# Basic Physics of Nuclear Medicine/Patient Dosimetry

This chapter overviews the **MIRD** method of estimating doses in
nuclear medicine and gives results of surveys of typical patient doses
for a small range of examinations. Doses from X-ray CT surveys are also
given for comparison purposes, and to aid with your appreciation of the
dosimetric impact of PET/CT and SPECT/CT.

## MIRD Calculations

The direct measurement of the absorbed
dose to
organs of the body from nuclear medicine procedures is rarely possible.
Most estimates depend on Monte Carlo
simulation. To compute a dose
distribution, many hundreds of thousands of virtual gamma rays
originating in numerous source organs must be traced through the body as
they interact with tissues by Compton
scattering
and photoelectric
absorption.
Contributions to the dose from Auger
electrons, internal conversion
electrons and beta
decays must also be included This latter part
of the simulation is usually quite straight-forward since these
radiations can be considered to be absorbed in the organ they originate
in.

The Medical Internal Radiation Dose
(MIRD) Committee in
the USA has provided a substantial body of data to assist in dose
estimation. The MIRD method of calculation allows estimation of the dose
to organs of significance from radioactivity distributed homogeneously
throughout a particular source organ. The method can be split into two
parts:

-   **Firstly**, some assumptions are made about or measurements are
    taken of the biological distribution of radioactivity throughout the
    body, taking into account both the biological and physical half-life
    of the labelled radionuclide.

:   The **cumulated activity**, *Ã*~*s*~, in any particular **source**
    organ is calculated by integrating the instantaneous activity over
    time. This data might be obtained from theoretical modelling (such
    as Compartmental
    Analysis)
    or from measurements using a gamma camera to measure the count rate
    as a function of time in the organ of interest - see the following
    figure:

!Estimation of the cumulated activity. The curve in this example could
for example represent the activity in part of the skeleton after
administration of
^99m^Tc-MDP.

:   Mathematically, this type of activity/time curve can be represented
    by the equation:

```{=html}
<div class="center">
```
$\tilde{A}_s = \int A_s (t) \cdot dt$

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

:

    :   where *A*~*s*~(*t*) is the activity in the source organ, *s* at
        time, *t*.

-   **Secondly**, the absorbed dose to a target organ, *D*~*t*~, is
    calculated from:

```{=html}
<div class="center">
```
$D_t = \sum_{s} \tilde{A}_s \cdot S(t,s)$

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

:   where the summation over *s* is intended to account for
    contributions from all source organs to the target organ. The
    **S-factor**, *S*(*t*,*s*) is the mean absorbed dose to the target
    organ, *t*, from unit activity of the relevant radioisotope
    distributed homogeneously within the source organ, *s*. Note that
    its possible for the source and target organ to be the same entity.

## The S-Factor

The challenging part of the MIRD method is the calculation of the
S-factor because it depends on accurate knowledge of the radiations
emitted by the radionuclide. Although the MIRD publications assist by
providing representative values of S-factors for most organs and for a
variety of radionuclides, it is important to understand the methodology
and hence the limitations of this technique.

The S-factor for each radiation type can be broken down into component
factors:

```{=html}
<div class="center">
```
$S(t,s) = D \frac{f(t,s)}{m}$

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

-   *D* is a measure of the total energy associated with the particular
    radiation type and is a physical entity known from the
    radioisotope\'s decay scheme,
-   *f*(*t*,*s*) is the **absorbed fraction** for the particular
    radiation emitted in the source organ, *s*, and absorbed by the
    target organ, *t*, and
-   *m* is the mass of the target organ.

The absorbed fraction represents the fraction of the total energy
emitted by radiation of a particular type that is absorbed in the target
organ. For beta particles, which have a short range in tissue, it can be
assumed that all the energy will be deposited in the source organ and
other target organs will not be irradiated. That is:

```{=html}
<div class="center">
```
$f(t,s) = 0\,\!$

```{=html}
</div>
```
Unless *t* = *s*, in which case:

```{=html}
<div class="center">
```
$f(t,s) = 1\,\!$

```{=html}
</div>
```
The determination of the S-factor is very simple in this case.

For gamma-rays, however, no such simple approach is possible since the
amount of energy deposited in the source organ and in the target organ
is dependent on:

-   the gamma ray energy, as well as
-   the size, shape and separation of the two organs in question.

Furthermore, we need to take into account not just the gamma-rays from
the nucleus of the radioisotope but also emissions from the whole atom,
i.e. emissions resulting from the electron shells when stimulated by
emissions from the nucleus. The situation for ^99m^Tc is illustrated in
the following table, as an example:

  Radiation                                             Radiations/Disintegration   Radiation Energy (keV)   Dose (keV/Disintegration)
  ----------------------------------------------------- --------------------------- ------------------------ ---------------------------
  Conversion electrons from M-shell, γ~1~               0.75                        1.63                     1.21
  γ~2~                                                  0.89                        140.5                    125
  Conversion electrons from K-shell, γ~2~               0.09                        119.5                    10.6
  Conversion electrons from L-shell, γ~2~               0.01                        137.5                    1.47
  Conversion electrons from M-shell, γ~2~               0.002                       140.0                    0.27
  Conversion electrons from N- and outer shells, γ~2~   0.0004                      140.4                    0.05
  Conversion electrons from K-shell, γ~3~               0.006                       121.6                    0.67
  Conversion electrons from L-shell, γ~3~               0.002                       139.6                    0.24
  Conversion electrons from M-shell, γ~3~               0.0003                      142.1                    0.05
  K~α1~ X-ray                                           0.04                        18.4                     0.74
  K~α2~ X-ray                                           0.02                        18.3                     0.39
  K~β~ X-ray                                            0.01                        20.6                     0.25
  K-Auger electron                                      0.02                        15.5                     0.32
  L-Auger electron                                      0.1                         2.2                      0.22

  : Adapted from the ENSDF Decay
  Data for
  ^99m^Tc (approximate values).

