A novel procedure for determining the optimal MLC configuration parameters in treatment planning systems based on measurements with a Farmer chamber

Let us first briefly re-examine the regular sweeping gap field (SG) in which a given 'sweeping gap' or 'sliding slit' with all leaf pairs perfectly aligned travels a given distance d at constant speed. The actual dose delivered at the centre of the sweeping gap field has two distinct contributions (Kim et al 2001, Siebers et al 2002): (i) the total primary fluence composed of the direct exposure to the source of primary radiation and the transmission through the MLC leaves and (ii) the scatter of the MLC, which contributes only a small fraction (Kim et al 2001) of the total leakage from the MLC and is usually not modelled in commercial TPSs. Consequently, the average dose $D_{\rm SG}$ at the centre of the field can be considered to be proportional to the total primary fluence $\phi_{\rm SG}$ produced in the area under a leaf pair (Arnfield et al 2000), that is,

$$\begin{equation} D_{\rm SG} = k\, \phi_{\rm SG} \, , \end{equation} \tag{ 1 }$$

where k is a constant that relates fluence values to dose units. The total primary fluence can be computed from transmission maps following an approach similar to Yu (1998). Thus, because of their symmetry and constant leaf speed, the delivered primary fluence for these tests $\phi_{\rm SG}$ can be approximated as the integral of the transmission map T(x, y) under a leaf pair with a given width $w_{\rm leaf}$ and across a distance d in the leaf motion direction:

$$\begin{equation} \phi_{\rm SG} = \int_0^{w_{\rm leaf}} \int_{-d/2} ^{d/2} T(x,y) \,dx \,dy \, . \end{equation} \tag{ 2 }$$

To characterise the sweeping gap beams the dosimetric leaf separation or dosimetric leaf gap (DLG) is often used. This parameter accounts for the increased transmission through the rounded leaf ends as well as for the MLC calibration and can be modelled by retracting the leaves a certain offset δ equal to half the DLG, which results in an effective gap that can be expressed as:

$$\begin{equation} {\rm gap}_{\rm eff} = {\rm gap}+ {\rm DLG} = {\rm gap} + 2\,\delta \ . \end{equation} \tag{ 3 }$$

The fluence $\phi_{\rm SG}$ can be split into two different contributions. The first contribution corresponds to the fluence across open leaves with an effective gap size gap$_{\rm eff}$. The second contribution is the integrated transmission T across the closed leaf section, which depends on both the distance d travelled by the sweeping gap and the gap size. Hence, for a certain leaf of width $w_{\rm leaf}$, $\phi_{\rm SG}$ can be expressed as:

$$\begin{equation} \phi_{\rm SG} = w_{\rm leaf} \, {\rm gap}_{\rm eff} + w_{\rm leaf} \left( d - {\rm gap}_{\rm eff} \right) T \, . \end{equation} \tag{ 4 }$$

Consequently, $D_{\rm SG}$ depends linearly on the effective leaf gap, as described by many authors:

$$\begin{equation} {D_{{\rm{SG}}}} = k{\mkern 1mu} {\phi _{{\rm{SG}}}} = k{\mkern 1mu} {w_{{\rm{leaf}}}}{\mkern 1mu} \left( {{\rm{ga}}{{\rm{p}}_{{\rm{eff}}}}{\mkern 1mu} \left( {1 - T} \right) + d{\mkern 1mu} T} \right)\;. \end{equation} \tag{ 5 }$$

Therefore, $D_{\rm SG}$ also depends linearly on the nominal leaf gap:

$$\begin{equation} D_{\rm SG} = \lambda \,{\rm gap} + \gamma \ , \end{equation} \tag{ 6 }$$

where

$$\begin{eqnarray} \lambda & = k \, w_{\rm leaf} \,\left( 1-T \right) \ , \end{eqnarray} \tag{ 7a }$$

$$\begin{eqnarray} &&\gamma {\rm{ }} = k{\mkern 1mu} {w_{{\rm{leaf}}}}{\mkern 1mu} \left( {2\delta \left( {1 - T} \right) + d{\mkern 1mu} T} \right)\;. \end{eqnarray} \tag{ 7b }$$

