Biological Modeling Reference Guide. Eclipse

Transcription

1 Biological Modeling Reference Guide Eclipse P/N B502697R01A OCTOBER 2009

2 Abstract Biological Modeling Reference Guide (P/N B502697R01A), document version 1.0, provides reference information about biological modeling in Eclipse. Manufacturer and Authorized European Representative Notice FDA 21 CFR 820 Quality System Regulations (CGMPs) ISO CE HIPAA Trademarks Copyrights Manufacturer: European Authorized Representative: Varian Medical Systems, Inc. Varian Medical Systems UK Ltd Hansen Way Gatwick Road, Crawley Palo Alto, CA , USA West Sussex RH10 9RG United Kingdom Information in this user guide is subject to change without notice and does not represent a commitment on the part of Varian. Varian is not liable for errors contained in this user guide or for incidental or consequential damages in connection with furnishing or use of this material. This document contains proprietary information protected by copyright. No part of this document may be reproduced, translated, or transmitted without the express written permission of Varian Medical Systems, Inc. Varian Medical Systems, Oncology Systems products are designed and manufactured in accordance with the requirements specified within this federal regulation. Varian Medical Systems, Oncology Systems products are designed and manufactured in accordance with the requirements specified within ISO quality standards. Varian Medical Systems, Oncology Systems products meet the requirements of Council Directive MDD 93/42/EEC. Varian s products and services are specifically designed to include features that help our customers comply with the Health Insurance Portability and Accountability Act of 1996 (HIPAA). The software application uses a secure login process, requiring a user name and password, that supports role based access. Users are assigned to groups, each with certain access rights, which may include the ability to edit and add data or may limit access to data. When a user adds or modifies data within the database, a record is made that includes which data were changed, the user ID, and the date and time the changes were made. This establishes an audit trail that can be examined by authorized system administrators. Aria and Varian are registered trademarks, Acuity, Eclipse, Eclipse Dose Volume Optimizer, Eclipse Proton, Enhanced Dynamic Wedge, MLC Shaper, On Board Imager, Portal Imaging, Smart Segmentation, Ximatron CT Imaging and Ximatron Digital Imaging are trademarks of Varian Medical Systems, Inc. The Independent JPEG Groupʹs JPEG software: Copyright , Thomas G. Lane. This software is based in part on the work of the Independent JPEG Group. 2

3 WHO ICD O codes and terms used by permission of WHO, from: International Classification of Diseases for Oncology, (ICD O) 3 rd edition, Geneva, World Health Organization, ICD 10 codes and terms used by permission of WHO, from: International Statistical Classification of Diseases and Related Health Problems, Tenth Revision (ICD 10). Vols 1 3, Geneva, World Health Organization, CAUTION: US Federal law restricts this device to sale by or on the order of a physician Varian Medical Systems, Inc. All rights reserved. Produced in Finland. 3

4 4 Biological Modeling Reference Guide

5 Contents CHAPTER 1 INTRODUCTION... 7 About This Manual... 7 Before You Begin... 7 Who Should Read This Manual... 7 Visual Cues... 8 Bold text...8 Courier font...8 Quotation marks...8 Italics...9 Related Publications... 9 Contacting Support Ordering Additional Documents Communicating Via the World Wide Web Sending CHAPTER 2 BIOLOGICAL MODELING Introduction to Biological Models...15 Cell Survival Models The Linear Cell Survival Model The Linear-Quadratic (LQ) Cell Survival Model...17 The LQ Cell Survival Model Including Incomplete Repair...17 The LQ Cell Survival Model Including Accelerated Repopulation The LQ Cell Survival Model Including both Incomplete Repair and Accelerated Repopulation EQD EQD 2 Including Incomplete Repair...22 EQD 2 Including Accelerated Repopulation EQD 2 Including both Incomplete Repair and Accelerated Repopulation...24 Dose-Response Models...25 The Linear Dose-Response Model The Linear-Quadratic (LQ) Dose-Response Model The LQ Dose-Response Model Including Incomplete Repair

6 The LQ Dose-Response Model Including Accelerated Repopulation The LQ Dose-Response Model Including both Incomplete Repair and Accelerated Repopulation Tumor and Normal Tissue Response...31 TCP-PoissonLQ...31 NTCP...32 NTCP-PoissonLQ Model NTCP-LKB Model...34 Composite TCP and NTCP Models Composite TCP Composite NTCP P Parameters of the Biological Models...36 APPENDIX A EQUIVALENCE BETWEEN LINEAR EQD 2 AND LQ DOSE-RESPONSE MODELS Biological Modeling Reference Guide

