Physikalische Grundlagen der Nuklearmedizin/ Dynamische Studien in Nuklearmedizin

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Inhaltsverzeichnis 1 Einleitung 2 Abteilungsanalyse 3 Einabteilungsmodell 4 Zwei Abteilungs Modell -(geschlossenes System) 5 Two Compartment Model - Open Catenary System 6 Zweiabteilungsmodell - offenes mamillares system 7 Models with Three Compartments 8 Glomerular Filtration Rate 9 Renography 10 Background Subtraction in Renography 11 Relative Renal Function

[Bearbeiten] Einleitung

Die ist das 13te Kaptitel des Wikibooks Physikalische Grundlagen der Nuklearmedizin

Der Stoffwechsel einer Substanz im menschlichen Körper ist das Ergebniss verwobener dynamischer Prozesse welche die Absorption, Verteilung, Verwendung, den Abbau bis hin zur Ausscheidung der Substanz umfassen. Die Messung nur eines dieser Paramter kann zu Ergenissen führen die eine Erkrankung anzeigen, jedoch nicht ihre genaue Ursache. Detailiertere Information über die Ursachen kann aus der Kenntniss des gesamten Stoffwechselssystems erhalten werden. Eine Methode besteht in der rechnerischen Simulation des Physiologischen Systems. Die Ergebnisse dieser Ansätze schliessen die Erzeugung einer Darstellung des gesammten Systems sowie ein Verständniss der Interaktionen zwischen seinen Komponenten ein. Dieser Weg besteht üblicherweise in folgenden Schritten.

1. Beschaffung Experimenteller Messdaten, über das zu simulierte System, durch Verwendung von mit entsprechenden Tracern markierten Substanzen.
2. Vergleich der Experimentellern Ergebniss mit den aus der Simulation erhaltenen Daten
3. Ändern der Parameter der Simulation bis diese beiden Datensätzte so gut wie möglich übereinstimmen, wobei Verfahren wie die Methode der kleinsten Quadrate, Maximum-Likelihood-Methoden

und ]Monte-Carlo-Simulationen eingesetzt werden.

Die allgemeinen Annahmen für diesen Ansatz sind:

• Die Hinzufügung des Tracers verändert das Verhalten des Systems nicht
• Die zu untersuchende Substanz ist während des gesammten Prozesses erhalten
• Der Tracer ist während des Prozesses, bis auf seinen Radioaktiven Zerfall, erhalten
• Das System ist im stationären Gleichgewicht.

Es gibt zwei wesentliche Modelarten:

• • Deterministisch wobei analysitsche ausdrücke verwendet werden um das exakte Zeiterhalten eines Tracers in jedem Teil des Systems zu beschreiben. Die mathematischen Ausdrücke sind meist Exponentialfuinktionen.

• • Stochastisch wobei das Verhalten des Systems durch Zufallsprozesse bestimmt wird die durch Wahrscheinlichkeitsfunktionen beschrieben werden.

There are two major types of model:

Wir werden uns weiter unten mit deterministischen Modellen im Detail beschäftigen.

[Bearbeiten] Abteilungsanalyse

Diese Art der deterministischen Analyse verucht das Stoffwechselsystem in eine Anzahl mit einander verbundener Ableitungen zu unterteilen. Wobei eine Abteilung als anatomische, physiologische, chemische oder physikalische Unterteilung der Systems verstanden werden kann. Man nimmt vereinfachend an, dass der traces innerhalb eines jeden Abteils gleichmässig verteilt ist. Das einfachstes System ist das Einabteilungsmodell. Wir beginnen unsere Behandlung des Themas mit diesem einfachen Modell und denen es später auf Komplexere Modelle aus und arbeiten uns dann zu den komplexeren vor. Die ersten modelle brachen wir im wesentliche um das Handwerkszeug zu erlernen, die späteren jedoch sind unmittelbar für die Nuklearmedizin relevant für dynamische Studien in der Nuklearmedizin, die Messdatenefassung und Analyse.

Man beachte das eine Tabelle heruntergeladen werden kann, mit deren Hilfe man mit den einzelnen Modellen interaktiv spielen kann.

[Bearbeiten] Einabteilungsmodell

In der folgenden Abbildung der ist der Fluss eines Tracers durch die Blutgefässe nach einer idealen Bolusinjektion beispielhaft gezeigt. Die Abteilung ist geschlossen bis auf den Einfluss und Ausfluss der zu Untersuchen Substanz (hier Blut) und der Tracer wir in die Abteilung injeziert. In userer Theorie nehmen wir das sich der Tracer sofort gleichmässig in der Abteilung verteilt sobald er injeziert wird. Seine Menge in der Abteilung wird mit der Zeit abnehmen, abhängig von der Rate des Ausflusses. Die in der Abbildung benutzten Variablen sind:

q: die Menge des Tracers innerhalb der Abteilung zu einer gegebenen Zeit t und
der Ausfluss F

Wir können die relative Abnahme k als Verhältniss dieser beiden Größen definieren und erhalten somit:

dies können wir umschreiben zu:

Ohne auf die mathematische Details einzugehen, (die denen bei der Ableitung des Gesetzes der radioaktiven Zerfall sehr ähnlich sind) lautet die Lösung dieser Gleichung:

q = q₀e ^{− kt}

wobei q₀< die Menge des Tracers bezeichnet die zum Zeitpunkt, t = 0 vorhanden ist.