These parameters λ and γ can be experimentally obtained and k and δ can be determined as:

$$\begin{eqnarray} k & = \frac{\lambda}{w_{\rm leaf} \,\left( 1-T \right)} \ , \end{eqnarray} \tag{ 8a }$$

$$\begin{eqnarray} \delta & = \frac{1}{2} \left( \frac{\gamma}{\lambda} - \frac{d \, T}{1 - T} \right) \ . \end{eqnarray} \tag{ 8b }$$

Let us consider next the asynchronous sweeping gap field (aSG) in which adjacent leaves are displaced from each other a certain amount s as illustrated in figure 1, and all leaf pairs move at constant speed while keeping the same nominal gap and MLC shape (Hernandez et al 2017). By introducing this shift between adjacent leaves, the tongue and groove regions are exposed, effectively reducing the total primary fluence with respect to the SG test by a certain amount $\Delta \phi_{\rm TG}(s)$ at each side of the leaves:

Zoom In Zoom Out Reset image size

$$\begin{equation} \phi_{\rm aSG} (s) = \phi_{\rm SG} - 2 \Delta \phi_{\rm TG}(s) \, . \end{equation} \tag{ 9 }$$

Thus, for the aSG fields:

$$\begin{equation} D_{\rm aSG}(s) = k \left( \phi_{\rm SG} - 2 \Delta \phi_{\rm TG} \right) = D_{\rm SG} - 2 k \Delta \phi_{\rm TG}(s) \, . \end{equation} \tag{ 10 }$$

For s = 0 no tongue-and-groove effect is present and $D_{\rm aSG}(0)$ corresponds identically to the usual sweeping gap test $D_{\rm SG}$. The fluence reduction caused by the tongue-and-groove can be readily obtained from equation (10) as:

$$\begin{equation} \Delta \phi_{\rm TG} (s) = \frac{D_{\rm aSG}(0)-D_{\rm aSG}(s)}{2 \, k} \, . \end{equation} \tag{ 11 }$$

The previous expression is the generalisation of the function A(s) used previously (Hernandez et al 2018) in terms of fluence to facilitate a more comprehensive and general interpretation.

The MLC model in the RayStation TPS is characterised by a set of user-definable configuration parameters (RaySearch Labs 2017). A brief description of these parameters is given here:

• Transmission T: A single configuration value T for the transmission is used as input for defining the model. Thus, this transmission value should be taken as an average of the interleaf and intraleaf transmissions.
• Tongue-and-groove width $w_{\rm TG}$: The tongue-and-groove region is centered at the leaf side and the width $w_{\rm TG}$ is added both outwards (tongue) and inwards (groove). A transmission value of $\sqrt{T}$ is assigned to this tongue-and-groove region, which has a total width of twice the $w_{\rm TG}$.
• Leaf tip width $l_{\rm tip}$: The width of a region starting at the leaf tip where a higher transmission value is used to model the increased transmission through the rounded leaf end. In this region a transmission of $\sqrt{T}$ is assigned and neither the tongue nor the groove are modelled.
• Position offset: The distance between the DICOM positions of the leaf tip and radiological position used for dose calculations can be adjusted with the parameters MLC x-position offset, gain and curvature. The x-position offset ($x_{\rm off}$) shifts leaf positions a constant amount regardless of its position, while the parameters gain and curvature allow for the introduction of a second order correction that depends on the leaf position. Thus, at a certain off-axis distance x the position offset can be expressed as (RaySearch Labs 2017):

  $$\begin{equation} \text{Position offset} = \pm x_{\rm off} + {\rm gain} * \, x \pm {\rm curvature} * \, x^2 \, , \end{equation} \tag{ 12 }$$

  where the ± sign depends on the MLC bank considered. At the central axis (x = 0) the position offset is equal to $x_{\rm off}$.

