In the proposed approach past treated cases collected from Nottingham City Hospital are stored in a database. Each case consists of two parts; condition of the patients, which describe the outcome of different clinical examinations and dose plan (dose in phase I and II) prescribed by the oncologist. The overall framework of our proposed methodology is illustrated in Fig. 2. In the first step, cases similar to the new case are extracted from the database using case based reasoning. Thereafter, extracted cases are evaluated using TOSPIS multi-criteria decision-making process to make a tradeoff between success rate (maximum dose to the cancerous cells) and side effects of the treatment. Afterward, based on parameters of the most ideal case selected by TOPSIS method dose in phase I and II of the treatment is determined by multi-objective integer goal programming (GP) method. The proposed approach can help oncologist to maximize the total dose without affecting the surrounding organs. The detailed description of each stage is given in the following sub sections.
Representation of the case
In order to retrieve cases similar to the new case from the database, two groups of features i.e. clinical stage and geometry of the prostate are used to measure the similarity between a new case and cases in database. Attributes associated with these features are listed in Table 3 and “Appendix”, describes these features in more details.
Table 3 Features of prostate cancer treatment The data type, measurement unit and scale of aforementioned parameters are different. Thus in order to develop a comprehensive similarity measure in case based reasoning we have used fuzzy set theory to normalize these parameters in the interval of [0 1]. The parameters and values for membership functions have been set and coordinated through Nottingham University Hospital oncologists, which reflect their judgments and perceptions about Gleason Score and PSA values.
The membership functions of each feature l (Gleason Score (l = 1), PSA (l = 2)) of case \( C_{p} \) are represented by a triplet (\( \nu_{pl1} ,\nu_{pl2} ,v_{pl3} ) \), where \( \nu_{plm} \), m = 1, 2, 3 are membership degrees of feature l to fuzzy sets low (m = 1), medium (m = 2) and high (m = 3).
The membership functions of PSA and Gleason Score are shown in Figs. 3 and 4 respectively.
Retrieval process of similar cases
During the discussion with oncologist it was found that clinical stage is an important decision making criteria. Usually, patient with similar clinical stage share same kind of treatment. Clinical stage can be arranged in the following sequence {T1a, T1b, T1c, T2a, T2b, T3a, T3b}. Corresponding to each new patient firstly cases having same clinical stage or adjacent to the new case are extracted from the database. Thereafter, from the filtered list cases similar to the new case are retrieved.
Distance between new case \( C_{p} \) and case in the database \( C_{q} \) is calculated using Eq. (1). It takes into consideration fuzzy membership values of Gleason Score (l = 1) and PSA value (l = 2).
$$ d_{1} \left( {C_{p} ,C_{q} } \right) = \left( {\mathop \sum \limits_{l = 1}^{2} \mathop \sum \limits_{m = 1}^{3} \left( {\nu_{plm} - \nu_{qlm} } \right)^{2} } \right)^{{\frac{1}{2}}} $$
(1)
Taking into account numerical values of different DVH volume percentage of rectum \( (b = 1) \) 66%, \( (b = 2) \) 50%, \( (b = 3) \) 25% and \( (b = 4) \) 10% distance between new case \( C_{p} \) and case in the database \( C_{q} \) is calculated using Eq. (2). In this equation (h = 1, 2) represents the phase of treatment. \( u_{phb} \) and \( u_{qhb} \) are the values of dose received for each percentage of the rectum in phase 1 and 2 of the treatment and is calculated based on DVH values.
$$ d_{2} \left( {C_{p} ,C_{q} } \right) = \left( {\mathop \sum \limits_{h = 1}^{2} \mathop \sum \limits_{b = 1}^{4} \left( {u_{phb} - u_{qhb} } \right)^{2} } \right)^{{\frac{1}{2}}} $$
(2)
Overall similarity measure between cases \( C_{p} \) and \( C_{q} \) is measured using Eq. (3).
