## Abstract

A new approach to characterize the kinetics of intravascular mixing process is presented. The mixing time, defined as the time required for achieving 95% homogeneity, is calculated by numerical simulations using a circulatory model applied to the intravascular marker indocyanine green (ICG). The results suggest that the mixing time is determined by cardiac output and the relative dispersion of transit time distribution across the systemic circulation, whereby the rate of mixing increases with increasing cardiac output and decreasing transit time dispersion, and vice versa. The estimation of plasma volume from simulated ICG dilution data using the backextrapolation method shows that slow mixing is accompanied by an overestimation of blood volume. This error may be negligible for mixing times of less than ∼3 min but high in disease states characterized by low cardiac output and/or high transit time dispersion. In view of the role of transit time dispersion as determinant of intravascular mixing, it would be interesting to know more about the effect of disease states on systemic transit time dispersion.

- plasma volume
- indocyanine green
- intravascular distribution
- extrapolation

mixing plays a key role in the circulatory transport of hormones or drugs. An empirical parameter for the intravascular mixing kinetics is the mixing time (*t*_{m}). An adequate characterization of the initial mixing phase is of special importance for evaluating the distribution kinetics of permeating indicators (exchange between the plasma and the extravascular space) (33, 34) and for fast-acting drugs like intravenous anesthetics (for a recent review, see Ref. 6). The mixing time plays a crucial role in estimating plasma (or blood) volume by dilution techniques with labeled albumin (or red blood cells), and when the backextrapolation method is used in case of eliminated indicator indocyanine green (ICG) (see, e.g., Ref. 10 and the references cited therein). Since complete mixing is regarded as a prerequisite for the application of these methods, the time range used for monoexponential extrapolation should start with *t*_{m}. The reported values of intravascular mixing time vary between 3 min (8, 24) and 15 min (1), and mixing appears to be further prolonged up to 45 min in critically ill children (15). Unfortunately, less is known about the physiological determinants of the mixing process. Furthermore, it is generally doubtful whether the commonly used backextrapolation method allows a correct estimation of circulating plasma volume (9, 23).

The aims of this paper are twofold: first, to quantitatively explain the determinants of intravascular mixing using a validated “whole body” model of circulatory transport (33), and second, to check the accuracy of the backextrapolation method and its dependency on the rate of mixing.

## METHODS

#### Definition of mixing time.

The mixing time has been previously defined in reactor theory as the time in which the measured concentration of the indicator in a noneliminating system reaches 95% of the final (completely mixed) concentration (*C*_{m}) (4, 16): (1) where *t*_{m} is mixing time. Note that, theoretically, a homogeneous concentration is reached after infinite time (*C*_{m} *C*_{∞}) and that *C*_{m} determines the whole body distribution volume, *V*_{B} = *D*_{iv}/*C*_{m} for an indicator dose (*D*_{iv}). The meaning *t*_{m} is illustrated in Fig. 2 with a concentration-time curve of the vascular marker indocyanine green (ICG) simulated using the circulatory model described below.