We can see in the table that a range of radiations are generated within
the atom as a whole. For instance, on the first line the energy of a
gamma-ray emitted by the nucleus is
given to an M-shell electron in
about 75% of one type of isomeric
transition giving it a maximum energy
of 1.63 keV, i.e. about 1.2 keV per nuclear disintegration. On the
second line, a gamma-ray resulting from another isomeric transition in
89% of instances emerges unscathed from the atom with an energy of 140.5
keV, or about 125 keV per disintegration. In a third case, in about 9%
of these latter isomeric transitions, an electron in the K-shell gains
energy of 119.5 keV, which is sufficient for it to overcome the binding
energy (about 21
keV)
of that shell and leave the atom. Notice that fluorescent
X-rays, resulting from electronic
transitions to fill vacancies in the K-shell are also produced - as well
as Auger electrons.

Part of a resultant MIRD table of S-factors for ^99m^Tc is reproduced
below as examples of typical values:\

  Target Organ       Liver as Source   Spleen as Source
  ------------------ ----------------- ------------------
  **Bladder Wall**   4.3 x 10^-5^      3.2 x 10^-5^
  **Stomach Wall**   5.1 x 10^-4^      2.7 x 10^-3^
  **Liver**          1.2 x 10^-2^      2.6 x 10^-4^
  **Lungs**          1.4 x 10^-4^      6.2 x 10^-4^
  **Pancreas**       1.1 x 10^-3^      5.1 x 10^-3^
  **Spleen**         2.5 x 10^-4^      8.9 x 10^-2^

  : The absorbed dose per unit cumulated activity, *S*(*t*,*s*), for
  ^99m^Tc (in mGy/MBq/hour)

The unit of the S-factor is typically mGy/MBq/hour. Note that
substantial discrepancies between actual and calculated values can be
expected depending on differences in anatomy and physiological function
between individual patients.

## The Dose in a Liver Scan

To illustrate how the table above can be used in a hypothetical
situation, suppose a liver scan is performed in which 37 MBq of
^99m^Tc-labelled sulphur colloid is administered to a patient. Suppose
also that it is found from metabolic modelling that:

-   80% of the cumulated activity goes to the patient\'s liver and
-   20% goes to their spleen.

When it is then assumed that the biological half-life for the colloid is
infinite in these two organs, which is a reasonable approximation, the
calculation of cumulated total activity is much simplified as
illustrated in the box below:

:   {\| class=\"toc\" style=\"background:transparent; text-align:left;\"

\| rowspan=\"5\" style=\"vertical-align:top; padding-right:5px;\" \|
$\tilde{A}_{\text{liver}}$ \| $= \int A(t) \cdot dt$ \|- \|
$= \int A_0\ \text{exp} \left ( \frac{-0.693t}{T_\frac{1}{2}} \right ) \cdot dt$
\|- \| $= A_0 \cdot 1.44 \cdot T_{\frac{1}{2}}$ \|- \|
$= 0.8 \cdot 37 \cdot 1.44 \cdot 6.03 \cdot \text{MBq.h}\,\!$ \|- \|
$= 256.6\ \text{MBq.h}\,\!$ \|- \|   \|\|   \|- \| rowspan=\"2\"
style=\"vertical-align:top; padding-right:5px;\" \|
$\tilde{A}_{\text{spleen}}$ \|
$= 0.2 \cdot 37 \cdot 1.44 \cdot 6.03\ \text{MBq.h}\,\!$ \|- \|
$= 64.1\ \text{MBq.h}\,\!$ \|- \|   \|\|   \|- \| rowspan=\"2\"
style=\"vertical-align:top; padding-right:5px;\" \|
$D_{\text{liver}}\,\!$ \|
$= (1.2 \cdot 10^{-2} \cdot 256.6) + (2.6 \cdot 10^{-4} \cdot 64.1)\ \text{mGy}\,\!$
\|- \| $= 3.2\ \text{mGy}\,\!$ \|}

Note that the liver dose arises almost exclusively from the presence of
radioactivity in the liver and that the activity in the spleen
contributes little to this dose - just 5%.

In similar way, the dose to the spleen can be calculated to be
`<span style="color:red;">`{=html}5.8 mGy`</span>`{=html}. Note that
even though less activity accumulates in the spleen the absorbed dose is
higher. This apparent paradox can be resolved however when you recall
that the absorbed dose is the energy absorbed **per unit mass**.

The International Commission on Radiological Protection (**ICRP**) have
published more accurate organ absorbed doses derived on the basis of
knowledge of physiological data for what is known as the **Reference
Man** and from the MIRD data. Note that the absorbed doses to the liver
and to the spleen in the following table are
`<span style="color:red;">`{=html}2.74`</span>`{=html} and
`<span style="color:red;">`{=html}2.85`</span>`{=html} mGy,
respectively - somewhat lower than we calculated in our simple,
hypothetical situation above.

