IOP PUBLISHING
PHYSIOLOGICAL MEASUREMENT
Physiol. Meas. 29 (2008) 281–294
doi:10.1088/0967-3334/29/3/001
On the physical and stochastic representation of an indicator dilution curve as a gamma variate M Mischi1, J A den Boer1 and H H M Korsten1,2 1 Department of Electrical Engineering, Eindhoven University of Technology, Den Dolech 2, 5612 AZ Eindhoven, The Netherlands 2 Department of Anesthesiology, Catharina Hospital, Michelangelolaan 2, 5623 EJ Eindhoven, The Netherlands
E-mail:
[email protected]
Received 1 October 2007, accepted for publication 19 December 2007 Published 11 February 2008 Online at stacks.iop.org/PM/29/281 Abstract The analysis of intravascular indicator dynamics is important for cardiovascular diagnostics as well as for the assessment of tissue perfusion, aimed at the detection of ischemic regions or cancer hypervascularization. To this end, indicator dilution curves are measured after the intravenous injection of an indicator bolus and fitted by parametric models for the estimation of the hemodynamic parameters of interest. Based on heuristic reasoning, the dilution process is often modeled by a gamma variate. In this paper, we provide both a physical and stochastic interpretation of the gamma variate model. The accuracy of the model is compared with the local density random walk model, a known model based on physics principles. Dilution curves were measured by contrast ultrasonography both in vitro and in vivo (20 patients). Blood volume measurements were used to test the accuracy and clinical relevance of the estimated parameters. Both models provided accurate curve fits and volume estimates. In conclusion, the proposed interpretations of the gamma variate model describe physics aspects of the dilution process and lead to a better understanding of the observed parameters, increasing the value and credibility of the model, and possibly expanding its diagnostic applications. Keywords: indicator dilution, physiological modeling, hemodynamics, biomedical ultrasonics, cardiovascular system
1. Introduction Hemodynamic quantification by means of indicators injected in the bloodstream has a long clinical history. An indicator bolus is injected upstream and its concentration-versus-time curve, referred to as indicator dilution curve (IDC), is measured downstream. Several methods 0967-3334/08/030281+14$30.00
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are available for IDC measurements, ranging from the classic invasive methods requiring patient catheterization, such as dye- or thermodilution (Fegler 1954, Murray Kinsman et al 1929), to the minimally invasive contrast imaging methods, such as contrast magnetic resonance imaging (Tofts 1997), contrast ultrasonography (Feinstein 2004) and scintigraphy (Anger 1964). Indicator dilution measurements have several applications in clinical diagnostics. Initially, indicators were used for cardiovascular quantification, aiming mainly at the assessment of blood volumes and cardiac output (Hamilton et al 1928, Zierler 2000). However, recent developments in contrast imaging have also permitted novel applications for the detection of myocardial or brain ischemia (Donnemiller et al 1997, Wei et al 1998), as well as angiogenetic processes related to cancer growth (Eckersley et al 2002, Ruediger et al 1999). The assessment of blood volumes and perfusion is based on the estimation of the indicator mean transit time (MTT). Therefore, the accurate estimation of this parameter is of major clinical importance. Owing to the indicator recirculation in the circulatory system and the low signal-to-noise ratio (SNR) of measured IDCs, the use of proper models for IDC fitting and extraction of hemodynamic parameters of clinical interest is necessary. The first model that was proposed is a monoexponential decay, representing the washout IDC of one mixing chamber (compartment) (Hamilton et al 1928, Zierler 2000). Since then, several models have been proposed for a closer representation of the IDC. Although some models, such as the local density random walk (LDRW) model (Sheppard and Savage 1951), provide a physical interpretation of the dilution process, being a solution of the diffusion with drift equation, the models that are typically implemented in clinical devices are the lognormal and the gamma variate models (Evans 1959, Linton et al 1995, Stow and Hetzel 1954, Thompson et al 1964). These models are empirically chosen due to their resemblance with measured IDCs, while no physical background is provided (Evans 1959, Linton et al 1995, Stow and Hetzel 1954, Thompson et al 1964). In this paper, we show that for a simplified dilution system the probability of detecting an indicator particle at a certain site and time can be modeled by a gamma statistics, and the gamma variate model can be adopted as the representation of the impulse response (IR) of this dilution system. Eventually, we present both a physical and stochastic representation of the dilution process by a gamma variate model. The model is fitted to the IDC by the Levenberg–Marquardt (LM) method (Marquardt 1963). A multivariate linear regression in the logarithmic domain (see the appendix) is used for its initialization. For validation, we compared the indicator MTT value from in vitro experimental data using the LDRW and gamma variate model fits. When multiplied by the flow (Q), the MTT estimation permits the volume (V) assessment as (Norwich 1977, Sheppard 1962) V = MTT · Q.
