Errors and Margins in your clinical practice

Learning Objectives

1. Understand the concept of systematic and random errors

2. Understand the nature and source of these errors

3. Understand how these errors are incorporated into margins.

Definitions

ICRU Volume Concepts

Classification of Errors

Consequences of Errors

Example

Systematic Error Calculation

• Errors for each fraction over the course of the treatment can be averaged to get the individual mean error.

\[ \mu = \frac {x_1+x_2+x_3+...+x_n} {n} \qquad(2)\]

• The overall mean error is the mean of individual mean errors. Note that this is usually equal to 0 unless something is systematically incorrect about the protocol.

\[ \mu_{pop} = \frac {\mu_1 + \mu_2 + \mu_3 + ... + \mu_n}{n_{pop}} \qquad(3)\]

Systematic Error Calculation

• Population systematic error (𝚺) is the STANDARD DEVIATION of the individual mean errors around the population mean error.

\[ \Sigma = \sqrt\frac {(\mu_1 - \mu_{pop})^2 + (\mu_2 - \mu_{pop})^2 + ... + (\mu_n - \mu_{pop})^2}{n-1} \qquad(4)\]

Also that for a sample, the standard deviation of the mean is an estimate of the population standard deviation.

Random Error Calculation

• For each individual, the standard deviation of the measurements is a representation of the standard deviation of the random error for the patient.

\[ sd_r = \sqrt \frac {(x_1-\mu)^2 + (x_2-\mu)^2 + ... + (x_3-\mu)^2} {n} \qquad(5)\]

• It is important to appreciate that in the above formula mean value is being subtracted or “removed”. In other words, the “systematic error” component is being removed.

Random Error Calculation

• For getting the population random error (𝛔) we obtain the root mean square (RMS) of these standard deviations.

  \[ \sigma = \sqrt \frac { (sd_1)^2 + (sd_2)^2 + ... + (sd_n)^2 } {n} \qquad(6)\]

  Note the critical difference between this and the previous formula is that in RMS mean is not being subtracted.

• Also note that for small sample size this value will be influenced by outliers where the random error is large for one / few patients.

Margins from Errors

Biological Consequences of Margins

Margins for OARs

• Organs at risk: Defined as organs in the irradiated volume that will receive a dose exceeding the tolerance.

• Additional issue is with the organization of the FSU : Parallel / Serial. For serial organ aim is to reduce the Dmax while for parallel dosimetric goal is to reduce volume receiving a dose.

• Key difference from PTV :

  • For OARs adjacent to the target volume “risk” comes when dose moves into the OAR (that is from 1 direction). Systematic errors are multiplied by 1.3 (one - tailed gaussian distribution)

  • For rare OARs enclosed by the target in 2 dimensions “risk” comes from two dimensions. Here we multiply by 2.2

• PRV around a parallel organ increases the volume receiving a dose and therefore evaluating the DVH of a PRV misrepresents the NTCP.

Geometric errors in IGRT

Delineation Errors

Delineation Errors Measurement

• Usual metrics of delineation accuracy (e.g. Dice Similarity Coefficient / Jaccard Coefficient) cannot be used to determine error magnitude.

• Measurements of errors should be done on axial planes by overlaying contours from multiple (blinded observers).

• Planes should be selected without significant change in the gradient in the supero-inferior direction.

• One of the observer contours should be considered as a reference contour and other contours measured w.r.t that to obtain the error.

• Standard deviation of the measured error (𝚺_delineation) should be calculated.

Delineation Error : Example

Delineation on prostate cancer done by different observers. Note that distances in panel B should not be used for error calculation. Image from Tudor et al. 2020

Delineation Error : Margin Calculation

For small number of samples (in this case observers) a rapid approximation of the standard deviation using the following equation:

\[ S = \frac {R} {d_2(N)} \qquad(7)\]

In the above equation, d₂(N) is dependent on the sample size as given by the following table.

Peer Review

P-Charts of major modification during peer review.

• Possibly the only QA process available for target volume in centers with limited number of delineation experts.

• Literature reported margins for delineation error are not easily transportable.

• Centers may not have sufficient trained staff to perform a delineation study.

Automatic Segmentation: Margins

• Automatic segmentation software should be commissioned before implementation.

• Commissioning involves:

  • Description of the dataset used for model training (if done in-house) along with description of the training process and training model metrics.

  • Reporting of the quantitative metrics and qualitative metrics (rating provided by observers) for segmentation accuracy and quality.