  Target Organ        w~T~    Absorbed Dose (mGy)                                      w~T~.H~T~ (mSv)
  ------------------- ------- -------------------------------------------------------- -----------------
  **Gonads (F)**      0.2     0.081                                                    0.016
  **Red Marrow**      0.12    0.41                                                     0.049
  **Colon**           0.12    0.07                                                     0.008
  **Lung**            0.12    0.2                                                      0.024
  **Stomach**         0.12    0.23                                                     0.028
  **Bladder**         0.05    0.04                                                     0.002
  **Breast**          0.05    0.1                                                      0.005
  **Liver**           0.05    `<span style="color:red;">`{=html}2.74`</span>`{=html}   0.137
  **Oesophagus**      0.05    0.1                                                      0.005
  **Thyroid**         0.05    0.03                                                     0.0015
  **Bone Surfaces**   0.01    0.24                                                     0.0024
  **Spleen**          0.025   `<span style="color:red;">`{=html}2.85`</span>`{=html}   0.071
  **Remainder**       0.05    0.21                                                     0.005

  : Data for estimation of effective dose, *E*, for a liver scan using
  37 MBq ^99m^Tc labelled colloid, where *w*~*T*~ is the tissue
  weighting factor and *H*~*T*~ is the equivalent dose for tissue, *T*.

Note that an **effective dose** of 0.35 mSv can be obtained by summing
the values in the right hand column of the table.

Finally, note that a radiation weighting factor, *w*~*R*~, of unity has
been assumed for simplicity throughout the above discussion for all the
radiation energies arising from the decay of ^99m^Tc.

## Typical Nuclear Medicine Doses

The following table lists the effective dose from a number of nuclear
medicine procedures (adapted from ICRP 53):

+----------+----------+----------+----------+----------+----------+
| Exa      | Rad      | Activity | Organ    | Absorbed | E        |
| mination | iopharma | (MBq)    |          | Dose     | ffective |
|          | ceutical |          |          | (mGy)    | Dose     |
|          |          |          |          |          | (mSv)    |
+==========+==========+==========+==========+==========+==========+
| **Bone   | ^99      | 740      | Bone\    | 46\      | 4.2      |
| Scan**   | m^Tc-MDP |          | Bladder  | 38       |          |
+----------+----------+----------+----------+----------+----------+
| **Heart  | ^201^Tl- | 120      | Testes\  | 68\      | 26(M)    |
| Wall**   | Thallous |          | Colon\   | 44\      | 16(F)    |
|          | Chloride |          | Kidney   | 65       |          |
+----------+----------+----------+----------+----------+----------+
| **Inf    | ^67^Ga   | 150      | Bone\    | 94\      | 15       |
| ection** | -Gallium |          | Colon\   | 25\      |          |
|          | Citrate  |          | Red      | 31\      |          |
|          |          |          | Marrow\  | 14\      |          |
|          |          |          | Adrenal  | 15       |          |
|          |          |          | Glands\  |          |          |
|          |          |          | Spleen   |          |          |
+----------+----------+----------+----------+----------+----------+
| **Re     | ^99m     | 150      | Bladder  | 9.3      | 0.73     |
| nogram** | ^Tc-DTPA |          |          |          |          |
+----------+----------+----------+----------+----------+----------+
| **Liver  | ^99m^Tc  | 74       | Liver\   | 5.2\     | 0.7      |
| Scan**   | -Sulphur |          | Spleen   | 5.6      |          |
|          | Colloid  |          |          |          |          |
+----------+----------+----------+----------+----------+----------+
| **       | ^99      | 74       | Lung\    | 4.9\     | 0.8      |
| Lung-Per | m^Tc-MAA |          | Liver    | 1.2      |          |
| fusion** |          |          |          |          |          |
+----------+----------+----------+----------+----------+----------+
| **Lu     | ^        | 20       | Lung     | 2.2      | 0.3      |
| ng-Venti | 99m^Tc-T |          |          |          |          |
| lation** | echnigas |          |          |          |          |
+----------+----------+----------+----------+----------+----------+
| *        | ^99m^    | 185      | Colon\   | 8.0\     | 1.9      |
| *Thyroid | Tc-Perte |          | Stomach\ | 4.8\     |          |
| Scan**   | chnetate |          | Thyroid  | 4.1      |          |
+----------+----------+----------+----------+----------+----------+
| *        | ^131^    | 150      | Bladder\ | 91\      | 9.2      |
| *Thyroid | I-Sodium |          | Stomach  | 5        |          |
| Scan**   | Iodide   |          |          |          |          |
+----------+----------+----------+----------+----------+----------+
| **PET    | ^18^     | 180      | Bladder\ | 29\      | 3.4      |
| Scan**   | F-2-Deox |          | Heart    | 11       |          |
|          | yglucose |          |          |          |          |
+----------+----------+----------+----------+----------+----------+

Note that typical effective doses from ^99m^Tc scans are of the order of
a few mSv. Also note the high effective dose from a ^201^Tl scan -
arising in part because of the high uptake of activity in the testes -
and also those from ^67^Ga and ^131^I. The administered activities
listed in the table are consistent with the most commonly used
activities found in a survey of Australasian Nuclear Medicine practices
conducted in 2000.

## Typical Doses in X-Ray CT

The radiation dose delivered to the patient in X-ray CT is determined by
factors which include the radiation energy and intensity
characteristics, the size of the patient, the efficiency of the
radiation detectors as well as factors derived from the specific form of
image acquisition used (axial versus helical scanning, for example).