(1)
It is important to note that the MTT adopted for the volume assessment in (1) is the transit time of an indicator particle that travels at the velocity of the carrier flow, and this can be viewed as the transit time of the median particle or, equivalently, the mean of the spatial distribution of the particles (Wise 1966). The fits of both models to IDC data measured in patients were also compared. The IDCs were measured in the right ventricle (RV) and left ventricle (LV). The assessment of the MTT between these two measurement sites permits interesting clinical applications, such as the estimation of the pulmonary blood volume (PBV) (Mischi et al 2004a), which is closely related to the cardiac function (Bindels et al 2000, Buhre et al 2001, Wiesenack et al 2001).
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Figure 1. Scheme of a multisection tube model for the simplified representation of the dilution dynamics. The distance between the indicator injection and measurement sites xi and xm is divided into k − 1 sections (compartments) of equal length x.
2. Methodology 2.1. Gamma variate model The gamma variate model for IDC fitting is given as C (t) = A · t α · e− β , t
(2)
where α and β are the shape and scale parameters, respectively, and the factor A defines the IDC amplitude (Thompson et al 1964). This model can be viewed as a formal simplification of a gamma probability density function (PDF), which is given as λ e−λt (λt)k−1 , (3) (k) where λ and k are the scale and the shape factor, respectively (Lindgren 1968). The operator ∞ (·) represents the Gamma operator, i.e., (x) = 0 y x−1 e−y dy with x and y ∈ R, which is a generalization of the factorial for non-integer numbers. The gamma variate model in (2) can be easily derived from (3) for k − 1 = α, λ = β −1 and A = λk(k)−1. In this paper we focus on the derivation of the model for integer numbers, which can then be generalized to non-integer numbers by introducing the operator (·). Therefore, we mainly refer to the Erlang PDF, which is given as f (t; λ, k) =
λ e−λt (λt)k−1 . (4) (k − 1)! (k) in (3) is substituted in (4) by the factorial (k − 1)!. In fact, the gamma distribution is a generalization of the Erlang distribution for non-integer numbers (Hillier and Lieberman 2005). The PDF in (4) represents the waiting-time probability for the occurrence of k events and is typically employed for traffic or production control problems (Hillier and Lieberman 2005). fE (t; λ, k) =
2.2. Multisection tube model For dilution modeling purposes, the hemodynamic system can be represented in simplified approximation by an infinite tube along the x-axis, as shown in figure 1. A carrier fluid flows through the tube and transports the indicator from the injection site xi to the measurement site xm. For simplicity and without loss of generality, we can assume xi = 0. The tube can
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be viewed as a multicompartment model, whose compartments are constituted by adjacent sections of equal length x and volume Vc = Ax, A being the tube cross-section area. The compartments can be enumerated as x(x)−1 . The use of this multisection tube model reduces the dilution process description to the representation of one-dimensional dynamics, i.e., the indicator transport and diffusion along the tube axis, defined by the spatial coordinate x. 2.3. Physical representation With reference to the model in figure 1, we characterize the complete dilution system as a cascade of equal compartments (tube sections) with a constant flow rate. Each compartment is seen as a mixing chamber with a unidirectional indicator washout and can be represented by its IR, given as (Chen et al 1998) fC (t) =
1 −t e τ, τ
(5)
where τ is the time constant and equals the ratio QVC−1 between the flow through the compartment Q and the compartment volume VC. As a result, the IR of the complete system from the first to the kth section (k = x(x)−1 ) is given by the k-fold convolution of the compartment IR, which results in the same expression as the Erlang PDF in (4). The physical meaning of the coefficients in (4) therefore becomes clear. The scale parameter λ represents the inverse τ −1 of the time constant of the compartment IR, while the parameter k represents the compartment rank number. Back to the multisection tube representation, (4) can then be written as t
e− τ τ
fE (t; τ, x) =
t xx −1 τ x x
. −1 !
