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This paper describes the algorithms used by the PC3D® for the calculation of doses in patients being treated by radiation therapy. A fine matrix of points is selected and the dose calculated at each point to determine the dose distribution throughout the entire defined 3D matrix volume. The dose to each of these matrix points is calculated by the algorithmic method described in the following sections.
Many planning systems define their algorithms in mathematical short hand symbolism that may mask the actual calculations that are being performed. What may in fact be a simple and trivially inconsequential mathematical procedure and not necessarily an accurate algorithm may be expressed quite impressively as a convoluted symbolic mathematical expression such as the following equation:
D(x,y,z) = N å( W(x’,y’) K(x-x’, y-y’,z) dx’dy’ (1)
V x V
where:
2
K(x-x’,y-y’,z) = Kp(x-x’,y-y’,z)+ å Ri(z)*Ks,i(x-x’,y-y’,z) (2)
i=1
These mathematical expressions shown above can be used to represent the overall algorithms used by the PC3DÒ and are intended to indicate that the algorithm determines the dose D at all (x,y) positions and for all depths z by modifying a normalization factor N by the photon fluence factor W and the dose modifying kernel K, and where K is a further a functions of the equally symbolic mathematical algorithms defined in the then following “where” equation.
However, this very exacting mathematical expression does in fact not give any indication of what actually goes into the “normalization factor” or the “photon fluence factor” or the “dose modifying kernel” and therefor tells you nothing in fact about the actual calculations that are necessary or involved. Upon first viewing this page, one might have noticed the above symbolic representation in the middle of the page and thought “Oh wow, this is going to be a very good, exact mathematical explanation of everything.”
However this elegant symbolic mathematical representation tells nothing of the accuracy or correctness of the “normalization factor” or the “modifying kernel” or the “fluence factor” or what factors are included in the consideration and, without further explanation, the only convolution involved is the convoluted mathematics which has only convoluted and perhaps confused the impressionable mind with spectacular mathematical symbolism which implies an accuracy that may not be there, but which no-one wants to admit to not understanding or recognizing.
Other planning systems may just throw around the word “convolution” as much as possible and mention “pencil beam” every so often and then mask the whole thing with a tirade of mathematical symbolism and an impressive hierarchy of highly paid salesmen and chief executive officers and everyone (well, almost everyone) seems adequately impressed ... some even impressed enough to cause them to spend hundreds of thousands of dollars. Think! Could that be you?
Instead, this paper will try to explain the PC3D® algorithms not only in comfortable conventional mathematical symbolism as presented at the beginning of this paper, but also in a more understandable, precise and clear textual method so that you can clearly understand all of the parameters that are considered and included, their derivation and the inherent accuracy of the algorithmic procedures. This is so that you can more correctly appreciate the elegance, accuracy and completeness of the PC3D® algorithms.
In actual fact, the algorithms are not ultimately manifested in formal mathematical symbolism anyway, but rather in compiled computer code which may be equally (or maybe even more) obscure to many people and which we at Medicalibration intend to keep as proprietary trade secrets anyway, because of the education, research, experience, and the years of time and great expense involved combined with the expertise and professional knowledge and education and decades of man-years of programming that is needed to accomplished the translating of the mathematical description of the physical processes into concrete, functional and reliably accurate computer treatment planning code.
This algorithmic explaination is to help you understand what we include and mathematically what is involved. How we accomplish this task is encoded in many lines of carefully generated computer code. We can tell you what we do, but certainly we do not intend to completely tell you how we do it. The “how” involves millions of bytes of code generated over many decades of man-years of hard professional work and expense.
However, we have often been asked about the algorithms that are used in the PC3D®, so we want to let you know that we have had those years of experience and professional expertise to enable us to select and develop the very best and most accurate procedures needed to quickly and accurately calculate clinical three-dimensional dose distributions and we want to demonstrate that to you and explain those algorithms with this paper.
The approach used by the PC3D® is to take very accurately measured and carefully checked beam information and calculate a modified dose distribution for any specified clinical field and to thus produce results which are exactly as would be measured in a identically shaped clinical field. For this terminology we call that true measurable clinical field dose distribution the “perfect profile”™. We then to further modify that “perfect profile” dose distribution for all of the various parameters associated with the patient and the geometric setup to exactly match each specific clinical situation.
The PC3D® calculations start with and rely on accurate measured data. The algorithms that are used to modify the measured data are based on the physical parameters involved and are carefully designed to reproduce the dose distribution that would be measurable and therefore verifiable in the real-world clinical situation.
No other more simple mathematical calculation procedure, whether using statistical probabilities (Monte-Carlo), pencil beams, or any other arbitrary convolution kernel, can possibly be more exact in recreating a “perfect profile”™ that reproduces the true measured data than the PC3D®.