The absorbed dose is far from uniform within the patient - being
greatest at the skin and a factor of about three lower in the centre of
the irradiated region. Typical values for skin absorbed doses range from
about 5 mGy to 50 mGy. Head scan absorbed doses are usually higher than
body scan doses because of the substantial attenuation in the skull. In
addition, the lens of the eye can receive a relatively large dose,
typically 50 to 100 mGy, and if repeat examinations are undertaken the
deterministic limit for cataract induction may be exceeded. Protocols,
using appropriate gantry angulation, should be developed to minimise the
possibility for direct irradiation of the eye. Likewise, relatively
large uterine doses may arise from abdominal or pelvic CT scans and this
may be a concern in scanning women of child bearing age.

A number of surveys have been published in recent years in which the
effective dose from CT scans have been estimated - see the table below.
The data indicates that the effective dose is typically 2 to 20 mSv with
the higher values relating to CT scans of the abdomen and pelvis and the
lower values relating to the head and cervical spine. This variation is
largely a reflection of the number of radiosensitive organs that may be
irradiated by the primary beam. The dose variation between countries is
a different matter!

  Examination        New Zealand   Australia   United Kingdom
  ------------------ ------------- ----------- ----------------
  **Head**           2.2           2.7         1.8
  **Chest**          9.8           10.7        7.8
  **Abdomen**        11.6          17.5        7.6
  **Liver**          6.5           13.4        7.2
  **Lumbar Spine**   5.0           5.4         3.3
  **Pelvis**         7.2           11.2        7.1

  : CT effective doses (mSv)

Notice that effective doses from nuclear medicine procedures are of a
similar order of magnitude, but generally lower, than that from CT
examinations. To assist with your comparison: the effective dose from a
chest X-ray is about 0.02 mSv, while that
from a mammogram is about 0.4 mSv and from an
intravenous pyelogram (**IVP**)
its about 3 mSv. How does this last figure compare with the dose from a
renogram? You might like to speculate on the reasons for the large
difference as an exercise to further develop your understanding of this
topic.


# Basic Physics of Nuclear Medicine/Production of Radioisotopes

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

Most of the radioisotopes found in nature have relatively long half
lives. They also belong to elements which are not handled well by the
human body. As a result medical applications generally require the use
of radioisotopes which are produced artificially.

We have looked at the subject of radioactivity in earlier chapters of
this wikibook and have then progressed to cover the interaction of
radiation with matter, radiation detectors and imaging systems. We
return to sources of radioactivity in this chapter in order to learn
about methods which are used to make radioisotopes.

The type of radioisotope of value to nuclear medicine imaging should
have characteristics which keep the radiation dose to the patient as low
as possible. For this reason they generally have a short half life and
emit only gamma-rays - that is no alpha-particle or beta-particle
emissions. From an energy point of view the gamma-ray energy should not
be so low that the radiation gets completely absorbed before emerging
from the patient\'s body and not too high that it is difficult to
detect. For this reason most of the radioisotopes used emit gamma-rays
of medium energy, that is between about 100 and 200 keV. Finally since
the radioisotope needs to be incorporated into some form of
radiopharmaceutical it should also be capable of being produced in a
form which is amenable to chemical, pharmaceutical and sterile
processing.

The production methods we will consider are nuclear fission, nuclear
bombardment and the radioisotope generator.

## Nuclear Fission

We were introduced to spontaneous fission in chapter
2
where we saw that a heavy nucleus can break into a number of fragments.
This disintegration process can be induced to occur when certain heavy
nuclei absorb neutrons. Following absorption of a neutron such nuclei
break into smaller fragments with atomic numbers between about 30 and
65. Some of these new nuclei are of value to nuclear medicine and can be
separated from other fission fragments using chemical processes.

The fission process is controlled inside a device called a nuclear
reactor. One such reactor exists in
Australia at Lucas Heights in New South Wales and
many others exist throughout the world.

## Nuclear Bombardment

In this method of radioisotope production charged particles are
accelerated up to very high energies and caused to collide into a target
material. Examples of such charged particles are protons, alpha
particles and deuterons. New nuclei can be formed when these particles
collide with nuclei in the target material. Some of these nuclei are of
value to nuclear medicine.

An example of this method is the production of ^22^Na where a target of
^24^Mg is bombarded with deuterons, that is:
`{{nowrap|{{chem|24|Mg}} + {{chem|2|H}} → {{chem|22|Na}} + {{chem|4|He}}.}}`{=mediawiki}

A deuteron you will remember from chapter
1 is the second most common
isotope of hydrogen, that is ^2^H. When it collides with a ^24^Mg
nucleus a ^22^Na nucleus plus an alpha particle is produced. The target
is exposed to the deuterons for a period of time and is subsequently
processed chemically in order to separate out the ^22^Na nuclei.

The type of device commonly used for this method of radioisotope
production is called a cyclotron. It consists
of an ion gun for producing the charged particles, electrodes for
accelerating them to high energies and a magnet for steering them
towards the target material. All arranged in a circular structure.

## Radioisotope Generator

![](NM11_2.gif "NM11_2.gif"){width="140"}

This method is widely used to
produce certain short-lived radioisotopes in a hospital or clinic. It
involves obtaining a relatively long-lived radioisotope which decays
into the short-lived isotope of interest.

A good example is ^99m^Tc which as we have noted before is the most
widely used radioisotope in nuclear medicine today. This isotope has a
half-life of six hours which is rather short if we wish to have it
delivered directly from a nuclear facility. Instead the nuclear facility
supplies the isotope ^99^Mo which decays into ^99m^Tc with a half life
of about 2.75 days. The ^99^Mo is called the **parent** isotope and
^99m^Tc is called the **daughter** isotope.