(6)
In this simplified model, the flow is not completely realistic. Real flow types such as laminar or turbulent flow could lead to imperfect mixing and deviation from (6). Here the radial distribution of a laminar (Poiseuille) flow is reduced to the monodimensional dispersion of the indicator along the tube axis, which is determined by the compartment size x and the time constant τ . The presence of turbulent flow would also be reduced to the same monodimensional interpretation. 2.4. Stochastic representation As an alternative to the deterministic solution of a multicompartment model, a random variable can be associated with the position of the indicator particle along the tube. In particular, a binomial distribution can be employed for the representation of the indicator concentration (probability density) as a result of the particle transitions (washout) from one section to the following. The movement (transition) of the indicator particle is represented as a result of a number nt of trials, with n the number of trials per unit time and t the time interval. For each trial we associate a probability p with the washout of the particle to the following section and a probability 1 − p with the complementary case, i.e., the continued residence of the particle in the same section. We are therefore assuming a unidirectional motion. For n np p, i.e., large n and small p, the Poisson distribution is a good approximation of a binomial distribution
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with λ = np constant (Papoulis 1991, Sheppard 1954). The probability of having k washouts during a time t can therefore be represented by a Poisson process as e−λt (λt)k . (7) k! The parameter λ can be referred to as washout rate. The IR of the multisection tube dilution system at a distance x = kx can be viewed in stochastic terms as the time PDF of having k washouts. With T being the random variable associated with the time t and P the probability function, this IR, h(t), can be derived as k−1 ∂ ∂ fP (i; t) = fE (t; λ, k). (8) h(t) = (1 − P [T > t|k]) = 1− ∂t ∂t i=0 fP (k; t) =
Therefore, the Erlang PDF represents the dilution IR at distance kx from the injection site when the parameter λ represents the washout rate, i.e., the expected number of transitions in n trials. By comparing the physical and statistical derivations of the model, we can easily conclude that the washout rate λ = np, which corresponds to the scale parameter of the gamma variate model, can be interpreted as the inverse of the time constant τ of the tube sections (compartments). As a result, the MTT of the indicator from the injection to the detection site equals kτ . By defining µ = kτ = kλ−1, the IR h(t) of the dilution system expressed by the PDF in (4) and (6) can be written as µτ −1 t t 1 e− τ . h(t) = · · µ (9) τ τ −1 ! τ The IDC at the detection site can be obtained by convolution of the IR with an input IDC. For an input IDC modeled as a Dirac impulse of area mQ−1 at time 0, the resulting IDC at the measurement compartment equals µτ −1 t t e− τ m , · · µ C(t) = (10) Qτ τ −1 ! τ where Q is the flow and m the injected indicator mass. According to the mass conservation law, the integral of (10) equals mQ−1. As the MTT µ represents the transit time of the median indicator particle, which flows with the velocity of the carrier fluid, it can be multiplied by the flow Q for the estimation of the volume between the injection and detection site. It is interesting to note that µ is also equal to the first statistical moment of (10). The constraint of having an integer k in the Erlang model can be again relaxed by employing the Gamma operator (·). As a result, the factorial (µτ −1 − 1)! can be expressed as (µτ −1). The integral and first statistical moment of (10) remain equal to mQ−1 and µ, respectively. The derivation of the gamma variate formulation in (2) is straightforward. The MTT is equal to (α + 1)β = µ. For A = ((µτ −1))−1τ −µ/τ , the time integral of (2) from 0 to ∞ is equal to 1. As (2) represents the IR of the dilution system, the integral of the IDC, i.e., the convolution of (2) with an input IDC, is equal to mQ−1, in agreement with the mass conservation law. This derivation of the gamma variate model results from the condition of a small value of p in the binomial representation of the multisection tube. When this condition is not satisfied and the probability p is sufficiently large, the binomial PDF can be approximated by a normal PDF (Sheppard 1954, 1962). This approximation leads to a random walk type of description of the dilution process, resulting eventually in the LDRW model. This model describes the contrast
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Figure 2. Scheme of the in vitro setup for IDC measurements. A centrifugal pump generates a water flow in a tube before the UCA injection site. The flow is measured by an electromagnetic flowmeter. The injected UCA bolus is then detected by an ultrasound transducer before and after its passage through a network of tubes that produces the volume to be estimated. Before the output of the open fluid dynamic circuit, the static pressure is stabilized.
