A comparison of deconvolution and the Rutland-Patlak plot in parenchymal renal uptake rate.

INTRODUCTION: Deconvolution and the Rutland-Patlak (R-P) plot are two of the most commonly used methods for analyzing dynamic radionuclide renography. Both methods allow estimation of absolute and relative renal uptake of radiopharmaceutical and of its rate of transit through the kidney.
MATERIALS AND METHODS: Seventeen patients (32 kidneys) were referred for further evaluation by renal scanning. All patients were positioned supine with their backs to the scintillation gamma camera, so that the kidneys and the heart are both in the field of view. Approximately 5-7 mCi of (99m)Tc-DTPA (diethylinetriamine penta-acetic acid) in about 0.5 ml of saline is injected intravenously and sequential 20 s frames were acquired, the study on each patient lasts for approximately 20 min. The time-activity curves of the parenchymal region of interest of each kidney, as well as the heart were obtained for analysis. The data were then analyzed with deconvolution and the R-P plot.
RESULTS: A strong positive association (n = 32; r = 0.83; R (2) = 0.68) was found between the values that obtained by applying the two methods. Bland-Altman statistical analysis demonstrated that ninety seven percent of the values in the study (31 cases from 32 cases, 97% of the cases) were within limits of agreement (mean ± 1.96 standard deviation).
CONCLUSION: We believe that R-P analysis method is expected to be more reproducible than iterative deconvolution method, because the deconvolution technique (the iterative method) relies heavily on the accuracy of the first point analyzed, as any errors are carried forward into the calculations of all the subsequent points, whereas R-P technique is based on an initial analysis of the data by means of the R-P plot, and it can be considered as an alternative technique to find and calculate the renal uptake rate.

INTRODUCTION

Deconvolution and the Rutland-Patlak (R-P) plot are two of the most commonly used methods for analyzing dynamic radionuclide renography. Both methods allow estimation of absolute and relative renal uptake of radiopharmaceutical and of its rate of transit through the kidney,[1] Gamma camera renography has been used widely for the assessment of renal function over the last 20 years.[2] Many methods have been used to derive quantitative parameters from the measurements.[3456] The uptake of activity in the kidney before the minimum transit time of the radiopharmaceutical is taken as a measure of renal function. Both relative and absolute uptake may be calculated, the latter by relating renal activity to the injected dose.[7] Measures of the rate of transit of radiopharmaceuticals through the kidney, such as the peak time,[8] mean transit time (MTT)[9] or renal outflow efficiency (ROE),[10] may also be calculated. These parameters are useful in the evaluation of upper urinary tract obstruction, in which they have shown improvement to diagnostic accuracy in patients with impaired renal function,[10] and in detecting renovascular hypertension using angio-tension-converting enzyme inhibitor renography.[8] Two of the most commonly used approaches to analysis are deconvolution[11] and the R-P plot.[5] Both methods attempt to use information on the time variation of the input to the kidney to obtain functional parameters that are independent of the shape of the renogram. Both methods also provide a means of subtraction of intrarenal vascular activity and therefore, of estimation of the true renal uptake. Deconvolution is widely used and considered as the gold standard for the estimation of renal transit in conditions in which the transit time is assumed to be prolonged, such as renovascular disease, transplant rejection, and obstructive uropathy.[111213141516] Theoretically, at least, using the renogram as the output function and the plasma disappearance curve as the input function, the spectrum of intra-renal transit times can be determined exactly,[517] In practice, there are many factors which raise questions about the validity of this approach.[1819] One of these, probably the most important, is the relationship between the spectrum of intrarenal transit times and the duration of data acquisition. It is mathematically obvious that deconvolution can only be applied correctly if the maximal transit time is shorter than the duration of data acquisition. Unfortunately, based on a renogram, one cannot tell whether the maximal transit time is longer or shorter than the acquisition time.[20] Deconvolution analysis has been a useful technique for analyzing organ function in nuclear medicine[4921] for a considerable time. There are three established techniques plus one more recent approach.[22] The three main methods are iterative deconvolution (also known as matrix inversion), Laplace transforms and Fourier transforms.[23] The iterative deconvolution analysis has been previously applied by many authers,[41622242526] in nuclear medicine investigations. The deconvolution technique has been extensively and most widely applied to renal studies.[91624252627282930] Deconvolution is a mathematical technique, which overcomes the influence of the tracer input curve on the renogram. The result of deconvoluting the renogram yields is a function termed the “retention function,” which represents the form of the renogram that would be obtained if an injection is given directly into the renal artery. An important advantage of the retention function and R-P plot, which were employed to determine the uptake rate, over inspection of the renogram is that the parameters derived from the retention function and R-P plot have physiological significance, unlike those derived from the renogram such as slope of the second phase, time to peak, and so on. This is, because the renogram is a complex curve which combines both renal and extra-renal factors. Measures of the rate of transit of radiopharmaceuticals through the kidney, such as the peak time,[8] MTT,[9] or ROE,[10] may also be calculated. These parameters are useful in the evaluation of upper urinary tract obstruction, in which they have been shown to improve diagnostic accuracy in patients with impaired renal function,[10] and in detecting renovascular hypertension using angio-tension-converting enzyme inhibitor renography.[8] The aim of the present study was to compare between the values of the renal uptake rate obtained by two methods, R-P plot (multiple time graphical analysis method,[3132] and the iterative deconvolution (matrix inversion) using 99mTc-DTPA (diethylinetriamine penta-acetic acid). To our knowledge, this research is the first study that compared the values of renal uptake rate applying these two methods using 99mTc-DTPA.