All other mathematical methods can only be equal to the PC3D® or else they must be less correct.
There are two major contributing sources of dose modification: 1) Beam Originating Modifiers where the source of the modification is within the beam itself and 2) Patient Originating Modifiers where the source of the modification is the variations within the patient to which the beam is being applied.
1) Beam Originating Modifiers (beam configuration and geometry variations that change the dose distribution) that are considered include such things as
a) the beam’s energy and spectrum,
b) the radiation modality (electrons, photons or gamma )
c) focal spot size,
d) head scatter,
e) the effect of various of beam shaping items such as
collimators,
blocks
block penumbra,
edge factors,
block transmission factors,
trays,
wedges,
profile shape
beam hardening (also a function of depth in patient)
edge wedges
compensators
(determined from the patient but applied to the beam)
dose compensators
missing tissue compensators
multi-leaf collimators
independent jaws
upper jaws
lower jaws
2) Patient Originating Modifiers (modifications to the dose distribution caused by the patient and the patient’s setup geometry) that are considered include such things as :
a) the distance from the source to the surface,
b) the skin surface obliquity,
c) the patient’s internal structure composition,
d) the depth into the patient,
e) the densities of the intervening materials and
f) the local density gradients within the patient,
g) the absorption and scattering power of materials in the body,
h) any bolus that may be applied to the patient (considered to be a modifiable part of the patient), plus
I) the effect of missing scatter from areas where the beam may “flash” last the edge of the patient in air and
j) other such dose distribution modifying items that may be associated with the patient or with the setup geometry.
k) the beam spectrum hardening with depth
BEAM ORIGINATING DOSE MODIFYING PARAMETERS
The procedure used by the PC3D® to determine the effect of the beam related beam profiles which is a FOUR step approach that entails:
1) first, the calculation of a family of 3D dose profiles calculated for a standard open field in both the width and length direction of the beam and at multiple depths using the PC3D® four dose kernel integration algorithms (completely described following in this paper),
D(x,y,z) ref = pencil beam integration, reference beam (1)
2) secondly, similar calculation of another 3D family of dose profiles for the same three-dimensions of profile calculation points, but this time for the irregular shaped clinical field using the irregular field or collimators that are to be used in the clinical application and these profiles are calculated using the same PC3D® algorithms,
D(x,y,z)clin = pencil beam integration, clinical beam (2)
3) and finally the ratio of the calculated clinical field, (2) above, to the calculated standard open field, (1) above, is determined at all points and at all depths.
R (x,y,z)clin/ref = D(x,y,z)clin / D(x,y,z)ref (3)
Ratio of PC3D® calculated profiles
4) This ratio, (3) above, is applied to selected actual physically measured library of beam profiles to yield a “perfect profile”™ dose profile specific to the clinical beam to be used in the clinical application of a treatment to the patient..
D(x,y,z)perfect = D(x,y,z)meas * R (x,y,z)clin/ref (4)
where: D(x,y,z)meas is the measured dose profiles for a field of the same configuration as the specified reference field.
This dimensionless ratio (3) is reapplied to the physically measured profiles to yield the actual clinical dose profiles (4) which will then be further modified by the various patient associated parameters.
This four step ratio method to determine the 3D dose profiles accomplishes only the first portion of the overall 3D dose distribution calculation for 3D planning, namely the determination of the “perfect” profiles”™ These profiles are a function of only those parameters that are associated with the beam itself.
Although the dose profiles that are determined as described in the section above yield cross axis profiles that are almost identical to the measured profiles, we use a ratio method of determining the initial profiles which produces results where any possible anomalies that might be in the PC3D® algorithm, no matter how small, are automatically and completely canceled out leaving only the true actual effect of the beam shaping modifiers (i.e.: almost “perfect”). It can be seen that this method will return the exact measured beam profiles when an open field is used and will similarly yield an exact beam profile for any modified field and will thus always return a calculated beam profile which exactly reproduces the modified dose distribution that would be measured in the real physical situation, our “perfect profile”™. No other planning system in the world can possible be better than the PC3D® in reproducing measured data.
Each portion of the ratio described in the preceding discussion requires the mathematical calculation of a 3D beam profile based on various physical parameters. The mathematical procedures and the physical parameters involved are explained following:
The PC3D® profile determining algorithms rely on a pencil beam approach applying multiple dose kernels each of which is determined from factors which are based on physical parameters of the radiation and derived from standard measurements or standardized physical data.
The PC3D® model used for the generation of the beam profiles consists of four separate contributing kernels which are used in a convolution equation (multiplied together at each point in the array as a function of relative position of the kernel with respect to each other kernels and to the dose calculation point) to determine the overall beam profile at all 3D positions.