7 Chapter 1 Introduction This chapter covers the notation used throughout this manual and lists other publications related to this manual. About This Manual Before You Begin This section provides background information necessary for using the application, who this manual is meant for, and the notation used throughout this manual. This manual assumes that you are familiar with Microsoft Windows XP. Before you begin using the application, read your Microsoft Windows documentation to become familiar with the Windows environment. Who Should Read This Manual This guide is written mainly for medical personnel who may have specific tasks in the treatment planning process, for instance, physicists or any other personnel responsible for setting up fields, calculating the dose distribution and evaluating plans. WARNING: Before or when using Eclipse, notice the following: Eclipse intended use is for dose estimations in the planning of radiation therapy treatments for patients having malignant or benign disease. Eclipse should only be used by qualified medical professionals. Ensure that individuals authorized to perform treatment planning functions are appropriately trained for the functions they perform. All treatment plan reports shall be approved by a qualified person before the information in them is used for radiotherapy treatment purposes. Always be aware that the quality of the output depends critically on the quality of the input data, and any irregularities or uncertainties about input data units, identification, or quality of any other nature shall be thoroughly investigated before the data are used. 7

8 Visual Cues This section presents the types of notes and precautionary notices used in the manual, along with their icons. The following notational conventions are used: Note: A note describes actions or conditions that can help the user to obtain optimum performance from the equipment or software. CAUTION: A caution describes actions or conditions that could result in minor or moderate injury or damage to equipment or loss of data. WARNING: A warning describes actions or conditions that could result in serious injury or death. Bold text This guide uses bold text for Menus and menu commands. Example: Choose Edit > Delete. Command buttons. Example: To confirm, click Yes. Option buttons. Example: In the dialog box, select the 3D option button. Check boxes. Example: In the dialog box, select the Import Structures check box. Text boxes. Example: In the ID and Name text boxes, type the appropriate information. Courier font Programming code, text in export files, names of environment variables, files and other items related to programming are written with the Courier font. Quotation marks This guide uses quotation marks for Internal cross references. Example: See Topic of Interest on page X. Text that you type in the user interface. Example: In the cell, type Data. 8 Biological Modeling Reference Guide

9 Italics This guide uses italics for References to other publications. Example: Refer to External Beam Planning Reference Guide. Emphasis. Example: Always verify plans visually. Related Publications For additional information, refer to the following sources: Biological Optimization, Biological Evaluation and Conformal Optimization Instructions for Use Provides basic information and procedures for using biological optimization, biological evaluation and conformal optimization in Eclipse. Reference Guide for External Beam Planning Provides background information and detailed instructions for preparing CT images for use in treatment planning and for creating external treatment plans in Eclipse. External Beam Planning Instructions for Use Provides basic information and procedures for using Eclipse for photon and electron therapy planning in the daily workflow. Beam Configuration Reference Guide Provides reference information and procedures for beam data configuration required for performing dose calculation for external treatment plans in Eclipse. Eclipse Algorithms Reference Guide Describes algorithms supported in the Eclipse treatment planning system. RT and Imaging Online Help Provides background information and detailed instructions for how to use ARIA and Eclipse, and describes all functions available in the ARIA and Eclipse applications. Introduction 9

10 Contacting Support Support services are available without charge during the initial warranty period. If you seek information not included in this publication, call Varian Medical Systems support at the following locations: United States and Canada telephone support United States and Canada Direct telephone support European telephone Support Fax (US) Fax (Service Europe) All other countries please call your local service office. To contact the support location nearest you for Service, Parts, or Support, see the list at the Varian Medical Systems website: Worldwide Listing Ordering Additional Documents To order additional documents, call the following: North America (Press 2 for parts) Global Customer Technical Bulletins for Varian Medical Systems may be found on the internet at: then select Support Communicating Via the World Wide Web If you have access to the Internet, you will find Varian Medical Systems support at the following location: Oncology Systems 10 Biological Modeling Reference Guide

11 Sending If you have a Varian account, enter your username and password. Otherwise, first click Create new account to get a username and password. From MyVarian home page, click Contact Us from the Support list along the left side of the window. If possible, please send all e mail inquiries through the my.varian.com web site at otherwise, use the following e mail addresses for support: North America (North America and Canada) support americas@varian.com Central & South America soporte.al@varian.com Europe (Europe, Middle East, Africa) Australia (Australia, New Zealand, Australasia) support emea@varian.com support anz@varian.com China / Asia (China, Asia) support china@varian.com Japan support japan@varian.com Brachytherapy Systems brachyhelp@varian.com (please leave the subject line blank) Introduction 11