• For estimation of error, at least 2 independent experts are required.

• Again measurement in axial plane along the cardinal axes should be done.

• Standard deviation of the error should be calculated for the delineation pairs between automatic segmentation and each expert.

Fusion Errors: Margins

• Again an example of a systematic error unless being used as an adaptive radiotherapy protocol.

• Before determining these errors image registration methodology and image quality metrics for registration should be finalized.

• For deformable image registration, the error calculation will be different as this process forces contours to have the same shape.

• Methodology for rigid registration follows the same approach as in manual / automatic segmentation with the exception that in this case delineation should be standardized (using reference contours). Image registration conditions should be varied.

• Standard deviation should be calculated from the observed values (𝚺_fusion).

Deformation Errors

Deformation Error: Margins

• Delineation of volumes needs to be done on multiple images.

• Displacement of contour between the planning and verification images quantified at various points.

• Mean and standard deviation of the displacement can then be calculated.

  • Systematic error: Standard deviation of means of each case

  • Random error: Mean of the standard deviations

\[ \Sigma_{deform} = \sqrt {\sigma_{deform}^2 (1 + \frac {1}{N_{frac}})}{} \]

In the above equation (𝛔) is the random error component calculated above.

Rotational Errors

Rotational Error Measurement

In the table below we show the relationship between rotational error in each axis and the corresponding translational displacements in the volume.

An important point to keep in mind is that for complex shapes rotational errors can magnify / reduce based on the shape. A “shape factor” may therefore be required.

Rotational Error Margins

• Rotational errors can be decomposed to systematic and random components.

  • Systematic error : Standard deviation of the means of the individual patients.

  • Random error : Means of the standard deviations of the individual patients.

• Rotational errors in the three axes may not be “independant”. In such a situation, it is best to add the errors linearly.

Example calculation of the rotational systematic error in the x axis for rotational error:

\[ \Sigma_x = \sqrt {(d_z*sin(\theta_z))^2 + (d_y*sin(\theta_y))^2} \qquad(8)\]

Surrogate Errors

• These errors occur when a surrogate is used for matching.

• Examples: clips / fiducials placed in tumor.

• Additionally applicable for multi-volume situations were one volume is preferentially matched (e.g. pelvic nodal volume is matched as a surrogate for the primary volume in cervical cancer).

• These errors will have a systematic and random component

Matching Errors

• Matching of verification images can be done automatically / manually / hybrid. This is an important source of error as it is dependant on individuals.

• Ideal registration is very difficult to define and parameters used for matching the results can vary.

• Two approaches:

  • Reference observer based error measurement : Here a single experiencedd observer or a grouped average of observers is considered to be the reference data.

  • Mean registration correction of the observers is considered as the reference instead of a reference standard.

• Requires a dedicated image matching study where in several observers are asked to do matching of images (offline).

Matching Errors: Margins

• Margin for matching errors depends on the measured inter- and intra- observer variance.

• For a multi-patient multi-observer study, there are two variances:

  • Multiple observers for a single patient (𝚻)

  • Single observer for multiple fractions (𝛕)

• The random error depends on the 𝛕 parameter as follows

\[ \sigma_{match} = \sqrt{ \tau^2 \frac{N_{rad}}{1-N_{rad}}} \qquad(9)\]

• The systematic error depends on the T² parameter as follows:

  \[ \Sigma_{match} = T^2 - \sigma_{match}^2 \qquad(10)\]

Technical Delivery Errors

• Technical delivery accuracy is influenced by machine and image guidance system related errors.

• Examples are : MLC / Jaw accuracy, Machine isocenter stability, machine - imager isocenter concordance etc.

• Ensuring the accuracy of these systems is performed by end-to-end quality assurance.

• Generally tolerances of the order of 1 mm or less (corresponding to standard deviations of 0.58 mm) are required for SRS / SRT treatment.

Adding Errors for Margins

• As already highlighted the different errors need to be added in quadrature as they are considered independant.

\[ \Sigma^2 = \Sigma_{delineation}^2+\Sigma_{fusion}^2+\Sigma_{deform}^2+..+\Sigma_{tech}^2 \qquad(11)\]

The above equation shows how systematic errors will need to be added to get a overall value of Systematic error. Similarly the random error will be calculated.

Deriving CTV-PTV Margins

After the systematic and random errors have been calculated then the CTV-PTV margin can be calculated.