The overall beam profiles for the standard open field and for the custom clinical field are then used to determine the profile ratio factor that is applied to the measured data. The profiles determined using this method, however, can be seen to be so close to the measured data that this last step might not in fact always be necessary, but the PC3DÒ nevertheless uses this last step of applying the ratio to the measured data to assure the most accurate results at all times.
The four individual contributing kernels are kept separate and used differently in the patient modifiers section to determine the effect of the patient modifier factors on the final dose distribution produced as the beam traverses the various patient structures. The four convolution kernels used for beam profile determination are described in the following four sections:
1) the Primary radiation kernel
2) the Collimator edge integration kernel
3) the Irregular field edge integration kernel
4) the Scatter contribution kernel
This fourth mentioned scatter kernel is utilized in the PC3D® algorithms in a unique and very special manner to give near Monte-Carlo accuracy which no other convolution is yet able to accomplish. Unique to the PC3D® algorithms, is that some portion of the incident radiation at each interaction point is absorbed and some portion is scattered from that interaction point. The amount absorbed depends on the energy of the incident radiation (an average is easily able to be used because the absorbtion is a slowly changing factor over the energy range where Compton interactions predominate, which is by far the largest portion of the interactions in radiation therapy energy ranges) and the material and its density (which is derived from the calibrated CT Hounsfield numbers) at the local point of interaction.
The scatter portion produced at each point in the scattering medium subsequently becomes the new “primary” radiation which continues on and subsequently interacts at some point “downstream” in the total 3D radiation absorbtion process where again some portion is absorbed and some portion again scattered on to again essentially becomes the new “primary” source of the next sequence of scattering.
In the PC3D® this cascade of sequential scatter and integration of absorbtion is repeated for several cycles of partial absorbtion and partial scatter and then the radiation is finally considered to be fully absorbed only after the third scattering interaction.
NOTE: this differs from the more complex and time consuming Monte-Carlo statistical procedures only in that the radiation is considered to be fully absorbed on the third interaction and not carried on, ad-infinitum, until each and every electron has been calculated and recalculated until each electron has been completely “thermalized”. This PC3D® algorithm results in much faster convergence on determining true absorbed dose in a three-D medium with essentially very similar results to a full Monte-Carlo calculation. i.e.: Faster than Monte-Carlo and more accurate than any other 3D algorithm.
The correctness of this procedure can be seen in the isodose curves that it produces in areas of higher density gradients. Build-up can be seen to occur in the regions where the beam traverses from within a lung into more solid tissues downstream in the beam. Dose can be seen to decrease before the beam enters a lung as lesser “Backscatter” from the lung decreases the dose in the tissue immediately overlying the lung. Also the dose can be seen to increase in areas lateral to material which are highly scattering and this effect is able to be seen in regions where materials with large differences in scattering and absorbtion are laterally adjacent where increased dose, such as is seen in lungs lateral to bones.
These 3D effects ARE NOT SEEN in some computer systems that are claiming to have superior calculation algorithms. However, no matter what their advertising claims may be, their algorithms cannot be correct if these effect are not seen and cannot be shown. The superiority of the PC3D® calculations can be seen in that only the PC3D® (and Monte-Carlo calculations) show these effects and thus the PC3D® is uniquely correct.
The four contributing calculation kernels will now be presented separately:
Primary Radiation KernelSection 1) the primary radiation kernel P(x,y,z)
This kernel is a function of the nominal, in air, off axis ratio factor (OAF) as a function of the radial distance (R) from the central axis of the beam. This is subsequently multiplied by the primary beam transmission (T) through any intervening collimators or blocks which is a function of the thickness (h) of the primary absorber.
This parameter can be determined either as a measured in air Off Axis Factor, normalized to a value of 1.00 at the position on the central axis of the beam or it can be derived from measured in-water phantom cross axis scans. If the data originates in scans which were initially determined in a water phantom, then the appropriate nominal “in-air” values are obtained by mathematically extracting the values by applying the appropriate Backscatter factors and removing any effect of the collimator blades.
The water scan used for this determination is the largest measured scan in the available data set. Some planning systems use a corner-to-corner scan to obtain such similar data, however they often tend to forget that as the scanning probe approaches into the corner of the field that the profile that is measured shows the effect of both collimator jaw edges and decreased scatter from the shrinking equivalent field as the position being measured approaches the corner
Primary radiation kernel:
P(x,y,z) = OAF(R) * T(h) (5)
where: P is the initial primary kernel
(x,y,z) is the position in space at 100 cm.
