the righthand side of equations (1) and (2) the following identitites are apparent: MO = Gt(O -Q,,(t)
(3)
356
K.
Conservation
H.
NORWICH
of fluid volume is expressed by the equation (Slat&y, iv*” dS = QinW-QA)
1972) (5)
where the integral is taken over the entire surface of the system and n is the unit outward-drawn normal at the surface. The surface is, in general, expanding and contracting with each pulsation or surge of fluid. Integrating equation (1) over the entire volume of distribution at any time, t,
,t$v+ JWCv-DVC,dV= C,,(t).Qi,(t>-c,,,
(6)
which, upon applying the divergence theorem becomes [EdI’+
Jcv*ndS-DjVC*r~ dS = Ci,(t)*Ql”(t)-C,,,(t).Q,“,(t) s s since D is constant. Let us now try the solution C = a constant in space and time = C,
(7)
63)
corresponding to the indicator infusion rate R,,(t). That is, C, Ci” and CoUt all assume the constant value C,. As we shall see below in equation (13), the time-variation of this infusion rate matches the time-variation of the fluid inflow rate, Q,,(t), so that what we have perhaps done is adhered very strictly to the principle of “equivalent supply” of tracer (Bergner, 1964). Substituting equation (8) into (7) gives 6~.n dS = QdO-Qout
Deter~&Wm
of Pdsatile Flow Rata by Indicabdilution Method!!! KENNETH H. NORWICH
Department of Physiology ond Imtitute of Biomedical Engineering, University of Toronto, Toronto, Ontorio, iU5S 1AS, Cona& (Received 7 February 1974, and in revtied fom
6 June 1974)
A new method is proposed for measuring no@steady flow rates when this flow is pulsatile in nature. This method invol-&s the use of indicators and does not require direct access to the vessel carrying the fluid. No knowledge of the associated mathematics is required for its application. The investigator infuses indicator into the vessel leading into a capillary or similar labyrinth at a time-varying rate such that theI indicator concentration at the outflow from the labyrinth remains constant in time. When this
condition at the outflow has heen achieved, t&e puhatile flow rate at the inflow is given simply as the ratio of the varying infusion rate to the
constant outfiow indicator concentration. 1. rntrodnl!tion Almost invariably, mathematical formulae for calculating flow-related parameters using indicators can be associated ‘with analogous formulae for calculating metabolism-related parameters using isotopic tracers. For example, the “direct Fick” method for calculat$ng steady flow rates has as its analogue the constant tracer infusion method for determining steady state appearance rate.7 The Stewart-Hamilton formula for computing steady flow rate by applying an indicator injection and measuring the area under the indicator-dilution curve has as its analogue the formula for computing steady state appearance rate by applying a tracer injection and measuring the area under the specific activity-time curve (Shipley & Clark, 1972). when appearance rate is not constant, this rate may be calculated using a variation of the constant infusion method (Norwich, 1973). If tracer is infused at a varying rate in such a way that plasma spccik activity is held constant, then the nonsteady rate of appearance of tracee (unlabelled metabolite) is t The rute of appemmce of a metabolite is the rate (mass/time) at which the substance entcaa its distribution space due to a process of se&t@ production from chunkal prscurson, etc. 333
354
K.
H.
NORWICH
given by the ratio of the varying infusion rate to the constant specific activity. Such experiments are rather difficult to execute because specific activity is usually measured in all samples after the experiment has been terminated, denying the hapless investigator the knowledge of specific activity which he is trying to control during the course of the experiment. The nonsteady flowrelated analogue of this procedure consists of infusing indicator at a timevarying rate at the fluid inlet to a labyrinth of a system in cyclical steady state (to be defined precisely below) in such a way that the indicator concentration at the outlet remains constant-in the absence of recirculation of indicator. The nonsteady flow rate is then obtained as the ratio of the time-varying indicator infusion rate to the constant indicator concentration at the inlet. This flow-related analogue is, however, more readily applicable than its metabolic counterpart because certain indicators, such as thermal indicators or optical indicators with fibre optic detectors, can be detected nearly instantly-i.e. effectively zero time elapses between a change in indicator concentration in the fluid stream and its detection in the laboratory. Problems do arise with the recirculation of indicator and these will be discussed below. Little mathematical labour is required to deduce this formula for a rather general system.
2. Mathematical
Demonstration
Consider a capillary labyrinth with a single inlet and a single outlet (Fig. 1). Flow must be in a cyclical steady state-i.e. a series of identical pulses-and the various conduits and capillaries may be elastic or distensible. We shall assume that the indicator is injected into the inflow conduit in such a way that it rapidly distributes itself uniformly over a cross section of the conduit. (Such a condition is unlikely to be met experimentally and this eventuality will be discussed below.) The fluid density is taken to be uniform throughout, and both indicator mass and fluid volume are conserved within the system. Conservation of indicator mass is expressed by the equation of convective diffusion (Levich, 1962; Perl, 1962; Norwich, 1971; Slattery, 1972).
where the indicator is infused into a very small volume, and the flow of the i&sate is very small in comparison with the total fluid flow; where C(r, t) is
INDICATOR
METHODS
FOR
PULSATILE
FLOWS
355
FW.1. Aschanatiocdviewisgivenofacapillarylabyrinthwitha~inflowanda single outflow channel. The horizontal arrows dcaig@te the diroztion “f fluid flow; the verticd arrow represents the infusion of indicator. An origin of co+rdmatcs is taken at some arhitrary point in space. The vectors rh (just downstmam from the site of infusion) and rmt designate reapa$ivcly the pc&k.ms of the inflow and out&v or&es.
the indicator concentration at a point designatedby position vector r at time t; v(r, t) is the velocity of the fluid point at (r, t); D is a constant coefficient of diffusion; &(t) and C,&) are the indicator concentrations (mass/volume) at the inlet and outlet respectively; Q,,(t) and Q,,&) are the periodic functions describing the intiow and outflow rates (volume/time) (not necessarily equal if the capillary bed is elastic; 6(r - r,,,) and 6(r - r,,) are delta functions in three dimensions; and h(t) and R,,(t) are the net rates of appearanceat the inlet and disappearancefrom the outlet respectively (massftime).R&) is the variable rate of infusion of indicator by the experimenter. From the righthand side of equations (1) and (2) the following identitites are apparent: MO = Gt(O -Q,,(t)
(3)
356
K.
Conservation
H.
NORWICH
of fluid volume is expressed by the equation (Slat&y, iv*” dS = QinW-QA)
1972) (5)
where the integral is taken over the entire surface of the system and n is the unit outward-drawn normal at the surface. The surface is, in general, expanding and contracting with each pulsation or surge of fluid. Integrating equation (1) over the entire volume of distribution at any time, t,
,t$v+ JWCv-DVC,dV= C,,(t).Qi,(t>-c,,,
(6)
which, upon applying the divergence theorem becomes [EdI’+
Jcv*ndS-DjVC*r~ dS = Ci,(t)*Ql”(t)-C,,,(t).Q,“,(t) s s since D is constant. Let us now try the solution C = a constant in space and time = C,
(7)
63)
corresponding to the indicator infusion rate R,,(t). That is, C, Ci” and CoUt all assume the constant value C,. As we shall see below in equation (13), the time-variation of this infusion rate matches the time-variation of the fluid inflow rate, Q,,(t), so that what we have perhaps done is adhered very strictly to the principle of “equivalent supply” of tracer (Bergner, 1964). Substituting equation (8) into (7) gives 6~.n dS = QdO-Qout
