Basic Physics of Nuclear Medicine/Deconvolution Analysis

Introduction.
The Nuclear Medicine renogram represents the response of the kidney to an injection of radiotracer into a peripheral vein. The arrival of the tracer in the kidney is therefore smeared out in time and its instantaneous response cannot be measured directly. analysis, as you will see below, is a method which allows the instantaneous response to be derived from a renogram, so that the functioning of kidneys in the same patient or between different patients can be more reliably compared.
Renography.
In renography, the background-corrected curves indicate how the quantity of tracer in each kidney varies with time. In addition, the shape of the each curve is dependent on the rate at which the tracer:
The rate of entry of the tracer into the kidney is called the input rate, I(t), and depends on:
In the case of a tracer which is extracted rapidly from the plasma (e.g. OIH, MAG3), the rate of extraction is dependent on the effective renal plasma flow (ERPF), so that:
I(t) = (ERPF) P(t).
In the case of a tracer which is extracted via glomerular filtration (e.g. DTPA), the rate of extraction is dependent on the glomerular filtration rate (GFR), so that:
I(t) = (GFR) P(t).
The variation in plasma concentration with time, P(t), consists initially of a spike whose amplitude and duration depend on:
Following this initial spike, the plasma concentration, P(t), decreases bi-exponentially with time depending on:
The plasma concentration, P(t), is therefore dependent on factors which are not related directly to renal function. As a result, renograms from different patients, or from the same patient at different times, may be difficult to compare directly.
An Ideal Renogram.
An ideal renogram could be obtained if an ideal injection of the tracer could be delivered directly into a kidney. Theoretically, such an ideal injection should give an input rate, I(t), which consists of a delta function, and is called an impulse input.
The resulting ideal renogram is referred to as the impulse response of the kidney, H(t).
In practice, the response of the kidney to a non-ideal input rate is measured. This real response, R(t), can be considered to arise from a convolution of the actual input rate and the impulse response of the kidney:
R(t) = I(t) * H(t).
The process of determining, H(t), on the basis of measuring R(t) and I(t) is called deconvolution. In other words, deconvolution of the renogram curves allows the response of the kidneys to an ideal injection to be determined. As a result, improved inter- and intra-patient comparisons can be obtained – in theory, at least!
Convolution.
Before treating deconvolution analysis, we will first of all consider the process.
Consider the model illustrated in the following figure of a grossly oversimplified representation of a kidney. This renal model considers the kidney to consist of four pathways through which the tracer flows equally. Let's say that it takes 3 minutes for the tracer to pass through the shortest pathway, 6 minutes for the longest pathway – and (you guessed it!) 4 and 5 minutes for the other two pathways.
When a delta function, "I", is fed as input to this model, the response of the kidney will be given by the impulse response, "H", as illustrated above. The form of the impulse response can be understood by considering the time course of the tracer through the model:
The important point to notice here is that the fall-off in the impulse response function gives information about the transit time for each pathway. This fall-off can be characterized by the mean transit time and the maximum transit time.
Note that this simple model can ideally be extended to include a very large number of pathways and so generate transit times which are more representative of the response of real kidneys. Note also that this form of modelling can be applied separately to the renal parenchyma, to the renal pelvis and to the whole kidney.
Unfortunately however, ideal impulse inputs are virtually impossible to generate in practice - except possibly in the case of a bolus injection into a renal artery! Nevertheless, the above thinking can be modified to include the effects of a non-ideal injection, as illustrated below:
Here, the input to our grossly oversimplified renal model is not an ideal impulse injection but is spread out so that the quantity of tracer decreases exponentially with time. When it is assumed that the response of the whole kidney to a succession of impulse inputs is equal to the sum of the individual responses to each impulse, we can infer that:
where "Dt" is the time interval used for each measurement of the quantity of the tracer.
Therefore, in general:
formula_1
where "i" = 1, 2, 3, 4, ...
Expressing this relationship as a continuous, as opposed to a discrete, function gives us what is called the convolution integral:
formula_2
which is usually written in the form:
formula_3
In other words, the real renogram curve, "R"("t"), can be considered to result from the ideal renogram curve, "H"("t"), being convolved with a real input function, "I"("t"). 
The model just described is an example of what is often referred to as a , i.e. a system where the response to a train of impulse inputs is given by the sum of the individual responses to each impulse. A linear system also assumes that the flow of tracer is constant through the system during the measurement period, i.e. the system is assumed to be stationary. In other words, for renography, it is assumed that the renal clearance rate and the urine flow rate remain constant during the study. The renal clearance may be assumed to remain constant after the initial phase of the renogram since the small quantity of tracer used (be it O123IH, 99mTc-MAG3 or 99mTc-DTPA) does not affect renal function. The same cannot be said for the urine flow rate, however, where excretion from the renal pelvis is frequently observed to occur in "bursts", as opposed to a continuous flow. Note that this lack of stationarity can give rise to major problems in the application of deconvolution analysis.
Deconvolution.
Deconvolution is the mathematical process used to determine "H" from measurements of "I" and "R". Deconvolution techniques which have been applied to renography include:
formula_4
which can be solved to give:
formula_5
where "I"-1 is the inverse matrix of matrix "I".
formula_6
"H"("t") can be determined from:
formula_7
i.e. dividing the Fourier transform of "R"("t") by the Fourier transform of "I"("t"), and calculating the inverse Fourier transform ("IFT") of the result.
Both of these techniques are highly sensitive to statistical fluctuations in the measurements, and the measured curves for "R"("t") and "I"("t") are therefore generally smoothed prior to Fourier transformation.
In clinical practice, a background corrected renogram curve can be used for "R"("t"), and a vascular region of interest (RoI) over, for example, the heart, can be used as a measure of "I"("t"). The , "H"("t"), obtained following deconvolution has a form similar to that shown in the following figure: 
"H"0 is the impulse response at time, "t" = 0, and can be shown to be equal to the kidney uptake constant. Indicators of the time taken for the tracer to pass through the kidneys can include the minimum, mean and maximum transit times, as well as the range of transit times. It is also possible to generate a transit time spectrum by differentiating "H"("t").
Finally, it should be noted that a background subtracted renogram is not strictly required in deconvolution analysis, and the raw renogram curve may be used directly. The reason becomes apparent by considering that the overall response function measured from the raw data arises from the sum of the kidney response function and the tissue response function, as illustrated in the figure below:
The renal transit times can still be obtained from the combined response function, and it can be seen that the uptake constant can be determined by .