# Basic Physics of Nuclear Medicine/Dynamic Studies in Nuclear Medicine

## Introduction

This is a developing chapter of a Wikibook entitled Basic Physics of
Nuclear Medicine.

The metabolism of a substance in the human
body is the result of a number of inter-related dynamic processes which
include the absorption, distribution, utilization, degradation and
excretion of the substance. The measurement of just one of these
parameters can give a result which is indicative of a disease, but may
not identify the actual cause of the disease. More detailed information
about the cause may be determined when knowledge of the complete
metabolic system is obtained. One method of gaining such knowledge is
through mathematical simulation of the physiological system. The
outcomes of this approach include generating a representation of the
entire system as well as an understanding of interactions between its
component parts. The approach typically involves:

1\. obtaining experimental data following stimulation of the system by
addition of a suitable tracer,\
2. comparing experimental data with data predicted by the mathematical
simulation, and\
3. varying parameters of the simulation until the two sets of data agree
as closely as possible using methods such as least
squares, maximum
likelihood and Monte Carlo
simulation.

The general assumptions for this approach are that:

-   the addition of the tracer does not perturb the system,
-   the tracee (i.e. the substance under investigation) is conserved
    throughout the process,
-   the tracer is conserved throughout the process -- allowing for
    radioactive decay, and
-   the system is in a steady state (i.e. the amount of tracee in each
    compartment of the system remains constant as does the exchange of
    tracee between each compartment).

There are two major types of mathematical model in use:

-   -   **Deterministic**: where analytical expressions are used to
        describe the exact behaviour of the tracer in each part of the
        system with time. The mathematical expressions used are usually
        exponential or power functions,

```{=html}
<!-- -->
```
-   -   **Stochastic**: where the behaviour of the system is determined
        by random processes which are described by probability
        functions.

Deterministic models are considered in some detail below.

## Compartmental Analysis

This form of deterministic analysis involves dividing the physiological
system into a number of interconnected compartments -- where a
compartment is defined as any **anatomical**, **physiological**,
**chemical** or **physical** subdivision of a system. A basic assumption
is that the tracer is uniformly distributed throughout a compartment.
The simplest of such systems to consider is the single compartment
model. We will start our treatment with this simple model and then
extend it to more complex ones -- the initial ones being considered
simply to develop the framework with the later ones providing direct
relevance to nuclear medicine dynamic studies; their acquisition and
analysis.

There is an ImageJ plug-in available, named
Compartments_TP,
which provides simulations of a number of additional models.

## Single Compartment Model

The flow of a tracer through a blood vessel following an ideal bolus
injection is shown in the following figure as an illustration of a
single compartment model. The compartment illustrated is closed except
for the inflow and outflow of the tracee, and the tracer is injected as
indicated. In these theoretical conditions, the tracer will mix
immediately and uniformly throughout the compartment following its
injection. And its quantity will reduce with time depending on the rate
of outflow. The variables used in the figure are:

```{=html}
<div class="center">
```
*q*: the quantity of tracer in the compartment at time, *t*, and\
*F*: the outflow.

```{=html}
</div>
```
!The single compartment
model

We can define the **fractional turnover**, *k*, as the ratio of these
two parameters, i.e.

```{=html}
<div class="center">
```
$k = \frac{-dq/dt}{q}$

```{=html}
</div>
```
which can be rewritten as:

```{=html}
<div class="center">
```
$\frac{dq}{dt} = -kq$

```{=html}
</div>
```
Without going into the mathematical details (which are similar to the
derivation of the radioactive decay
law!), the solution to this
equation is:

```{=html}
<div class="center">
```
$q = q_0\ exp(-kt)\,\!$

```{=html}
</div>
```
where *q~o~* is the quantity of tracer present at time, *t* = 0.

This equation is plotted below to illustrate the influence of the value
of the fractional turnover, *k*:

!Graphical illustration of the quantity of tracer,
`<span style="color:red;">`{=html}q`</span>`{=html} versus time for
relatively high and low values of the fractional turnover,
`<span style="color:red;">`{=html}k`</span>`{=html}.\<

The graph indicates that the quantity of tracer in the compartment will
decrease exponentially with time following injection at a rate dependent
on the outflow, as might be intuitively expected.

## Two Compartment Model -- Closed System

A more complex, and yet still relatively simple, set of models are those
based on two compartments. In a closed system the tracer simply moves
between the two compartments without any overall loss or gain -- see the
following figure:

!Closed two compartment
model

Therefore,

  --------------------------------------------- ----- ----------------------------------------------
  $\frac{dq_1}{dt} = k_{21} q_2 - k_{12} q_1$   and   $\frac{dq_2}{dt} = k_{12} q_1 - k_{21} q_2$.
  --------------------------------------------- ----- ----------------------------------------------

Since there is no loss of tracer from the system,

```{=html}
<div class="center">
```
$q_1 + q_2 = \text{constant} = q_0\,\!$

```{=html}
</div>
```
Therefore,

```{=html}
<div class="center">
```
$\frac{dq_1}{dt} = -\frac{dq_2}{dt}\,,$

```{=html}
</div>
```
indicating that as the quantity of tracer in Compartment #1 decreases,
the quantity in Compartment #2 increases, and vice versa. Now, consider
the situation illustrated in the figure above, where the tracer is
injected into Compartment #1 at time, *t* = 0. At this time,

  ----------------- ----- ------------------
  $q_1 = q_0\,\!$   and   $q_2 = 0\,,\,\!$
  ----------------- ----- ------------------

and, initially,

  --------------------------------- ----- --------------------------------
  $\frac{dq_1}{dt} = -k_{12} q_0$   and   $\frac{dq_2}{dt} = k_{12} q_0$
  --------------------------------- ----- --------------------------------

The solutions to these equations are:

```{=html}
<div class="center">
```
$q_1 = q_0 \left \lbrack 1 - \frac{k_{12}}{k_{12} + k_{21}} \left \lbrace 1 -\ \text{exp}\ - (k_{12} + k_{21}) t \right \rbrace \right \rbrack$

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

```{=html}
<div class="center">
```
$q_2 = q_0 \left \lbrack \frac{k_{12}}{k_{12} + k_{21}} \left \lbrace 1 -\ \text{exp}\ - (k_{12} + k_{21}) t \right \rbrace \right \rbrack$

```{=html}
</div>
```
and their behaviour in the special case when *k*~12~ = *k*~21~, and the
volume of the two compartments is the same, is illustrated below:

!Graphical illustration of the change in the quantity of tracer in
Compartments #1 and #2 versus
time.

Note that this model predicts that a steady state will be reached as the
quantity of tracer in Compartment #1 decreases exponentially and the
quantity in Compartment #2 increases exponentially, with the rate of
each change controlled by the sum of the turnover rates.\

## Two Compartment Model -- Open Catenary System

This is an extension of the single compartment model considered earlier
with two compartments connected in series, as shown in the following
figure:

!Open catenary two compartment
model.

In this model,

  --------------------------------- ----- ---------------------------------------------
  $\frac{dq_1}{dt} = -k_{12} q_1$   and   $\frac{dq_2}{dt} = k_{12} q_1 - k_{20} q_2$
  --------------------------------- ----- ---------------------------------------------

The solutions to these equations are:

```{=html}
<div class="center">
```
$q_1 = q_0\ \text{exp}(-k_{12}t)$

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

```{=html}
<div class="center">
```
$q_2 = q_0 \frac{k_{12}}{k_{12} - k_{20}}\lbrack \text{exp}(-k_{20}t) - \text{exp}(-k_{12}t)\rbrack\,,$

```{=html}
</div>
```
and the behaviour of *q*~1~ and *q*~2~ is shown in the figure below for
the special case of *k*~20~ being three times the value of *k*~12~:

!Graphical illustration of the quantity of tracer versus time in the
open catenary two compartment
model.

Note that the behaviour of *q*~2~ in this figure is similar to arterial
tracer flow following an intravenous injection, and to the cumulated
activity parameter
used in radiation dosimetry.

## Two Compartment Model -- Open Mamillary System

This model is equivalent to the closed two compartment system considered
above with the addition of an outflow from one compartment:

!Open mamillary two compartment
model.