OAF is the (in air) Off Axis Factor
derived using measured fields and
Reference book Backscatter Factors
R is the radial distance from the central axis
T is the primary transmission factor which is a function of
h which is the HVL thickness of the primary absorber
The primary kernel is convoluted with an infinitesimal primary beam profile radial penumbra (Rp) distance that is a function of source focal spot size (FS) and distance of the calculation point distance from the focal spot position (d). The changing shape of this primary radiation kernel is further also a function of the primary dose deposition spread caused by angular deflection resulting from the primary ionization interactions (Compton scattering and some pair production at higher energies) which is a function of the amount and the kind of material being traversed (t).
The penumbra spread is thus a function of both the distance from the source and the distance traveled through the medium
Pen(FS,d,t) (6)
where: Pen is the penumbra factor which is function of
FS which is the nominal focal spot size,
d is the distance from the source, and
t is the intervening tissue thickness
This results in a primary kernel correct for all (x,y,z) positions that is a function of radial distance from the central axis of the beam and a function of the distance of the kernels position from the focal spot of the source of the radiation and a function of the amount of material causing additional primary beam kernel spread.
Collimator edge integration kernelSection 2) the Collimator edge integration kernel C(x,y,z)
This kernel is the combined edge effect influence which is caused by the penumbra of the collimators or beam shaping blocks. This kernel varies with the radial distance (r) of the point of calculation to the projection of the edge of the field and the type of collimator edge projecting that edge and the distance of the calculation point below the surface of the patient.
The penumbra edge integration kernel is the edge effect as determined for the various block edge type which may be a standard or alloy block or may be standard or multi-leaf collimators. It is also a function of the depth below the surface and the distance from the penumbra causing edge.
The parameters defining the spread of this edge effect kernel express the dose as a function of the perpendicular radial (r) distance of the edge from the projected block edge relative to the Focal Spot radial penumbra factor (Rp) described earlier and whether the edge is diverging or non-diverging and if non-diverging then also as a function of the distance of the edge from the central axis.
Both edge integration kernels have parameters that define the shape of the profile both under the block edge and also in the area on the open side of the field near a block edge. This “edge-effect” kernel is composed of four exponential factors which are convoluted with the primary transmission of the collimator or the block to define the penumbra on the edge of any portion of the field. Each of the exponential factors is further a function of depth. Furthermore, because the edge effect kernel changes (spreads) with depth, the values of each of these exponential parameters themselves a function of the depth below the surface of the material being irradiated.
Since a picture is generally considered to be worth a thousand words, the following diagram is included to help to graphically explain the mathematics behind both the collimator edge factors and the block edge factors. The mathematics is the same for both, but the actual parameters vary for each and further vary with block type and depth below the surface.
PENUMBRA ALGORITHM DIAGRAM
(insert block edge diagram here )
Figure 1: Penumbra Factor Diagram. The factors used for the penumbra calculations for the collimators and block edge factors are diagrammed above and are labeled in the equations as B, T, L, H and S as explained following:
B Base, This is a constant and is a function of the basic transmission factor of the block or collimator . This base level is the same without regard to the distance from any edge and is a function of the HVL of the collimator or block.
T Tail, the extended exponential tail portion of the penumbra
T varies with depth and the Tvalue varies with radial distance(r) from the projected block or collimator edge
Tvalue (d,r) = T (d)* e ^(0.693 r / Tcoefficient (d))
where the Tcoefficient is a further function of depth
Tcoefficient (d) = Tconst + Tbase * depth
L Low Half, the portion of the penumbra below the Break level less the Tail portion and the Base portion
L varies with depth and the Lvalue varies with radial distance from the block edge
Lvalue(d,r) = L(d) * e ^(0.693 r / Lcoefficient (d))
where the Lcoefficient is a function of depth
Lcoefficient = Lconst + Lbase * depth
H High Half, the portion of the penumbra above the Break level less the shoulder portion
H varies with depth and the Hvalue varies with the distance from the block edge (r)
Hvalue(d,r) = H(d) * e ^(0.693 r / Hcoefficient (d))
where the Hcoefficient is a function of depth
Hcoefficient (d) = Hconst + Hbase * depth
S Shoulder portion, the portion of the penumbra above the Break level less the High Half portion.