(This figure is in colour only in the electronic version)
dilution process by a normal spatial distribution of the indicator, whose mean translates with the velocity of the carrier flow and whose variance increases as a linear function of time. The model formulation in the continuous domain can be given as (Bogaard et al 1984, Wise 1966) m η ηµ − η2 ( µt + µt ) e e , (11) CLDRW (t) = Qµ 2π t with µ equal to the transit time between the injection and detection sites of the spatial mean of the indicator distribution (median indicator particle) and η being an adimensional parameter equal to the ratio between convection and diffusion in the dilution system. The parameter µ of the LDRW model, which is also referred to as MTT, can therefore be multiplied by the flow in order to obtain volume estimates (Wise 1966). For the LDRW model, the MTT µ is smaller than the first statistical moment of the model, which is equal to µ(1 + η−1) and can be referred to as mean residence time (MRT) (Bogaard et al 1984, Mischi et al 2004a). The LDRW model can also be derived by solving the diffusion with drift differential equation (Norwich 1977). This derivation clarifies the physical meaning of the parameter η, which for a tube model equals µQ2(2D)−1A−2, with D and A representing the longitudinal diffusion coefficient of the system and the tube cross-sectional area, respectively. The longitudinal diffusion coefficient D is related to the molecular diffusion coefficient and depends on the type of flow (Bogaard et al 1984). 2.5. Experimentation A dedicated set of measurements was performed aimed at testing the validity of the gamma variate hypotheses. For this purpose, the in vitro setup described in Mischi et al (2004a) for IDC measurements by contrast ultrasonography was used (figure 2). A network of tubes of equal diameter simulated the dilution volume. The flow was generated by a calibrated
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Figure 3. Example of gamma variate (dashed line) and LDRW (solid line) fits to two IDCs (gray lines) measured in vitro by contrast ultrasonography before and after a volume network of 1080 mL with a flow of 1 L min−1. The correlation coefficients of the fits are 0.996 and 0.997, respectively.
550 bio-console centrifugal pump (Medtronic, Minneapolis, MN) and measured by the embedded electromagnetic flowmeter. Different flows, ranging from 1 to 5 L min−1 in R (Bracco, Milan, Italy) five equal steps, were tested. A bolus of 5 mL of UCA SonoVue diluted 1:100 was injected in the system. Rapid injections, which could be approximated by a Dirac function, were performed manually. Two measurement sites were placed before and after the dilution volume. The adopted ultrasound scanner was a Sonos 5500 (Philips Medical Systems, Andover, MA) equipped with an S3 probe. The scanner was set in power modulation mode at 1.9 MHz with a low mechanical index (0.1) in order to limit the collapse of UCA microbubbles (Chang et al 2001). The injected UCA dose for the adopted setting of the scanner ensured an approximately linear relationship between UCA concentration and backscattered acoustic intensity (Mischi et al 2004a). As a result, the IDC shape was preserved by the measuring system. The acoustic intensity was measured by Q-Lab software (Philips Medical Systems, Andover, MA) for acoustic quantification, which permits extracting the backscattered acoustic intensity from any region of interest without the signal distortion due to logarithmic compression, typically adopted by ultrasound imaging systems. In general, if the compression function is known, the video output of the ultrasound scanner is also suitable for the measurement. The adopted sampling frequency was 25 Hz. The aim of this experimentation was the assessment of the dilution volumes based on the IDC MTT estimations. Different volumes were tested, each with a carefully calibrated value, ranging in four steps from 310 to 1080 mL. In total, 40 IDCs were obtained. Figure 3 shows, as an example, a typical observed IDC. The experimental in vitro IDCs were comparable to real IDCs measured in patients. In order to illustrate this, we used data from 20 IDCs measured in patients from the RV and LV at the Department of Cardiology of the Catharina Hospital in Eindhoven (The Netherlands). These data were obtained after consent of the local ethical committee by trans-thoracic echocardiography. These measurements were performed using the same ultrasound scanner and setting adopted for the in vitro experiments. The same dose of UCA that was used for the in vitro measurements was manually injected in an antecubital vein. In order to fit the first pass IDC and exclude recirculation curves, the IDC was fitted up to 30% of the peak concentration along the downslope (Millard 1997). Figure 4 shows an example of RV and LV IDCs measured in vivo from a four-chamber view. The MTT estimation from these two IDCs,
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Figure 4. Example of gamma variate (dashed line) and LDRW (solid line) fits to two IDCs (gray lines) measured in a patient by contrast ultrasonography in the right ventricle (RV) and the left ventricle (LV), respectively. Both fits have a correlation coefficient higher than 0.99 before recirculation occurs.