In this case,

  -------------------------------------------------------- ----- -------------------------------------------
  $\frac{dq_1}{dt} = -k_{10}q_1 - k_{12}q_1 + k_{21}q_2$   and   $\frac{dq_2}{dt} = k_{12}q_1 - k_{21}q_2$
  -------------------------------------------------------- ----- -------------------------------------------

At *t* = 0:

  ----------------- ----- ---------------
  $q_1 = q_0\,\!$   and   $q_2 = 0\,\!$
  ----------------- ----- ---------------

and, initially

  ------------------------------------------- ----- -------------------------------
  $\frac{dq_1}{dt} = -(k_{10} + k_{12})q_1$   and   $\frac{dq_2}{dt} = k_{12}q_0$
  ------------------------------------------- ----- -------------------------------

The solutions to these equations are:

```{=html}
<div class="center">
```
$q_1 = q_0 \left \lbrack \frac{k_{21} - a_1}{a_2 - a_1} \text{exp}(-a_1t) + \frac{k_{21} - a_2}{a_1 - a_2}\text{exp}(-a_2t) \right \rbrack$

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

```{=html}
<div class="center">
```
$q_2 = \frac{q_0 k_{12}}{a_2 - a_1} \lbrack \text{exp}(-a_1t) - \text{exp}(-a_2t) \rbrack$

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

  ----------------------------- ----- --------------------------------------------
  $a_1a_2 = k_{10}k_{21}\,\!$   and   $a_1 + a_2 = k_{10} + k_{21} + k_{12}\,\!$
  ----------------------------- ----- --------------------------------------------

The behaviour of *q*~1~ and *q*~2~ is illustrated in the figure below:

!Graphical illustration of the quantity of tracer versus time in the
open mamillary two compartment
model.

This model has been widely adopted in the study of:

-   metabolism of plasma proteins, where
    Compartment #1 is the plasma and Compartment #2 is the extravascular
    space,
-   trapping of pertechnetate ion in the thyroid
    gland, where:
    -   Compartment #1: the plasma,
    -   Compartment #2: the thyroid gland,
    -   *k*~12~: clearance rate from plasma into the gland, and
    -   *k*~21~: leakage rate from the gland into the plasma.

## Models with Three Compartments

The open mamillary model above has been extended to study **iodine
uptake** using a third compartment which is fed by an irreversible flow,
*k*~23~, from Compartment #2:

!Thyroid iodine uptake
model.

where:

-   Compartment #1: the plasma,
-   Compartment #2: the trapping of inorganic iodide in the thyroid
    gland, and
-   Compartment #3: iodide within the gland which has become organically
    bound as part of hormone systhesis processes.

The open mamillary type of model has also been applied to renal
clearance with the system consisting of
an intravascular compartment, with an extravascular compartment
exchanging with it and connected irreversibly with a urine compartment:

!Renal clearance model.

The intravascular compartment (#1) in the figure above represents tracer
which is exchangeable with the renal parenchyma and the extravascular
space. The urine compartment (#2) represents tracer which has been
cleared by the kidneys and is therefore associated with the renal pelvis
and the bladder. The extravascular compartment (#3) represents the
tracer which has not been cleared, e.g. tracer which becomes bound to
other molecules or tracer in extrarenal tissues.

When the tracer is injected into the intravascular compartment via a
peripheral vein, the initial distribution will not be uniform throughout
the body -- but this non-uniformity will even out as the blood
circulates. For a highly vascular region, a plot of the quantity of
tracer versus time will show an initial sharp rise which will rapidly
fall off. The magnitude of this spike will vary with:

-   the anatomical region,
-   the site of the injection, and
-   the speed of the injection.

Compartmental analysis cannot therefore be applied to this phase of a
renogram since the basic assumption of uniform tracer distribution,
implicit in compartmental analysis, cannot be applied.

Following this phase, the quantity of tracer in the intravascular
compartment begins to fall because of:

-   uptake by the kidneys -- represented by *k*~12~ in the figure above,
-   diffusion into the extravascular space -- represented by *k*~13~.

As the quantity of tracer in the extravascular compartment builds up,
exchange in the opposite direction begins to occur (represented by
*k*~31~), and so a maximum is reached before its quantity of tracer
falls off. This is illustrated in the figure below for a situation
where:

  --------------------------- --------------------------- ---------------------------
  *k*~12~ = 0.05 per minute   *k*~13~ = 0.04 per minute   *k*~31~ = 0.06 per minute
  *l*~1~ = 0.13 per minute    *l*~2~ = 0.024 per minute
  *A*~1~ = 0.65               *A*~2~ = 0.35
  --------------------------- --------------------------- ---------------------------

!Predictions of the renal clearance
model.

Ultimately, all the tracer will end up in the urine compartment.

The equations used for the figure above are:

```{=html}
<div class="center">
```
$q_1 = A_1\ \text{exp}(-l_1t) + A_2\ \text{exp}(-l_2t)\,\!$

```{=html}
</div>
```
```{=html}
<div class="center">
```
$q_2 = A_3\ \text{exp}(-l_1t) - A_4\ \text{exp}(-l_2t)\,\!$

```{=html}
</div>
```
```{=html}
<div class="center">
```
$q_3 = A_5\ \lbrack \text{exp}(-l_1t) - A_2\ \text{exp}(-l_2t)\rbrack\,\!$

```{=html}
</div>
```
where *l*~1~ and *l*~2~ are constants related to the fractional
turnovers, and *A*~1~ through *A*~5~ are also constants such that:

  --------------------- ----- ---------------------
  $A_1 + A_2 = 1\,\!$   and   $A_3 + A_4 = 1\,\!$
  --------------------- ----- ---------------------

In practice, the renal clearance can be obtained by monitoring the
quantity of tracer in the intravascular compartment, e.g. the blood
plasma concentration, *P*, where:

```{=html}
<div class="center">
```
$P = \frac{\text{Quantity of Tracer in Intravascular Space}}{\text{Volume of Intravascular Space}}$

```{=html}
</div>
```
The time dependence of this plasma concentration will vary in the same
way as *q*~1~, so that:

```{=html}
<div class="center">
```
$P(t) = C_1\ \text{exp}(-l_1t) + C_2\ \text{exp}(l_2t)\,\!$

```{=html}
</div>
```
where *C*~1~ and *C*~2~ are related to *A*~1~ and *A*~2~, respectively.
The renal clearance, which is related to *k*~12~, can therefore be
determined by characterizing the biexponential fall off in the quantity
of tracer in the intravascular compartment.

## Glomerular Filtration Rate

The Glomerular Filtration Rate
(GFR) is generally regarded
as one of the most important single indicator of renal function. It is
particularly important in assessing the presence and severity of kidney
failure.

There are three major methods of determining a patient\'s GFR:

-   Inulin clearance,
-   Creatinine clearance,
-   Radiotracer clearance.

Inulin clearance has been used for many years and
is often regarded as the most reliable and accurate of the three
methods. Its major disadvantages however include the need for continuous
intravenous infusion, timed urine collections via a bladder catheter and
protracted chemical analysis. Creatinine
clearance has been widely used for
routine GFR assessment as a result. However, while this method gives
similar results as inulin clearance under normal conditions, the
validity of its results is questionable in patients who have moderate to
advanced renal failure because of an increasing significance of tubular
secretion.

The third method, radiotracer clearance has been widely adopted using
^51^Cr-EDTA. This tracer is known to be physiologically inert, not bound
to plasma proteins and not metabolized by erythrocytes or organs other
than the kidneys. It is normally excreted within 24 hours of injection,
98% via the kidneys. ^51^Cr has a half-life of about 28 days and decays
by 100% electron capture into stable vanadium, emitting monoenergetic
(320 keV) gamma-rays in about 10% of the transformations. In addition,
^51^Cr-EDTA determination of GFR can be used in conjunction with OIH
renal plasma flow assessment for the differential diagnosis of various
renal conditions.

The typical radioactivity administered for ^51^Cr-EDTA clearance is 1-10
MBq and the radiopharmaceutical is generally administered via
intravenous injection. This *Single Shot* technique assesses the GFR
through venous blood sampling, in the simplest case, or by continuous
external monitoring of the gamma-rays from ^51^Cr in the more
sophisticated approach. When the patient counts are plotted against time
on a log/linear axis, a curve is generated which falls off rapidly at
first and thereafter decreases at a constant rate, representing the
behaviors of *q*~1~ is our last figure. This initial fall-off arises as
a result of the establishment of an equilibrium between the radiotracer
and the extravascular, extracellular fluids. The slower second phase
reflects renal excretion and contains the information necessary for GFR
assessment.

!The plasma clearance of ^51^Cr-EDTA predicted using the three
compartment model discussed
above.

A quick and simple technique is to obtain two blood samples from the
patient, one at two hours and the other at four hours post injection.
The counts per unit volume in the plasma of each sample are determined
using a scintillation counter and compared with the counts from a
*standard* solution. The standard solution is made by diluting an
injection, identical to the patient\'s, in a known volume of water, e.g.
1 liter.

The slope, m, of the second portion of the above curve can be determined
from:

```{=html}
<div class="center">
```
$m = \frac{\ln b_1 - \ln b_2}{t_2 - t_1}$

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

-   *t*~1~: time from injection for the first blood sample, usually 120
    minutes,
-   *b*~1~: counts per
    milliliter
    (mL) in the plasma from the first sample (corrected for background
    counts),
-   *t*~2~: time from injection for the second blood sample, usually 240
    minutes,
-   *b*~2~: counts/mL in the plasma from the second sample (also
    corrected for background).