S also varies with depth as the break point varies with depth
and varies with the radial distance (r) from the block edge
Svalue(r,d) = S(d) * e ^(0.693 r / Scoefficient (d) )
where the Scoefficient is a function of depth
Scoefficient (d) = Sconst + Sbase * depth
The Collimator edge effect kernel becomes:
4
C(r,d) = B + å e u (r u ,d u ,F I ) + e o (r o, d o , F I ) (7)
i=1
where: “u” is under and “o” is open outside the edge
d is the depth below the surface of the material
1 to 4 are parts of the equation (relating to T,L,S and H)
B is the baseline primary after a specified HVL
r is the radial distance from the projected edge
e is exponential of the radial distance r relative to p
and further set e = 0 (zero) if negative
p is the penumbra factor which is a function of edge type
p= Pen as defined in equation (6)
F are four factors that are included in the integration:
Irregular Field Edge Integration KernelSection 3) the Irregular field edge integration kernel E(x,y,z)
This kernel is the combined edge effect influence which is caused by the penumbra of the various beam shaping blocks. This kernel also varies with the radial distance (r) of the point to the projection of the edge of the block and the type of block that is projecting the edge (divergent or straight sided) and the distance of the calculation point below the surface of the patient. If the Block type is “straight sided” then the block edge integration kernel is also a function of the distance from the central axis of the beam.
Other than these additional considerations for the irregular field shaping block type, the algorithm that determines the kernel that is applied it to the generation of the profile values is essentially identical to the algorithm that generates the collimator edge kernel.
The block Edge effect kernel:
4
E(r,d) = B + å e u (r u ,d u ,F I ) + e o (r o, d o , F I ) (8)
i=1
where: the parameters are the same as for equation 7, except the T,L,S and H penumbra are as defined for the applicable edge type
Scatter Contribution KernelSection 4) the scatter contribution kernel
This kernel is influenced by the shape of the open areas of the field and determined from measured data and the shape of the open areas of any irregular field. The resulting scatter at each point is determined by integrating the scatter at that point for all contribution of any portions of the open field and at all radial distances from the projected calculation point. This scatter contribution is additive for radials which detect the near side of an block or edge and subtractive for radials which detect the far side of an edge or block.
The scatter that is produced from any areas under any sort of a block has already been determined from the block’s transmission factor and included in the various edge effect kernels. The dose under the block is therefore determined from the primary transmission which is already included in section (1) above and is a function of the various edge factors calculated according to sections (2) and (3) above which apply to areas both under a block and inside the field next to a block.
The scatter that is in effect under a block is calculated to have originated only in the open areas of the field beyond the projected block edges. The scatter that may originate in the volume under the block is assumed to be nil and the scatter contribution under a block instead included in the consideration of the edge kernel widths.
This algorithm using the edge effect kernel for determination of scatter contribution under blocks is also used for partial transmission blocks. The actual physical production of the partial transmission block is generally considered to be greater source of uncertainty than any minor differences in scatter that may have actually been generated originating in the area under the block. The inclusion of the scatter under a block in the edge effects kernel instead is determined to be acceptable in the clinical situation.
Scatter integration Kernel:
ó n
S(x,y,z) = ô S(r,z)open - S(r,z)blocked (9)
r = õ 0
where: n is the number of radial steps
S(r,z) is the scatter air ratio for r and z
r is the radial distance to the projected edge
z is the depth of overlying material
THE “PERFECT PROFILE”™
Each of the profiles that go into the determination of the ratio that is used to determine the “perfect profile”Ô is the results of a combination of all of the described kernels integrated for each pencil beam position over the entire area of the projected clinical field.
ó((
D(x,y,z) =ô(([P(x,y,z) InvSqr*C(x,y,z)*E(x,y,z)]*[(S(x,y,z)+TAR(0)]
õ((
x,y,z (10)
PATIENT ORIGINATING DOSE MODIFYING PARAMETERS:
Once this “perfect” beam 3D dose distribution profile has been determined, it needs to be applied to a physical patient and this perfect dose distribution is thus consequently modified by a multitude of patient related dose modifying parameters.
Because some of the patient related modifying parameters are related to the primary and some are related to the secondary scatter radiation, each of those two contributing kernels have been kept separate and were only combined briefly earlier to determine the “ratio factor” that was applied to the measured data to produce the “perfect profiles”Ô for the beam configuration that is used for the clinical application.
The physical parameters associated with the determination of the dose modifying patient related factors at each selected matrix point are determined by means of an elaborate and elegant ray tracing algorithm. For a complete discussion of ray tracing algorithms, see any good post graduate mathematical textbook for graphics algorithms. The raytracing algorithm used in the PC3D can be found in such texts as the Wolfenstein algorithm, which is well accepted as being superior for its speed and accuracy.
Medicalibration has further added special modifications to account for specific needs in radiation therapy physics such as accounting for the density and absorbtion coefficients along those ray many trace paths.
This mathematical procedure follows the radiation’s photon paths pixel by pixel through the patient’s entire CT data set from the focal spot of the source of the radiation to each point in the calculation matrix and then also from each matrix point to all other matrix points. This mathematical ray tracing algorithm integrates the densities and continuously calculates an weighted average along the entire path length and at all calculation points and also detects various interaction “events” as it traces all of the paths of the rays and keeps track of all of those various values that it determines in tracing between the various points.