$$ S\left( {C_{p} ,C_{q} } \right) = \frac{1}{{1 + d_{1} \left( {C_{p} ,C_{q} } \right) + d_{2} \left( {C_{p} ,C_{q} } \right)}} $$
(3)
Solution methodology for improving CBR
In simple case based reasoning usually decision is based on the most similar extracted case. However, sometime in radiotherapy dose-planning process the most similar case may not be the most appropriate case to make decision. In addition to similarity there are other criteria, which can have more impact on the preference of a case over other cases. In the experiment, it was found that the case which has a high similarity measure is not always the most appropriate case to make decision, because sometimes they have low success rate or dose received by different volume percentages of rectum surpass the restrictions as shown in Table 2. In order to solve the aforementioned problem firstly, cases similar to new case are retrieved from the database thereafter cases are evaluated using TOPSIS technique to make a trade-off between similarity measure, success rate and side effect of treatment.
TOPSIS methodology
TOPSIS (Technique for Order Preference by Similarity to an Ideal Solution) is a MCDM method developed by Hwang and Yoon (1981). The main purpose of this technique is to rank different alternatives based on their distances from ideal positive and negative solution. TOPSIS can be performed using following steps:
At the beginning of the process a decision Matrix D is constructed. The row of each matrix represents alternatives, while column represents different criteria.
$$ DM = \left[ {y_{ij} } \right] = \left[ {\begin{array}{*{20}c} {y_{11} } & \cdots & {y_{1r} } \\ \vdots & \ddots & \vdots \\ {y_{n1} } & \cdots & {y_{nr} } \\ \end{array} } \right] $$
(4)
where \( y_{ij} \) \( (i = 1, \ldots ,n ;j = 1, \ldots ,r) \) are the elements of the decision matrix D.
Thereafter, following steps are performed to select best alternative:
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Step 1 Decision Matrix is normalized using Eq. (5):
$$ R_{ij} = \frac{{y_{ij} }}{{\sqrt {\mathop \sum \nolimits_{j = 1}^{j} y_{ij} } }} $$
(5)
\( R_{ij} \) is the normalized value of element \( y_{ij} \) in decision matrix.
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Step 2 Weighted normalized decision matrix is calculated using Eq. (6).
$$ v_{ij} = w_{i} R_{ij} $$
(6)
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Step 3 Positive and negative ideal solutions are specified using Eqs. (7) and (8) respectively:
$$ PIS = \left\{ {v_{1}^{*} , \ldots ,v_{r}^{*} } \right\} = \left\{ {\begin{array}{*{20}l} {\left. {\mathop {\hbox{max} }\limits_{j} v_{ij} } \right|i\, \in \,benefit} \hfill \\ {\left. {\mathop {\hbox{min} }\limits_{j} v_{ij} } \right|i\, \in \,cost} \hfill \\ \end{array} } \right. $$
(7)
$$ NIS = \left\{ {v_{1}^{ - } , \ldots ,v_{r}^{ - } } \right\} = \left\{ {\begin{array}{*{20}l} {\left. {\mathop {\hbox{min} }\limits_{j} v_{ij} } \right|i\, \in \,benefit} \hfill \\ {\left. {\mathop {\hbox{max} }\limits_{j} v_{ij} } \right|i\, \in \,cost} \hfill \\ \end{array} } \right. $$
(8)
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Step 4 Distance of each alternative from Positive Ideal Solutions (PIS) and Negative Ideal Solutions (NIS) are calculated using Eqs. (9) and (10) respectively.
$$ D_{j}^{ + } = \sqrt {\mathop \sum \limits_{i = 1}^{n} \left( {v_{ij} - v_{i}^{*} } \right)^{2} } $$
(9)
$$ D_{j}^{ - } = \sqrt {\mathop \sum \limits_{i = 1}^{n} \left( {v_{ij} - v_{i}^{ - } } \right)^{2} } $$
(10)
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Step 5 Finally, relative closeness coefficient is calculated using Eq. (11) and alternative with higher coefficient is ranked better.