If using the Van Herk Formula then the margin is :

\[ PTV = 2.5\Sigma + 0.7\sigma \qquad(12)\]

This will give a 90% confidence of coverage of CTV by 95% isodose.

Corollary:

1. Coverage of the remaining 10% cannot be predicted by this.
2. If the isodose coverage required is different then this formula is not useful.

Factors affecting Margin Calculation

The actual formula for calculation of the margin (CTV - PTV) is given by the following formula:

\[ M = \alpha\Sigma_{total} + \beta[\sqrt {\sigma_{total}^2+\sigma_p^2} - \sigma_p] \qquad(13)\]

Here \(\alpha\) is the constant which is multiplied by the standard deviation of the total systematic error (\(\Sigma\)), \(\beta\) is the constant to be multiplied by the standard deviation of the random error (\(\sigma\)). \(\sigma_p\) is the penumbral width.

Usually the value of \(\alpha\) is 2.5 and \(\beta\) is 1.64

Parameter ⍺

Parameter 𝛃

The parameter corresponds to the covering isodose. For 95% isodose this parameter is equal to 1.64. If another isodose coverage is desired then the value changes.

For modern IMRT measuring the \(\sigma_p\) may not be simple as it cannot be derived from primary beam configurations.

Recommended to measure the distance between the 95% - 80% isodose at various points around the PTV perimeter.

\[ \sigma_p = \frac {d_{95-80}}{0.80} \qquad(14)\]

Hypofractionation

• When fewer fractions are employed the effects of random errors become more pronounced (for e.g. in single fraction treatments, random errors behave as systematic errors).

• However if the standard deviations of systematic and random errors are obtained from large patient studies then these can be used for these treatments also.

• If the prescription isodose coverage at the periphery is < 95% then the dose gradient at the periphery of the PTV becomes steeper. As a result the dose can become substantially more or less with random errors at the periphery. Hence for random errors the constant reduces.

• For very high dose single fraction treatments, zero margins mean that there is going to be a reduced dose at the periphery - however it is understood that even lower dose levels are sufficiently tumoricidal.

Margins from Few Observations

• When the number of fractions used for derivation of margins is less then the standard deviation is larger as the number of observations is less. Hence a correction is needed for the systematic error :

\[ \Sigma_{treat}^2 = \Sigma_{obsv}^2 + \frac{\sigma^2} {N_{patient}} - \frac {\sigma^2} {N_{fractions}} \]

• Similarly for random errors, the actual random error will be less than that calculated (due to correlations between the mean error and values of the random error).

  \[ \sigma_{treat}^2 = \sigma_{pop}^2\frac{N_{fractions}-1}{N_{fractions}} \]

CTV-PTV Margin: Limitations

1. Assumes that the dose cloud is not affected by the changes in the geometric errors - just the position of the dose cloud with respect to the coordinates change.
2. Unless the plan is perfectly conformal the margins around the non-conformal areas are larger than required.
3. When using biological optimization, under-dose to PTV may give low predicted TCP values resulting in unwanted dose at CTV periphery.
4. Tumor clonogen density is not modeled by these margins. Especially in presence of concave target volumes, doses to the periphery or edges of the target volumes may be “enhanced” when they do not need to be.

Conclusions

• Modern IGRT practice requires a thorough understanding of the uncertainties and the source.

• Image guidance reduces but does not eliminate error and inadequate attention to margins may paradoxically worsen tumor control.

• Margin computation in daily IGRT is not a straightforward affair and needs a team based approach for optimal margin reduction strategies to be adopted.

Resources & Acknowledgements

1. Tudor GSJ, Bernstein D, Riley S, Rimmer Y, Thomas SJ, van Herk M, et al. Geometric Uncertainties in Daily Online IGRT Refining the CTV–PTV Margin Calculation for Contemporary Photon Radiotherapy. British Institute of Radiology; 2020. https://doi.org/10.1259/geo-unc-igrt.

2. van Herk M. Errors and margins in radiotherapy. Semin Radiat Oncol 2004;14:52–64. https://doi.org/10.1053/j.semradonc.2003.10.003.

3. The Royal College of Radiologists. On target 2: updated guidance for image-guided radiotherapy. The Royal College of Radiologists; 2021.

4. McKenzie A, van Herk M, Mijnheer B. Margins for geometric uncertainty around organs at risk in radiotherapy. Radiother Oncol 2002;62:299–307