So the nuclear facility produces the parent isotope which decays
relatively slowly into the daughter isotope and the daughter is
separated chemically from the parent at the hospital/clinic. The
chemical separation device is called, in this example, a **^99m^Tc
Generator**:

It consists of a ceramic column with ^99^Mo
adsorbed onto its top surface. A solution
called an **eluent** is passed through the column, reacts chemically
with any ^99m^Tc and emerges in a chemical form which is suitable for
combining with a pharmaceutical to produce a radiopharmaceutical. The
arrangement shown in the figure on the right is called a **Positive
Pressure** system where the eluent is forced through the ceramic column
by a pressure, slightly above atmospheric pressure, in the eluent vial.

The ceramic column and collection vials need to be surrounded by lead
shielding for radiation protection purposes. In addition all components
are produced and need to be maintained in a sterile condition since the
collected solution will be administered to patients.

Finally an Isotope
Calibrator
is needed when a ^99m^Tc Generator is used to determine the
radioactivity for preparation of patient doses and to check whether any
^99^Mo is present in the collected solution.\

## Operation of a 99m-Tc Generator

Suppose we have a sample of ^99^Mo and suppose that at time $t=0$ there
are $N_0$ nuclei in our sample and nothing else. The number $N(t)$ of
^99^Mo nuclei decreases with time according to radioactive decay law as
discussed in Chapter
3:\
\

```{=html}
<div class="center">
```
$N(t)=N_\mathrm{0}e^{-\lambda_\mathrm{Mo}t}$

```{=html}
</div>
```
\
where $\lambda_\mathrm{Mo}$ is the decay constant for ^99^Mo.\
\
Thus the number of ^99^Mo nuclei that decay during a small time interval
$dt$ is given by\
\

```{=html}
<div class="center">
```
$\mathrm{d}N(t)=-\lambda_\mathrm{Mo} N_\mathrm{0} e^{-\lambda_\mathrm{Mo}t}\mathrm{d}t$

```{=html}
</div>
```
\
Since ^99^Mo decays into ^99m^Tc, the same number of ^99m^Tc nuclei are
formed during the time period $dt$. At a time $t'$, only a fraction
$\mathrm{d}n(t')$ of these nuclei will still be around since the ^99m^Tc
is also decaying. The time for ^99m^Tc to decay is given by $t'-t$.
Plugging this into radioactive the decay law we arrive at:\
\

```{=html}
<div class="center">
```
$\mathrm{d}n(t')=-\mathrm{d}N(t)e^{-\lambda_\mathrm{Tc}\left(t'-t\right)}=\lambda_\mathrm{Mo} N_\mathrm{0}e^{-\lambda_\mathrm{Mo}t}e^{-\lambda_\mathrm{Tc}\left(t'-t\right)}\mathrm{d}t$

```{=html}
</div>
```
\
Now we sum up the little contributions $\mathrm{d}n(t')$. In other words
we integrate over $t$ in order to find the number $n(t')$, that is the
number of all ^99m^Tc nuclei present at the time $t'$:\
\

```{=html}
<div class="center">
```
$n(t')=\int_0^{t'}-\mathrm{d}N(t')e^{-\lambda_\mathrm{Tc}\left(t'-t\right)}=\lambda_\mathrm{Mo}N_\mathrm{0}e^{-\lambda_\mathrm{Tc}t'} \int_0^{t'}e^{\left( \lambda_\mathrm{Tc}-\lambda_\mathrm{Mo}\right)t} \mathrm{d}t$

```{=html}
</div>
```
\
Finally solving this integral we find:

```{=html}
<div class="center">
```
$\begin{matrix}
\Rightarrow n(t')&=& \frac{\lambda_\mathrm{Mo}}{\lambda_\mathrm{Tc}-\lambda_\mathrm{Mo}}N_\mathrm{0} e^{-\lambda_\mathrm{Tc}t'}\left(e^{\left( \lambda_\mathrm{Tc}-\lambda_\mathrm{Mo}\right)t'} -1\right) \\
\end{matrix}$

```{=html}
</div>
```
The figure below illustrates the outcome of this calculation. The
horizontal axis represents time (in days), while the vertical one
represents the number of nuclei present (in arbitrary units). The green
curve illustrates the exponential decay of a sample of pure ^99m^Tc. The
red curve shows the number of ^99m^Tc nuclei present in a ^99m^Tc
generator that is never eluted. Finally, the blue curve shows the
situation for a ^99m^Tc generator that is eluted every 12 hours.

!Comparison of the physical decay of ^99m^Tc with its activity arising
from ^99^Mo decay in a radioisotope generator with and without elution
at 12 hour
intervals.

Photographs taken in a nuclear medicine hot lab are shown below:

  ------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- ----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
  !Pb shielding surrounding the laminar flow cabinet, where radiopharmaceuticals are prepared.{width="180"}   !Another view of the Pb shielding surrounding a laminar flow cabinet.{width="180"}

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

## External links

-   Concerns over Molybdenum
    Supplies
    -- news from 2008 compiled by the British Nuclear Medicine Society.
-   Cyclotron Java
    Applet -- a
    Java-based interactive demonstration of the operation of a cyclotron
    from GFu-Kwun Hwang, Dept. of Physics, National Taiwan Normal
    University, Virtual Physics Laboratory.
-   Nuclear Power Plant
    Demonstration -- a
    Java-based interactive demonstration of controlling a nuclear
    reactor. Also contains nuclear power Information links.
-   ANSTO -- information about Australia\'s
    nuclear organization.
-   Medical
    Valley --
    contains information on what nuclear medicine is, production of
    nuclear pharmaceuticals, molybdenum and technetium -- from The
    Netherlands Energy Research Foundation Petten.


