# 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.
Deconvolution 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:

-   enters the kidneys via the renal arterial system, and
-   leaves the kidneys via the ureters.

The rate of entry of the tracer into the kidney is called the **input
rate**, `<span style="color:red;">`{=html}I(t)`</span>`{=html}, and
depends on:

-   the plasma concentration of the tracer in the renal artery,
    `<span style="color:red;">`{=html}P(t)`</span>`{=html},
-   the rate at which the tracer is extracted from the plasma by the
    kidneys.

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**
(`<span style="color:red;">`{=html}ERPF`</span>`{=html}), so that:

```{=html}
<div class="center">
```
`<span style="color:red;">`{=html}`<big>`{=html}I(t) = (ERPF)
P(t)`</big>`{=html}`</span>`{=html}

```{=html}
</div>
```
.

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**
(`<span style="color:red;">`{=html}GFR`</span>`{=html}), so that:

```{=html}
<div class="center">
```
`<span style="color:red;">`{=html}`<big>`{=html}I(t) = (GFR)
P(t)`</big>`{=html}`</span>`{=html}

```{=html}
</div>
```
.

The variation in plasma concentration with time,
`<span style="color:red;">`{=html}P(t)`</span>`{=html}, consists
initially of a spike whose amplitude and duration depend on:

-   the rate of injection,
-   the site of injection, as well as
-   the rate of mixing of the tracer in the blood.

Following this initial spike, the plasma concentration,
`<span style="color:red;">`{=html}P(t)`</span>`{=html}, decreases
bi-exponentially with time depending on:

-   the exchange rates with the extravascular space, and
-   the uptake of the tracer by the kidneys.

The plasma concentration,
`<span style="color:red;">`{=html}P(t)`</span>`{=html}, 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,
`<span style="color:red;">`{=html}I(t)`</span>`{=html}, 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, `<span style="color:red;">`{=html}H(t)`</span>`{=html}.

In practice, the response of the kidney to a non-ideal input rate is
measured. This real response,
`<span style="color:red;">`{=html}R(t)`</span>`{=html}, can be
considered to arise from a convolution of the actual input rate and the
impulse response of the kidney:

```{=html}
<div class="center">
```
`<span style="color:red;">`{=html}`<big>`{=html}R(t) = I(t) \*
H(t)`</big>`{=html}`</span>`{=html}.

```{=html}
</div>
```
The process of determining,
`<span style="color:red;">`{=html}H(t)`</span>`{=html}, on the basis of
measuring `<span style="color:red;">`{=html}R(t)`</span>`{=html} and
`<span style="color:red;">`{=html}I(t)`</span>`{=html} 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 convolution 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.

!Illustration of a gamma camera imaging a hypothetical
kidney.

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 impulse response, *H*, in the first three minutes following the
    impulse injection will be equal to 1, because all of the input
    quantity of the tracer remains in the field of view of the gamma
    camera.
-   In the 4th minute, the quantity of tracer will decrease because 25%
    of it will have passed through the shortest pathway and will have
    left the field of view, so that *H* will decrease to 0.75.
-   Similarly, in the 5th minute, it will decrease to 0.5.
-   And finally to 0.25 in the 6th minute - and zero in the 7th minute.

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:

!Illustration of the response to a non-ideal
input.

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:

  --------------------------------------------------------
  $R_1 = I_1 H_1 Dt\,\!$
  $R_2 = (I_1 H_2 + I_2 H_1) Dt\,\!$
  $R_3 = (I_1 H_3 + I_2 H_2 + I_3 H_1) Dt\,\!$
  $R_4 = (I_1 H_4 + I_2 H_3 + I_3 H_2 + I_4 H_1) Dt\,\!$
  etc. etc.
  --------------------------------------------------------

where *Dt* is the time interval used for each measurement of the
quantity of the tracer.

Therefore, in general:

```{=html}
<div class="center">
```
$R_i = \sum_{j=1}^i I_j H_{i-j+1} \Delta t$

```{=html}
</div>
```
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**:

```{=html}
<div class="center">
```
$R(t) = \int_{0}^t I(\tau)H(t-\tau)d\tau$

```{=html}
</div>
```
which is usually written in the form:

```{=html}
<div class="center">
```
$R(t) = I (t) \cdot H(t)\,\!$

```{=html}
</div>
```
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
linear system, 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 O^123^IH,
^99m^Tc-MAG3 or ^99m^Tc-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:

-   **Matrix Inversion** - where the convolution integral is expressed
    in matrix form as:

```{=html}
<div class="center">
```
$R = I \cdot H Dt\,\!$

```{=html}
</div>
```
which can be solved to give:

```{=html}
<div class="center">
```
$H = \frac{1}{\Delta t} I^{-1} \cdot R$

```{=html}
</div>
```
where *I*^-1^ is the inverse matrix of matrix *I*.

-   **Fourier Transformation** - where the Fourier
    transform (*FT*)
    of the convolution integral is calculated, i.e.

```{=html}
<div class="center">
```
$FT [R(t)] = FT [I(t)] \cdot FT [H(t)]\,\!$

```{=html}
</div>
```
*H*(*t*) can be determined from:

```{=html}
<div class="center">
```
$H(t) = IFT \left \lbrack \frac{FT\,[R(t)]}{FT\,[I(t)]} \right \rbrack$

```{=html}
</div>
```
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 impulse
response, *H*(*t*), obtained following
deconvolution has a form similar to that shown in the following figure:

!General form of the impulse response obtained using deconvolution
renography.

*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:

!Illustration of the addition of kidney and tissue response
functions.

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 extrapolation.