The ray tracing algorithm determines all of the following items for all rays from the source of the radiation as they travel to each matrix calculation point. This ray tracing algorithm can handle any oblique entry into any stack of CT information from any direction:
1) The distance from the focal spot of the source to the first interception of the ray by any part of the patient as the ray enters the patient. (if a bolus is present, for ray tracing purposes it is here considered to be part of the patient).
2) The distance from the focal spot of the source to the last point of exit of the ray as it leaves from the patient.
3) The distance from the focal spot of the source to the matrix calculation point, isocenter point and all other user defined points of special calculation.
4) The distance from every matrix calculation point to every other matrix calculation point, see item (7) following .
5) The occurrence of any ray which “flashes” past the periphery of the patient.
6) The total average density of all of the material overlying the calculation point between the calculation point and the point of the first interception of the ray by any part of the patient as the ray enters the patient (see (1) above). This is determined from the pixel Hounsfield densities at the points that are adjusted for the Hounsfield to density correction for each and every point along the ray path and corrected for oblique transit through the CT planes.
7) The total average density of all of the material that exists along each path between each of individual matrix calculation point to each other matrix calculation point.
8) The actual physical density of the material at the point of interaction of the radiation in the patient at the matrix calculation point.
9) The density gradient of the material at the exact matrix point of calculation. This is NOT the gradient to other surrounding matrix calculation points, which may be at variable and excessive distances on an absolute physical interaction distance scale but rather the instantaneous density gradient (the density slope) at the point.
10 It further determines if the first intersection of the ray trace path, as it encounters the data set, is into the open end of a stack of CT slices and not across an acceptable external surface boundary between two CT slices of the patient’s data set.
NOTE: The dose volume histogram calculations also use the ray tracing algorithm except that the limiting parameters are the surface of the individual structures rather that the surfaces of the external contour of the patient.
This 3D ray tracing algorithm thus determines everything that is needed for the 3D determination of the modifying effect that the patient has on the beam being projected into the patient. These various items are used to determine the modification to the “perfect profile”™ as follows:
The unmodified dose distribution is based on a modification of the “perfect profiles”™ and starts with the assumption that the “perfect profile”™ impinges on a “perfect” phantom. This “perfect” phantom is assumed to be positioned at a distance of the nominal axis distance of the machine. and is considered to consist of a flat presenting surface of unit density material water. The 3D dose distribution that would be produced in this “perfect” phantom is subsequently modified by applying all of the modifying influences that are patient related and determined from information derived from the ray tracing algorithms to yield the dose distribution that would then actually result in the actual clinical patient.
The dose contribution at each matrix point is determined by
A) The first step in determining the dose at any point calculates the amount of the primary that reaches each point. This simple first step is done by using the inverse square correction for the distance to the point (3) determined by the ray trace algorithms and the primary dose spread at that distance, times the attenuation caused by the overlying material which is the product of the amount of material [(3)-(1)] times its average density (6). This is multiplied by the primary in-air portion of the Off Axis Factor determined from the “perfect profile”Ô (which would include any effects of collimator edges, beam shaping blocks, compensators or wedges).
B) Some portion of this amount of primary radiation (A) that reaches this point is absorbed. This is considered to be the “primary” contribution at that point. The amount absorbed depends on the Hounsfield density of the material at the point (8) times the Hounsfield to density correction table that is appropriate for the CT scan data set times the absorption coefficient for materials of that density. This factor is also used in section (D) following.
This makes the assumption that the most significant interaction is by Compton scatter which is primarily a function of the density and the energy of the material and not dependent on the atomic number of the material.
Some other portion of the primary that reaches this point is scattered. The amount scattered depends on the scatter ratio at the point (determined from the scatter portion of the “perfect profile”Ô as determined earlier). The balance of the radiation continues on down-stream.
C) The amount of radiation scattered from the primary point of interaction is then transmitted to other points and attenuated on the way to that point by an amount determined by the average density (7)and distance (4) to that point.
In fact, It can be seen that conversely this enables each point to instead be considered to receive some portion of dose from scatter emanating from other previous interaction points “upstream” in the absorbtion cascade sequence. This consideration enables the effect of lateral scatter and the effect of build-up on the entry to regions of higher density to be considered and manifested. Furthermore, no dose is considered to come from points in the matrix that are in the “flash” (5) portion of the beam that do not impinge on the patient but instead “flash” fast the lateral edges of the patient.
D) This is repeated a second time and the amount of radiation that reaches the third interaction point is, for the PC3D algorithms, considered to be completely absorbed. This is also a function of the absorption conversion factor for the material at this point which has been determined by the Hounsfield density at the point times the Hounsfield-to-density conversion for the CT scan data set that is used in section (B) above.