# Basic Physics of Nuclear Medicine/Chapter Review

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## Chapter Review: Atomic & Nuclear Structure

-   The atom consists of two components -- a nucleus (positively
    charged) and an electron cloud (negatively charged);
-   The radius of the nucleus is about 10,000 times smaller than that of
    the atom;
-   The nucleus can have two component particles -- **neutrons** (no
    charge) and **protons** (positively charged) -- collectively called
    **nucleons**;
-   The mass of a proton is about equal to that of a neutron -- and is
    about 1,840 times that of an electron;
-   The number of protons equals the number of electrons in an isolated
    atom;
-   The **Atomic Number** specifies the number of protons in a nucleus;
-   The **Mass Number** specifies the number of nucleons in a nucleus;
-   **Isotopes** of elements have the same atomic number but different
    mass numbers;
-   Isotopes are classified by specifying the element\'s chemical symbol
    preceded by a superscript giving the mass number and a subscript
    giving the atomic number;
-   The **atomic mass unit** is defined as 1/12th the mass of the
    stable, most commonly occurring isotope of carbon (i.e. C-12);
-   **Binding energy** is the energy which holds the nucleons together
    in a nucleus and is measured in **electron volts** (eV);
-   To combat the effect of the increase in electrostatic repulsion as
    the number of protons increases, the number of neutrons increases
    more rapidly -- giving rise to the **Nuclear Stability Curve**;
-   There are \~2450 isotopes of \~100 elements and the unstable
    isotopes lie above or below the Nuclear Stability Curve;
-   Unstable isotopes attempt to reach the stability curve by splitting
    into fragments (**fission**) or by emitting particles/energy
    (**radioactivity**);
-   Unstable isotopes \<=\> radioactive isotopes \<=\> radioisotopes
    \<=\> radionuclides;
-   \~300 of the \~2450 isotopes are found in nature -- the rest are
    produced artificially.

## Chapter Review: Radioactive Decay

-   **Fission**: Some heavy nuclei decay by splitting into 2 or 3
    fragments plus some neutrons. These fragments form new nuclei which
    are usually radioactive;
-   **Alpha Decay**: Two protons and two neutrons leave the nucleus
    together in an assembly known as an **alpha-particle**;
-   An alpha-particle is a He-4 nucleus;
-   **Beta Decay -- Electron Emission**: Certain nuclei with an excess
    of neutrons may reach stability by converting a neutron into a
    proton with the emission of a **beta-minus particle**;
-   A beta-minus particle is an electron;
-   **Beta Decay -- Positron Emission**: When the number of protons in a
    nucleus is in excess, the nucleus may reach stability by converting
    a proton into a neutron with the emission of a **beta-plus
    particle**;
-   A beta-plus particle is a **positron**;
-   Positrons **annihilate** with electrons to produce two back-to-back
    gamma-rays;
-   **Beta Decay -- Electron Capture**: An inner orbital electron is
    attracted into the nucleus where it combines with a proton to form a
    neutron;
-   Electron capture is also known as **K-capture**;
-   Following electron capture, the excited nucleus may give off some
    gamma-rays. In addition, as the vacant electron site is filled, an
    X-ray is emitted;
-   **Gamma Decay -- Isomeric Transition**: A nucleus in an excited
    state may reach its ground state by the emission of a gamma-ray;
-   A gamma-ray is an electromagnetic photon of high energy;
-   **Gamma Decay -- Internal Conversion**: the excitation energy of an
    excited nucleus is given to an atomic electron.

## Chapter Review: The Radioactive Decay Law

-   The radioactive decay law in equation form;
-   Radioactivity is the number of radioactive decays per unit time;
-   The **decay constant** is defined as the fraction of the initial
    number of radioactive nuclei which decay in unit time;
-   **Half Life**: The time taken for the number of radioactive nuclei
    in the sample to reduce by a factor of two;
-   Half Life = (0.693)/(Decay Constant);
-   The SI Unit of radioactivity is the **becquerel** (Bq)
-   1 Bq = one radioactive decay per second;
-   The traditional unit of radioactivity is the **curie** (Ci);
-   1 Ci = 3.7 x 10^10^ radioactive decays per second.

#

Chapter Review: Units of Radiation Measurement ==

-   **Exposure** expresses the intensity of an X- or gamma-ray beam;
-   The SI unit of exposure is the **coulomb per kilogram** (C/kg);
-   1 C/kg = The quantity of X- or gamma-rays such that the associated
    electrons emitted per kg of air at STP produce in air ions carrying
    1 coulomb of electric charge;
-   The traditional unit of exposure is the **roentgen** (R);
-   1 R = The quantity of X- or gamma-rays such that the associated
    electrons emitted per kg of air at STP produce in air ions carrying
    2.58 x 10^-4^ coulombs of electric charge;
-   The **exposure rate** is the exposure per unit time, e.g. C/kg/s;
-   **Absorbed dose** is the radiation energy absorbed per unit mass of
    absorbing material;
-   The SI unit of absorbed dose is the **gray** (Gy);
-   1 Gy = The absorption of 1 joule of radiation energy per kilogram of
    material;
-   The traditional unit of absorbed dose is the **rad**;
-   1 rad = The absorption of 10^-2^ joules of radiation energy per
    kilogram of material;
-   The **Specific Gamma-Ray Constant** expresses the exposure rate
    produced by the gamma-rays from a radioisotope;
-   The Specific Gamma-Ray Constant is expressed in SI units in
    C/kg/s/Bq at 1 m;
-   Exposure from an X- or gamma-ray source follows the **Inverse Square
    Law** and decreases with the square of the distance from the source.