We can now extrapolate this straight line back to the time of injection,
*t*~0~, to determine what the plasma counts would be upon instantaneous
mixing of the tracer throughout the patient\'s plasma compartment, i.e.

```{=html}
<div class="center">
```
$\ln b_0 = \ln b_1 + m(t_1 - t_0)\,\!$

```{=html}
</div>
```
as illustrated in the following figure:

!Counts for the two plasma samples are fit to a straight line which is
back-extrapolated to the time of injection (dashed line) to determine
the logarithm of b~0~. The plasma clearance, q~1~, predicted using the
three compartment model is shown
shaded. to determine the logarithm of b0. The plasma clearance, q1, predicted using the three compartment model is shown shaded."){width="512"}

Therefore, we can write:

```{=html}
<div class="center">
```
$b_0 = \text{exp}\ \big(\ln b_1 + m(t_1 - t_0)\big)$

```{=html}
</div>
```
The **Dilution Principle** can now be used to determine the volume of
this plasma compartment by comparing the plasma counts with those from
the standard solution, i.e.

```{=html}
<div class="center">
```
$V = \frac{S \cdot V_s}{b_0}$

```{=html}
</div>
```
which results in

```{=html}
<div class="center">
```
$V = \frac{S \cdot 1000}{b_0}$

```{=html}
</div>
```
when the standard injection is diluted in 1 liter. The clearance (in
ml/min) is then given by the following equation:

```{=html}
<div class="center">
```
$\text{Clearance} = V \cdot m\,\!$

```{=html}
</div>
```
Results for two patients are shown below to illustrate this technique.

**Patient A**

  Sample               Counts/mL
  -------------------- -----------
  Background           477
  *b*~1~ at 119 mins   11,438
  *b*~2~ at 238 mins   6,235
  Standard, *S*        150,020

This patient\'s ^51^Cr-EDTA clearance was determined to be 38.8 mL/min.
This result was assessed to be indicative of chronic renal failure,
which was later found to be due to lupus
nephritis. The patient was then placed on
steroid therapy.

Two months later the patient was re-tested and the clearance was found
to have risen to 52.7 mL/min. For the patient\'s age, this clearance was
gauged as within the normal range indicating that the therapy was having
a positive effect. The therapy was then ceased. Two months further, the
patient was again tested having been without steroid therapy for this
period. The result was 54.2 mL/min reflecting successful treatment.

**Patient B**

  Sample               Counts/mL
  -------------------- -----------
  Background           425
  *b*~1~ at 122 mins   3,103
  *b*~2~ at 250 mins   1,390
  Standard, *S*        104,600

This patient had a high blood pressure and a renal involvement required
confirmation. The clearance however was 117.3 mL/min, which is well
within the normal range. The kidneys were therefore excluded from the
investigation of this patient\'s condition.

------------------------------------------------------------------------

Note that the number of blood samples is not limited to two, with some
methods requiring three, four or more samples, and other methods using
external monitoring of the clearance. Each method is nevertheless based
on the form of analysis outlined above where the rate constant of the
second phase of the clearance curve is determined along with the volume
of distribution of the radiotracer. The timing of blood sampling is
therefore after the first phase has finished, i.e. more than about two
hours following injection, with the volume of distribution determined
using a single-exponential fit to this later phase.

------------------------------------------------------------------------

It is important to appreciate that the clearance of ^51^Cr-EDTA
determined as described does not equate directly with the Glomerular
Filtration Rate (GFR) since the method assumes a single exponential
dependence. ^51^Cr-EDTA clearance results are therefore typically
corrected by a factor, either empirically- or theoretically-derived, to
force them to express the true GFR. Empirically-derived corrections
include those of:

-   Chantler (1969), where
    the ^51^Cr-EDTA clearance is multiplied by a factor of 0.8;
-   Brien (1969), where
    clearances above 50 mL/min are multiplied by 0.82 and added to 6;
-   Brochner-Mortensen
    (1972), where a
    second-order polynomial is applied

to obtain the GFR. A correction based on a theoretical consideration of
the relationship between true GFR and single-exponential clearance
values based on compartmental analysis has been introduced (Fleming,
2007) which gives improved
corrections, especially at high GFRs. This correction is of the form:

```{=html}
<div class="center">
```
$GFR = \frac{\text{Clearance}}{(1 + f \cdot \text{Clearance})}$

```{=html}
</div>
```
where $f$ = 0.0017 min/mL.

As a final step, corrected clearance measurements are generally
standardized to the body surface area (BSA) of the Standard
Man, i.e. 1.73 m^2^. This is typically done
using estimates of the BSA based on the patient\'s height and weight --
as derived from DuBois
(1916) or Haycock
(1978), for instance. A
single-exponential correction technique, based on BSA-scaled clearances
has also been introduced (Jodal & Brochner-Mortensen,
2008) which is similar to
that of Fleming (2007) but provides improved correction in paediatric
studies.

## Renography

It should be apparent from the discussion above that the urine
compartment (#2) consists of the quantity of the tracer in the urine,
without distinguishing whether the urine is in the renal
pelvis, the ureters
or the bladder. These anatomical spaces
can be incorporated by extending the three compartment mamillary model
to five compartments:

!Compartmental analysis applied to
renography.

Note that the passage of the tracer through the renal parenchyma can be
characterized by a transit time, *t*~0~, and that *k*~56~ is related to
the rate of urine production.

The solutions to the resultant differential equations for the quantity
of tracer in the renal parenchyma, the renal pelvis and the bladder
incorporate consideration of the time delay, *t*~0~, so that:

-   When *t* \< *t*~0~:

  -------------------------------------------------------------------------------
  ${\color{Blue}q_4 = 1 - A_3\ \text{exp}(-l_1t) - A_4\ \text{exp}(-l_2t)}\,\!$
  ${\color{Red}q_5 = 0}\,\!$
  ${\color{OliveGreen}q_6 = 0}\,\!$
  -------------------------------------------------------------------------------

-   When *t* \> *t*~0~:

  --------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
  ${\color{Blue}q_4 = A_3[1 - \text{exp}(-l_1t_0)] \text{exp}(-l_1\lbrace t-t_0\rbrace) + A_4 [1-\text{exp}(-l_2t_0)]\text{exp}(-l_2\lbrace t-t_0\rbrace)}\,\!$
  ${\color{Red}q_5 = A_7\ \text{exp}(-l_1\lbrace t-t_0 \rbrace) + A_8\ \text{exp}(-l_2\lbrace t-t_0\rbrace) - A_9\ \text{exp}(-l_3\lbrace t-t_0\rbrace)}\,\!$
  ${\color{OliveGreen}q_6 = 1 - A_{10}\ \text{exp}(-l_1\lbrace t-t_0\rbrace) - A_{11}\ \text{exp}(-l_2\lbrace t-t_0\rbrace) + A_{12}\ \text{exp}(-l_3\lbrace t-t_0\rbrace)}\,\!$
  --------------------------------------------------------------------------------------------------------------------------------------------------------------------------------

where *l*~3~ is related to *k*~56~. The time course of the quantity of
tracer in each compartment is shown below:

!The parenchymal (*q*~4~), renal pelvis (*q*~5~) and bladder (*q*~6~)
curves, generated using *t*~0~ = 2 minutes and *k*~56~ = 1 per
minute., renal pelvis (q5) and bladder (q6) curves, generated using t0 = 2 minutes and k56 = 1 per minute.")

The quantity of tracer in the overall kidney can be obtained by summing
the renal parenchyma and renal pelvis curves, so that:

```{=html}
<div class="center">
```
$q_{\text{kidney}} = q_4 + q_5\,\!$

```{=html}
</div>
```
as shown below:

!The outcome of summing the renal pelvis and parenchyma
curves.

What is recorded in a renogram in practice is not just this kidney
curve, but also the quantity of tracer in:

-   overlapping and underlying tissues, and in
-   the intravascular space of the kidney itself.

These contributions add a **background** upon which the true renogram is
superimposed. The quantity of tracer in this background varies with
time, but not in the same way as the true renal curve. The time course
of this background is likely to behave in a manner similar to the sum of
the intravascular (*q*~1~) and extravascular (*q*~3~) curves derived
earlier using this five compartment model.

The following equation can be derived on this basis:

```{=html}
<div class="center">
```
$q_{\text{Bgd}} = b_1 q_1 + b_3 q_3\,\!$

```{=html}
</div>
```
where *b*~1~ and *b*~3~ represent the contributions to the detected
renogram curve from the tracer in the intravascular and extravascular
spaces, respectively. For example, the curves below were generated using
*b*~1~ = 0.05 and *b*~3~ = 0.02 and

```{=html}
<div class="center">
```
$q_{\text{Renogram}} = 0.5\,q_{\text{Kidney}} + q_{\text{Bgd}}\,\!$

```{=html}
</div>
```
!Renogram and background curves typical of those acquired in
practice.

In practice, this background curve should be subtracted from the raw
renogram data to obtain a curve which reflects the true quantity of
tracer in the kidney (see the previous figure). This process is
sometimes referred to as **blood background subtraction** -- although
you should now be able to appreciate that this is a bit of a misnomer!