#### Model simulation.

We have shown previously that the plasma concentration of ICG after bolus venous injection can be well fitted by a minimal circulatory model using the inverse Gaussian density (Brownian passage time density) (26) as transit time density function (TTD) for the pulmonary and systemic circulation (Fig. 1) (33). The equation for the arterial concentration-time curve after rapid bolus injection (dose *D*_{iv}) of ICG is only available in the Laplace domain: (2) where *Q* denotes cardiac output, *E* is the systemic extraction of ICG, and *f̂*_{p}(*s*) and *f̂*_{s}(*s*) are the Laplace transforms of pulmonary and systemic TTD. The Laplace transform of the inverse Gaussian TTD is given by (25) (3) where the index *i* stands for *p* (pulmonary) and *s* (systemic). The parameters *V*_{i} and *RD*_{i}^{2} denote the vascular volume and relative dispersion, respectively, of each subsystem. Note that the pulmonary subsystem includes the lungs and the heart chambers. The mean transit times of ICG through the subsystems are given by *MTT*_{i} = *V*_{i}/*Q*, and the mean circulation time is obtained as *MCT* = *MTT*_{p} + *MTT*_{s}. Simulations were performed based on *Eqs. 2* and *3* using MAPLE 8 (Waterloo Maple, Waterloo, ON, Canada) after a numerical method of inverse Laplace transformation was implemented (22). To evaluate the effect of *Q* and systemic transit time dispersion (*RD*_{s}^{2}) on the *t*_{m}, *C*(*t*) data were simulated for different values of *Q* and *RD*_{s}^{2} without elimination (*E* = 0 in *Eq. 2*), and *t*_{m} was calculated according to *Eq. 1*. The plasma volume was estimated from the data simulated with *Eq. 2* by applying the backextrapolation method, i.e., monoexponential extrapolation back to *time 0* to obtain an estimate of *C*_{m}, the concentration that would be obtained after instantaneous mixing. The range of backextrapolation (time window) started when complete mixing was achieved, i.e., at time *t*_{m} after injection. The calculations were performed for three different time windows (1, 3, and 5 min) using five equally spaced time points within each window (e.g., *t*_{m}, *t*_{m} + 0.2, *t*_{m} + 0.4, *t*_{m} + 0.6, and *t*_{m} + 0.8 min for the 1-min window). Data simulation was based on the following parameters (mean values) estimated in humans (liver donors) by Niemann et al. (18): *Q* = 7 l/min, *V*_{p} = 1.5 liters, *V*_{s} = 4.6 liters (i.e., a whole body plasma volume of *V*_{B} = *V*_{p} + *V*_{s} = 6.1 liters) and ICG clearance (*CL*) of 1 l/min. The latter value and *Q* were used to estimated the intrinsic hepatic clearance (*CL*_{int}) using the well stirred liver model (e.g., Ref. 30) (4) assuming a constant fractional liver blood flow of *Q*_{H}*/Q* = 0.25. The resulting value of *CL*_{int} = 1.9 l/min is then used in *Eq. 4* to calculate the systemic extraction ratio *E* = *CL/Q* as a function of *Q*.

To get an idea of the range of relative dispersions of TTDs, transit time dispersion *RD*_{p}^{2} and *RD*_{s}^{2} were estimated by fitting *Eqs. 2* and *3* to published ICG data using SCIENTIST 3.0 (Micromath, St. Louis, MO). The values of *RD*_{p}^{2} = 0.15 and *RD*_{s}^{2} = 0.81 estimated for a patient undergoing coronary bypass surgery (Fig. 1 in Ref. 7) were similar to those of a liver donor (*RD*_{p}^{2} = 0.3, *RD*_{s}^{2} = 0.2) (Figs. 2 and 3 in Ref. 18) and a healthy volunteer (*RD*_{p}^{2} = 0.06, *RD*_{s}^{2} = 0.5) (Fig. 2 in Ref. 17), whereas a much higher value of dispersion across the systemic circulation (*RD*_{s}^{2} = 4.6) was found in this subject under propranolol (Fig. 2 in Ref. 17). Note that, from the TTD of ICG through the human pulmonary circulation, a *RD*_{p}^{2} value of 0.5 was reported (14).

## RESULTS

#### Mixing time.

As shown in Fig. 2, high *Q* (7 l/min) and low systemic transit time dispersion (*RD*_{s}^{2} = 0.2) is characterized by rapid mixing (*t*_{m} = 1.5 min). The dependence of mixing time from *Q* and systemic transit time dispersion could be described by the following empirical equation (Fig. 3) : (5)

The role of *RD*_{s}^{2} and *Q* as determinants of intravascular mixing time (*Eq. 5*) is illustrated in Fig. 4. Dividing *Eq. 5* by the mean circulation time (*MCT = V*_{B}*/Q*), we obtain *t*_{m}/*MCT* = 0.02*Q* + 1.1*RD*_{s}^{2}, and it becomes obvious that the dimensionless mixing time *t*_{m}/*MCT* increases proportionally to the systemic transit time dispersion for *RD*_{s}^{2} > 2: (6)

We are not dealing here with the situation of relatively rapid mixing, which is characterized by *RD*_{s}^{2} < 2, since we are primarily interested in understanding the slow down of mixing under physiologically realistic conditions. A complex, non-monotonous dependence of *t*_{m} on *RD*_{s}^{2} is found for *RD*_{s}^{2} < 2, i.e., for sufficiently low *RD*_{s}^{2} values (after reaching maximum mixedness), the mixing time may theoretically increase with decreasing *RD*_{s}^{2} (see discussion). For the example shown in Fig. 2 (*Q* = 7 l/min), the *t*_{m}-*RD*_{s}^{2} curve shows a flat minimum at *RD*_{s}^{2} = 0.7, and the mixing time of 1.5 min obtained for *RD*_{s}^{2} = 0.2 is slightly larger than this minimal value. For *RD*_{s}^{2} = 1, mixing times of 0.95 and 2.2 min are found for *Q* = 7 and 3 l/min, respectively. In both cases, one obtains for *t*_{m}/*MCT* a value of 1.1, which shows that, for rapid mixing, *t*_{m} is practically identical to *MCT*, suggesting that *Eq. 6* also holds for *RD*_{s}^{2} = 1. Although all simulations addressed so far were based on arterial sampling, it appears interesting to look also at the mixing behavior after passage through the whole circulation, i.e., the effect on the mixed venous concentration upstream of the injection site. From the resulting curve (Fig. 5), simulated with the same parameters as in Fig. 2, except for *RD*_{s}^{2} = 0.7 instead of 0.2, it is obvious that for optimal mixing (see above) a mixing time of <1 min can be achieved.