Die Gleichung ist unten aufgetragen um den Einfluss der reltiven Abnahme k darzustellen:

Graphische Darstellung des zeitlichen Verlaufs der enthaltenen Menge des Tracers, q für einen großen und einen kleinen Wert der relativen Abnahme k.

Der Graph zeigt, dass die Menge des Tracers in der Abteilung nach der Injektion exponentiell mit der Zeit abnimmt, wobei die Abnahmerate vom Ausfluss abhängig ist, wie man intuitiv erwarten kann.

[Bearbeiten] Zwei Abteilungs Modell -(geschlossenes System)

Ein etwas aufwendigere, doch immernoch recht einfache, Klasse von Modellen sind diejenigen die auf zwei Abteilungen aufbauen. In einem geschlossenem System bewegt sich der Tracer zwischen zwei System ohne das insgesamt eine zu oder Abnahme der Menge des Tracers auftritt. Dies ist in der folgenden Abbildung dargestellt.

Hieraus ergeben sich sofort zwei das System beschreibende Gleichungen,

und

Da die Gesamtmenge des Tracers erhalten ist gilt ferner,

q₁ + q₂ = konstant = q₀

und somit,

Wobei das Vorzeichen andeutet, dass eine Abnahme des Tracers in Abteilung #1 eine Zunahme in Abteilung #2 und umgekehrt bedeutet. Man betrachte nun die Situation wie sie in der obigen Abbildung gezeigt ist, wobei der Tracer zum Zeitpunkt t = 0 injeziert wird. Zu diesem Zeitpunkt gilt daher:

and, initially,

und

Die Lösungen dieser Gleichungen sind:

und

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

[Bearbeiten] 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:

In this model,

Die Lösungen dieses Gleichungssystems sind:

und

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:

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.

[Bearbeiten] Zweiabteilungsmodell - offenes mamillares system

Dieses Modell ist equvatent zu dem oben betrachteten Zweiabteilungsmodell, wobei ein zusätzlicher Abfluß aus einer Abteilung vorhanden ist:

In diesem Falle gilt,

Bei t = 0:

und am Anfang

Die Lösungen dieser Gleichungen sind:

und

wobei

The behaviour of q₁ and q₂ 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,
  • k₁₂: clearance rate from plasma into the gland, and
  • k₂₁: leakage rate from the gland into the plasma.

[Bearbeiten] 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:

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

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

The equations used for the figure above are:

where l₁ and l₂ are constants related to the fractional turnovers, and A₁ thru A₅ are also constants such that:

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 q₁, so that:

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.

[Bearbeiten] Glomerular Filtration Rate

The Glomerular Filtration Rate (GFR) is generally regarded as the most important single index of renal function. It is particularly important in assessing the presence and severity of kidney failure, and hence can indicate for instance whether haemodialysis therapy should be considered.

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, in addition to the disturbing and potentially hazardous impact on the patient. Creatinine clearance has been widely used for routine GFR assessment, as a result. However, while the latter method gives similar results as the former 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 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 conjuction with ¹²³I-PAH 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 behavious 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.

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 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:

• t₁: time from injection for the first blood sample, usually 120 minutes,
• b₁: counts 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 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 the what the plasma counts would be upon instantaneous mixing of the tracer throughout the patient's plasma compartment, i.e.

ln(b₀) = ln(b₁) + m(t₁ − t₀)

und

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 is then given by the following equation:

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

Patient A

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

This patient's ⁵¹Cr-EDTA clearance was determined to be 38.8 ml/min, which equals 25.9 ml/m²/min when corrected for the area of their body surface - for standarization purposes. 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 35.2 ml/m²/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 haaving been without steroid therapy for this period. The result was 36.2 ml/m²/min indicating successful treatment.

Patient B

Sample	Counts
Background	425
b1 at 122 mins	3,103
b2 at 250 mins	1,390
Standard	104,600

This patient had a high blood pressure and a renal involvement required confirmation. The clearance however was 117.3 ml/min, or 59 ml/m²/min, which is well within the normal range. The kidneys were therefore excluded from the investigation of the patient's condition.

[Bearbeiten] 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 plevis, 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, 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:

• When t > t₀:

where l₃ is related to k₅₆. 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:

q_Niere = q₄ + q₅

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

The following equation can be derived on this basis:

q_Bdg = b₁q₁ + b₃q₃

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 und b₃ = 0.02 und

q_Renogramm = 0.5b_Niere + q_Bdg

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 renogram is shown in the following figure, to assist you in comparing them with the predictions from compartmental analysis:

Datei:NM14 58.jpg

Datei:NM14 59.gif

[Bearbeiten] 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, or 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.

[Bearbeiten] 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.

wobei LN and RN sich entsprechend auf die rechte und linke Niere beziehen.

Suppose that:

• N_Niere(t): background corrected renal count rate, and

• N_Bdg(t): count rate from an intravascular region of interest.

It should be apparent at this stage that:

We can therefore conclude that in the initial phase of the renogram, i.e. when t < t₀:

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

und

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:

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.