THEORY

Theory and derivation of the retention function

In a renogram study, the renal curve (R (t)), is considered to be a convolution of the input function (blood curve), which will be called B (t), and the renal retention function (also known as the impulse response function), which will be called H (t). The retention function represents the curve that would be obtained if a spike bolus of tracer were delivered at one point in time into the renal artery. The shape of H (t) may be analyzed to produce information about the function of the kidney being studied, and one of the most frequently used parameters is the MTT. The renogram curve, R (t), after corrections for blood and tissue background, is a convolution of the input function from the blood to the kidney, I (t), and the retention function of the kidney, H (t). The main assumption made in this approach is that the kidney can be modeled as a linear, stationary system.[3334] In practice, the linearity of the system is generally maintained, but stationarity may often be violated.[16] The technique adopted in this study is that of discrete deconvolution using the matrix algorithm.[24] This algorithm has been applied previously in renal deconvolution.[41625] In this method, the linear matrix H is evaluated in a successive manner, starting with H (1) and working through to the final element H (n). Thus, the value of the retention function in the ith interval is given by:

where Δt is the sampling interval which is taken to be 20 s in this study, R and I are obtained by selecting regions of interest over the kidney and over the heart, respectively, and creating time-activity curves for the duration of the study. Before evaluating H from Eq. 1, it is necessary to reduce the effect of statistical variations inherent in R and I. This is achieved by applying once a 1:2:1 linear non-stationary smooth with appropriate end point constraints to both kidneys, and input curves since this has been shown to be an appropriate degree of data filtering in this study.[35]

Principles of Gjedde-Patlak analysis

The original idea of Patlak and Blasberg was to create a model independent graphical analysis method: Whatever the tracer is facing in the tissue, there must be at least one irreversible reaction or transport step, where the tracer or its labeled product cannot escape.[36] It is assumed that all the reversible compartments must be in equilibrium with plasma, i.e. the ratio of the concentrations of tracer in plasma and in reversible tissue compartments must remain stable. In these circumstances only the accumulation of tracer in irreversible compartments is affecting the apparent distribution volume. In practice, this can happen only after the initial sharp concentration changes when the plasma curve descends slow enough for tissue compartments to follow.[36] When the steady state is achieved, the Gjedde-Patlak plot becomes linear. The slope of the linear phase represents the net transfer rate K (influx constant). To make it simple, K represents the amount of accumulated tracer in the kidney to the amount of tracer that has been available in plasma.[36] The y-axis of plot contains apparent distribution volumes, that is the ratio of activities of tracer in the kidney and in plasma. On x-axis is normalized plasma integral, that is, the ratio of the integral of plasma activity and the plasma activity.[222336] The Patlak plot is given by the expression:

This means that the measured kidney activity is divided by plasma activity, and plotted at a “normalized time” (integral of input curve from injection divided by instantaneous plasma activity).[37] For systems with irreversible compartments, this plot will result in a straight line after sufficient equilibration time [Figure 1]. The slope and the intercept must be interpreted according to the underlying compartment model. For the 99mTc-DTPA, the slope represents the kidney uptake rate, while the intercept V (as appeared in the equation on the previous page) or F, which is due to the blood within the organ (the blood background subtraction factor). The value of F will be equal to the ratio of volumes of the blood activity in the organ to the blood contributing to the blood curve.[23] For every scintigraphic examination, the renal uptake rate of 99mTc-DTPA was calculated twice, applying the two methods R-P plot[3132] and deconvolution (matrix inversion) by the same operator. A R-P plot[39] is applied to the first few minutes of the renal and blood curves. The intercept of that plot (F) allows completion of the background subtraction process, and the slope (K) indicates what proportion of the blood curve is entering the kidney each second [Figure 1]. H (0) is the initial retention function value, after elimination of blood background activity. In practice, it is best obtained either as the plateau value of the retention function [Figure 2], or as the slope of R-P plot (i.e., “K”) [Figure 1].[21] The R-P plot starts with an initial value and rises with a slope. The first proof of the R-P plot[38] actually demonstrated that the initial value was equal to the blood background subtraction factor “F” (the interception of the straight line with y-axis as shown in Figure 1), and that the slope was similar to the uptake constant “H (0)” (the plateau value as shown in Figure 2).

Figure 1

Renal Rutland-Patlak plot for left and right kidneys

Figure 2

Renal retention function obtained by applying iterative deconvolution (matrix inversion) method, for left (obstructed) and right (normal) kidneys

MATERIALS AND METHODS

Seventeen patients (32 kidneys), 10 male and 7 female (their ages range from 15 to 62 year), were referred for further evaluation by renal scanning. Eight kidneys were diagnosed to be obstructed by both radiological investigations (intravenous pyelo-graphy) and diuretic renography. As part of the preparation procedure, the patient should be well hydrated and the urinary bladder is emptied before the study. All patients were positioned supine with their backs to the scintillation gamma camera, so that the kidneys and the heart are both in the field of view. The camera used in this investigation is a Siemens type camera with 16″ diameter Nal (Tl) crystal used in conjunction with a high-sensitivity parallel-hole collimator. Approximately 5-7 mCi of 99mTc-DTPA in about 0.5 ml of saline is injected intravenously and dynamic sequential 20 s frames are acquired (64 × 64 matrix) and stored in a computer equipped with data analysis software. The study on each patient lasts for approximately 20 min. The time-activity curves of the parenchymal region of each kidney, as well as the heart were obtained for analysis. The data were then analyzed with deconvolution and the R-P plot. Both deconvolution and R-P plot approaches applied for the assessment of renal uptake, are based on the assumption that up to a given time after injection, corresponding to the minimum transit time, there is no output of activity from the renal region of interest (ROI).[35]

RESULTS

Figures 2 and 3 represent the renograms and renal retention functions obtained by applying iterative deconvolution (matrix inversion) method, for left and right kidneys for one patient. Figure 1 demonstrates renal R-P plot. The x-axis is the integrated radioactivity in the blood for an ROI around the heart, divided by the radioactivity in the blood for that ROI during a certain time. The y-axis is the radioactivity in the kidney ROI during a certain time, divided by the radioactivity in the blood during the same time. Figure 4 demonstrates the relationship between the values of the renal uptake rate obtained by applying R-P method and the iterative deconvolution (matrix inversion) method. The regression equation of the R-P against iterative deconvolution (matrix inversion) was Y = 1.18 X − 0.55 (r = 0.83).

Figure 3

Renograms for left (obstructed) and right (normal) kidneys

Figure 4

Scatter plots of renal uptake rate determined by Rutland-Patlak method and by the, iterative deconvolution (matrix inversion) method

Results (statistical analysis)

The scintigraphy uptake rate values determined by R-P method and by the iterative deconvolution (matrix inversion) method, were further analyzed to the method of Bland-Altman, which is a supplementary method to compare two different methods when the true value is unknown. The data was plotted as scatter plot of the mean values versus the difference of both calculations [Figure 5]. A plot of the mean of kidney uptake function calculations obtained by applying the two methods on 32 99mTc-DTPA renography (horizontal axis), versus the differences in the two calculations (vertical axis). The horizontal solid lines [Figure 5] indicate the mean difference between the two calculations. The horizontal dashed lines indicate the 95% limits of agreement (mean ± 1.96 standard deviation). Ninety seven percent of the values in the study (31 cases from 32 cases, 97% of the cases) were within limits of agreement. A strong positive association (n = 32; r = 0.83; R2 = 0.68) was found between the values that obtained by applying the two methods [Figure 4]. The value of the uptake rate can be calculated from the slope of the straight line [Figure 1], for this patient the slopes are equal to 4%/min for the left kidney and is equal to 8%/min for the right kidney. Whereas the values of the uptake rate obtained by applying the deconvolution method are equal 3%/min and 5%/min for the left and right kidneys, respectively, which are equal the value of the plateau of the retention function [Figure 2]. From Figure 2, we noticed that the excretion of the radioactivity from the right kidney (normal) began earlier than that of the left kidney (obstructed). It took approximately 4 min to begin for the right kidney, whereas it took approximately 12 min for the left kidney.