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13 Chapter 2 Biological Modeling Radiobiological models are used in radiobiological treatment planning to estimate the tumor control probability (TCP) and the normal tissue complication probability (NTCP). One TCP model, TCP PoissonLQ, and two NTCP models, NTCP PoissonLQ and NTCP LKB, are presented in this manual and are summarized here. The TCP PoissonLQ model is defined as: Eq. 1 M n 2 TCP(D) = N exp 0 exp ad k,i d k,i i = 1 k = 1 v i V ref N v = exp exp EQD 2, i e e lnln2 D 50 i = 1 M = Total number of voxels. D = Total dose. d k,i = Dose of the kth fraction to voxel i. n = Total number of fractions. N 0 = Initial number of cells. and = Parameters of the LQ model. v Relative volume of voxel i compared to the i V = reference volume for which the parameters are ref obtained. D 50 = Dose giving a 50% response probability. = Maximum normalized gradient of the dose response curve. EQD 2,i = Equivalent dose in voxel i given in 2 Gy fractions. The equality in the equation is derived in Appendix A on page 39. v i V ref 13

14 The general EQD 2 is defined as: Eq. 2 n, d k,i, and are defined as in Equation 1 on page 13. The NTCP PoissonLQ model, based on the relative seriality model by Källman et al. 1, is expressed as: Eq. 3 n d k = 1 k,i 1 + d k,i EQD 2,i,general = v i M n s s V ref 2 NTCP P LQ D = 1 1 exp N 0 exp ad k,i d k,i i = 1 i = k v M EQD 2,i 1 1 exp exp e e lnln2 s i 1 s V ref = D 50 i = 1 s = Relative seriality v i V ref M, D, d k,i, n, N 0,,, , and are defined as in Equation 1 on page Källman P, Ågren A, Brahme A. Tumour and normal tissue responses to fractionated non uniform dose delivery, Int J Radiat Biol. 1992; 62: Biological Modeling Reference Guide

15 The NTCP LKB model is expressed as: Eq. 4 t 1 NTCP LKB (D) exp u 2 = du 2 D t eff D = m D 50 and M v i 1 n D eff = EQD n V 2,1 ref i = 1 D 50 = Dose giving a response probability of 50%. M = Total number of voxels. v i V ref = Relative volume of voxel i compared to the reference volume. EQD 2,i = Equivalent dose in voxel i given 2 Gy fractions. Introduction to Biological Models The main goal in radiation therapy is to deliver a sufficiently high dose to the tumor so that all tumor cells are killed while avoiding radiation induced damage to the surrounding normal tissue. Radiobiological models are used in radiobiological treatment planning to estimate the probability of eradicating the tumor and the probability of inducing damage to the normal tissue. In this reference guide, one model for estimating tumor control probability, TCP PoissonLQ, and two models for estimating normal tissue complication probability, NTCP PoissonLQ and NTCP LKB, are presented. The TCP PoissonLQ model and the NTCP PoissonLQ model are based on cell survival models and Poisson statistics and some background to these models is given. The NTCP LKB model, on the other hand, is based on a probit function. Biological Modeling 15

16 Cell Survival Models A cell survival curve describes the relationship between the absorbed dose and the fraction of cells that survive. The meaning of cell survival is different in different contexts. For proliferating cells, the term clonogenic cell survival is often used and in this context surviving cells are cells that have retained their reproductive capacity, that is, cells still capable of undergoing cell divisions. Conventionally, the survival fraction is plotted on a log linear scale with dose on the linear x axis. The clonogenic cell survival as a function of dose shows an exponential decrease at high doses. At low doses (around 2 Gy) there is usually a curvature or, as it is commonly called, a shoulder region on a log linear plot which is explained by cellular repair mechanisms. Cell survival models aim to describe the shape of survival curves and two of the most commonly used models are described here. The Linear Cell Survival Model The clonogenic cell survival as a function of dose is in a first approximation an entirely exponential process. The simple linear cell survival model is expressed as: Eq D D 0 S L (D) = e D = Total dose. D 0 = Dose that reduces the survival by a factor 0.37 (e -1 ). In this model, cellular repair is not considered as it is done in the linear quadratic cell survival model (see The Linear Quadratic (LQ) Cell Survival Model on page 17). 16 Biological Modeling Reference Guide