$$ C_{j}^{*} = \frac{{D_{j}^{ - } }}{{D_{j}^{ - } + D_{j}^{ + } }} $$
(11)
In our proposed method extracted cases are evaluated based on similarity measures, success rate (total prescribed dose, dose in phase I and II of treatment) and side effects of treatment (deviation from recommended dose limit of different volume percentage of rectum as given in Table 2). Similarity measure, total dose and dose in phase I of the treatment is considered as our positive weighted criteria and are given more importance than others. Dose in phase II of the treatment is considered as negative weighted criteria which means that cases having higher value of dose in phase II of treatment is given less importance in decision making process. The weights for all the criteria have been considered the same for this research. The full description of criteria can be found in Malekpoor et al. (2017).
Optimizing the final dose plan
During the study it was found that sometimes dose plan suggested by TOPSIS–CBR is not an optimal dose plan and there is a scope for improvement. On the other hand, sometimes-calculated dose plan is not suitable for a new case. It violates recommended dose limits associated with different volume percentages of the rectum. To solve the above problem optimization of dose planning is performed using integer goal programming mathematical model, where the deviation from DVH recommended values is calculated by the help of best similar case suggested by CBR–TOPSIS. Thereafter, deviations corresponding to different volume percentages of rectum are calculated using Eq. (12):
$$ S_{v}^{p} = d_{q1} DVH_{v}^{1p} + d_{q2} DVH_{v}^{2p} - Recomended\,standard $$
(12)
where \( S_{v}^{p} \) represents deviation of a new case p corresponding to different volume percentage of rectum v (v = 66, 50, 25, 10%). \( d_{q1} \) and \( d_{q2} \) represent dose of extracted case in phase I and II of treatment respectively. \( DVH^{1} \) and \( DVH^{2} \) are the values for Dose Volume Histograms of the new case which are available for each new case and have been also used in finding similarity measure of the case based reasoning. An example of these values for each case is available in Table 4, columns 3–10.
Table 4 Features value of five retrieved cases This value determines the suitability of solution prescribed by CBR–TOPSIS method by calculating amount of dose received by different volume percentage of the rectum.
To treat cancerous cells, in real life sometime oncologists overlook recommended dose limit. The amount of deviation from recommended limit is usually based on oncologists’ past experiences. To employ the knowledge and expertise of oncologist, in this article, deviations are calculated based on extracted past treated patients’ information stored in the database.
Goal programming methodology
Goal programming is basically a multi-objective linear optimization tool, which helps solution to move towards ideal goal. Goal programming consists of following attributes: an objective function, a set of limitations related to goals and systematic constraints. The aim of objective function is to minimize deviations from the given goals as far as possible. The deviation in the objective function is usually weighted to define the priority of some of them to the others. Mathematical formulation of the goal programming is as follows:
$$ \hbox{min} Z = \mathop \sum \limits_{j = 1}^{n} \left( {w_{k} d_{k}^{ + } + w_{k} d_{k}^{ - } } \right) $$
(13)
s.t.
$$ f\left( {x_{1} , \ldots ,x_{i} , \ldots ,x_{M} } \right)_{k} - d_{k}^{ + } + d_{k}^{ - } = g_{k} \quad k = 1,2, \ldots , n; \,\,and\,\, i = 1,2, \ldots ,M; $$
(14)
$$ AX \le or \ge B $$
(15)
where X is a set of variables, that is, \( X = \left\{ {x_{1} ,x_{2} , \ldots ,x_{M} } \right\} \), A is a matrix consisting of coefficient for variables in our systematic constraints, B is a matrix for right side values of systematic constraints and \( g_{k} \) represents the goal corresponding to constraint k.
\( d_{k}^{ + } \) and \( d_{k}^{ - } \) are the auxiliary variables that demonstrate the upper and lower deviations from the goal \( g_{k} \). In objective function, we try to minimize deviations to satisfy the goals. \( w_{k} \) is the importance of the kth goal compared to other goals. As can be seen, the deviations include positive and negative deviations from the considered goal. Minimization procedure is done only on the undesired deviation and thus only the undesired deviation appears in the final objective function of the problem.