2.3.1. Sweeping gap tests (SG)

Let us investigate the doses for the sweeping gap tests as calculated by the MLC model described in the previous section. These measurements are typically carried out at the central axis, therefore the gain and curvature parameters can be omitted and the position offset equals $x_{\rm off}$. The fluence $\phi_{\rm SG}$ delivered by the SG field tests can be obtained integrating the transmission map for the central leaf (as indicated in Eq 2). This is equivalent to calculating $\phi_{\rm SG}$ as the sum of the products of the area and the transmission for all the regions under the central leaf as depicted in Fig 2(a):

Zoom In Zoom Out Reset image size

$$\begin{equation} \phi_{\rm SG} = w_{\rm leaf} \left( {\rm gap} + 2 \, x_{\rm off} + 2 \, \sqrt{T} \, l_{\rm tip} + T \left( d - {\rm gap} - 2 \, x_{\rm off} - 2 \, l_{\rm tip} \right) \right) \ . \end{equation} \tag{ 13 }$$

As introduced in equation (4) the dose produced by a sweeping gap without tongue-and-groove is indistinguishable from a simple MLC model that uses an effective leaf gap to account for the increased transmission through the rounded leaf tip. From equations (4) and (13) it can be deduced that (see details in Supplementary Materials (stacks.iop.org/PMB/65/155006/mmedia)):

$$\begin{equation} {\rm gap}_{\rm eff} = {\rm gap} + 2 \,x_{\rm off} + 2 \, \, \frac{\sqrt{T}-T}{1-T}\,l_{\rm tip} \ \ , \end{equation} \tag{ 14 }$$

where the last term can be interpreted as twice an offset δ_i introduced by the leaf tip model:

$$\begin{equation} \delta_i = \, \frac{\sqrt{T}-T}{1-T}\,l_{\rm tip} \end{equation} \tag{ 15 }$$

And the total offset δ, as defined in equation (3), can be related to the configuration parameters in the RayStation MLC model as:

$$\begin{equation} \delta = x_{\rm off} + \delta_i \, = \,x_{\rm off} + \frac{\sqrt{T}-T}{1-T}\,l_{\rm tip} \ \ . \end{equation} \tag{ 16 }$$

Hence, the calculated doses for the SG tests are equal to the doses predicted when no leaf tip model is considered and the leaf positions are shifted by a distance $\delta = x_{\rm off} + \delta_i$. The parameter $x_{\rm off}$ can be understood as an offset due to the MLC positioning calibration ($\delta_{\rm cal}$) and δ_i can be interpreted as an intrinsic offset providing the same equivalent fluence as the additional transmission at the leaf tip considered by the MLC model in the TPS.

2.3.2. Asynchronous sweeping gap tests (aSG)

To investigate the fluence reduction caused by the TG effect, the fluence for the aSG fields is evaluated. In this model three different regions can be distinguished depending on the relative amounts of s, $l_{\rm tip}$ and gap^*, defined as ${\rm gap}^* = {\rm gap}+2 x_{\rm off}$.

• (a)  
  $0 < s \leq l_{\rm tip}$: since no TG is modelled at the leaf tip, there is no fluence reduction for distances between adjacent leaves smaller than the leaf tip width, hence $\phi_{\rm aSG}(s) = \phi_{\rm SG}$ and $\Delta \phi_{\rm TG}(s) = 0$  .
• (b)  
  $l_{\rm tip} < s \leq {\rm gap}^* + l_{\rm tip}$. This situation corresponds to the illustration in figure 2(b). Using equation (2) and accumulating the fluence under the central leaf pair, the fluence can be computed with respect to the $\phi_{\rm SG}(s)$ given in equation (13) as (see details in Supplementary Materials):

  $$\begin{equation} \phi_{\rm aSG}(s) = \phi_{\rm SG} - 2(s - l_{\rm tip}) \left( 1 - 2 \, \sqrt{T} + T \right) w_{\rm TG} \ . \end{equation} \tag{ 17 }$$

  And the fluence reduction results into:

  $$\begin{equation} \Delta \phi_{\rm TG}(s) = \frac{\phi_{\rm SG}-\phi_{\rm aSG}(s)}{2} = (s - l_{\rm tip}) \left( 1 - 2 \, \sqrt{T} + T \right) w_{\rm TG} \,. \end{equation} \tag{ 18 }$$