## Chapter Review: Interaction of Radiation with Matter

-   **Alpha-Particles**:
    -   exert considerable electrostatic attraction on the outer orbital
        electrons of atoms near which they pass and cause ionisations;
    -   travel in straight lines -- except for rare direct collisions
        with nuclei of atoms in their path;
    -   energy is always discrete.
-   **Beta-Minus Particles**:
    -   attracted by nuclei and repelled by electron clouds as they pass
        through matter and cause ionisations;
    -   have a tortuous path;
    -   have a range of energies;
    -   range of energies results because two particles are emitted -- a
        beta-particle and a **neutrino**.
-   **Gamma-Rays**:
    -   energy is always discrete;
    -   have many modes of interaction with matter;
    -   important interactions for nuclear medicine imaging (and
        radiography) are the Photoelectric Effect and the Compton
        Effect.
-   **Photoelectric Effect**:
    -   when a gamma-ray collides with an orbital electron, it may
        transfer all its energy to the electron and cease to exist;
    -   the electron can leave the atom with a kinetic energy equal to
        the energy of the gamma-ray less the orbital binding energy;
    -   a positive ion is formed when the electron leaves the atom;
    -   the electron is called a **photoelectron**;
    -   the photoelectron can cause further ionisations;
    -   subsequent X-ray emission as the orbital vacancy is filled.
-   **Compton Effect**:
    -   A gamma-ray may transfer only part of its energy to a valence
        electron which is essentially free; \*\* gives rise to a
        scattered gamma-ray;
    -   is sometimes called Compton Scatter;
    -   a positive ion results;
-   **Attenuation** is term used to describe both absorption and
    scattering of radiation.

## Chapter Review: Attenuation of Gamma-Rays

-   Attenuation of a narrow-beam of gamma-rays increases as the
    thickness, the density and the atomic number of the absorber
    increases;
-   Attenuation of a narrow-beam of gamma-rays decreases as the energy
    of the gamma-rays increases;
-   Attenuation of a narrow beam is described by an equation;
-   the **Linear Attenuation Coefficient** is defined as the fraction of
    the incident intensity absorbed in a unit distance of the absorber;
-   Linear attenuation coefficients are usually expressed in units of
    cm^-1^;
-   the **Half Value Layer** is the thickness of absorber required to
    reduce the intensity of a radiation beam by a factor of 2;
-   Half Value Layer = (0.693)/(Linear Attenuation Coefficient);
-   the **Mass Attenuation Coefficient** is given by the linear
    attenuation coefficient divided by the density of the absorber;
-   Mass attenuation coefficients are usually expressed in units of
    cm^2^ g^-1^.

## Chapter Review: Gas-Filled Detectors

-   Gas-filled detectors include the **ionisation chamber**, the
    **proportional counter** and the **Geiger counter**;
-   They operate on the basis of ionisation of gas atoms by the incident
    radiation, where the positive ions and electrons produced are
    collected by electrodes;
-   An **ion pair** is the term used to describe a positive ion and an
    electron;
-   The operation of gas-filled detectors is critically dependent on the
    magnitude of the applied dc voltage;
-   The output voltage of an ionisation chamber can be calculated on the
    basis of the capacitance of the chamber;
-   A very sensitive amplifier is required to measure voltage pulses
    produced by an ionisation chamber;
-   The gas in ionisation chambers is usually air;
-   Ionisation chambers are typically used to measure radiation exposure
    (in a device called an **Exposure Meter**) and radioactivity (in a
    device called an **Isotope Calibrator**);
-   The total charge collected in a proportional counter may be up to
    1000 times the charge produced initially by the radiation;
-   The initial ionisation triggers a complete gas breakdown in a Geiger
    counter;
-   The gas in a Geiger counter is usually an inert gas;
-   The gas breakdown must be stopped in order to prepare the Geiger
    counter for a new event by a process called **quenching**;
-   Two types of quenching are possible: electronic quenching and the
    use of a **quenching gas**;
-   Geiger counters suffer from **dead time**, a small period of time
    following the gas breakdown when the counter is inoperative;
-   The true count rate can be determined from the actual count rate and
    the dead time using an equation;
-   The value of the applied dc voltage in a Geiger counter is critical,
    but high stability is not required.

## Chapter Review: Scintillation Detectors

-   NaI(Tl) is a scintillation crystal widely used in nuclear medicine;
-   The crystal is coupled to a **photomultiplier tube** to generate a
    voltage pulse representing the energy deposited in the crystal by
    the radiation;
-   A very sensitive amplifier is needed to measure such voltage pulses;
-   The voltages pulses range in amplitude depending on how the
    radiation interacts with the crystal, i.e. the pulses form a
    spectrum whose shape depends on the interaction mechanisms involved,
    e.g. for medium-energy gamma-rays used in in-vivo nuclear medicine:
    the Compton effect and the Photoelectric effect;
-   A **Gamma-Ray Energy Spectrum** for a medium-energy, monoenergetic
    gamma-ray emitter consists (simply) of a Compton Smear and a
    Photopeak;
-   **Pulse Height Analysis** is used to discriminate the amplitude of
    voltage pulses;
-   A pulse height analyser (PHA) consists of a **lower level
    discriminator** (which passes voltage pulses which are than its
    setting) and an **upper level discriminator** (which passes voltage
    pulses lower than its setting);
-   The result is a variable width window which can be placed anywhere
    along a spectrum, or used to scan a spectrum;
-   A **single channel analyser** (SCA) consists of a single PHA with a
    scaler and a ratemeter;
-   A **multi-channel analyser** (MCA) is a computer-controlled device
    which can acquire data from many windows simultaneously.