The uncorrected and corrected curves are shown below to assist with
direct comparison:

!Renogram curves pre- and post-background
correction.

and an example from a patient\'s ^99m^Tc-DTPA renogram is shown in the
following figure, to assist you in comparing them with the predictions
from compartmental analysis:

  -------------------------------- --------------------------------
  ![](NM14_58.jpg "NM14_58.jpg")   ![](NM14_59.gif "NM14_59.gif")
  -------------------------------- --------------------------------

A final figure illustrates a form of analysis that can be used in
^99m^Tc-MAG3 renography in a patient with obstructive uropathy:

!Analysis of a MAG3
renogram{width="512"}

## Background Subtraction in Renography

In practice, the background activity in a renogram must be taken into
account when interpreting a renogram. This is generally achieved by
estimating the background activity and subtracting it from the raw
renogram data. The question is: how can this background activity be
measured?

One method has been based on recording the activity at
nephrectomy sites in patients whose
remaining kidney is being examined. However, it should be noted that
removal of a kidney also removes an intravascular source of the
background activity. As a result nephrectomy sites commonly appear
colder than the extra-renal tissues in renogram images.

A potentially better method is to record the activity in the region of a
non-functioning kidney.

In most patients, however, a non-renal region must be used for
background estimation. Ideally, the choice of region should reflect the
same intra- and extravascular background as the kidney itself. There
appears to be no standardization in this area, with practices including
the use of a region between the kidneys, above the kidneys, over the
heart and below each kidney.

A comparison of such methods is provided in Taylor et al (1997) [^1] and
results of a similar, but more basic, analysis are summarised below:

```{=html}
<div id="bgdPlacement">
```
!centre\|Bgd RoI placement

```{=html}
</div>
```
!centre !centre

These renographic images were generated using Monte Carlo simulation[^2]
and analysed using OsiriX. Split function was
calculated using the Integral of counts in the 90-150 second time
period. The Mean Absolute % Error was computed from a series of such
simulated images where the left kidney to right kidney split function
varied from 100%:0%, through 50%:50% to 0%:100% in steps of 5
percent[^3].

It is seen that the Lateral RoIs yielded the lowest % Error and that the
Central RoI, which was used in the early days of gamma-camera renography
using ^99m^Tc-DTPA, yielded a substantially greater error of \~12%. This
is in line with current clinical practice where central RoIs are not
longer used and lateral or oblique, or perirenal, RoIs are preferred.

In comparison, no background correction generated an error of 20.7%,
which highlights the overall importance of background correction.

The variation of background activity is illustrated below for a real
LK:RK split function ratio of 50%:50% and for the extremes of 100%:0%
and 0%:100%. !centre\|frame\|567x567px\|6th degree polynomial fits to
the Lateral background
data
!Monte Carlo simulated AP renogram smoothed in time using a 32-frame
average. Raw data courtesy of
dynamicrenalstudy.org
The first point to note is that the background curves vary with time in
a manner which reflects the renal curves themselves, quickly rising
during the uptake phase and falling slowly during the excretion phase.
Secondly, note that the right kidney (RK) curves decrease more slowly
that those of the left kidney (LK) -- possibly as a result of retention
of the MAG3 tracer in the liver. This is quite different to the
assumption used in the earlier analytical
model. The extent of liver involvement can be gauged from the Monte
Carlo simulated AP renogram[^4] on the left where a reasonable view of
the liver and its dynamics can be observed throughout the study. This is
consistent with the pharmacokinetic model which was used to generate the
virtual study[^5].

Thirdly, note that the RK curves are of greater magnitude than those of
the LK at 50%:50% and 0%:100%, while the opposite is the case at
100%:0%. This highlights the requirement to use separate background RoIs
for each kidney.

The final point to note is in relation to the 90-150 second time window,
where the Integral or Slope method is
typically applied. It can be seen that both the sum of background counts
(the **Integral**) and the rate of change in these counts (the
**Slope**) vary during this period, which indicates that both methods
will be influenced by these features of the background curves.

Once the background region is selected and the activity/time curves are
generated, the background curve should be scaled by a factor dependent
on the relative areas of the background and renal regions, prior to
subtraction from the raw renogram curve. In addition, note that some
practices also involve further scaling of the background curve depending
on the kidney location. Finally, more sophisticated methods of
background correction have been developed and include:

-   the generation of interpolated background regions from samples of
    the background around the kidney,
-   the estimation of background correction factors using extrapolation
    techniques, and
-   deconvolution
    analysis.

It should be noted that caution should be used in translating these
results directly to real patient studies in that the raw data is derived
from a two-dimensional representation of a compartmental model.

## Relative Renal Function

The relative function of a patient\'s kidney is generally defined as
that kidney\'s renal clearance rate expressed as a percentage of the
patient\'s overall renal clearance rate, i.e.

  ---------------------------------------------------------------------------------------- ----- ----------------------------------------------------------------------------------------
  $\text{LK Relative Fn.} = \frac{\text{LK Clearance}}{\text{LK} + \text{RK Clearance}}$   and   $\text{RK Relative Fn.} = \frac{\text{RK Clearance}}{\text{LK} + \text{RK Clearance}}$
  ---------------------------------------------------------------------------------------- ----- ----------------------------------------------------------------------------------------

where LK and RK refer to the left and right kidneys, respectively.

Suppose that:

-   *N*~Kidney~(t): background corrected renal count rate, and

```{=html}
<!-- -->
```
-   *N*~Bgd~(t): count rate from an intravascular region of interest.

It should be apparent at this stage that:

  -------------------------------------------- ----- --------------------------------------
  $N_{\text{Kidney}}(t)\ \alpha\ q_4(t)\,\!$   and   $N_{\text{Bgd}}\ \alpha\ q_1(t)\,\!$
  -------------------------------------------- ----- --------------------------------------

We can therefore conclude that in the initial phase of the renogram,
i.e. when *t* \< *t*~0~:

```{=html}
<div class="center">
```
$N_{\text{Kidney}}(t) = UC \int_0^t N_{\text{Bgd}}(t) \cdot dt$

```{=html}
</div>
```
where *UC* in the kidney uptake constant. This constant is related to
that kidney\'s clearance rate, and we can therefore write:

```{=html}
<div class="center">
```
$\text{LK Relative Fn.} = \frac{\text{LK Update Constant}}{\text{LK} + \text{RK Uptake Constant}}$

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

```{=html}
<div class="center">
```
$\text{RK Relative Fn.} = \frac{\text{RK Uptake Constant}}{\text{LK} + \text{RK Uptake Constant}}$

```{=html}
</div>
```
However, we have already seen above that the background corrected renal
count rate is directly related to the uptake constant and we can
therefore conclude that:

  ---------------------------------------------------------------------------------- ----- ----------------------------------------------------------------------------------
  $\text{LK Relative Fn.} = \frac{\text{LK Counts}}{\text{LK} + \text{RK Counts}}$   and   $\text{RK Relative Fn.} = \frac{\text{RK Counts}}{\text{LK} + \text{RK Counts}}$
  ---------------------------------------------------------------------------------- ----- ----------------------------------------------------------------------------------

Note that this analysis indicates that relative renal function can be
determined from measurement of the relative counts in each kidney
following the initial vascular spike but prior to the commencement of
the excretion phase.

## Kidney Depth Correction

Given that two kidneys can be at different depths within a patient, a
correction for the difference in photon attenuation is generally
applied. The conventional way of doing this is to acquire lateral views
following a renogram so as to measure the depth of each kidney and to
apply an exponential correction based on an assumed linear attenuation
coefficient for the gamma ray energy (e.g. 0.153 cm^-1^ for ^99m^Tc
gamma rays).

The situation is illustrated by the following study where the right
kidney is slightly over 1 cm deeper than the left. The study is actually
of images generated by Monte Carlo simulation of a digital phantom
(XCAT) which includes the effects of photon attenuation, scattering,
limited spatial resolution and noise[^6]. The left kidney is positioned
7.31 cm deep while the right is at a depth of 8.64 cm, and the split
function is 50:50.

  ----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- ------------------------------------ ---------------------------------------------------------------------------------------------------------------------------------
  \\<File:nm14> 301e.gif\|thumb\|center\|A virtual renogram acquired at 10 seconds per frame for 120 frames. Raw data courtesy of [dynamicrenalstudy.org\|alt=A virtual renogram.   ![   !Regions of interest used for renogram analysis.{width="192"}
  ----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- ------------------------------------ ---------------------------------------------------------------------------------------------------------------------------------

Note that the two curves do not overlap, as would be expected for a
50:50 split function, as a result of the difference in attenuation. The
split function estimated by the integral of counts between 90 and 150
seconds is 54:46. When this attenuation difference is corrected,
however, the following curves are obtained and the split function
estimate is now a more realistic 49:51.

!A virtual renogram, attenuation corrected.\|alt=A virtual renogram,
attenuation
corrected.