similar to the in vitro experiments, permits the assessment of the blood volume between the measurement sites, i.e., the PBV (Mischi et al 2004a). The data from the in vitro as well as the in vivo experiments were used to obtain model fits for both the gamma variate and the LDWR model. Before fitting, the IDC data were filtered by a linear phase filter (50 taps) with a cut-off frequency at 2 Hz. The fitting was obtained in two steps. The first step was a multivariate linear regression in the logarithmic domain, as proposed in Mischi et al (2004b) for the LDRW fitting. This method includes an iterative search for the ‘injection time’ t = 0 that results in the minimum mean squared error of the IDC fit. The fitting parameters obtained in this step were used for the initialization of the second step: a LM nonlinear iterative fitting. Examples of fitted curves are given in figures 3 (in vitro data) and 4 (in vivo data). The MTT of the indicator was derived as the parameter µ of the IDC model fits. The value of the dilution volume was estimated as the product between the flow and the observed MTT difference (MTT) derived from the IDC fits as MTT = µ2 − µ1 ,
(12)
where µ1 and µ2 are the µ parameters of the first and the second IDC model fit, respectively. Even though the injection time of the two IDCs was the same, usually their estimates did not coincide, and the injection time estimate for the second IDC was delayed. In fact, the injection time estimates that provide the best IDC fits do not always correspond to the real injection time (Bogaard et al 1984), resulting typically in delayed estimates (Wise 1966). This time difference was therefore added to the MTT in (12). The values of µ1 and µ2 were estimated by both the gamma variate and the LDRW model fits. In vitro, the flow was measured by the electromagnetic flowmeter embedded in the system. In vivo, ultrasound Doppler time integration across the aorta was used for the cardiac output estimation (Huntsman et al 1983).
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Figure 5. In vitro volume estimates by the LDRW (squares) and the gamma variate (triangles) models. The solid lines indicate the real volumes. The measurements are reported for flows ranging between 1 and 5 L min−1.
3. Results The fitting of all in vitro IDCs provided excellent results, with average correlation coefficients R = 0.981 (standard deviation equal to 0.03) and R = 0.988 (standard deviation equal to 0.02) for the LDRW and the gamma variate model, respectively. The improvement of the correlation coefficient R between the initial linear regression and the subsequent LM fitting was 0.034 and 0.003 for the LDRW and the gamma variate model, respectively. In order to evaluate whether the linear regression initialization was necessary, the sensitivity of the LM fitting to the parameter initialization was tested. A variation of 5% to the initial values estimated by the linear regression was therefore applied. Due to the low SNR of the measured IDCs, this resulted in a decrease of the fit accuracy to an average correlation coefficient of 0.936 and 0.981 for the LDRW and the gamma variate fits, respectively. As a result, we concluded that, especially for the LDRW model, an accurate parameter initialization was required. The ranges of the observed parameters in the in vitro experiments were α ∈ [0.9, 20] and β ∈ [0.05 s, 5.8 s] for the gamma variate model, and η ∈ [1.76, 179.5] and µ ∈ [0.53 s, 19.2 s] for the LDRW model. The in vitro dilution volume estimates for all flows and volumes are reported in figure 5. The correlation between the estimated and the real volumes resulted in determination coefficients R2 = 0.998 and R2 = 0.999 for LDRW and the gamma variate model, respectively. The average standard deviation of each volume estimate over the five different flows was equal to 2.0% and 1.0% of the calibrated (real) volume for the LDRW and the gamma variate model, respectively. Both models underestimated the largest volume. However, for larger volumes the gamma variate model provided larger volume estimates than the LDRW model. As a result, the average underestimation of the largest volume was more limited for the gamma variate than for the LDRW model (3.4% against 7.4%). The average values of the model parameters estimated in patients were α = 1.3 ± 1.0, β = 3.3 ± 2.4 s, η = 2.9 ± 2.5 and µ = 5.2 ± 3.1 s, for the 20 RV IDCs, and α = 2.1 ± 1.1, β = 5.2 ± 3.0 s, η = 5.0 ± 2.6 and µ = 13.5 ± 5.2 s for the 20 LV IDCs. The average
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Figure 6. Bland–Altman plot of the PBV estimates in patients by the two models. The difference between the gamma variate and the LDRW volume estimates (Y-axis) is reported against the average value of the two estimates (X-axis). The mean difference (black line) and the standard deviation (dashed lines) are also shown.
