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It is estimated that about four in ten cancer patients have radiotherapy as part of their treatment6 and as this figure continues to grow, so does the importance of treatment with minimal complications from the radiation.
In terms of frequency of radiotherapy accidents resulting from overdosing and underdosing , treatment planning accounts for 28% of these accidents and treatment delivery 20%. Such accidents can have devastating effects, ranging from the irreparable damage of normal cells and organs and in some cases even in fatalities14. This highlights the need for optimization techniques that can improve the quality of treatment plans and thus the accuracy of delivery of radiotherapy.
Cancer is the leading cause of death worldwide, with a projected 12 million deaths in 20306. Although radiotherapy is a well attested means of treating cancer it does carry with it a number of risks and side effects from the irreparable damage of cells peripheral to the tumour to a sever reduction in the quality of life9. This highlights the need for advancements in the radiotherapy arena, which can both reduce the damage to peripheral cells and increase the chance of completely destroying the tumour.
Such advances include, Tomotherapy, which is a "computer-controlled rotational radiotherapy delivered using an intensity-modulated fan beam of radiation"1 and combines a CT scanner with an external beam radiotherapy machine; and Intensity Modulated Radiation Therapy(IMRT). However to make the most effective use of these new techniques, a suitable approach in the optimization of each patient's treatment plan is required1.
Section 2 of this paper provides an overview of radiotherapy and highlights two particularly important treatment machines. Seaction three introduces the optimization techniques and section 4 introduces iterative approaches as an alternative to optimization techniques. Finally section 5 compares the different techniques and further areas of development.
2. Radiation therapy
2.1What is radiation therapy:
Radiation therapy involves "delivering a concentrated radiation dose to a cancerous region"1. The success of radiotherapy relies upon the fact that for any given radiation dose, normal cells repair themselves better than cancerous cells.
In creating a treatment plan the radiation oncologist's main objective is to eradicate all the cancer cells, avoiding critical organs, near which the tumour may be located while also limiting damage to the surrounding healthy cells. A number of dose prediction models can be used to reach this decision; Monte Carlo simulations are regularly used to determine the radiation properties of treatment beams5.
The complex nature of the treatments calls for an optimization approach in order to develop the best plan for treating each patient. The optimization approach should provide the oncologist with sufficient flexibility so that he can always produce an acceptable treatment plan1.
According to Roijin et al (2002) the peripheral organs and tissues are referred to as critical structures and tumours are referred to as target97.
In conformal radiotherapy, the goal is to deliver a volume of high dose that closely conforms to the shape of the patient's tumour volume13.
2.2How is radiation therapy delivered to treat a cancer patient:
Romijin et al (2002) noted that although, it may be possible to kill all the tumour cells by treating a patient with a single beam of radiation this would also incur the risk of severe damage to any normal tissues located within the path of the beam of radiation. To avoid this, beams are delivered from a number of different angles spaced around the patient so that the intersection of these beams includes the targets; hence the targets receive the highest dose of radiation while critical structures receive radiation from some source, but not all beams and thus can be spared9.
Intensity Modulated Radiation Therapy (IMRT) uses hundreds of tiny radiation beam-shaping devices, called collimators to deliver a single dose of radiation7. The IMRT technique is considered to be the most effective radiation therapy for many forms of cancer9.The radiation intensity is varied across the beam, allowing a very high degree of conformation to the tumour and allowing the creation of very complex dose distributions. These dose distributions are obtained by dynamically blocking different parts of the eam9,13. However, due to the large number of beams and the complexity of the beam intensity, a computer-based optimization algorithm is required to determine and optimal treatment plan that allows the delivery of sufficiently high radiation doses to targets while limiting the radiation dose delivered to healthy tissues1,13.
2.5.Radiation Treatment Planning:
The planning process begins when the patient is diagnosed with a tumour mass and radiation is selected as the treatment regime. A 3D image of the affected region, which contains the tumour mass and surrounding areas is acquired via computed tomography (CT)13. From these data, the location of the oncologist outlines the target, critical structures that need to be held to a low dose and radiosensitive organs which may be unavoidably irradiated19,22.Next, the dose to the patient from each individual beamlet is computed1.
Dose computation is achieved through convolution/superposition techniques; whereby Monte-Carlo generated energy deposition kernels and the superposition of beamlets are used to convolve the total energy release in the patient from the radiation source.
3. Optimization Techniques:
"The following optimization techniques are designed for IMRT when the orientations of the beams is given. The objective is to design a radiation treatment plan that delivers a specific level of radiation, the prescription dose, to the targets and for the critical structures a radiation dose does not exceed tolerance dose. However, if the targets are located within close proximity of the critical structures this goal is violated; hence a common approach is to search for a radiation treatment plan that satisfies the prescription and tolerance dose requirements to the largest extent possible9.