Commercial renal analysis software typically provide options for the
automatic computation of kidney depth based on empirically derived
relationships between the patient\'s height and weight and the depth of
each kidney. One such package provides the following options (with
weight expressed in kg and height in cm):

+--------------+--------------------------+--------------------------+
| Method       | Left Kidney Depth (cm)   | Right Kidney Depth (cm)  |
+==============+==========================+==========================+
| S            | 1                        | 1                        |
| tandard[^11] | 3.2\*(Weight/Height)+0.7 | 3.3\*(Weight/Height)+0.7 |
+--------------+--------------------------+--------------------------+
| Emory[^12]   | 16.17\*(Weight/Height    | 15.13\*(Weight/Height)   |
|              | )+0.027\*Age(years)-0.94 | +0.022\*Age(years)+0.077 |
|              |                          |                          |
|              |                          | `<small>`{=ht            |
|              |                          | ml}`<small>`{=html}Note: |
|              |                          | contains a typographical |
|              |                          | error[^13]`</small>`     |
|              |                          | {=html}`</small>`{=html} |
+--------------+--------------------------+--------------------------+
| Itoh[^14]    | 14.0285\                 | 13.6361\                 |
|              | *(Weight/Height)^0.7554^ | *(Weight/Height)^0.6996^ |
+--------------+--------------------------+--------------------------+
| T.Itoh       | 17.                      | 16.                      |
|              | 05\*(Weight/Height)+0.13 | 55\*(Weight/Height)+0.66 |
+--------------+--------------------------+--------------------------+

A graphical representation of the difference in kidney depth generated
by these equations for a broad range of weight-to-height ratios is shown
in the figure below (note: an age of 50 years was assumed for the Emory
formulae):

!Difference in kidney depths (left kidney depth minus right kidney
depth).\|alt=Difference in kidney
depths..|alt=Difference in kidney depths."){width="460"}

It can be seen that little change in depth is predicted by the Standard
formulae and that derived attenuation corrections are likely to have
little effect on relative function estimates. It can also be seen that
the T.Itoh formulae indicate that the right kidney is always deeper than
the left. The final two sets of formulae indicate that the right kidney
is deeper for weight-to-height ratios of less than 0.6 to 0.7 and the
left kidney is deeper above this ratio range. The overall range of depth
differences predicted by these latter three formulae is seen to be no
more than ±5 mm. Such differences yield an anticipated attenuation
correction factors in the range 0.95 to 1.05.

The predictions of these formulae should be contrasted with the results
of studies of kidney depth which have shown that differences of greater
than 5 mm can be expected in about 60% of patients, and greater than
1 cm in about a third of patients[^15]. It can therefore be inferred
that the above formulae will only provide sufficient attenuation
correction in less than half of patients, and that attenuation
correction factors as low as 0.6 can occasionally be expected.

It is therefore apparent that accurate kidney depth measurement is
imperative for the proper interpretation of all renograms, and that
automatic kidney depth formulae should be applied with caution.

## Semi-Automatic Methods of ^99m^Tc-MAG3 Renogram Analysis

Commercial software for renogram analysis generally provides a number of
methods for what is called semi-automatic analysis -- where some
features of the analysis require user interaction (e.g. kidney
identification) and others are performed automatically (e.g.
computations) to determine split renal function and Effective Renal
Plasma Flow (ERPF). Here the
software might request the user to, for instance:

-   input the height and weight of the patient -- so that body surface
    area and kidney depths can be computed using empirically-derived
    formulae, and
-   define regions of interest (RoIs) around each kidney -- with
    background regions automatically placed
    adjacent to these kidney RoIs.

Counts/time curves are generated for each of these four RoIs. The counts
in the background RoIs are then scaled automatically to those of
background areas of the same size as the respective kidney RoI and the
scaled background curves are subtracted from the kidney curves to
generate a net renogram curve for each kidney.

The counts in a pre-defined time interval in the uptake phase of each
renogram (e.g. 1 to 2 minutes post-injection) are then used to compute:

-   the sum of counts (called the **Integral**) and/or
-   the rate of increase in counts (called the **Slope**)

for each kidney.

A comparison of these two methods was performed using the Lateral RoIs
for background subtraction of the renographic series used
earlier, and the 90 to 150 second period for
analysis. Both methods yielded very similar results:

  ---------------------------------------------------------------------------- ---------------------------------------------------------------
  ![](IntegralVsSlope1.png "IntegralVsSlope1.png"){width="330" height="330"}   !centre\|337x337px
  ---------------------------------------------------------------------------- ---------------------------------------------------------------

The Integral method, however, generated a mean absolute % error of just
2.6%, while the Slope method generated a substantially greater error of
9.4% - mainly due to inaccuracies at the lower and higher LK/RK ratios.

In comparison, no background correction generated an error of 20.7%
using the Integral method while with the Slope method the error was just
9.2%. This highlights, once again, the importance of such corrections.

The Integral can be corrected for tissue attenuation using the computed
depth of each kidney and an assumed linear attenuation coefficient for
^99m^Tc gamma rays. Computation of the Integral in this manner is a
major source of error in this form of analysis since the use of the
patient's height and weight (along with their age in one method) to
determine kidney depth is likely to be inaccurate -- see the previous
section.
In addition, no standard exists for the value of the linear attenuation
coefficient (μ) in the renographic scanning situation, with values such
as 0.12[^16], 0.14[^17] and 0.153[^18] per cm in clinical use. For a
kidney depth of 6 cm, for instance, these values of μ would result in
correction factors of between 2 and 2.5 being applied, i.e. to give a
variation of \~20%.

**Split Function** can be determined from the percentage contribution of
each kidney's Attenuation-Corrected (AC) Integral to the sum of the two
kidneys' AC Integrals.

In addition, **Kidney Uptake** can be determined by expressing the AC
Integrals as a percentage of the injected dose.

**Effective Renal Plasma Flow** (ERPF) may then be calculated on the
basis of empirically-derived formulae such as:

```{=html}
<div class="center">
```
Left Kidney ERPF (mL/min) = -21.579 + (11.621\*Left Kidney Uptake) [^19]

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

```{=html}
<div class="center">
```
ERPF Value (mL/min) = 14.98 \* (Left Kidney Uptake + Right Kidney
Uptake) + 8.28 [^20]

```{=html}
</div>
```
Alternatively, a Washout Coefficient can be defined as:

```{=html}
<div class="center">
```
0.48026\*(exp(1.35315\*Sum of Kidney Uptakes)/100)-1

```{=html}
</div>
```
to be multiplied by an estimate of the Plasma Volume, such as:

```{=html}
<div class="center">
```
Body Surface Area \* 1,000 + 0.608 mL

```{=html}
</div>
```
to give the Total MAG3 Clearance. This final factor is used to calculate
the Total ERPF (mL/min) using the formula:

```{=html}
<div class="center">
```
1.25821\* (Total MAG3 Clearance) + 18.34451 [^21]

```{=html}
</div>
```
A figure of 0.53 has also been used as an ERPF Conversion Factor[^22],
i.e. to convert from ERPF to MAG3 Clearance. However, it should be noted
that values of this Factor between 0.61 and 0.9 have been reported
[^23].

As an alternative to these Integral-based methods, the slopes of the
curves for each kidney during the uptake phase of the renogram have been
used to define a Slope Index for each kidney, based on the Slope of each
renogram, attenuation-corrected for kidney depth, and expressed as a
percentage of the injected dose multiplied by 10^6^! Split Function can
then presumably be obtained by relating the Slope Index for each kidney
to the sum of Slope Indices -- and the Total MAG3 Clearance calculated
using the formula:

```{=html}
<div class="center">
```
(12.725 \* Sum of Kidney Slope Indices + 6.24)/Body Surface Area

```{=html}
</div>
```
**Body Surface Area** (BSA) estimation
is another source of variation in these computations. It is widely used
to reference a patient's ERPF to a standard BSA, e.g. 1.73 m^2^. It is
also used within calculations using formulae such as the one above. At
least two BSA computation methods are used in this field, as detailed
below:

```{=html}
<div class="center">
```
Du Bois Formula: BSA = 0.007184 \* W^0.425^ \* H^0.725^ [^24]

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

```{=html}
<div class="center">
```
Haycock Formula: BSA = 0.024265 \* W^0.5378^ \* H^0.3964^ [^25]

```{=html}
</div>
```
where W is the patient's weight in kg and H is their height is cm, to
give the BSA in m^2^. These formulae give reasonably similar BSA
estimates for adults, e.g. a difference of \~2%, for a 69 kg, 163 cm
patient, and their choice as a result is reasonably inconsequential in
this application, irrespective of their accuracy.

In summary, commercial semi-automatic renal analysis software is
generally based on one of the following two approaches:

-   determine the absolute uptake of MAG3 for each kidney and relate
    these uptakes to the ERPF using empirical relationships derived from
    limited patient studies (36 presumably-Japanese patients in the case
    of Itoh et al.[^26] for instance);
-   determine the rate of MAG3 uptake for each kidney and relate these
    rates to MAG3 clearances empirically-derived from limited patient
    studies (12 presumbly-Japanese patients in the case of Oriuchi et
    al.[^27]).