17 The Linear-Quadratic (LQ) Cell Survival Model Eq. 6 The fraction of surviving cells as a function of dose is usually not an entirely exponential process as it is assumed by the linear cell survival model (see The Linear Cell Survival Model on page 16), but there is a curvature or shoulder on the log linear survival curve at low doses due to the influence of cellular repair mechanisms. The linear quadratic (LQ) model can handle this type of response and takes fractionation effects into account. The model is expressed as: n n n n S LQ (D) S LQ d k exp d k d k 2 exp d k d k 2 d = = = = exp d k 1 + k k = 1 k = 1 k = 1 k = 1 D = Total dose. d k = Dose of the kth fraction. n = Total number of fractions. and = Parameters of the LQ model. The α/β ratio describes the fractionation sensitivity of the tissue. For late responding tissues, the α/β value is low (commonly around 3 Gy) while acutely responding normal tissue and most tumors have a larger α/β value (around 10 Gy). A small α/β value indicates that large fractional doses are not well tolerated. When the fractional doses are equal, this expression is simplified to: Eq. 7 S LQ (D) exp nd nd 2 d = = exp D The LQ Cell Survival Model Including Incomplete Repair In the simple LQ cell survival model (see The Linear Quadratic (LQ) Cell Survival Model on page 17) it is assumed that the time between dose fractions is sufficient for completely repairing the sublethal damage. In bi exponential repair models, incomplete repair is Biological Modeling 17

18 accounted for and it is assumed that the repair process has a fast and a slow component. The LQ model is then expanded to the following expression (Levin Plotnik et al. 2 ): Eq. 8 lns LQrepair (D) = n k 1 k 1 k 1 k 1 d k d 2 k 21 ld k d p s,q 2ld k d p l,q p = 1 q = p p = 1 q = p k = 1 n k 1 k 1 k 1 k 1 d d k k l d p + 2 s,q l d = p l,q p = 1 q = p p = 1 q = p k = 1 D = Total dose. dk = Dose of the kth fraction. n = Total number of fractions. and = Parameters of the LQ model. l = Fraction of the total repair that is due to long repair times. And ln2 s,q = exp t T q 1 2,s l,q = exp t ln2 T q 1 2,l t q = Time between fraction q and q + 1. T (1/2),s and T (1/2),l = Repair half times for the short and long repair, respectively. 2. Levin Plotnik D, Hamilton RJ, Niemierko A, Akselrod S. A model for optimizing normal tissue complication probability in the spinal cord using a generalized incomplete repair scheme. Radiat Res. 2001; 155 (4): Biological Modeling Reference Guide

19 The LQ Cell Survival Model Including Accelerated Repopulation There are studies showing that cells start to divide at a higher rate after some time following irradiation. This phenomenon can be taken into account by modifying the LQ cell survival model as (Fowler 3 ): Eq. 9 n T T start n 2 T S LQrepop (D) exp d k d k p = 2 = exp d k d 2 k k = 1 k = 1 After some rewriting this gives: T T start T p expln2 Eq. 10 n d lns LQrepop (D) d k 1 + k T T start = ln2 T p k = 1 D = Total dose. d k = Dose of the kth fraction. n = Total number of fractions. and = Parameters of the LQ model. T = Total treatment time. T start = Time point when the accelerated repopulation sets in. T p = Potential cell doubling time. When T < T start, it is assumed that no repopulation occurs. The fraction of surviving cells is not allowed to become larger that one even if there is a major contribution of accelerated repopulation. 3. Fowler JF. The linear quadratic formula and progress in fractionated radiotherapy. Br J Radiol Aug; 62 (740): Biological Modeling 19

20 The LQ Cell Survival Model Including both Incomplete Repair and Accelerated Repopulation When both incomplete repair and accelerated repopulation is taken into account the expression for cell survival becomes: Eq. 11 lns LQr&r (D) = k = 1 D = Total dose. d k = Dose of the kth fraction. n = Total number of fractions. and = Parameters of the LQ model. l = Fraction of the total repair that is due to long repair times. And n k 1 k 1 k 1 k 1 d d k k d 1 l p 2 s,q l d p l,q p = 1 q = p p = 1 q = p T T start ln2 T p ln2 s,q = exp t T q 1 2,s ln2 l,q = exp t T q 1 2,l t q = Time between fraction q and q + 1. t (1,2),s and t (1/2),1 = Repair half times for the short and long repair, respectively. T = Total treatment time. T start = Time point when the accelerated repopulation sets in. T p = Potential cell doubling time. When T < T start, it is assumed that no repopulation occurs. The fraction of surviving cells is not allowed to become larger that one even if there is a major contribution of accelerated repopulation. 20 Biological Modeling Reference Guide

21 EQD 2 EQD 2, the equivalent dose in 2 Gy fractions, is based on the linear quadratic (LQ) cell survival model and can be used for comparing fractionation schedules. It estimates the total dose delivered in 2 Gy fractions that gives the same biological effect as the total dose obtained using the fractionation schedule of interest. The general EQD is expressed as: Eq. 12 n k = 1 d k 1 lns EQD LQ D + d k ,simple = d = ref d ref D = Total dose. d k = Dose of the kth fraction. n = Total number of fractions. and = Parameters of the LQ model. d ref = Reference fractional dose that is usually 2 Gy. Henceforth, d ref is set to 2. If equal fractional doses are used, this expression reduces to: Eq. 13 D1 + d EQD 2,simple = EQD 2 is a tool for comparing treatment plans qualitatively. If, for example, two fractionation schedules are to be compared and both are rescaled using EQD 2, the treatment plan with the highest EQD 2 will according to the model give more damage than the other plan. Biological Modeling 21

