A. Direct-Integration Method

Direct integration in which directly integrate under the actual measured values by a number of methods. It does not give very much information but it does allow to calculate the number of disintegrations rather easily. The most common method used is the trapezoidal method, simply approximating the area by a series of trapezoids.

#### B. Least-square Analysis

An alternative to direct integration of a data set is to attempt to fit curves of a given shape to the data. The curves are represented by mathematical expressions that can be directly integrated. The most common approach is to attempt to characterize a set of data by a series of exponential terms, as many systems are well represented by this form, and exponential terms are easy to integrate. In general, the approach is to minimize the sum of the squared distance of the data points from the fitted curve. The curve will have the form:

$$A\_t = a\_1 e^{-b\_1} + a\_2 e^{-b\_2 t} + \dots \tag{12}$$

The difference of the square between each point and the result of the fitted curve at that point and minimizing this quantity by taking the derivative of this expression with respect to each of the unknowns, ai and bi, and setting it equal to zero. Once the ideal estimates of ai and bi are obtained, the integration of A(t) from zero to infinity is expressed as:

$$\int\_0^\infty A(t)d(t) = \frac{a\_1}{b\_1} + \frac{a\_2}{b\_2} + \dots \tag{13}$$

Here the units of coefficients ai are the same as the units of activity and the integration of it is the cumulated activity and the units of the bi are time�<sup>1</sup> . If the coefficients are the fractions of the administered activity or radiopharmaceuticals, then the area under the curve represents the normalized cumulated activity (e.g. Bq-h/Bq).

#### C. Compartmental Analysis

Biological system where a group of compartments interconnected through transfer rate coefficient. Cumulated activity of the various compartments requires a system of coupled differential equations expressing transfer of the radiopharmaceutical between compartments and elimination from the system. The result to the time-activity curve for each compartment will be a sum of exponentials, not obtained by least-squares fitting each compartment separately, but obtained by varying the transfer rate coefficients between compartments until the data are well fit by the model.

Absorbed dose calculation: There are several ways to calculate absorbed dose, for instance, the analytical, local energy deposition, dose point kernel, dose voxel kernels, Monte Carlo and Tabular approaches. All these methods will produce

different types of outputs, average absorbed dose, absorbed dose maps and voxelbased absorbed dose in which absorbed dose volume histogram can be generated.

### **6.8 Activity quantification**

Activity quantification is very important and this should be determined for a particular ROI/VOI/Structure before Time Activity Curve estimations can generated.