Note that: Targets + Critical structures =Structures
Structures are eradicated using a predetermined set of beams, each corresponding to a particular beam angle".(This section is based on pp. of Romijin et al )
3.1. Dose Computation
The dose computation is obtained as follows:
Each beam is composed of beamlets and each pixel in an image is called a bixel; is the decision variable denoting the radiation intensity of beamlet and is dose received by voxel from beamlet at unit intensity.
We can express the total radiation dose at a voxel as a linear function of the radiation intensities as follows:
These functions are referred to as the dose calculation functions
Where N is the set of all bixels contained in all beamlets, , S is the set of all structures with targets corresponding to and critical structures, .
Each of the structures is discretized ( made more suitable for numerical evaluation and implementation on a computer) into a finite set of voxels; the sets are not necessarily disjoint, for examples, if a tumour has invaded a critical structure, there will be an overlap between a target and a critical structure9.
Various dose constraints are involved in the design of treatment plans.
Clinically prescribed lower and upper bounds, say, and , for dose at voxel j are incorporated with (1) to form the dosimetric constraints13:
These constraints are called full volume constraints because they need to be satisfied everywhere in a particular structure13. "The dose of any given voxel may be subject to multiple lower and upper bound constraints.T he bound choice reflects the intuition that saving a critical structure should never be at the expense of not curing the disease".
Partial volume constraints: Romijin et al(2002) states that "In most practical situations there does not exist a feasible treatment plan as defined above, as it is usually impossible to satisfy the full volume constraints for each voxel in the target volumes as well as all critical structures.
There will be some voxels in the target volumes that have to be underused and some voxels in the critical structures that have to be overdosed.
A powerful approach is to relax the full value constraints and then add partial volume constraints that constrain the shape and location of the DVHs of all structures."
The dose-volume histogram displays the fraction of each region of the patient that receives at least a specified dose level. In some cases, the radiation oncologist is willing to sacrifice a portion of a critical structure in order to improve the probability of curing the disease. Oncologists often specify constraints of the form "No more than % of this region at risk can exceed a dose of ".Thus for a particular region at risk, the oncologist determines both a dose limit and a fraction of the structure that can exceed the dose limit.This type of requirement is called a partial volume constraint.
For our formulae, the dose limit will be denoted by and the fraction of the volume allowed to exceed this limit will be denoted by .
In these formulae, is the number of pixels in the region at risk and is the number of pixels in the normal tissue.
Advantages: in contrast with established mixed-integer and global optimization formulations they do so while retaining linearity of the optimization problem thereby ensuring that the problem can be solved efficiently.(put in discussion)
3.2. Dose-Volume Histograms
"Most useful and most popular plan evaluation tool in 3D conformal radiation therapy.DVHs can indicate at a glance the potential for undesired consequences by identifying the existence of hot spots in a critical structure and cold spots in a target volume5.Radiation oncologists often use a cumulative dose volume histogram (DVH) to judge the quality of a treatment plan. A cumulative dose volume histogram displays the fraction of the patient that receives at least a specified dose level1.
A DVH, graphically summarizes the simulated radiation distribution within a volume of interest of a patient which would result from a proposed radiation treatment plan. It is a useful means of comparing rival treatment plans for a specific patient by clearly presenting the uniformity of dose in the target volume and any hot spots in adjacent normal organs or tissues. However, because of the loss of positional information in the volume(s) under consideration, it should not be the sole criterion for plan evaluation. DVHs can also be used as input data to estimate tumour control probability (TCP) and normal tissue complication probability (NTCP). The sensitivity of TCP and NTCP calculations to small changes in the DVH shape points to the need for an accurate method for computing DVHs8.
That is, for a given structure, for any given structure , such a histogram is a nonincreasing function17:
Here, (d) is the relative volume of the part of the structure that receives units of radiation or more, clearly:
Using the finite representation of all structures using voxels, the DVH for structures under radiation intensies can be expressed as:
e.g in the figure below, the point on the DVH for the target ( indicated by vertical dotted line) indicates that 90% of the target volume receives at least 70 units of radiation(Gray(Gy)). Typically the goal is to control or constrain the values of the DVH.
The following section is based on Shepard et al(1999) pp.731-733
3.3 Linear programming techniques:
According to Wright(1997), "the properties of a linear programme are:
A vector of real variables, ,whose optimal values are found by solving the problem;
A linear objective function,
Linear constraints, ,both inequalities and equalities,"
The standard form of a linear programme is :
Where and are vectors of lower and upper bounds on the variables.