22 EQD 2 Including Incomplete Repair Eq. 14 In the general EQD 2, it is assumed that the time between dose fractions is sufficient for completely repairing the sublethal damage. When incomplete repair is taken into account, the EQD 2 is then expanded to: lns EQD LQrepair D 2,repair = n k = 1 d d k 1 k l k p = 1 d k 1 1 p q = p s,q k 1 d k 1 l p = 1 p q = p l,q = d k = Dose of the kth fraction. n = Total number of fractions. and = Parameters of the LQ model. l = Fraction of the total repair that is due to long repair times. And ln2 s,q = exp t T q 1 2,s ln2 l,q = exp t T q 1 2,l t q = Time between fraction q and q + 1. t (1/2),s and t (1/2),1 = Repair half times for the short and long repair, respectively. 22 Biological Modeling Reference Guide

23 EQD 2 Including Accelerated Repopulation When accelerated repopulation is taken into account, the EQD 2 becomes (Fowler 4 ): Eq. 15 n k = 1 d k 1 + d k ln T T start T p lns EQD LQrepop D 2 2,repop = = 0,if ln T T n start d d T p k 1 + k k = 1 d k = Dose of the kth fraction. n = Total number of fractions. and = Parameters of the LQ model. T = Total treatment time. T start = Time point when the accelerated repopulation sets in. T p = Potential cell doubling time. When T < T start, it is assumed that no repopulation occurs. If the accelerated repopulation term becomes very large, this would give a negative EQD 2 value. In this case EQD 2 is set to zero. 4. Fowler JF. The linear quadratic formula and progress in fractionated radiotherapy. Br J Radiol Aug; 62 (740): Biological Modeling 23

24 24 Biological Modeling Reference Guide EQD 2 Including both Incomplete Repair and Accelerated Repopulation Eq. 16 When both incomplete repair and accelerated repopulation is taken into account, the EQD 2 becomes: EQD 2,r&r = n d d k 1 k 1 k k 1 k = 1 1 l d p p = 1 1 k 1 k s,q q = p l d p p = 1 ln2 l,q q = p T T start T p = d k = Dose of the kth fraction. n = Total number of fractions. and = Parameters of the LQ model. l = Fraction of the total repair that is due to long repair times. And ln2 s,q = exp t T q 1 2,s ln2 l,q = exp t T q 1 2,l t q = Time between fraction q and q+1. t (1/2),s and t (1/2),1 = Repair half times for the short and long repair, respectively. T = Total treatment time. T start = Time point when the accelerated repopulation sets in. T p = Potential cell doubling time. When T < T start, it is assumed that no repopulation occurs. If the accelerated repopulation term becomes very large, this would give a negative EQD 2 value, and in this case EQD 2 is set to zero.

25 Dose-Response Models A dose response model describes the probability of response (biological effect) as a function of dose. The dose response curve usually has a sigmoid shape. At low doses the probability of response is small but increases with increasing dose. A dose response model can be derived from a cell survival model using the Poisson distribution resulting in the following relation: Eq. 17 PD = exp N 0 SD N 0 = Initial number of cells. S(D) = Arbitrary cell survival model. The sigmoid shape is frequently described by the dose giving a 50% response, D 50, and the maximum normalized gradient of the dose response curve,. The parameter is defined as (Brahme 5 ): Eq. 18 = max P D The initial number of cells can be expressed as N 0 (Lind et al. 6 ). = exp(e) The Linear Dose-Response Model Using the linear cell survival model (see The Linear Cell Survival Model on page 16), we obtain the following expression for the response as a function of dose: 5. Brahme A. Dosimetric precision requirements in radiation therapy. Acta Radiol Oncol 1984; 23: Lind BK, Mavroidis P, Hyödynmaa S, Kappas C. Optimization of the dose level for a given treatment plan to maximize the complication free tumor cure. Acta Oncol. 1999; 38 (6) : Biological Modeling 25