#### Backextrapolation.

The backextrapolation method was applied to simulated (error free) data. Using *t*_{m} as starting point of the exponential fit, five data points in windows of 1, 3, and 5 min, respectively, were used for extrapolation to estimate the intercept *C*(0) at *t* = 0. The calculated value of the plasma volume, *D*_{ICG}/*C*(0), was then compared with the value used in data simulation (*V*_{B} = 6.1 liters). The results obtained for a fitting window of 1 min (Fig. 6) show that the degree of *V*_{B} overestimation increases if mixing is prolonged on the one hand with increasing systemic transit time dispersion and on the other hand with decreasing *Q*. It can be seen from Fig. 6 that the error is low (<4%) for rapid mixing (*t*_{m} < 3 min). One can assume that this result obtained for *Q* ≥ 5 l/min and *RD*_{s}^{2} = 2 generally holds for *RD*_{s}^{2} ≤ 2 and *Q* ≥ 5 l/min. For example, an error of <3% was found for *RD*_{s}^{2} = 1 and *Q* = 7 as well as 3 l/min (with mixing times of 0.95 and 2.2 min) and for the case shown in Fig. 2. That the estimation error further increases with increasing fitting window is demonstrated in Fig. 7 for *RD*_{s}^{2} = 2. However, it is important to note that for rapid mixing (*t*_{m} < 3 min) slightly better results were obtained with a time window of 3 min for monoexponential extrapolation. To evaluate the effect of a change in fractional liver blood flow, the calculations were also performed for *Q*_{H}*/Q* = 0.1 and 0.4. The results differ slightly from those shown in Fig. 3 for *Q*_{H}*/Q* = 0.25 but show the same overall pattern. For example, in case of *RD*_{s}^{2} = 5, a decrease in extraction (*Q*_{H}*/Q* = 0.1) led to a 25% (or 10%) decrease in the estimation error for *Q* = 3 l/min (or 10 l/min), whereas an increase in extraction (*Q*_{H}*/Q* = 0.4) increased it by ∼14 and 4%, respectively.

## DISCUSSION

#### Mixing time.

The present results reveal that, besides *Q* (or circulation time), the relative dispersion of transit time through the systemic circulation is an important determinant of circulatory mixing, whereby the slope of the increase in mixing time with *RD*_{s}^{2} increases with decreasing *Q* (Fig. 4). This effect of transit time dispersion is in principal accordance with the finding that, for inverse Gaussian distributed circulation times (one subsystem), the degree of mixing first steeply increases with increasing *RD*^{2} up to a flat maximum between 1 and 2 and then decreases with increasing relative dispersion (30). Although this latter part is relevant for the physiological system considered here, maximum mixedness cannot be defined as clearly since the circulation consists of two subsystems with concentration measurement in arterial blood after the first passage through the lungs.

Interestingly, the dependence of *t*_{m} on *Q* and *RD*_{s}^{2} derived here for intravascular mixing resembles that in chemical reactors (2, 4, 16), with the difference that in reactors only the first phase is observed, i.e., the degree of mixing increases with increasing *RD*^{2} until maximum mixedness is achieved (*t*_{m} ∼ 1/*RD*^{2}). [Note that for inverse Gaussian TTD, *RD*^{2}/2 is for *RD*^{2} ≪ 1 identical to the dispersion number of the advection-dispersion model (31)]. In other words, mixing is brought about by circulation time dispersion (unmixed plug flow as extreme case with very low *RD*_{s}^{2}), but after maximum mixedness (for *RD*_{s}^{2} between 0.5 and 1, Fig. 5) mixing time increases again with increasing dispersion in physiological systems.

