Determining Margins
Margins account for residual uncertainties present during treatment. An appropriate margin accounts for systematic and random treatment errors while minimizing the treated volume to the degree possible.
Systematic Error  Random Error  

Definition  Systematic errors are those errors which have a nonzero mean and accumulate linearly over the course of treatment.  Random errors are those with a mean of zero and the amplitude of which varies from period to another. 
Effect on Dose Distribution  Systematic errors shift the dose distribution relative geometrically. They are much more important than random errors because they result can result in significant, localized under dosing.  Random errors result in "blurring" of the dose distribution with maximal effect at volume margin. 
Common Examples 


Quantifying Error Sources
Errors may be determined on the clinical scale evaluation of the error present in a few cases. See the below example in which 3 patients undergo the same simulation and treatment technique.
Mean Group (Treatment type) systematic error, M, is determined as the mean of each treatment’s error.
Standard Deviation (SD) of group systematic error, Σ, is determined as the standard deviation of the each patient’s mean error.
Mean of random error is taken to be zero by definition.
Standard deviation (SD) of random error, σ, is determined as the root mean square (RMS) of the standard deviation of each patients treatment error.
\begin{equation} \tag{Root Mean Square, RMS} x_{RMS} = \sqrt{\frac{1}{n}(x_1^2 + x_2^2 + … + x_n^2)}\end{equation}
Example: Quantifying Error
Patient 1 Setup Error  Patient 2 Setup Error  Patient 3 Setup Error  

Day 1  2  1  2  
Day 2  1  0  1  
Day 3  3  1  1  
Mean  SD  2  1  0  1  1.33  0.58  Group Systematic Error = Mean of means = 1.11 SD of Group Error = SD of means = 1.02  SD of Random Error = RMS of SDs = 0.88 
Biological Factors Influencing Margins
Several biological factors influence TCP and probability of achieving prescription dose. Although technological limitations mean these factors are not routinely monitored, it is important to be aware of the effects. As biological imaging improves, these factors will become important in creating patient specific margins.
 Probability of tumor cell presence outside gross tumor volume (in CTV)
 Density of tumor cells
 Greater tumor cell density requires a higher dose for control
 Tumor cell radiosensitivity
 Tumor cells with more oxygen are more radiosensitive
 Proximity of normal tissue to target
 A nearby radiosensitive normal structure may merit reduction to prevent unacceptable complications.
Margin Formulas
Several Authors have generated formulae (recipes) for establishing margins based on probability of achieving prescription or tumor control probability (TCP) change.^{1}^{2}^{3}^{4} These generally weight the systematic error component, Σ, more heavily than the random error component, σ, because of their relative impacts on plan quality. Below is one representative margin recipe put forth by Van Herk et al. to assure a minimum dose to CTV is 95% for 90% of patients assuming a dose distribution with perfect conformation.
\begin{equation} \textrm{Margin} \geq 2.5 \Sigma + 0.7 \sigma \end{equation}
Knowledge Test
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