  This expression shows that in this region the fluence reduction increases linearly with the excess distance over the leaf tip. The slope of the linear relationship is a known function of the transmission T and the width $w_{\rm TG}$. Note that the fluence reduction caused by the TG effect does not depend on the gap size of the sweeping gap under consideration.
• (c)  
  ${\rm gap}^* + l_{\rm tip} < s$. In this region opposing leaves begin to interdigitate and the fluence reduction reaches a maximum that no longer depends on the distance s between neighbouring leaves:

  $$\begin{equation} \phi_{\rm aSG}(s) = \phi_{\rm SG} - 2\,{\rm gap}^* \left( 1 - 2 \, \sqrt{T} + T \right) w_{\rm TG} \ . \end{equation}$$

  Therefore, the fluence reduction is constant and equal to

  $$\begin{equation} \Delta \phi_{\rm TG}(s) = {\rm gap}^* \left( 1 - 2 \sqrt{T} + T \right) w_{\rm TG} \,. \end{equation}$$

The calculated fluence reduction for any gap as contiguous leaves are shifted a distance s is represented in figure 3. The curve starts with a flat segment where there is no fluence reduction since the MLC model does not take into account the TG at the leaf tip. As the distance from the leaf tip increases, the fluence reduction increases linearly with a slope that depends on the transmission and the tongue-and-groove width. Finally, a plateau of fluence reduction is reached when leaves interdigitate.

Zoom In Zoom Out Reset image size

In this section the procedure for the determination of the optimal MLC configuration parameters in RayStation is provided. The procedure is based on the measurements of the SG and aSG tests using a Farmer-type ionisation chamber placed at the central axis and the previously derived equations. The length of the Farmer chamber makes it notably suited for these measurements because we want to determine the average doses and this chamber spans several leaves (Hernandez et al 2017).

• (a)  
  The average transmission T is obtained as the ratio of measurements of a field with all leaves closed and an open field.
• (b)  
  A set of measurements of sweeping gaps with different gap sizes is measured to obtain the constant of proportionality between the dose and the nominal gap (λ) and its y-intercept (γ). The constant k that relates fluence to dose can be obtained from λ using equation (8a) and the parameter δ can be computed using equation (8b).
• (c)  
  The asynchronous sweeping gap fields are measured with different distances between neighbouring leaves. The experimental fluence reduction $\Delta \phi^{\rm \textrm{exp}}_{\rm TG}(s)$ can be obtained from the sweeping gap doses and the previously obtained constant k (see equation (11)).
• (d)  
  The parameters $l_{\rm tip}$ and $w_{\rm TG}$ are obtained by fitting the fluence reduction curve $\Delta \phi_{\rm TG}$, with the characteristics shown in figure 3, to the experimental fluence reduction $\Delta \phi^{\rm \textrm{exp}}_{\rm TG}(s)$.
• (e)  
  Finally, the x-offset is computed using equation (16).

In the last step we omitted the gain and curvature because dose calculations at the central axis are independent of these parameters. Gain and curvature can be set to zero as a first approximation, but very small values can be carefully used to fine tune the MLC positions. The gain parameter has no influence on the doses from the sweeping gaps because it shifts the leaf positions in the same direction for both MLC banks and it can be used to compensate for linear discrepancies in the MLC positioning calibration. The curvature parameter directly affects the sweeping gap doses because of the different sign for each MLC bank in equation (12), which for a positive curvature produces a gradual increment in the offset (and in the DLG) at increasing off-axis positions. Thus, the curvature parameter can be assessed in several ways:

• (a)  
  From the dose measurement of the SG tests at different off-axis positions. If a different δ value is obtained, this will result in different $x_{\rm off}$ values (see equation (16)) and these differences can be accounted for using equation (12) and an appropriate curvature parameter.
• (b)  
  From a measured dose profile along the leaf movement direction (for instance with a detector array or film dosimetry) for a SG beam with a small gap size, e.g. a 2 mm or 5 mm gap. The curvature parameter can then be adjusted to fit the calculated profile to the measured profile.

