# 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,
• 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:

q: the quantity of tracer in the compartment at time, t, and
F: the outflow.
The single compartment model

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

$k = \frac{-dq/dt}{q}$

which can be rewritten as:

$\frac{dq}{dt} = -kq$

Without going into the mathematical details (which are similar to the derivation of the radioactive decay law!), the solution to this equation is:

$q = q_0\ exp(-kt)\,\!$

where qo 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, q versus time for relatively high and low values of the fractional turnover, k.

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,

$q_1 + q_2 = \text{constant} = q_0\,\!$

Therefore,

$\frac{dq_1}{dt} = -\frac{dq_2}{dt}\,,$

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:

$q_1 = q_0 \left \lbrack 1 - \frac{k_{12}}{k_{12} + k_{21}} \left \lbrace 1 -\ \text{exp}\ - (k_{12} + k_{21}) t \right \rbrace \right \rbrack$

and

$q_2 = q_0 \left \lbrack \frac{k_{12}}{k_{12} + k_{21}} \left \lbrace 1 -\ \text{exp}\ - (k_{12} + k_{21}) t \right \rbrace \right \rbrack$

and their behaviour in the special case when k12 = k21, 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:

$q_1 = q_0\ \text{exp}(-k_{12}t)$

and

$q_2 = q_0 \frac{k_{12}}{k_{12} - k_{20}}\lbrack \text{exp}(-k_{20}t) - \text{exp}(-k_{12}t)\rbrack\,,$

and the behaviour of q1 and q2 is shown in the figure below for the special case of k20 being three times the value of k12:

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

Note that the behaviour of q2 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:

$q_1 = q_0 \left \lbrack \frac{k_{21} - a_1}{a_2 - a_1} \text{exp}(-a_1t) + \frac{k_{21} - a_2}{a_1 - a_2}\text{exp}(-a_2t) \right \rbrack$

and

$q_2 = \frac{q_0 k_{12}}{a_2 - a_1} \lbrack \text{exp}(-a_1t) - \text{exp}(-a_2t) \rbrack$

where

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

The behaviour of q1 and q2 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,
• k12: clearance rate from plasma into the gland, and
• k21: 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, k23, 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 k12 in the figure above,
• diffusion into the extravascular space - represented by k13.

As the quantity of tracer in the extravascular compartment builds up, exchange in the opposite direction begins to occur (represented by k31), and so a maximum is reached before its quantity of tracer falls off. This is illustrated in the figure below for a situation where:

 k12 = 0.05 per minute k13 = 0.04 per minute k31 = 0.06 per minute l1 = 0.13 per minute l2 = 0.024 per minute A1 = 0.65 A2 = 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:

$q_1 = A_1\ \text{exp}(-l_1t) + A_2\ \text{exp}(-l_2t)\,\!$
$q_2 = A_3\ \text{exp}(-l_1t) - A_4\ \text{exp}(-l_2t)\,\!$
$q_3 = A_5\ \lbrack \text{exp}(-l_1t) - A_2\ \text{exp}(-l_2t)\rbrack\,\!$

where l1 and l2 are constants related to the fractional turnovers, and A1 through A5 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:

$P = \frac{\text{Quantity of Tracer in Intravascular Space}}{\text{Volume of Intravascular Space}}$

The time dependence of this plasma concentration will vary in the same way as q1, so that:

$P(t) = C_1\ \text{exp}(-l_1t) + C_2\ \text{exp}(l_2t)\,\!$

where C1 and C2 are related to A1 and A2, respectively. The renal clearance, which is related to k12, 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,

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 51Cr-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. 51Cr 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, 51Cr-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 51Cr-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 51Cr 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 q1 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 51Cr-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:

$m = \frac{\ln b_1 - \ln b_2}{t_2 - t_1}$

where:

• t1: time from injection for the first blood sample, usually 120 minutes,
• b1: counts per milliliter (mL) in the plasma from the first sample (corrected for background counts),
• t2: time from injection for the second blood sample, usually 240 minutes,
• b2: 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, t0, to determine what the plasma counts would be upon instantaneous mixing of the tracer throughout the patient's plasma compartment, i.e.