An evaluation of the accuracy of these methods is warranted given the
strong empirical features which are evident and the very limited number
of patients surveyed. Such an evaluation can be conducted using Monte
Carlo simulated renograms [^28] and results of such are shown in the
following table:

!A comparison of estimates of MAG3 Clearance and Split Function using
five different analysis methods on two different commercial software
packages from six virtual renograms.\|alt=A comparison of estimates of
MAG3 Clearance and Split Function using five different analysis methods
on two different commercial software packages from six virtual
renograms.

For this evaluation, the formulae for the Itoh ERPF (MAG3), Inoue (MAG3
Without Samples) and Oriuchi Clearance (MAG3) methods [^29] were used to
calculate the MAG3 Clearance and Split Function for each renogram using
spreadsheet software. The values of these parameters for the ERPF
Modified Schlegel and ERPF Modified Gates methods were calculated using
the manufacturer\'s software package directly[^30] and assumed an ERPF
Conversion Factor of 0.76 (i.e. the mean value of Oriuchi's study [^31])
-- as did the Itoh[^32] method.

It is apparent that no correlation in any of the results exists except,
for instance, that when the MAG3 Clearance is halved from 260 to 130
mL/min, the computed clearances also reduce -- although the results for
the Modified Schlegel method are quite reasonable in some situations. A
second point is that the variation in split function estimates is likely
to result in part from the use of (potentially
inaccurate)
automatic methods of kidney depth determination.

So\.... should these automatic computations be used clinically without,
for instance, a confirming set of kidney depth measurements from lateral
scans, post-renogram, or from an ultrasound scan? This question was
answered in 1985\<ref name=\"gruenewald /\> but unfortunately has not
yet seemed to infiltrate the psyche of producers of modern renal
analysis software.

Another question: should conjugate imaging be considered for renogram
analysis in commercial dual-headed gamma camera software packages?

## Conjugate Imaging in Renography

The principle of conjugate imaging is based on computing the Geometric
Mean of activity measurements from two opposing detectors and is
illustrated in the following diagram where a homogeneous medium (yellow
rectangle) containing a point source of radioactivity (red dot). The
thickness of the medium is d and the point source is a distance t~1~
from Detector #1 and a distance t~2~ from Detector #2. !Conjugate
counting.\|thumb{width="311"
height="311"}

The activity detected by Detector #1, assuming no scatter is detected,
is given by:

```{=html}
<div class="center">
```
$A_1 = A \exp(-\mu*t_1)$

```{=html}
</div>
```
while that detected by Detector #2 is given by:

```{=html}
<div class="center">
```
$A_2 = A \exp(-\mu*t_2)$.

```{=html}
</div>
```
The Geometric Mean, GM, of the detected activities is given by:

```{=html}
<div class="center">
```
$GM = \sqrt{A_1*A_2}$

```{=html}
</div>
```
so that:

```{=html}
<div class="center">
```
$GM = \sqrt{A\exp(-\mu*t_1)*A\exp(-\mu*t_2)}$

```{=html}
</div>
```
and, therefore:

```{=html}
<div class="center">
```
$GM = \sqrt{A\exp(-\mu*t_1)*A\exp(-\mu*(d-t_1))}$

```{=html}
</div>
```
and, as a result:

```{=html}
<div class="center">
```
$GM = \sqrt{A \exp(-\mu*d)}$

```{=html}
</div>
```
This final equation indicates that the Geometric Mean is dependent
solely on the thickness of the medium, d, which contains the point
source and not on the distance of the source from either detector.

Translating this to renography implies that imaging with opposing gamma
camera heads (as with a dual camera SPECT system) would provide spilt
function measurements which are independent of renal depth.

Conjugate imaging in renography is illustrated by the following images
[^33] of Monte Carlo simulated renograms acquired simultaneously using
anterior-posterior (AP) and posterior-anterior (PA) projections. The
simulations had renal depths of 7.31 cm on the left and 8.64 on the
right and the real LK/RK split function ratio was 50%:50%.

  -------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- --------------------------------------------------------------------------------------------------------------------------------------------------- ----------------------------------------------------------------------------------------------------------------------
  !Anterior-posterior (AP), on the left, and posterior-anterior (PA) Monte Carlo simulated renograms. Raw data courtesy of dynamicrenalstudy.org\|frame, on the left, and posterior-anterior (PA) Monte Carlo simulated renograms. Raw data courtesy of dynamicrenalstudy.org|frame")   !Geometric mean renogram.\|256x256px `<small>`{=html}Geometric Mean renogram`</small>`{=html}   !Curves for the geometric mean renogram.\|thumb
  -------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- --------------------------------------------------------------------------------------------------------------------------------------------------- ----------------------------------------------------------------------------------------------------------------------

These images were used to compute the Geometric Mean using
ImageJ and the resultant images were analysed
using OsiriX. Image computation involved
multiplying the AP and PA renograms and taking the square root of the
result, at 32-bit accuracy. Computed images were the converted to 16-bit
for DICOM storage. The resultant Geometric Mean
renogram (see the centre panel above) provided an LK/RK split function
estimate of 51%:49%, while the PA-only data, not corrected for renal
depth, provided an inferior estimate of 54%:46%.

It is apparent from the above illustration that conjugate imaging may
have a role in clinical renography and may not be reliant on renal depth
measurements for attenuation correction. However, it should be noted
that the method assumes that the activity is located in a homogeneous
medium with a spatially-uniform linear attenuation coefficient, μ, which
indicates that the method may suffer in real patients from aberrations
generated by sources such as digestive gases overlying the kidneys.

It should be noted, as a concluding remark, that conjugate counting is
well established in quantitative renal analysis using static
^99m^Tc-DMSA images[^34].

## Blood Pool Compensation for Split Function Determination

A recent publication has introduced the technique of Blood Pool
Compensation for split function determination in renography[^35]. Here
the activity in each kidney and the liver is measured over time. The
liver activity is assumed to represent blood pool activity during the
uptake phase of the renogram. A linear fit to a plot of liver activity
versus kidney activities is then back-extrapolated to zero liver
activity to determine the kidney activity with zero blood pool
contribution. The split function is then calculated using the relative
contributions at this zero activity.

This simple method is illustrated in the following figures. Three
regions of interest (RoIs) defined for the liver, the left kidney and
the right kidney are shown in the top left panel and the three resulting
curves are shown in the top right panel of the table for a Monte
Carlo-simulated renogram with 50/50 split function. A liver counts
versus kidney counts plot for the uptake phase is shown in the bottom
left panel, where an intercept of 17,281 counts is found for the left
kidney and an intercept of 18,719 counts for the right kidney. Split
function is then estimated to be 48% on the left and 52% on the right.

  ------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- ------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
  !BPC RoIs{width="256" height="256"}                                                                                !BPC Curves{width="256" height="256"}
  !Liver counts versus kidney counts during the uptake phase of the renograms.{width="256" height="256"}   !Blood Pool Compensation: Mean Absolute Error for 50/50 Function{width="256" height="256"}
  ------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- ------------------------------------------------------------------------------------------------------------------------------------------------------------------------------

The bottom right panel shows the Mean Absolute % Error, computed from a
series of such simulated images where the left kidney to right kidney
split function varied from 100%:0%, through 50%:50% to 0%:100% in steps
of 5 percent[^36], to be 7.5%.

## References

```{=html}
<references />
```

[^1]: Taylor A Jr, Thakore K, Folks R, Halkar R & Manatunga A, 1997.
    Background subtraction in technetium-99m-MAG3
    renography. Journal of
    Nuclear Medicine, 38(1):74-9.

[^2]: Brolin G, Edenbrandt L, Granerus G, Olsson A, Afzelius D,
    Gustafsson A, Jonsson C, Hagerman J, Johansson L, Riklund K &
    Ljungberg M, 2014. The accuracy of quantitative parameters in 99 m
    Tc-MAG3 dynamic renography: a national audit based on virtual image
    data. Clin Physiol
    Funct Imaging. Oct 28. doi: 10.1111/cpf.12208. Epub ahead of print.

[^3]: Brolin, G, 2015. *Personal communication*.

[^4]: Brolin G, Edenbrandt L, Granerus G, Olsson A, Afzelius D,
    Gustafsson A, Jonsson C, Hagerman J, Johansson L, Riklund K &
    Ljungberg M, 2014. The accuracy of quantitative parameters in 99 m
    Tc-MAG3 dynamic renography: a national audit based on virtual image
    data. Clin Physiol
    Funct Imaging. Oct 28. doi: 10.1111/cpf.12208. Epub ahead of print.

[^5]: Brolin G, Gleaner KS & Ljungberg M, 2013. Dynamic (99 m)Tc-MAG3
    renography: images for quality control obtained by combining
    pharmacokinetic modelling, an anthropomorphic computer phantom and
    Monte Carlo simulated scintillation camera
    imaging. Physics in
    Medicine & Biology, 58(10):3145-61.