26 Eq. 19 P L D exp N 0 S L D exp D N 0 exp D = = = exp exp e D 0 D 0 D = Total dose. D 0 = Dose that reduces the survival by a factor 0.37 (e 1 ). N 0 = Initial number of cells. = Maximum normalized gradient of the dose response curve. By inserting the dose giving a response value of 0.5 (D 50 ) we obtain: Eq. 20 P L D exp D exp e expln 1 D D = = = exp exp e D 0 D lnln2 = e D 0 D D 50 0 = e lnln2 The linear dose response model can thus be expressed with the parameters D 50 and : Eq. 21 D P L D = exp exp e e lnln2 D 50 The parameters D 50 and are only valid if the dose D is delivered using the same reference dose per fraction, d ref, as the one that was used when these parameters were estimated. By using EQD 2 instead of the physical dose D, the physical dose is rescaled to an isoeffective dose delivered in fractions of 2 Gy. In this way the parameters of the linear dose response model derived for 2 Gy per fraction can be used even if the dose per fraction is not equal to 2 Gy. By replacing the total dose D by EQD 2 the following expression is obtained: Eq. 22 EQD 2 P L EQD 2 = exp exp e e lnln2 D 50 one of the four expressions for EQD 2 defined in EQD 2 on page 21 can be used. 26 Biological Modeling Reference Guide

27 The Linear-Quadratic (LQ) Dose-Response Model Eq. 23 When the dose response model is based on the linear quadratic (LQ) cell survival model (see The Linear Quadratic (LQ) Cell Survival Model on page 17), the following expression is obtained: n n P LQ D exp N 0 S LQ D exp N 0 exp d k d 2 d = = k = exp exp e d k 1 k k = 1 k = 1 N 0 = Initial number of cells. D = Total dose. d k = Dose of the kth fraction. n = Total number of fractions. and = Parameters of the LQ model. = Maximum normalized gradient of the dose response curve. When the fractional doses are equal, this expression can be simplified to: Eq. 24 d P LQ D = exp exp e D The clinically important D 50 parameter can be included in the LQ model by having P LQ (D) = P L (D) for a given fractional dose d ref, usually 2Gy, which gives (Lind et al. 7 ): Eq. 25 e lnln2 e lnln2 = D 50 1 d = ref D This equality holds only for the constant fractional dose d ref for which the parameters were estimated or when. Using this expression for the parameter, the LQ dose response model, P LQ (D), is equivalent to the linear dose response model using EQD 2, P L (EQD 2 ), as shown in Appendix A. 7. Lind BK, Mavroidis P, Hyödynmaa S, Kappas C. Optimization of the dose level for a given treatment plan to maximize the complication free tumor cure. Acta Oncol. 1999; 38 (6): Biological Modeling 27

28 The LQ Dose-Response Model Including Incomplete Repair The dose response model obtained using the LQ cell survival model including cellular repair is expressed as (Levin Plotnik 8 ): Eq. 26 P LQrepair D = n k 1 k 1 k 1 k 1 d exp exp ed k k l d p + 2 s,q l d = p l,q p = 1 q = p p = 1 q = p k = 1 d k = Dose of the kth fraction. n = Total number of fractions. and = Parameters of the LQ model. = Maximum normalized gradient of the dose response curve. l = Fraction of the total repair that is due to long repair times. And ln2 s,q = exp t T q 1 2,s ln2 l,q = exp t T q 1 2,l t q = Time between fraction q and q+1. t (1/2),s and t (1/2),1 = Repair half times for the short and long repair, respectively. 8. Levin Plotnik D, Hamilton RJ, Niemierko A, Akselrod S. A model for optimizing normal tissue complication probability in the spinal cord using a generalized incomplete repair scheme. Radiat Res. 2001; 155 (4): Biological Modeling Reference Guide

29 The LQ Dose-Response Model Including Accelerated Repopulation Using the LQ cell survival model including repopulation, the following dose response expression is obtained: Eq. 27 n d P LQrepop D exp exp ed k 1 + k T T start = ln2 T p k = 1 d k = Dose of the kth fraction. n = Total number of fractions. and = Parameters of the LQ model. = Maximum normalized gradient of the dose response curve. T = Total treatment time. T start = Time point when the accelerated repopulation sets in. T p = Potential cell doubling time. When T < T start, it is assumed that no repopulation occurs. Biological Modeling 29

30 30 Biological Modeling Reference Guide Eq. 28 The LQ Dose-Response Model Including both Incomplete Repair and Accelerated Repopulation When both incomplete repair and accelerated repopulation is taken into account, the dose response expression becomes: n k 1 k 1 k 1 k 1 d P LQr&r D exp exp ed k 1 k d p s,q s d j T T l,q q = p l start = + ln2 p = 1 p = 1 q = p T p k = 1 = d k = Dose of the kth fraction. n = Total number of fractions. and = Parameters of the LQ model. = Maximum normalized gradient of the dose response curve. l Fraction of the total repair that is due to long repair times. And ln2 s,q = exp t T q 1 2,s ln2 l,q = exp t T q 1 2,l = t q = Time between fraction q and q+1. t (1/2),s and t (1/2),l = Repair half times for the short and long repair, respectively. T = Total treatment time. T start = Time point when the accelerated repopulation sets in. T p Potential cell doubling time. When T < T start, it is assumed that no repopulation occurs.