The MLC model assessed corresponds to version 9A of the RayStation TPS and the dose calculation algorithm was a collapsed cone convolution. Dose calculations were carried out with both a 1 mm and a 2 mm grid size.

The measurement procedure for determining the optimal configuration parameters was carried out on two TrueBeam linear accelerators (Varian Medical Systems) equipped with a High Definition 120 MLC (HDMLC) and a Millennium120 MLC. Photons with nominal energies of 6 MV with flattening filter (6WFF) and without flattening filter (6FFF), 10 MV (10WFF) and 15 MV (15WFF) with flattening filter.

All measurements were carried out with a Farmer ionisation chamber placed in a water phantom at 10 cm depth and with a source to surface distance of 90 cm.

The following test fields were measured, with jaws set to a 10 cm x 10 cm square field:

• (a)  
  Transmission fields for both leaf carriages with leaves closed behind the jaws.
• (b)  
  Regular sweeping gap tests for gap sizes 2, 5, 10, 20 and 30 mm. The leaves moved from a position where the gap center was at -60 mm to a position where the gap centre was at +60 mm, travelling a total distance d  =  120 mm.
• (c)  
  Asynchronous sweeping gap tests for a 20 mm gap size with shifts between adjacent leaves s from 0 to 10 mm in 1 mm steps and from 10 to 30 mm in 2 mm steps.
• (d)  
  Asynchronous sweeping gap tests for gaps 5, 10 and 30 mm with several shifts s. These fields were used only to validate that the fluence reduction due to the tongue-and-groove is independent of the leaf gap used and that the same curve $\Delta \phi^{\rm \textrm{exp}}_{\rm TG} (s)$ is obtained regardless of the gap size.

From these measurements the optimal parameters T, $l_{\rm tip}$, $w_{\rm TG}$ and $x_{\rm off}$ were obtained and compared with a set of initial values obtained following the manufacturer recommended procedure (Savini et al 2016), which consisted of profile measurements for a set of static MLC defined fields containing square fields and MLC fields that abutted at the leaf side or with abutting leaf tips.

Finally, the optimal values obtained earlier were validated in a set of clinical plans. In particular, ten stereotactic clinical plans with small target volumes (ranging between 1 and 10 cc) were selected that were considered particularly challenging for the MLC model. The mean MLC gap ranged between 4 and 18 mm and the mean tongue and groove index (Vieillevigne et al 2019), defined as the ratio of the tongue and groove length exposed over the corresponding gap, ranged from 0.04 to 0.38. All plans were delivered using a 6FFF energy and the HDMLC onto a solid water plastic phantom (Phantom model 036A from CIRS Inc.) containing an EBT3 radiochromic film placed at the isocentre plane. The films were calibrated in absolute dose and handled according to a well establish protocol (Lewis et al 2012). All plans were then calculated with both the initial and optimal set of parameters. A planar dose analysis using the gamma method with a 2 %-2 mm global criteria and a dose threshold set at 10% of the maximum dose was used for comparison.

No differences were found between the 2 mm and 1 mm dose calculation grid sizes. The measured average transmission was 1.03 %, which is in close agreement to values reported in other studies (Kim et al 2018).

Results for the SG tests are shown in figure 4(a). A linear regression fit (r² > 0.999 9) was carried out on the dose of the SG tests versus the nominal gap, with a proportionality constant λ = 0.614 Gy/(MU mm) and a y-intercept γ = 1.015 Gy/MU. Using equation (8a) a constant k = 0.248 2 Gy/(MU mm²) was obtained. The offset δ was calculated from equation (8b) as 0.205 mm, which is equivalent to a DLG parameter of 0.41 mm.