Using sophisticated implications of the simplex algorithm or adaptations of primal-dual interior point methods, many timization large-scale optimization problems have easily been dolved.Primal-dual interior-point methods are particularly useful because they can be
There are several possible approaches to linear programming radiotherapy:
3.3.1. Minimizing the total integral dose
Minimise the total integral dose subject to a lower bound on the dose to the tumour. The integral dose is the total dose summed over all of the pixels. Using the same form as in (1), this can be reformulated as :
Where is the subset of the pixels located in the tumour, , is the number of beamlets, and is the lower bound on the dose of the tumour. A non-negativity constraint is placed on the beam weights (), ensuring that no negative solutions for the beam weights are produced. Then the above formulas are designed so that the optimizer will drive the tumour dose down to the specified lower bound1.
The objective function can be modified so as to minimize a weighted integral dose:
is the subset of pixels in the region at risk, and is the subset of normal tissue pixels(those located outside both the target and the sensitive structures).
In this formulation, a weighting factor is assigned to each region of the patient. is the target weight, is the region at risk weight, and is the normal tissue weight.
The objective function equals the sum over the entire volume of each pixel's volume multiplied by its weighting factor. By increasing the relative weight assigned to a region at risk, the user can place a greater emphasis upon reducing the dose to that region. By increasing the relative weight assigned to the tumour, the user can place a greater emphasis upon achieving a uniform target dose1.
3.3.2. Placing bounds on the dose to the tumour
Place both an upper and a lower bound on the dose to the tumour. In this case, the objective might be to minimize a weighted integral dose over all of the nontumour pixels:
and indicate the lower and upper bounds on the dose to the target. When minimizing the integral dose, the highest weights are assigned to the beamlets that deposit the greatest fraction of their integral dose within the tumour. The results can become unsatisfactory if any of the beam weights is made too large. Heavily weighted beams produce streaks of high dose through the patient, and this could lead to patient complications. A solution to this problem is to place an upper bound on the ratio between beam weight and the average beam weight:
In this case, the maximum beam weight is the product of and the the mean beam weight.This approacjh yields a more evenly distributed integral dose as compared to the previous linear programming optimizations.
3.3.3. Minimizing the maximum deviation
Minimize the maximum deviation from the prescribed target dose, , subject to one or more constraints.This can be accomplished using the following model:
In the above equation, is the number of pixels in .An upper limit of was placed on the mean dose to the region at risk.Again, the maximum beam weight is constrained to be less than the product of the mean beam weight and .
3.4. Nonlinear Programming Techniques:
A concise mathematical description of the nonlinear programming(NLP) problem is as follows:
Here, is a vector of variables that g(x) represents the set of constrain are continuous real numbers is the objective function, and represents the set of constraints. and are vectors of lower and upper bounds placed on the variables.
With a nonlinear formulation, there is an expanding range of possible objective functions and constraints as compared with linear programming.
For many of the simulations, a weighted least squares objective function is used. In these cases, the optimizer minimized the weighted squared differences between the prescribed and the actual doses summed over all the pixels. The objective function is:
is the target weight, is the region at risk weight, and is the normal tissue weight.The values of are determined using (1).The matrix describes the prescribed dose.Outside the target, is typically set to equal zero.This problem is a bound-constrained weighted least squares problem and can therefore be solved by various specialized large-scale optimization algorithms.(such as ?)
For a particular patient, the best choice of weighting factors is not always intuitive. Thus, in order to obtain an acceptable result, one may need to run a series of optimizations. Ration constraints on average and maximum beam intensities as well as other bounding constraint give rise to general constrained nonlinear programs.
Ideally this type f biological modelling could serve as a very useful tool in radiotherapy optimization. For example, an oncologist could choose to maximise the probability of tumour control subject to a cap placed upon the probability of complications for each critical structure. Unfortunately, the current biological models require input parameters that are not known with great certainty.
3.5. Mixed Integer Programming
Mixed Integer Programming (MIP) makes possible a second approach to the implementation of partial volume constraints. Mathematically, the MIP problem appears as follows:
is a vector of variables that are continuous real numbers and is a vector of variables that can only take integer values, is the linear objective function, and represents the set of constraints and are vectors of lowers and upper bounds placed on the continuous variables, and is the integrality requirement.
The goal of this optimization is to minimize the maximum deviation in dose over the target subject to a partial volume constraint.