[^6]: Brolin G, Edenbrandt L, Granerus G, Olsson A, Afzelius D,
    Gustafsson A, Jonsson C, Hagerman J, Johansson L, Riklund K &
    Ljungberg M, 2014. The accuracy of quantitative parameters in 99 m
    Tc-MAG3 dynamic renography: a national audit based on virtual image
    data. Clin Physiol
    Funct Imaging. Oct 28. doi: 10.1111/cpf.12208. Epub ahead of print.

[^7]: Tonnesen KH, Munck O, Hald T, Mogensen P & Wolf H, 1974. Influence
    of the renogram of variation in skin to kidney distance and the
    clinical importance hereof. Presented at the Internation Symposium
    Radionuclides in Nephrology, Berlin. Cited by Schlegel JU & Hamway
    SA, 1976. Individual renal plasma flow determination in 2
    minutes. Journal of
    Urology, 116, 282-5.

[^8]: Taylor A, Lewis C, Giacometti A, Hall EC & Barefield KP, 1993.
    Improved formulas for the estimation of renal depth in
    adults. Journal of
    Nuclear Medicine, 34, 1766-9.

[^9]: Taylor A, 1994. Formulas to estimate renal depth in
    adults. Journal of
    Nuclear Medicine, 35, 2054-5.

[^10]: Itoh K & Arakawa M, 1987. Re-estimation of renal function with
    99mTc-DTPA by the Gates\'
    method. Kaku Igaku, 24,
    389-96.

[^11]: Tonnesen KH, Munck O, Hald T, Mogensen P & Wolf H, 1974.
    Influence of the renogram of variation in skin to kidney distance
    and the clinical importance hereof. Presented at the Internation
    Symposium Radionuclides in Nephrology, Berlin. Cited by Schlegel JU
    & Hamway SA, 1976. Individual renal plasma flow determination in 2
    minutes. Journal of
    Urology, 116, 282-5.

[^12]: Taylor A, Lewis C, Giacometti A, Hall EC & Barefield KP, 1993.
    Improved formulas for the estimation of renal depth in
    adults. Journal of
    Nuclear Medicine, 34, 1766-9.

[^13]: Taylor A, 1994. Formulas to estimate renal depth in
    adults. Journal of
    Nuclear Medicine, 35, 2054-5.

[^14]: Itoh K & Arakawa M, 1987. Re-estimation of renal function with
    99mTc-DTPA by the Gates\'
    method. Kaku Igaku, 24,
    389-96.

[^15]: Gruenewald SM, Collins LT & Fawdry RM, 1985. Kidney depth
    measurement and its influence on quantitation of function from gamma
    camera renography.
    Clinical Nuclear Medicine 10, 398-401.

[^16]:

[^17]:

[^18]:

[^19]: `<small>`{=html}Itoh K, Tsukamoto E, Kato C, Shiga T, Yamashita T
    & Nonomura K, 1995. Non-invasive quantification of individual renal
    function by means of gamma camera renography with ^99^Tc^m^-MAG3.
    Abstract. Nuclear Medicine Communications, 16(5):
    413`</small>`{=html}

[^20]: GE Healthcare, 2007. Renal Analysis Operator Guide, B-12.

[^21]: `<small>`{=html}Oriuchi N, Onishi Y, Kitamura H, Inoue T, Tomaru
    Y, Higuchi T, Inoue T & Endo K, 1998. Noninvasive measurement of
    renal function with ^99m^Tc-MAG~3~ gamma-camera renography based on
    the one-compartment
    model. Clinical
    Nephrology, 50(5), 289--94`</small>`{=html}

[^22]: Russell CD & Dubovsky EV,1999. Reproducibility of single-sample
    clearance of 99mTc-mercaptoacetyltriglycine and
    131I-orthoiodohippurate.
    Journal of Nuclear Medicine, 40(7):1122-4.

[^23]:

[^24]: Du Bois D & Du Bois EF, 1916. A formula to estimate the
    approximate surface area if height and weight be known. Archives of
    Internal Medicine 17(6): 863--71

[^25]: Haycock GB, Schwartz GJ & Wisotsky DH, 1978. Geometric method
    for measuring body surface area: A height-weight formula validated
    in infants, children and
    adults. Journal of
    Pediatrics, 93:62--6.

[^26]:

[^27]:

[^28]:

[^29]: Siemens, 2012. Renal Processing Manual.

[^30]:

[^31]:

[^32]:

[^33]: Brolin G, Edenbrandt L, Granerus G, Olsson A, Afzelius D,
    Gustafsson A, Jonsson C, Hagerman J, Johansson L, Riklund K &
    Ljungberg M, 2014. The accuracy of quantitative parameters in 99 m
    Tc-MAG3 dynamic renography: a national audit based on virtual image
    data. Clin Physiol
    Funct Imaging. Oct 28. doi: 10.1111/cpf.12208. Epub ahead of print.

[^34]: Fleming JS, Cosgriff PS, Houston AS, Jarritt PH, Skrypniuk JV &
    Whalley DR, 1998. UK audit of relative renal function measurement
    using DMSA
    scintigraphy. Nuclear
    Medicine Communications, 19(10):989-97.

[^35]: Wesolowski MJ, Conrad GR, Šámal M, Watson G, Wanasundara SN,
    Babyn P, & Wesolowski CA, 2016. A simple method for determining
    split renal function from dynamic (99 m)Tc-MAG3 scintigraphic
    data. Eur J Nucl Med
    Mol Imaging, 43(3):550-8

[^36]: Brolin, G, 2015. *Personal communication*.


# 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.


# Basic Physics of Nuclear Medicine/Sonography & Nuclear Medicine

!A sonographic study of valves in a patient\'s
heart.

This is a developing chapter of a wikibook entitled Basic Physics of
Nuclear Medicine.

Sonography is an application of the SONAR
principle in medical imaging where the surfaces of internal organs and
their inner structure can be depicted (see the image below). This
imaging modality has the advantage of not using ionizing radiation and
is frequently used to provide correlative anatomical information for
nuclear medicine images. An overview of basic features of sonographic
imaging is provided in this chapter.

We\'ll start with a consideration of relevant characteristics of sound
waves before describing various imaging methods.

## Sound Waves

Sound waves travelling through air consist of
periodic fluctuations in air pressure, called compressions and
rarefactions, as illustrated below:

!Illustration of a vibrating tuning fork producing a sequence of
compressions and rarefactions in the surrounding
air.

These longitudinal waves travel with
a velocity, *v*, of about 330 m/s in air at STP, and at higher
velocities in denser media, such as water and soft tissue. Indeed a
medium is needed for the waves to propagate -- remember that the physics
behind the statement \'In space no one can hear you scream\', which was
used to promote the movie Alien "wikilink"), is quite
correct!

A sequence of compressions and rarefactions is referred to as one
**cycle**, as illustrated. The **wavelength**, *λ*, is defined as the
length of one cycle and the **frequency**, *f*, as the number of cycles
which pass a fixed point every second. These quantities are related
through the famous equation:

```{=html}
<div class="center">
```
*v* = *f* λ

```{=html}
</div>
```
The human ear is sensitive to sound frequencies up to about 20 kHz, and
waves of higher frequency are referred to as
ultrasound. Much higher frequencies are used
in diagnostic sonography, in the
range 1-15 MHz. Low frequencies in this range can be used to image large
deep structures, while high frequencies can be used for small,
superficial objects.

  Medium                      Velocity (m/s)
  --------------------------- ----------------
  **Air**                     331
  **Brain**                   1,541
  **Kidney**                  1,561
  **Liver**                   1,549
  **Muscle**                  1,585
  **Fat**                     1,450
  **Soft Tissue (average)**   1,540

The velocity of ultrasound waves is identical to that of sound waves in
the same medium, and is given in the following table for a range of
internal organs. Note that a velocity of 1,540 m/s is generally assumed
for soft tissue in sonographic imaging and represents an *average* of
that for a number of tissues, muscles and organs.

Ultrasound waves are generally produced in pulses for sonographic
imaging, with the time interval between pulses used to detect ultrasound
echoes produced within the body. This technique exploits what\'s known
as the **Pulse-Echo Principle**, as illustrated in the diagram below.
The upper half of the diagram depicts an ultrasound transducer emitting
one pulse of ultrasound into a hypothetical body, which is assumed to
consist of just two tissues. The lower half of the diagram depicts the
situation after the ultrasound pulse has encountered the interface
between the two tissues. A reflected pulse is shown travelling back
towards the transducer, i.e. the echo, and a transmitted pulse is seen
to continue into the second tissue.