Zoom In Zoom Out Reset image size

The measured doses for the aSG tests for different gap sizes are represented in figure 4(b) as a function of the distance between adjacent leaves s. The figure clearly illustrates the dose reduction caused by the tongue-and-groove effect as leaf sides are increasingly exposed. The experimental fluence reduction $\Delta \phi^{\rm exp}_{\rm TG}(s)$ can be readily computed for all the gap sizes using equation (11) and has been represented in figure 5(a). The curves obtained for different gap sizes coincide for s < gap, demonstrating that the fluence reduction caused by the tongue and groove is independent of the gap size and depends only on the amount of leaf side exposed.

Zoom In Zoom Out Reset image size

To derive the optimal configuration parameters we used the $\Delta \phi^{\rm exp}_{\rm TG}(s)$ for the 20 mm gap, depicted in figure 5(b). The experimental fluence reduction exhibits a linear segment for s values between 5 mm and 20 mm. A linear regression fit was carried out in this region and a and slope of 0.322 7 mm was obtained. Using this slope and the transmission value T  =  1.03 % the tongue and groove width parameter $w_{\rm TG}$ was computed with equation (18) as $w_{\rm TG}$  =  0.43 mm.

The leaf tip width parameter $l_{\rm tip}$ was also determined through the $\Delta \phi^{\rm exp}_{\rm TG}(s)$ curve. We estimated it as the intersection of the extrapolated linear fit (depicted with a dashed blue line in figure 5(b)) and the abscissa axis, which yielded a value of $l_{\rm tip}$  =  1.05 mm.

The obtained $l_{\rm tip}$ parameter produces an intrinsic offset δ_i = 0.107 mm (see equation (15)). Using equation (16) the offset parameter $x_{\rm off}$ was calculated from δ and δ_i as $x_{\rm off} = 0.12\ {\rm mm}$.

Finally, the curvature was assessed by repeating the sweeping gap test measurements at off-axis positions of -4 cm and +4 cm and recalculating the δ parameter. The values agreed with the δ value measured at the isocentre to within ± 0.03 mm, which corresponds to differences in the DLG parameter around ± 0.06 mm. These results indicate that the optimal value for gain and curvature can be considered zero within the experimental uncertainties.

The agreement between measured and calculated doses for both the original and the optimal set of configuration parameters was performed in two stages. First the dose agreement for the synchronous and the asynchronous sweeping gaps was evaluated. Then, the agreement in clinical plans was investigated.

3.2.1. Test cases

TPS calculations were carried out for the test fields with both (i) the original set of configuration parameters and (ii) the optimal parameters obtained with the proposed procedure. For each test field the calculated dose was computed as the average dose to a Farmer-type structure in the TPS and compared to the measured dose. Both the calculated and measured doses and the differences between calculations and measurements are represented in figure 6.

Zoom In Zoom Out Reset image size

The original set of parameters produced dose differences of up to 10 % and 15 % for the smaller sweeping gap tests of 10 mm and 5 mm gap, respectively. When no shift was applied to adjacent leaves (s = 0), the doses obtained with the original set of parameters differed only around 2 % from the measured doses except for the smallest sweeping gap of 5 mm, which showed a 5 % difference. However, as adjacent leaves were shifted the dose difference increased and reached a maximum of around s  =  3 mm. As the distance s increased the dose difference slowly decreased, indicating that the original set of parameter overestimated the dose reduction due to the tongue and groove effect.

On the other hand, the optimal set of parameters provided doses in close agreement with measurements, with differences smaller than 1 % for all gaps except for the 5 mm gap, which yielded a maximum difference of 2 %. The largest discrepancies were found for s values around the leaf tip width parameter ($l_{\rm tip}$  = 1.05 mm) and also when the opposing leaves begin to interdigitate ($s \gtrsim \, {\rm gap}$), coinciding with the regions where the fitted $\Delta \phi_{\rm TG}(s)$ curve differed the most from the experimental curve (see figure 5(b)).

Similar dose differences were obtained for the other combination of MLCs and energies, with the optimal parameters providing an excellent agreement with measured doses.

3.2.2. Clinical cases

When the MLC configuration parameters in table 1 for the 6FFF beam and the MLCHD were used in the clinical plans the gamma passing rate improved significantly with respect to the passing rates obtained with the initial configuration parameters (see figure 7).

Zoom In Zoom Out Reset image size