31 Tumor and Normal Tissue Response TCP-PoissonLQ The main goal in radiation therapy is to deliver a sufficiently high dose to the tumor so that all tumor cells are killed while avoiding damage to the surrounding normal tissue. The tumor control probability, TCP, and normal tissue complication probability, NTCP, can be estimated by the models presented here. The TCP PoissonLQ model and the NTCP PoissonLQ model are based on cell survival models and Poisson statistics, while the NTCP LKB model is based on a probit function. In order to eradicate a tumor, it is believed that all clonogenic tumor cells have to be killed. Using the LQ dose response model (with or without incomplete repair and/or repopulation) or, equivalently (see Equivalence Between Linear EQD 2 and LQ Dose Response Models on page 39), using the linear dose response model with EQD 2 (with or without incomplete repair and/or repopulation) the probability of tumor control, TCP, can be obtained. The TCP for target (tumor) j is calculated according to Equation 29 on page 32: Biological Modeling 31

32 Eq. 29 TCP j D M v i M n P V ref 2 = LQ j D i = exp N 0 exp d k,i d k,i i = 1 i = 1 k = 1 v i V ref v i M M V ref EQD TCP j D = P l,j EQD 2,1 = exp exp e ,i e lnln2 D 50 i = 1 i = 1 v i V ref P LQ (D) = LQ dose response model (see The Linear Quadratic (LQ) Cell Survival Model on page 17). P L (EQD 2 ) = Linear dose response model using EQD 2, the equivalent dose given in 2 Gy fractions (see The Linear Dose Response Model on page 25). D = Total dose. d k = Dose of the kth fraction to voxel i. n = Total number of fractions. M = Total number of voxels. N 0 = Initial number of cells. and = Parameters of the LQ model. v Relative volume of voxel i compared to the reference i = V ref volume for which the parameters are obtained. D 50 = Dose giving a 50% response probability. = Maximum normalized gradient of the dose response curve. NTCP NTCP-PoissonLQ Model The normal tissue complication probability for tissue j, NTCP j, can be calculated using the relative seriality model defined by Källman and co workers 9 together with the LQ dose response model (with or without incomplete repair). Equivalently (see Equivalence Between 9. Källman P, Ågren A, Brahme A. Tumour and normal tissue responses to fractionated non uniform dose delivery, Int J Radiat Biol. 1992; 62: Biological Modeling Reference Guide

33 Linear EQD 2 and LQ Dose Response Models on page 39), NTCPj can be calculated using the relative seriality model together with the linear dose response model with EQD 2 (with or without incomplete repair). Eq. 30 v i 1 s M NTCP j D 1 1 P LQ,j D s V ref = = j = 1 M n s V ref 1 1 exp N 0 exp d k d 2 k j = 1 k = 1 v i 1 s M NTCP j D 1 1 P L,j EQD 2 s V ref = j = 1 v i 1 s = M 1 1 exp EQD 2 exp e e lnln2 s V ref D 50 j = 1 P LQ (D) = LQ dose response model (see The Linear Quadratic (LQ) Cell Survival Model on page 17). P L (EQD 2 ) = Linear dose response model using EQD 2, the equivalent dose given in 2 Gy fractions (see The Linear Dose Response Model on page 25). D = Total dose. d k,i = Dose of the kth fraction. n = Total number of fractions. M = Total number of voxels. N 0 = Initial number of cells. and = Parameters of the LQ model. v i V ref = Relative volume of voxel i compared to the reference volume for which the parameters are obtained. D 50 = Dose giving a 50% response probability. = Maximum normalized gradient of the dose response curve. s = Relative seriality parameter describing the volume dependence of the dose response relation by specifying the internal organization of the organ. v i 1 s Biological Modeling 33

34 For a parallel organ, such as the lung, the parameter s is small. This means that the organ is not that sensitive to high local doses (hot spots) but both low and high doses are important to consider. On the other hand, a serial organ, such as the spinal cord, has a high s value and is very sensitive to high local doses even if the mean dose is low. NTCP-LKB Model The normal tissue complication probability, NTCP, based on the LKB model (Lyman 10 and Kutcher et al. 11 ) is defined as: Eq. 31 NTCP LKB D D t eff D = m D 50 and D eff = t exp = u2 du 2 M v i EQD 1 n n V ref 2,i i = 1 D 50 = Dose giving a 50% response probability. m = Slope of the response curve. n = Volume dependence. M = Total number of voxels. v Relative volume of voxel i compared to the reference i = V ref volume. EQD 2 = Equivalent dose in voxel i given in 2 Gy fractions. 10. Lyman JT. Complication probability as assessed from dose volume histograms. Radiat Res Suppl. 1985; 8: S Kutcher GJ, Burman C, Brewster L. Histogram reduction method for calculating complication probabilities for three dimensional treatment planning. Int J Radiat Oncol Biol Phys. 1991; 21: Biological Modeling Reference Guide