This assumes that the energy of the three-times-scattered radiation has been degraded to a degree that allows complete absorption by the third interaction point. This procedure makes the algorithm essentially become a three-step Monte-Carlo calculation which assumes that some radiation is absorbed in the first interaction and some is scattered and that some is absorbed in the second interaction and some is scattered, but that all of the scattered radiation that occurs in the second interaction is finally totally absorbed by the third interaction. Statistics would indicate that the fraction of radiation scattered at the third interaction point and continuing on is only a very small fraction and the amounts scatter on from any subsequent interactions are progressively orders of magnitude smaller. The isodode line curves produced with this assumption would indicate this is a valid assumption..
Any radiation that might have come from calculation points that are determined to be in the “flash” portion of the beam (5) and consequently outside of the patient, are considered to contribute NO lateral scatter to the dose at this point.
We have read in their literature that the PLATO and the HELAX and The Pinnacle (and most other planning systems) all use a simple effective depth or effective path length to determine the percent depth dose and have no real way to include the consideration of lateral scatter, missing “flash” or density gradient build-up. This is generally an acceptably method to determine the percent depth dose EXCEPT in cases near a heterogeneity gradient. This is the situation where the PC3D® algorithms prove superior in handling the effect of lateral scatter and internal buildup in regions of electron non-equilibrium.
Dose(at point x,y,z) =
(Primary contribution
=OAF(in air and radial distance)
Profile Ratio (at X,Y,Z)
=Calc’d profile for Clinical Field
=Calc’d profile for Equiv.Open
=3D Kernel integration (x,y at depth z)
*Block Edgefactor (radial r from edge)
*Collimator edge factors
*Block Transmission
*Measured Profile Data
*Standard Backscatter Factor Data
*Depth Dose Ratio)
=PDD Percent Depth Dose (at 3D ray traced depth)
=Sterling Coefficient
*InvSqr Correction factor (physical distance)
+Scatter Contribution)
=Scatter exposure (from adjacent 3D positions)
*Scatter abortion (for CT density at calc point)
The determination of a calculated beam profile as done by the PC3DÒ can be seen to be not based solely on any single, simple Gaussian distribution convolution kernel or even multiple fixed kernels as is typical for some other planning systems, but rather is determined from a convolution of several different kernels each derived from a wide variety of contributing physical factors which results in a generated beam profile which is seen to reproduce the measured data almost exactly.
Monitor Unit Calculations:
The determination of monitor unit settings is another calculation that depends first on the dose distribution produced be each beam that gives into the overall treatment plan. It further depends on the method of weighting :
1) weight to calibration dose
2) weight to isocenter
3) weight to other user defined point
4) weight by applied Monitor Unit Settings
5) weight to Dmax dose at entry SSD
prescription:
1) prescribe cGy to percent
2) prescribe cGy to some user specified raw Isodose Line
3) prescribe cGy to isocenter
4) prescribe cGy to other user defined point
and normalization:
1) normalize 100% to raw Isodose line
2) normalize 100% to matrix maximum
3) normalize 100% to the isocenter
4) normalize 100% to other user defined point
= 80 possible methods of weighting, prescribing and normalizing.
Plus the overall treatment can be subdivided into up to 9 different phases where each phase may have its own weighting, prescription and normalization = 80 raised to the 9th power! No other system has as much versatility as the PC3D.
The determination of monitor units also depends on several other factors particular to the overall setup such as :
1 Output Factor (Collimator Factors, upper and lower)
2 Tray Factor
3 Wedge Factor (function of field size)
4 Calibrated Output (cGy/min. or MU per min.)
The PC3D can accept calibrated output at either SSD + Dmax depth or at SAD isocenter with Dmax overlying material.
A complete discussion of the calculations necessary to accurately determine the required monitor unit setting from the calculated raw isodose line distributions are available in a separate paper.
The accuracy of these algorithms has been validated by entry of carefully measured data and comparison of the calculated results with the measured expected results The standard data set consists of PC3D® data files generated by the PC3D® utilities from the data provided by AAPM Task Group 23 for photon energies of 4MV and 18MV. The calculated results are compared to measured values provided by TG-23.
A complete sample data set for both the 4MV and the 18MV photons is provided as sample data on initial installation. The same data set and standard test case phantoms can be provided to users of PC3D® if they desire to make their own validation tests.
Results:
All calculated results when compared with measured results can be seen to be well with the stated limits of +/- 2% in areas of low dose gradient and within +/- 2 mm in areas of high dose gradient. It can further be seen that most differences are in the third decimal place and result in differences that are less than 1%. Some of these differences are so small that they may, in fact, be errors caused by the TG-23 physical measuring limitations and NOT in the PC3D’s calculation algorithms.