$\ln b_0 = \ln b_1 + m(t_1 - t_0)\,\!$

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 b0. The plasma clearance, q1, predicted using the three compartment model is shown shaded.

Therefore, we can write:

$b_0 = \text{exp}\ \big(\ln b_1 + m(t_1 - t_0)\big)$

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.

$V = \frac{S \cdot V_s}{b_0}$

which results in

$V = \frac{S \cdot 1000}{b_0}$

when the standard injection is diluted in 1 liter. The clearance (in ml/min) is then given by the following equation:

$\text{Clearance} = V \cdot m\,\!$

Results for two patients are shown below to illustrate this technique.

Patient A

Sample Counts/mL
Background 477
b1 at 119 mins 11,438
b2 at 238 mins 6,235
Standard, S 150,020

This patient's 51Cr-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
b1 at 122 mins 3,103
b2 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 51Cr-EDTA determined as described does not equate directly with the Glomerular Filtration Rate (GFR) since the method assumes a single exponential dependence. 51Cr-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:

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:

$GFR = \frac{\text{Clearance}}{(1 + f \cdot \text{Clearance})}$

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 m2. 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, t0, and that k56 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, t0, so that:

• When t < t0:
 ${\color{Blue}q_4 = 1 - A_3\ \text{exp}(-l_1t) - A_4\ \text{exp}(-l_2t)}\,\!$ ${\color{Red}q_5 = 0}\,\!$ ${\color{OliveGreen}q_6 = 0}\,\!$
• When t > t0:
 ${\color{Blue}q_4 = A_3[1 - \text{exp}(-l_1t_0)] \text{exp}(-l_1\lbrace t-t_0\rbrace) + A_4 [1-\text{exp}(-l_2t_0)]\text{exp}(-l_2\lbrace t-t_0\rbrace)}\,\!$ ${\color{Red}q_5 = A_7\ \text{exp}(-l_1\lbrace t-t_0 \rbrace) + A_8\ \text{exp}(-l_2\lbrace t-t_0\rbrace) - A_9\ \text{exp}(-l_3\lbrace t-t_0\rbrace)}\,\!$ ${\color{OliveGreen}q_6 = 1 - A_{10}\ \text{exp}(-l_1\lbrace t-t_0\rbrace) - A_{11}\ \text{exp}(-l_2\lbrace t-t_0\rbrace) + A_{12}\ \text{exp}(-l_3\lbrace t-t_0\rbrace)}\,\!$

where l3 is related to k56. The time course of the quantity of tracer in each compartment is shown below:

The parenchymal (q4), 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:

$q_{\text{kidney}} = q_4 + q_5\,\!$

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 (q1) and extravascular (q3) curves derived earlier using this five compartment model.

The following equation can be derived on this basis:

$q_{\text{Bgd}} = b_1 q_1 + b_3 q_3\,\!$

where b1 and b3 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 b1 = 0.05 and b3 = 0.02 and

$q_{\text{Renogram}} = 0.5\,q_{\text{Kidney}} + q_{\text{Bgd}}\,\!$
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 99mTc-DTPA renogram is shown in the following figure, to assist you in comparing them with the predictions from compartmental analysis:

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

Analysis of a MAG3 renogram

## 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.

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.

## 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:

• NKidney(t): background corrected renal count rate, and
• NBgd(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 < t0:

$N_{\text{Kidney}}(t) = UC \int_0^t N_{\text{Bgd}}(t) \cdot dt$

where UC in the kidney uptake constant. This constant is related to that kidney's clearance rate, and we can therefore write:

$\text{LK Relative Fn.} = \frac{\text{LK Update Constant}}{\text{LK} + \text{RK Uptake Constant}}$

and

$\text{RK Relative Fn.} = \frac{\text{RK Uptake Constant}}{\text{LK} + \text{RK Uptake Constant}}$

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.