In contrast to *Q*, very little is actually known about the systemic transit time dispersion and its changes in pathological conditions. Recalling that all organs of the systemic circulation are lumped into one subsystem, *RD*_{s}^{2} is determined by the heterogeneity of perfusion (*Q*_{i}) and vascular volumes (*V*_{s,i}) of organs as well as mixing within individual organs (*RD*_{s,i}^{2}) assumed to be arranged in parallel (29, 33) (7)

Since available information indicates that organs behave as nearly well mixed systems with *RD*_{s,i}^{2} between 0.2 and 0.8 (14, 20, 30), the distribution of organ transit times (*MTT*_{i} = *V*_{s,i}/*Q*_{i}) appears to be an important determinant of *RD*_{s}^{2}. This implies that a redistribution of *Q* may affect *RD*_{s}^{2}. It is as yet unclear, however, whether a change in the transit time dispersion across organs (*RD*_{s,i}^{2}) has to be taken also into account, for example, due to a possible flow dependency (13). Furthermore, the results obtained by Cousineau et al. (3) in dog heart suggest that β-adrenergic blockade may change organ transit time dispersion, besides its effect on *Q* distribution (17). Thus it is questionable whether *Eq. 7* can be used to predict changes in systemic transit time dispersion due to a redistribution of *Q*, assuming constant *RD*_{s,i}^{2} values.

Using the present model (*Eqs. 2* and *3*), an average *RD*_{s}^{2} value of 3 was estimated from ICG dilution curves in dogs (33), whereas, as noted above, in humans values between 0.2 and 5 were observed. Normally mixing times appear to be less than ∼3 min; extremely long mixing times up to 45 min, as reported in patients with shock (15, 21), could be explained by an increase in *RD*_{s}^{2}, i.e., greater heterogeneity of organ transit times (Fig. 4). The present results are consistent with the suggestion of a prolonged mixing time in patients with chronic heart failure (24). The importance of plasma volume estimation in these patients (12) underlines the practical relevance of our results. Interestingly, it has been hypothesized earlier that a redistribution of *Q* (due to vasoconstricted and vasodilated vascular beds) could be detected by ICG dilution curves (27) and that alterations in circulatory function by disease, drugs, and aging may affect the mixing characteristics (35).

#### Backextrapolation.

As one could expect, the results suggest that the error involved in applying the backextrapolation method to ICG data is very low for fast intravascular mixing and may remain less than ∼10% if the mixing time does not exceed 5 min (Fig. 6). For slow mixing, the error increases, indicating that the method appears not suitable for plasma volume estimation in disease states characterized by low *Q* and/or high systemic transit time dispersion. That the error increases with prolonged mixing due to increasing transit time heterogeneity is in principal accordance with the results obtained by Iijima et al. (9) using a model with two parallel systemic circuits (5). Regarding the role of the time window, a qualitatively similar effect as shown in Fig. 7 was observed in analyzing real data (11). Although the accuracy of the backextrapolation method also depends on the slope of the plasma concentration vs. time curve (i.e., ICG clearance), the results obtained here assuming that the splanchnic circulation receives 25% of *Q* (*Q*_{H}*/Q* = 0.25) reflect the typical situation, also because this ratio remains relatively constant in many disease states despite a change in *Q* (e.g., Refs. 19, 28). Furthermore, although a decrease in *Q*_{H}*/Q* increases the accuracy, the slight increase in error predicted for *Q*_{H}*/Q* = 0.4 does not limit the usefulness of the backextrapolation method if intravascular mixing is fast (see above).

In light of the present results, the effect of disease states on the backextrapolation error appears to be underestimated in a worthwhile discussion of factors affecting the accuracy of *V*_{B} measurement using ICG dilution (10). However, why should one use the extrapolation method rather than much better model-based methods that utilize the whole *C*(*t*) curve? Given the potential limitations of the backextrapolation method, circulatory models like *Eq. 2* (32, 33) or similar approaches to estimate *V*_{B} (5, 7) represent a reliable alternative for analyzing indicator dilution data. With these models, it is possible to estimate not only *V*_{B} (and its components *V*_{p} and *V*_{s}) but also *Q* and ICG clearance (i.e., liver function). Assuming inverse Gaussian transit time distributions, the present recirculatory model (33) additionally allows estimation of transit time dispersions.

In summary, these simulations reveal the roles played by *Q* and systemic transit time dispersion in determining the time to achieve complete mixing throughout the vascular space. These results are consistent with the suggestion that very fast mixing is brought about by high *Q* and low relative dispersion of systemic transit times; for example, as part of the “fight-or-flight” stress response. It will be important to further investigate the potential increase of transit time dispersion in disease states or due to drugs that affect blood flow distribution, especially in view of the predicted prolongation of mixing time.

- Copyright © 2009 the American Physiological Society