Further comparison of the TG-23 test case results has been made between the Medicalibration PC3D® and the ADAC Pinnacle. and also a comparison has been done comparing the Nucletron’s Plato and the Theratronics’ TheraPlan and the results were published recently in Medical Physics, December 1997. The HELAX and the CMS and Monte-Carlo algorithms were not provided for inclusion in the published comparison. The results show that the PC3D® calculations compare exceptionally well with the expected measured results The results also show the PC3DÒ to be by far the most accurate of all systems that were tested and a complete tabular report is available from Medicalibration..
No other more simple mathematical calculation procedure using Monte-Carlo, pencil beams, or any other single or multiple convolution kernel can possibly be more exact than the PC3D® in producing profile information that recreates the true distribution of measured profiles. All other mathematical methods can only be equal to the PC3D® or else they must be less correct.
The ray tracing algorithm is critical to the speed of the calculations and a great deal of programming effort and mathematical matrix theory has gone into streamlining the code and making it as efficient as possible without any sacrifice in accuracy.. With the increasing speed of computers and the functionality of multiprocessors, the calculation times are continually being reduced.
Currently, using the latest PC3D® programs on a single processor 233 megaHertz Pentium, 2000 points (40x50 matrix) can be calculated in just under 1/2 second! And almost at twice that speed when running at 450 megaHertz
This incredible speed lets the resulting isodose distribution be viewed in less than a second, which is essentially instantaneous for treatment planning purposes and with expected improvements the PC3D® will soon be producing isodose contour in “real-time”. This incredible capability and speed allows the dosimetrist to make minor changes and detailed adjustments in perfecting the dose distribution specifically for the patient and instantaneously seeing the results of those changes on the shape of the isodose curves. Minor changes in position, field size and weighting and even changes in wedge number can be made and the resulting effect can be critically evaluated continuously as the changes are made.
The initial basic setup can be determined and evaluated in the single plane mode described in the paragraph above and the complete 3D dose cloud distribution can be created in a proportionally quick time when desired. Twenty 3D planes would take about 10 seconds, 100 in less than a minute. The time required is also a function of the total number of points that are being calculated per matrix plane. Currently the time required to calculate a clinical dose distribution is not excessive and is not a limiting factor and furthermore, the times are constantly being reduced with faster processors.
Fast and Accurate:
The PC3D® algorithms are fast and extremely accurate. The speed of PC3D® hardware is certainly fast enough for clinical usefulness and is increasing with the evolution of the personal computer architecture and the CPU processor.
The accuracy of the results are, without a doubt, as good or better than any other treatment planning system in the world and certainly well within any clinical requirements.
Networkable with multiple modality input
The PC3D® can be networked to other PC3D’s, to local area networks or over wide area networks and can accept CT data from the widest possible source of digital data input modalities including those networks as well as Optical Disk, DAT tapes, reel-to-reel tapes, and even 8 inch floppy disks. The PC3DÒ can interpret all of the most customary CT scanner information including the currently latest DICOM RT format.
The PC3D® also has the widest ability to accept data from non-digital sources such as film scanners of various types and camera image grabbers connected to CT scanners, ultra sound units and film digitizers.
Ease of use and Clear output presentation
The PC3D® system has been specifically programmed for ease of use using both a mouse and instant access mnemonic keystrokes.
The output tables, plots and graphs are carefully designed for clear presentation of the results. In many cases, the development and augmentation of the system has evolved with the aid and support of our many satisfied customers.
Excellent initial training and continuing support The PC3D® system is installed with assistance in entering the necessary beam data parameters if so desired, followed by several days of initial training in the use of the PC3D®. The PC3D® system undergoes on-site validation testing with sample data and with the sites own data to provide the assurance of the ability to produce correct treatment plans for application to the patients.
There are also continuing education classes available for brush-up or advanced training available on a routine schedule at the PC3DÒ headquarters. Such additional training is available as an on-site service. The PC3D® is also supported by a professional staff of experts in the use of the PC3D® which are available for phone support “in one ring or less”.
Highest Value
All of the previous enumerated advantages combined with reasonable price and extreme reliability of hardware and software, combined with exceptional ease of use and upgradeability make the PC3D® a system that many facilities have been using happily and successfully for many years and with Medicalibration’s service, support and responsiveness combine to make the PC3D a value that simply cannot be beaten, by any other system, at any price.
Medicalibration Voice: (209) 524-6789
558 Van Dyken Way (800) 782-7360
Ripon, CA 95366
E-mail: info@medicalibration.com FAX: (209) 599-1111
ALGORITH.DOC (C) Copyright 1997-1998 MEDICALIBRATION
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