Figure 5

Bland-Altman analysis. Scatter plots of the mean renal uptake rate by, Rutland-Patlak method and the iterative deconvolution (matrix inversion) method, as the x-axis; against difference in renal uptake rate, as the y-axis, The solid line indicates the mean difference and the dotted lines indicate, the 95% of agreement (1.96 standard deviation)

DISCUSSION

Deconvolution and the R-P plot have been widely used for analysis of renography, each enabling the derivation of renal uptake corrected for vascular background contribution. Rutland[1] have shown that the two methods for uptake measurement are theoretically equivalent, in this article the two methods have validated in practice in a series of 32 renograms that cover a wide variety of ages. Theoretically, the deconvolution method is an ideal method for the estimation of renal transit.[16394041] In practice, however, there are many physiological and technical factors that hamper the proper application of deconvolution analysis to the renogram.[1019] Indeed, the conditions that need to be met for deconvolution are not entirely fulfilled. The linearity of the system is not respected when the precordial curve is used as the input function, as it differs from the true plasma curve. The required stationary condition, on the other hand, is violated by changes in renal emptying due to back-pressure of the bladder. An abrupt change in urine flow due to the injection of a diuretic during the acquisition totally invalidates the deconvolution analysis.[20] The R-P plot has now been around for over 20 years, which poses the question of why it has taken so long for this relationship to be made evident? There are two likely reasons. Firstly, that the other forms of deconvolution were available, and so there was no great pressure to look for alternatives. The second is that in many of cases the R-P plot did not appear to have a level, but actually appeared to decline once tracer started to leave the organ. This will occur when the blood curve being used is not a true measure of arterial blood. That will occur in the later stages of externally measured blood curves when there is an excess of counts due to soft-tissue activity also being measured in the “blood” curve. It is possible that looking at the later stages of the R-P plot is a way of actually seeing how truly the externally measured blood curve matches the input function to the organ.[23] The reasoning behind this approach is that in producing the R-P plot, both the content function and the integral of the input function are divided by the input function (i.e., blood curve), and that this has the effect of producing data equivalent to that which would be produced if the input function did not vary (i.e., if there were a constant blood level of tracer).[23] The convolution technique (the iterative method) relies heavily on the accuracy of the first point analyzed (first point of the retention function as shown in Figure 1), as any errors are carried forward into the calculations of all the subsequent points, whereas R-P technique is based on an initial analysis of the data (four points which constitute the straight line as shown in Figure 3) by means of the R-P plot.[3138] The conventional deconvolution methods are iterative (or matrix inversion) deconvolution; or using either Laplace or Fourier transforms. Whichever method is used, all are very sensitive to small variations in the input data, making it difficult to prevent large errors developing. Furthermore, all these conventional methods first generate a retention function and then use that retention function to measure the uptake rate from the plateau height.[22] Whereas in R-P plot method, the uptake rate was calculated directly from the slope of the R-P plot straight line. In R-P plot the tracer concentration curves of tissue ROI and arterial plasma are transformed and combined into a single curve that approaches linearity when certain conditions are reached. The data could be plotted in a graph, and a line can be fitted to the linear phase. The slope of the fitted line represents the net uptake rate of the tracer or volume of distribution.[11] The graphical analysis methods are independent of any particular model structure.[20]

CONCLUSION

There was a strong correlation between renal uptake values measured by R-P and iterative deconvolution methods. We believe that R-P analysis method is expected to be more reproducible than iterative deconvolution method, because the deconvolution technique (the iterative method) relies heavily on the accuracy of the first point analyzed, as any errors are carried forward into the calculations of all the subsequent points, whereas R-P technique is based on an initial analysis of the data by means of the R-P plot,[3831] and we also believe that this technique can be considered as an alternative technique to find and calculate the renal uptake rate.