The following formulation is used to generate the simulated treatment:
Is a standard mixed integer programming technique for modelling an "if-then" constraint. The value of M is chosen to be large enough so that the corresponding constraint is trivially satisfied when .
If exceeds , then must be set equalto 1.The variables can then be used to enforce the partial volume constraint.In the above case, the optimizer sums over all of the pixels in the region at risk, and it requires that this value be less than times the total number of pixels in the region at risk. Mixed-integer programming approach which allows optimization over beamlet fluence weights( beamlet radiation intensities) as well as beam couch angles.
Algorithmic design motivated by clinical cases
Numerical tests on real patient cases that good treatment plans are returned in 30 mins.
The MIP plans consistently provide superior tumour coverage and conformity, as well as dose homogeneity within the tumour region while maintaining a low irradiation to important critical and normal tissues.
The MIP model allows simultaneous optimization over the space of beamlet fluence weights and beam and couch angles.
Based on the experiments with clinical data, this approach can return good plans which are clinically acceptable and practical. The plans consistently provide homogenous and conformal dose to the tumour, while maintaining low irradiation to critical structures
Although the MIP instances are difficult to solve optimally, the specialized techniques implemented enable soling them to prove-optimality.
Compared to other systems which perform optimization over only a subset of beam parameters, this MIP approach allows consideration of a more comprehensive set of parameters.
Shepard et al present results from an investigation into a group of iterative approaches to treatment plan optimization with the view of developing an inverse appropriate for tomotherapy.
All the following optimization approaches rely on the following equation for the dose computation:
is the dose delivered to voxel by beamlet per unit intensity
is the beamlet intensity
is the total dose delivered to voxel
In the ratio method, the weight of each pencil beam is updated using a ratio of geometric means,the iterative formulation is given by:
Where, N is the total number of voxels included in update factor computation and is a matrix containing both he prescribed dose for each target voxel and the tolerance dose for each voxel located in a critical structure.
The ratio method corresponds to the optimization per ray of an underlying objective function.This objective function is given by:
A first order approximation of the logarithm of the objective function is:
This objective function applies on a beamlet basis.
Here, the update factor is equal to a ration of two summations.The iterative formula is:
The above formula is a least squares minimization.The objective function is:
Least -squares minimization minimizes the sum of the squared difference between the delivered and the prescribed/tolerance doses.
Because of the convex nature of this least-squares objective function, any local minimum is also the global minimum.
With the maximum likelihood estimator, a Poisson distribution is assumed for both the emission of photons from the source and the number interactions in each voxel.
We have that is equivalent tot maximising the logarithm of this function given by:
Through successive iterations, the iterative formula that maximises is:
Where, n is related to the damping or under damping, and k is the iteration index.
A primary advantage of these algorithms is their ability to perform large-scale dose optimizations while minimizing the memory requirements.
These methods can be made more flexible and robust through the addition of weighting factors assigned to each region of the patient or through the addition of dose-volume considerations.
The ratio method converges very quickly, and thus may serve as a useful technique for obtaining an initial guess for the beam weights.
Iterative least-squares minimization benefits from the fact that the objective function is intuitive and well established.
The three iterative approaches to optimization are stable with respect to the objective function value. The initial beam weight selection, however, can influence the final dose distribution. Because a number of plans can produce the same objective function value, one would want to choose the plan that is most easily delivered.
What are the new techniques being developed:
Discussion of the difficulty in quantifying optimality in radiotherapy:
"Development of delivery techniques with a high degree of computer control. These techniques offer many new opportunities in the delivery of radiation therapy. (use in discussion)
According to Shepard et al, the main advantages of using linear programming formulations is the speed of output of the results and the ease of formulation. These are particularly desirable features in radiation treatment planning. It does however have a lack of adaptability because only a limited number of objective functions and constraints can be found that meet the conditions of linearity. Thus an oncologist is not always guaranteed to be able to find a feasible treatment plan.
Shepard et al also mention that nonlinear programming has the desirable feature in that, unlike linear programming, one can devise an abundance of complex objective functions and constraints. However, this also poses the limitation that the complex nature of generating such functions leads to time-consuming optimizations.
Advantages of MIP
Physicians are experience with this kind of requirement
Partial volue constraints are much more in tuitive than assigning relative weights to each region of the patient.
Disadvantages of MIP
Only linear objective functions and linear constraints can be included in the optimization
Due to the complex formulation each constraint can add considerable time to the optimization.
Advantages of NLP
it is possible to devise numerous complicated objective functions and constraints.
Disadvantages of NLP
increasing the complexity of the formulation will often lead to mre time-consuming optimizations.
For general functions, the optimizer can only guarantee that the solution is locally optimal.