If the goal is to achieve more than a certain value, then GP tries to minimize the negative deviation from the goal and positive deviation is going to be maximized automatically as much as the hard constraints allow it to and the opposite happens if the goal is to achieve less than a certain value.
If the goal is to achieve precisely equal to a value, then both of the negative and positive deviations are considered as undesired deviations and objective function tries to minimize both of the deviations.
During the discussion with oncologists it was found that the main objective of dose planning process is to maximize overall prescribed total dose while respecting the dose corresponding to different volume percentage of rectum. If the two-dose plans have same value of total dose and dose received by different volume percentage of rectum is within the constraint the dose plan having higher amount of dose in first phase of treatment is considered as a better dose plan compared to others. In this article goals are set based on abovementioned criteria. Goal objectives are as follows:
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Objective 1: Goal objective of the total dose plan is to assign maximum amount of recommended dose in our case pool.
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Objective 2: Goal objective of the dose in Phase I of the treatment is to deliver maximum or amount of dose prescribed in phase I in our case pool or higher.
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Objective 3: Goal objective of the dose plan in Phase II of the treatment is to assign maximum amount of dose prescribed in phase II in our case pool or higher.
Figure 5 shows the Process of modeling the GP problem.
The first step is to running CBR–TOPSIS and obtain the best case corresponding to a new case. Thereafter, in the second step by using the dose prescribed by the best case from CBR–TOPSIS and DVH values related to the new case based on Eq. 12, allowed deviations from different volume percentages of the rectum (\( S_{v}^{p} \)) is calculated. Through Objectives 1–3 the goals for GP are set and with the help of the calculated allowed deviations the hard constraints of the GP are defined.
The mathematical formulation for integer goal programming related to prostate cancer dose planning process is as follows:
$$ \hbox{min} \,Z = \mathop \sum \limits_{j = 1}^{3} \left( {w_{k} d_{k}^{ + } + w_{k} d_{k}^{ - } } \right) $$
(16)
$$ x_{1} + x_{2} - d_{1}^{ + } + d_{1}^{ - } = g_{1} $$
(17)
$$ x_{1} - d_{2}^{ + } + d_{2}^{ - } = g_{2} $$
(18)
$$ x_{2} - d_{3}^{ + } + d_{3}^{ - } = g_{3} $$
(19)
$$ DVH_{66\% }^{1p} x_{1} + DVH_{66\% }^{2p} x_{2} + S_{66\% }^{p} \le 45; $$
(20)
$$ DVH_{50\% }^{1p} x_{1} + DVH_{50\% }^{2p} x_{2} + S_{50\% }^{p} \le 55; $$
(21)
$$ DVH_{25\% }^{1p} x_{1} + DVH_{25\% }^{2p} x_{2} + S_{25\% }^{p} \le 65; $$
(22)
$$ DVH_{10\% }^{1p} x_{1} + DVH_{10\% }^{2p} x_{2} + S_{10\% }^{p} \le 70; $$
(23)
$$ x_{1} ,x_{2} \ge 0\,{\text{and}}\,{\text{integer}}; $$
(24)
$$ d_{1}^{ + , - } ,d_{2}^{ + , - } ,d_{3}^{ + , - } \ge 0; $$
(25)
where \( k = 1,2,3 \), the goals; \( x_{1} \) and \( x_{2} \), dose plan in phase I and II of the treatment; \( w_{j} \) weight of kth goal; \( d_{k}^{ + } \), positive deviation from the kth goal; \( d_{k}^{ - } \), negative deviation from the kth goal; \( g_{1} , g_{2} \) and \( g_{3} \), goal objectives; \( DVH_{66, 50,25,10\% }^{1p} \), DVH values of the new case p, in the first phase of treatment corresponding to 66, 50, 25 and 10% of the rectum volume; \( DVH_{66, 50,25,10\% }^{2p} \), DVH values of the new case p, in the second phase of treatment corresponding to 66, 50, 25 and 10% of the rectum volume; \( S_{66,50,25,10\% }^{p} \), the amount of deviation oncologists committed corresponding to different volume percentage of rectum for the new case p.