values of α and β increased (p < 0.02) from the RV to the LV IDC. The increase of α is a clear consequence of the addition of a compartment (dilution volume) such as the lungs. The increase of β can be interpreted as a result of an increase, for the same flow (cardiac output), of the average time constant τ = β, due to the large volume of the added compartment. As expected, also the parameter µ of the LDRW model increased significantly (p < 10−6) between the two IDCs. The average correlation coefficients of the fits in patients were R = 0.967 and R = 0.963 (standard deviations equal to 0.037 and 0.041) for the LDRW and the gamma variate fits, respectively. Based on the measured IDCs and the two models, PBV estimates in patients were also derived. The comparison of the results provided by the two models is reported in the Bland– Altman plot in figure 6 (Bland and Altman 1986). The mean difference between the gamma and the LDRW PBV estimates was 63.1 mL, showing an increase for higher volumes. This result confirms the results obtained by the in vitro measurements. The standard deviation of the volume difference was 69.7 mL, i.e., 11.1% of the mean volume, which was equal to 623.1 mL and was estimated as the mean provided by the two models. 4. Discussion A new formalism for the physical and stochastic representation of indicator dilution dynamics by a gamma process is introduced. The model parameters are simultaneously interpreted as characterizing unidirectional indicator washout from tube sections or waiting times between successes, e.g., transitions of the indicator particles. These interlaced interpretations lead to a representation of the indicator dilution systems in terms of indicator IR or PDF, respectively. The dilution system identification reduces to the estimation of the time constant τ of the compartment exponential washout curve and the number of compartments k representing the dilution system, or, equivalently, to the estimate of the washout rate (success rate) λ and the mean waiting time (the MTT µ) to have a number of washouts (successes) equal to k.