The length of time taken for the pulse, once produced by the transducer,
to travel to the interface and the echoed pulse to return is termed the
**pulse-echo time**, *t*, and its measurement allows the depth, *d*, of
the interface to be determined using the following equation:

```{=html}
<div class="center">
```
$d = \frac{\overline{v}t}{2}$

```{=html}
</div>
```
Note that in this equation:

:\*the average velocity of ultrasound in the tissue is used, and

:\*the factor, `<span style="color:red;">`{=html}2`</span>`{=html},
arises because the pulse and its echo must travel the same distance, one
from the transducer to the interface and the other from the interface
back to the transducer:

!Illustration of the pulse-echo
principle.{width="256"}

Ultrasound transducers exploit the piezoelectric
effect to cause a crystal to vibrate
at ultrasound frequencies. The resultant vibrations generate pulses of
compressions and rarefactions which propagate through the tissues.
Echoes produced by tissue interfaces are then detected by the same
crystal -- exploiting the piezoelectric effect once again.

The ultrasound pulse becomes attenuated as it passes through tissue and
four phenomena result when it encounters a tissue interface, as
illustrated below:

!Illustration of phenomena which result when an ultrasound pulse
encounters a tissue
interface.{width="256"}

  Interface                Reflection Coefficient (%)
  ------------------------ ----------------------------
  **Soft Tissue -- Air**   99.9
  **Fat -- Muscle**        1.08
  **Fat -- Kidney**        0.64
  **Muscle -- Liver**      1.5

Some of the energy in the pulse is scattered through a process called
**non-specular reflection**, some of it generates an echo in a
**specular reflection** process, some of it is transmitted through the
interface to produce further echoes at other interfaces and a small
amount is absorbed. The reflectivity of an interface depends on the
acoustic impedance of the two tissues
involved, and representative values are shown in the table.

Notice in the table that a huge reflection can occur at a soft tissue --
air interface. Its for this reason that a coupling medium is used
between the transducer and the patient\'s skin. Internal reflections are
seen in the table to be of the order of 1%, yielding a useful
transparency for imaging purposes.

#top

## Ultrasound Scanner

A simplified block diagram of a sonography system is shown in the figure
below. The type of scanner shown operates using a linear array
transducer, which we\'ll learn more about shortly. We can see the
**Master Timer** in the top right of the figure. This circuit sets the
number of ultrasound pulses which the transducer generates every second
-- a factor referred to as the Pulse Repetition Frequency (**PRF**). Its
also seen that echo pulses picked up by the transducer are amplified by
a **Receiver Amplifier** whose output is
demodulated before being fed to a **Scan
Converter** so that the location and amplitude of detected echoes can be
displayed.

!Simplied block diagram of an ultrasound scanner which uses a linear
array
transducer.{width="256"}

The Time-Gain Compensation (**TGC**) circuit provides for selective
amplification of the echo signals so as to compensate for attenuation of
distant ultrasound echoes and suppress more proximal ones. The switch
array is used to excite the multiple crystals in the transducer as shown
below:

!Illustration of a linear array transducer interrogating different
lines of
tissue.{width="256"}

!B-mode image from a patient\'s liver
scan.

In the simplest arrangement, each crystal generates an ultrasound pulse
one after the other so that sequential lines of tissue can be rapidly
and continuously insonated.

The ultrasound image is referred to as a **B-Mode** scan and consists of
a 2D representation of the echo pattern in a cross-section of tissue
with the transducer position at the top of the image. The locations of
echo-producing tissue interfaces are generally represented by bright
pixels on a dark background, with the amplitude of each echo signal
being represented by the pixel value -- see the image on the right.

The image shown was actually acquired using a more sophisticated
transducer called a **phased array**, which generates a sector-shaped
scan. This type of transducer also uses a linear array of small
crystals, but with them excited in complex timing sequences, controlled
by delay circuitry -- as shown in the figure below. The ultrasound beam
can be steered to scan a region in this manner while being focussed at
different depths simultaneously.

!Illustration of a phased array
transducer.

There are numerous other transducer designs, each of which have
particular advantages in different clinical applications. Two designs of
mechanical transducer are shown below as examples. The left panel
illustrates a transducer with a single crystal which is rocked back and
forth during the scanning process, while the right panel illustrates a
rotating arrangement of single crystals:

!Illustration of two designs of mechanical
transducer.{width="512"}

Components of the scan conversion circuitry are illustrated in the
following figure:

!Simplified block diagram of the scan converter of an ultrasound
scanner.

The figure illustrates the process of digitizing the analogue echo
signals using an analogue-to-digital converter (**ADC**) and applying
pre-processing to the digital data using an input look-up table
(**ILUT**) prior to storage in random access memory. This memory is
filled in the sequence which was used to scan the patient and read out
in a manner suitable for the display device, which is typically an LCD
monitor. Prior to display, the image data can
be post-processed using an output look-up table (**OLUT**) so that
contrast enhancement and other processing functions can be applied. Note
that we\'ve encountered this type of digital image processing in a more
general form in another
chapter
of this wikibook. The box labelled
`<span style="color:blue;">`{=html}`<b>`{=html}μP`</b>`{=html}`</span>`{=html}
represents a microprocessor which is used to control this scan
conversion circuitry, as well as many other functions of the scanner,
e.g. the timing used for phased array emission and reception.

A digital image resolution widely used in sonography is 512 x 512 x
8-bits -- a magnified view of the central region of the liver scan shown
earlier is provided below to illustrate the digital nature of the data:

!Magnified view of the central region of the liver scan shown
earlier.

We conclude this section with photos of a sonography system and typical
transducers:

!`<span style="color:white;">`{=html}.`</span>`{=html}

#top

## Doppler Ultrasound

The Doppler Effect is widely exploited in
the remote measurement of moving objects, and can be used in medical
sonography to generate images (and sounds!) of flowing blood. The effect
is demonstrated by all wave-like phenomena, be they longitudinal or
transverse waves, and has been used with light, for instance, to reveal
that we live in an expanding universe! Its also exploited using radio
waves in highway speed traps, and can be experienced with sound waves
when an ambulance passes by with its siren blaring.

Let\'s take the example of a train engine sounding its whistle, as
illustrated in the diagram below:

!Illustration of the origin of the Doppler
Effect.

When the train is stationary, and there isn\'t a wind blowing, the sound
will emanate equally from the whistle in all directions, as illustrated
on the left. When the train is moving, however, sound frequencies will
be increased in the forward direction and reduced in the opposite
direction, as illustrated on the right, to an extent dependent on the
velocity of the train. This apparent change in frequency of the sound
waves experienced by a stationary listener is referred to as the Doppler
Shift.

The situation for exploiting the Doppler Effect for the detection of
blood flow is illustrated in the following diagram:

!Illustration of blood flow detection using the Doppler Effect with
ultrasound
waves.

The diagram shows a Doppler transducer placed on the skin and aimed at
an angle, θ, towards a blood vessel, which contains blood flowing with a
velocity of *u* m/s, at any instant. The transducer emits ultrasound
waves of frequency, *f*~o~, and echoes generated by moving reflectors in
the blood, e.g. red blood cells, have a frequency, *f*~r~. The
difference between these two frequencies, Δ*f*, is related to the
velocity of the flowing reflectors through the following equation:

```{=html}
<div class="center">
```
$\Delta f = f_0 - f = \frac{2f_0 u \cos \theta}{v}$

```{=html}
</div>
```
where *v* is the velocity of sound in the medium.

So, for instance, when ultrasound with a frequency in the range 2-10 MHz
is applied in medicine to detect blood flowing in arteries (where
typical velocities are 0-5 m/s), the equation above reveals that the
frequency differences will be in the audible range of sound frequencies,
i.e. 0-15 kHz. Their signals can therefore be fed through speakers so
that this sound can be heard.

Its also possible to examine the frequency content of Doppler shifts to
examine subtle details of the distribution of blood velocities during a
cardiac cycle by computing their Fourier
transforms.
Its more common however to produce images of the distribution of
frequency shifts within blood vessels using techniques such as
**Colour-Flow** or **Colour-Power** imaging. These techniques are used
to automatically fuse Doppler signals with B-Mode ultrasound images, as
illustrated below:

!A colour-flow image on the left with a colour-power image on the
right.

Colour-flow processing is sensitive to the direction of blood flow, i.e.
it can detect both positive and negative Doppler shifts, and uses a
colour look-up table
(CLUT)
so that shifts in one direction are displayed in shades of red with
those in the other direction in shades of blue -- as illustrated by a
patient\'s jugular vein and carotid artery depicted in the left panel of
the figure above. A simplified block diagram of a sonography system used
for such imaging is shown below:

!Block diagram of a colour-flow sonography imaging
system.

The system uses a beam former circuit to excite the crystals of the
phased array transducer for B-Mode imaging and Doppler shift detection
in a rapid alternating manner, with the echo signals being fed to B-Mode
scanning circuitry and the Doppler signals fed to an autocorrelation
detector for analysis. Output
data from these circuits are then blended within the scan conversion and
formatting circuitry, prior to display of the fused image.

As a final point, note that the colour-power image displayed above does
not contain any blood-flow direction information, since this technique
computes the power of reflected Doppler-shifted pulses instead of their
frequency content.

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