While \( x_{1} \) and \( x_{2} \) are the values of dose plan in phase I and II of the treatment respectively, \( x_{1} + x_{2} \) equals to total dose plan of the treatment. There are three goals (objectives) in our problem which are explained in this section above (Sect. 3.4.1) and are \( k = 1,2,3 \) respectively in the above model. DVH values for each phase of the treatment and each percentage of the rectum volume is available in our data set (total of 8 values for each case) and \( S^{p} \) for each volume of the rectum is being calculated by expression 12.
Equation (16) is the objective function for minimizing the deviations from our goals. Equations (20)–(25) are our goal related constraints, which determine deviations from total dose plan, dose plan in Phase I and II of the treatment respectively. Equations (17)–(19) are our systematic goals which restrict the optimization process to find solutions without violating the recommended doses based on oncologists suggestions and pre-prescribed standards. Equation (24) helps to achieve positive integer values for our dose plan.
In above GP model, we try to optimize (finding a solution as much as possible near to ideal goals) the total amount of dose plan, dose in phase I and Phase II of the treatment so that we can prescribe an optimal treatment. To achieve above objective firstly we have calculated possible deviations from recommended standards (as shown in Table 2) using CBR–TOPSIS method and consider it as constraints to our goals. Thereafter, optimal combination of total dose, dose in Phase I and II of treatment is generated while satisfy the considered constraints.
Maximization of dose plan within safe risk zone
If dose received by different volume percentages of rectum is within the constraint the dose plan is acceptable. The higher the total dose, more likely the probability is to kill the cancerous cells. Hence, where there are positive \( S_{v}^{p} \) it means more dose can be prescribe without deviating prescribed dose limit. Higher dose can kill the cancer cells without causing any significant damage to rectum and in exchange can increase the chance of eradicating cancerous cells. In the final step modification is performed to minimize the deviation from recommended standards as described in Eq. (26).
$$ S_{v}^{p} = \left\{ {\begin{array}{*{20}l} {S_{v}^{p} } \hfill &\quad {if\quad \ge 0} \hfill \\ 0 \hfill &\quad {if\quad \le 0} \hfill \\ \end{array} } \right. $$
(26)
In real life sometimes to treat cancerous cells oncologists overlook the recommended dose limit associated with different volume percentage of rectum. Similarly, in our proposed model the system will retrieve the past similar cases and based on extracted cases it will decide the dose limit associated with different volume percentage of rectum. The proposed model will overlook the recommended dose limit if oncologist has performed similar decision in the past. Once the dose limit is set, goal programing method will search for the optimal dose plan.
Modification rule for integer programming
Usually, dose is delivered in 2 Gy packs. Hence, dose in phase I and II of the treatment must be an even number. In order to solve the problem of odd numbers the following conditions are incorporated in programming:
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1.
If calculated dose in phase I or phase II of treatment is an odd number then:
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a.
Increase the dose by 1 Gy. If dose received by different volume percentages of rectum violate the constraint then decrease the dose by 1 Gy.
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2.
If dose plan in both Phases of the treatment is odd number then:
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a.
Increase the dose plan in the both phases of the treatment by 1 Gy and check the limitation suggested by oncologists. If violated go to step b.
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b.
Increase the dose plan in phase I by 1 Gy and decrease dose plan in phase II by 1 Gy. Check the limitation suggested by oncologists. If violated go to step c.
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c.
Decrease the dose plan in phase I by 1 Gy and increase dose plan in phase II by 1 Gy. Check the limitation suggested by oncologists. If violated go to step d.
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d.
Decrease the dose in phase I and II of treatment by 1 Gy