Basic Physics of Nuclear Medicine/Dynamic Studies in Nuclear Medicine
Contents
 1 Introduction
 2 Compartmental Analysis
 3 Single Compartment Model
 4 Two Compartment Model  Closed System
 5 Two Compartment Model  Open Catenary System
 6 Two Compartment Model  Open Mamillary System
 7 Models with Three Compartments
 8 Glomerular Filtration Rate
 9 Renography
 10 Background Subtraction in Renography
 11 Relative Renal Function
 12 Kidney Depth Correction
Introduction[edit]
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 interrelated 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[edit]
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 plugin available, named Compartments_TP, which provides simulations of a number of additional models.
Single Compartment Model[edit]
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:
F: the outflow.
We can define the fractional turnover, k, as the ratio of these two parameters, i.e.
which can be rewritten as:
Without going into the mathematical details (which are similar to the derivation of the radioactive decay law!), the solution to this equation is:
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:
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[edit]
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:
Therefore,
and  . 
Since there is no loss of tracer from the system,
Therefore,
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,
and 
and, initially,
and 
The solutions to these equations are:
and
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:
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[edit]
This is an extension of the single compartment model considered earlier with two compartments connected in series, as shown in the following figure:
In this model,
and 
The solutions to these equations are:
and
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}:
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[edit]
This model is equivalent to the closed two compartment system considered above with the addition of an outflow from one compartment:
In this case,
and 
At t = 0:
and 
and, initially
and 
The solutions to these equations are:
and
where
and 
The behaviour of q_{1} and q_{2} is illustrated in the figure below:
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[edit]
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:
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:
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 nonuniformity 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 
Ultimately, all the tracer will end up in the urine compartment.
The equations used for the figure above are:
where l_{1} and l_{2} are constants related to the fractional turnovers, and A_{1} through A_{5} are also constants such that:
and 
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:
The time dependence of this plasma concentration will vary in the same way as q_{1}, so that:
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[edit]
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}CrEDTA. 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 halflife of about 28 days and decays by 100% electron capture into stable vanadium, emitting monoenergetic (320 keV) gammarays in about 10% of the transformations. In addition, ^{51}CrEDTA 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}CrEDTA clearance is 110 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 gammarays 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 falloff 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.
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:
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.
as illustrated in the following figure:
Therefore, we can write:
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.
which results in
when the standard injection is diluted in 1 liter. The clearance (in ml/min) is then given by the following equation:
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}CrEDTA 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 retested 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 singleexponential fit to this later phase.
It is important to appreciate that the clearance of ^{51}CrEDTA determined as described does not equate directly with the Glomerular Filtration Rate (GFR) since the method assumes a single exponential dependence. ^{51}CrEDTA clearance results are therefore typically corrected by a factor, either empirically or theoreticallyderived, to force them to express the true GFR. Empiricallyderived corrections include those of:
 Chantler (1969), where the ^{51}CrEDTA 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;
 BrochnerMortensen (1972), where a secondorder polynomial is applied
to obtain the GFR. A correction based on a theoretical consideration of the relationship between true GFR and singleexponential 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:
where = 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 singleexponential correction technique, based on BSAscaled clearances has also been introduced (Jodal & BrochnerMortensen, 2008) which is similar to that of Fleming (2007) but provides improved correction in paediatric studies.
Renography[edit]
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:
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}:
 When t > t_{0}:
where l_{3} is related to k_{56}. The time course of the quantity of tracer in each compartment is shown below:
The quantity of tracer in the overall kidney can be obtained by summing the renal parenchyma and renal pelvis curves, so that:
as shown below:
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:
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
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:
and an example from a patient's ^{99m}TcDTPA 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 ^{99m}TcMAG3 renography in a patient with obstructive uropathy:
Background Subtraction in Renography[edit]
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 extrarenal tissues in renogram images.
A potentially better method is to record the activity in the region of a nonfunctioning kidney.
In most patients, however, a nonrenal 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[edit]
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.
and 
where LK and RK refer to the left and right kidneys, respectively.
Suppose that:
 N_{Kidney}(t): background corrected renal count rate, and
 N_{Bgd}(t): count rate from an intravascular region of interest.
It should be apparent at this stage that:
and 
We can therefore conclude that in the initial phase of the renogram, i.e. when t < t_{0}:
where UC in the kidney uptake constant. This constant is related to that kidney's clearance rate, and we can therefore write:
and
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:
and 
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[edit]
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. 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.
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.
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) 

Standard^{[1]}  13.2*(Weight/Height)+0.7  13.3*(Weight/Height)+0.7 
Emory^{[2]}  16.17*(Weight/Height)+0.027*Age(years)0.94  15.13*(Weight/Height)+0.022*Age(years)+0.077
Note: contains a typographical error^{[3]} 
Itoh^{[4]}  14.0285*(Weight/Height)^{0.7554}  13.6361*(Weight/Height)^{0.6996} 
T.Itoh  17.05*(Weight/Height)+0.13  16.55*(Weight/Height)+0.66 
A graphical representation of the difference in kidney depth generated by these equations for a broad range of weighttoheight ratios is shown on the left in the figure below (Note: an age of 50 years was assumed for the Emory formulae):
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 weighttoheight ratios of less than 0.6 to 0.7 and the left kidney is deeper above this ratio range. An important point however is that the overall range of depth differences predicted by these latter three formulae is seen to be no more than ±5 mm.
Using such a range of depth differences for attenuation correction with ^{99m}Tc gamma rays yields the following figure, where a range of correction factors of roughly ±5% can be seen.
It can therefore be concluded that application of the Standard method for kidney depth correction is unlikely to substantially affect estimates of split renal function and that the Emory method can influence raw data estimates by up to roughly ±5%.
 ↑ 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, 2825.
 ↑ 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, 17669.
 ↑ Taylor A, 1994. Formulas to estimate renal depth in adults. Journal of Nuclear Medicine, 35, 20545.
 ↑ Itoh K & Arakawa M, 1987. Reestimation of renal function with 99mTcDTPA by the Gates' method. Kaku Igaku, 24, 38996.