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While the use of the gamma variate model in previous publications was mainly based on its heuristic resemblance with an IDC (Thompson et al 1964), this paper presents a structured derivation of the model based on both physical and stochastic principles. Since the parameters of the LDRW model were already known to have a physical meaning, being the LDRW model derived as a solution of the diffusion equation, this brought us in a position that allows a comparison of the applicability of these physical principles to real dilution systems. Compared to the complexity of real hemodynamic systems, the proposed model derivation is based on extremely simple assumptions. However, even the use of the gamma variate model, a simple model with only three parameters, can already provide an accurate description of the human pulmonary hemodynamic system, permitting the estimation of clinical parameters of diagnostic relevance, such as for instance the PBV. A dedicated model fitting based on a linear regression in the logarithmic domain was used to initialize a subsequent nonlinear iterative regression. This is a critical issue as variations of 5% of the initial values already resulted in a clear deterioration of the fit accuracy. In our in vitro data, although the results of the linear regression were already accurate, the nonlinear regression always led to increased correlations. However, the improvement was in most cases not significant. In vivo, probably due to the lower SNR of the measured IDCs, a systematic improvement was no longer recognizable. In vitro and in vivo data were collected during controlled and careful measurements. The quality of the IDC fits (R > 0.9 for all in vivo as well as in vitro data) suggests that both models provide parameters that could be of clinical use. The shapes of the in vitro IDCs resembled those observed in vivo and the distribution of the parameters estimated in vivo was amply covered by the parameter intervals observed in vitro. Therefore, the in vitro parameters are suitable for further interpretation aimed at clinical applications. Obviously, the predictive value of our in vitro results for the hemodynamics in humans is limited, due to the complexity of physiological dilution systems. Conclusions pertaining to that system will require further validation by means of in vivo measurements. The presented measurements in patients, however, already suggest a meaningful relationship between the dilution system morphology and the estimated parameters. An example is the increase of the number of compartments α and the time constant β when a large compartment, such as the lungs, is included in the measured dilution system. In general, differently from other authors (Thompson et al 1964), we did not find any contradiction between the estimated values of the model parameters, both in vitro and in vivo, and the hypotheses from which the models are derived. In particular, values of β that are larger than 1 do not represent a contradiction according to the presented derivations. The two analyzed models appear to be equally suited as a means for IDC fitting. A comparison between the volume measurements based on the gamma variate and the LDRW model showed a slightly higher correlation for the gamma variate model estimates. This is a first confirmation of the gamma variate hypothesis, which, differently from the LDRW model, is derived under the assumption of unidirectional movement of the indicator particles. As a result, the transit time of the median indicator particle τ k = µ is equal to the mean of the transit time distribution, i.e., the first statistical moment of the model. This differs from the LDRW model, where the MTT µ is smaller than the first statistical moment of the model, i.e., the MRT. Although further investigation is required, this might explain the volume underestimation provided by the LDRW model for increasing volumes, confirmed by both our in vitro and in vivo measurements. The results of the gamma variate model in terms of accuracy of the IDC fits and MTT estimations are comparable to those obtained by the LDRW model, although the models are based on different physical assumptions. We may therefore conclude that the unidirectional
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movement condition is a good representation of the real dilution system. In the physiological system, the presence of valves in the heart as well as in the venous circulation might in fact support the hypothesis of unidirectional motion. 5. Conclusions This paper provides for the first time a combined physical and stochastic background to the derivation of the gamma variate model for the representation of indicator dilution dynamics. The theoretical derivations are supported by experimental data, involving a comparison with the LDRW model. As a result of this study, a meaningful interpretation of the model parameters can be achieved, possibly adding diagnostic potential to the clinical use of the gamma variate model for indicator dilution curve analysis. An immediate application, as shown in this paper, can be the measurement of the PBV based on the MTT analysis of the RV and LV IDCs. This and other possible applications require future work for the analysis of the parameters in vivo. Appendix The fitting of a model f (t; θ ), which is a function of the parameter vector θ , to an IDC C(t), is performed by finding the vector θ that minimizes the mean square error ε(θ ) between the real data and the model as
∞ (C(t) − f (t; θ ))2 dt. (A.1) ε(θ ) = 0
For simplicity and without loss of generality, we assume the injection time (beginning of the IDC) equal to 0. If the model f (t; θ ) can be represented as a linear combination of functions xi(t) with i = 1, 2, . . . , n, f (t; θ ) = θ0 + θ1 x1 (t) + θ2 x2 (t) + · · · + θn xn (t),
(A.2)
then the vector θ that minimizes the mean square error (A.1) can be derived in the discrete domain as (Krzanowski 1998) ⎡ ⎤ θ1 ⎥ ⎢ ⎢θ2 ⎥ θ = ([X]t [X])−1 [X]t C = ⎢ .. ⎥ , (A.3) ⎣.⎦ θn
where C is the vector containing the IDC samples and [X] is a matrix whose columns are filled by the samples of the model functions xn(t), i.e., ⎞ ⎛ 1 x11 . . . x1n .. .. ⎟ ⎜ .. (A.4) [X] = ⎝1 . . . ⎠. 1 xn1
···
xnn
Although the gamma variate is a nonlinear function, a transformation in the logarithmic domain ln(C(t)) allows the use of a linear model as given in (A.2) for its representation. Therefore, a linear regression can be applied for the model fitting, and the estimate for the vector θ can be
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