35 Composite TCP and NTCP Models Composite TCP Composite NTCP Individual TCPs can be combined into an overall (or composite) TCP if there are several tumors. Similarly, individual NTCPs for different organs can be combined into a composite NTCP. Composite TCP and NTCP models are used to estimate the overall probability of eradicating all tumors of interest and the overall probability of injuring any of the normal tissues of interest, respectively. The composite TCP for all tumors is calculated according to: Eq. 32 TCPD = j TCP j D assuming that all tumors j have to be eradicated in order to be cured. The overall probability of injury is obtained by combining NTCPs for organs at risk Eq. 33 NTCPD = 1 1 NTCP j D j assuming that it is sufficient for only one of the organs j included in the composite NTCP to collapse to severely harm the patient. It is important to carefully consider which NTCPs that should be included in the composite NTCP. The risk of inducing severe damage to normal tissue cannot be compared with the risk of minor complications. P + The probability of complication free tumor control, P +, is an attempt to combine the individual TCPs and NTCPs into a single measure of the quality of the treatment plan (Lind 12 ). 12. Lind BK, Mavroidis P, Hyödynmaa S, Kappas C. Optimization of the dose level for a given treatment plan to maximize the complication free tumor cure. Acta Oncol. 1999; 38 (6): Biological Modeling 35

36 Eq. 34 P + = TCPD1 NTCPD Which NTCPs that are included into the P + expression is highly essential. Loss of tumor control cannot be compared with the risk of minor normal tissue complications. Only NTCPs of severe complications should be considered to be included in the calculation of P +. Parameters of the Biological Models It is important to be aware of that the parameters listed as default parameters in Eclipse can only be considered for the endpoint and reference volume for which they have been derived from. Some of the parameters are derived from another linear dose response model, here labeled with a star (*), than the one presented in this manual. These parameters cannot be used by the models presented in this reference guide, unless they are transformed as described below. The linear dose response model used in this reference guide can be expressed as (see The Linear Dose Response Model on page 25): Eq. 35 D P L D = exp exp e e lnln2 D 50 as the other linear dose response model* has the following form: Eq. 36 expe D P L D * D 50 D = 2 = exp exp e e + lnln2 D 50 These two expressions are equal if: Eq. 37 * ln ln 2 = e Hence, by simply rescaling the * of the other linear dose response model* by the term lnln e, these parameters can be used by the linear dose response model used in this reference guide. 36 Biological Modeling Reference Guide

37 If the estimated parameters D 50 and are not derived from a dose per fraction of 2 Gy in the entire volume of interest (that is, uniform dose), a fraction size correction to 2 Gy per fraction is performed for these parameters using EQD 2. Biological Modeling 37

38 38 Biological Modeling Reference Guide

39 Appendix A Equivalence Between Linear EQD 2 and LQ Dose-Response Models The linear dose response model using EQD 2 and the linear quadratic (LQ) dose response model using a rewriting of the parameter α are equivalent. This can be shown by the following steps: The linear dose response model can be expressed as: Eq. 1 Eq. 2 P L EQD 2 exp e EQD 2 = exp e lnln2 D 50 D1 + d EQD 2 = d ref By inserting Equation 2 on page 39 in Equation 1 on page 39 the following expression is obtained: Eq. 3 1 D + d P L EQD 2 = exp exp e D 50 1 d e lnln2 ref The LQ dose response model is defined as: Eq. 4 Eq. 5 P LQ D = exp exp e ""D1 + d = e lnln2 D 50 1 d ref a. Lind et al in Acta Oncol. 1999;38(6): a 39

40 By inserting Equation 5 on page 39 in Equation 4 on page 39 the following expression is obtained: Eq. 6 1 D + d P LQ D = exp exp e D 50 1 d e lnln2 ref which is the same expression as the one obtained for the linear dose response model in Equation 3 on page 39. Here the equivalence between the linear dose response model and the LQ dose response model was only shown for the simple case incomplete repair and accelerated repopulation were not considered. However, by using the same approach using the EQD 2 including repair and/or repopulation and the corresponding LQ dose response model, the equivalence in these cases can be showed as well. 40 Biological Modeling Reference Guide