## Chapter Review: Nuclear Medicine Imaging Systems

-   A **gamma camera** consists of a large diameter (25-40 cm) NaI(Tl)
    crystal, \~1 cm thick;
-   The crystal is viewed by an array of 37-91 PM tubes;
-   PM tubes signals are processed by a position circuit which generates
    +/- X and +/- Y signals;
-   These position signals are summed to form a **Z signal** which is
    fed to a pulse height analyser;
-   The +/- X, +/- Y and discriminated Z signals are sent to a computer
    for digital image processing;
-   A **collimator** is used to improve the spatial resolution of a
    gamma-camera;
-   Collimators typically consist of a Pb plate containing a large
    number of small holes;
-   The most common type is a **parallel** multi-hole collimator;
-   The most resolvable area is directly in front of a collimator;
-   Parallel-hole collimators vary in terms of the number of holes, the
    hole diameter, the length of each hole and the septum thickness --
    the combination of which affect the sensitivity and spatial
    resolution of the imaging system;
-   Other types include the **diverging**-hole collimator (which
    generates minified images), the **converging**-hole collimator
    (which generates magnified images) and the **pin-hole** collimator
    (which generates magnified inverted images);
-   Conventional imaging with a gamma camera is referred to as **Planar
    Imaging**, i.e. a 2D image portraying a 3D object giving
    superimposed details and no depth information;
-   **Single Photon Emission Computed Tomography** (SPECT) produces
    images of slices through the body;
-   SPECT uses a gamma camera to record images at a series of angles
    around the patient;
-   The resultant data can be processed using **Filtered Back
    Projection** and **Iterative Reconstruction**;
-   SPECT gamma-cameras can have one, two or three camera heads;
-   **Positron Emission Tomography** (PET) also produces images of
    slices through the body;
-   PET exploits the positron annihilation process where two 0.51 MeV
    back-to-back gamma-rays are produced;
-   If these gamma-rays are detected, their origin will lie on a line
    joining two of the detectors of the ring of detectors which
    encircles the patient;
-   A **Time-of-Flight** method can be used to localise their origin;
-   PET systems require on-site or nearby **cyclotron** to produce
    short-lived radioisotopes, such as C-11, N-13, O-15 and F-18.

## Chapter Review: Production of Radioisotopes

-   Naturally-occurring radioisotopes generally have long half lives and
    belong to relatively heavy elements -- and are therefore unsuitable
    for medical diagnostic applications;
-   Medical diagnostic radioisotopes are generally produced
    artificially;
-   The **fission** process can be exploited so that radioisotopes of
    interest can be separated chemically from fission products;
-   A **cyclotron** can be used to accelerate charged particles up to
    high energies so that they to collide into a target of the material
    to be activated;
-   A radioisotope generator is generally used in hospitals to produce
    short-lived radioisotopes;
-   A **technetium-99m generator** consists of an alumina column
    containing Mo-99, which decays into Tc-99m;
-   Saline is passed through the generator to elute the Tc-99m -- the
    resulting solution is called **sodium pertechnetate**;
-   Both positive pressure and negative pressure generators are in use;
-   An **isotope calibrator** is needed when a Tc-99m generator is used
    in order to determine the activity for preparation of patient doses
    and to test whether any Mo-99 is present in the collected solution.

## Exercise Questions

1\. Discuss the process of radioactive decay from the perspective of the
nuclear stability curve.

2\. Describe in detail FOUR common forms of radioactive decay.

3\. Give the equation which expresses the Radioactive Decay Law, and
explain the meaning of each of its terms.

4\. Define each of the following:

:   \(a\) Half life;
:   \(b\) Decay Constant;
:   \(c\) Becquerel.

5\. A sample of radioactive substance is found to have an activity of
100 kBq. Its radioactivity is measured again 82 days later and is found
to be 15 kBq. Calculate:

:   \(a\) the half-life;
:   \(b\) the decay constant.

6\. Define each of the following radiation units:

:   \(a\) Roentgen;
:   \(b\) Becquerel;
:   \(c\) Gray.

7\. Estimate the exposure rate at 1 metre from a 100 MBq source of
radioactivity which has a Specific Gamma Ray Constant of 50 mR per hour
per MBq at 1 cm.

8\. Briefly describe the basic principle of operation of gas-filled
radiation detectors.

9\. Illustrate using a graph how the magnitude of the voltage pulses
from a gas-filled radiation detector varies with applied voltage and
identify on the graph the regions associated with the operation of
Ionisation Chambers and the Geiger Counters.

10\. Describe the construction and principles of operation of a
scintillation spectrometer.

11\. Discuss the components of the energy spectrum from a monoenergetic,
medium energy gamma- emitting radioisotope obtained using a
scintillation spectrometer on the basis of how the gamma-rays interact
with the scintillation crystal.

12\. Describe the construction and principles of operation of a Gamma
Camera.

13\. Compare features of three types of collimator which can be used
with a Gamma Camera.