A model to accumulate fractionated dose in a deforming organ

$: A model to accumulate fractionated dose in a deforming organ$
PII S0360-3016(99)00007-3

BIOLOGY CONTRIBUTION

A MODEL TO ACCUMULATE FRACTIONATED DOSE IN ADEFORMING ORGAN

DI YAN, D.Sc., D. A. JAFFRAY, Ph.D.,AND J. W. WONG, Ph.D.

Department of Radiation Oncology, William Beaumont Hospital, Royal Oak, MI

Purpose: Measurements of internal organ motion have demonstrated that daily organ deformation existsthroughout the course of radiation treatment. However, a method of constructing the resultant dose delivered tothe organ volume remains a difficult challenge. In this study, a model to quantify internal organ motion and amethod to construct a cumulative dose in a deforming organ are introduced.Methods and Materials: A biomechanical model of an elastic body is used to quantify patient organ motion in theprocess of radiation therapy. Intertreatment displacements of volume elements in an organ of interest iscalculated by applying an finite element method with boundary conditions, obtained from multiple dailycomputed tomography (CT) measurements. Therefore, by incorporating also the measurements of daily setuperror, daily dose delivered to a deforming organ can be accumulated by tracking the position of volume elementsin the organ. Furthermore, distribution of patient-specific organ motion is also predicted during the early phaseof treatment delivery using the daily measurements, and the cumulative dose distribution in the organ can thenbe estimated. This dose distribution will be updated whenever a new measurement becomes available, and usedto reoptimize the ongoing treatment.Results: An integrated process to accumulate dosage in a daily deforming organ was implemented. In thisprocess, intertreatment organ motion and setup error were systematically quantified, and incorporated in thecalculation of the cumulative dose. An example of the rectal wall motion in a prostate treatment was applied totest the model. The displacements of volume elements in the rectal wall, as well as the resultant doses, werecalculated.Conclusion: This study is intended to provide a systematic framework to incorporate daily patient-specific organmotion and setup error in the reconstruction of the cumulative dose distribution in an organ of interest. Therealistic dose distribution in an organ of interest gives the true dose–volume relationship, and may play animportant role in the evaluation of the dose response of human organs. Dose reconstruction during the course oftreatment delivery can also be used as an important feedback for the online optimization of individual treatmentplans. © 1999 Elsevier Science Inc.

Internal organ motion, Biomechanical model of elastic body, Construction of the cumulative dose distribution inan organ of interest.

INTRODUCTION

Internal organ motion throughout the course of radiationtherapy has long been suspected to be problematic to theresultant dose delivered. Intensive studies of internal organmotion in the thorax (1–3), abdomen (4, 5), and pelvic(6–10) regions have been made using either fast computedtomography (CT) scans or multiple CT scans through thetreatment course. Significant variations in organ geometry,which could happen either during radiation dose delivery orbetween fractions of dose delivery, have been demonstrated.Due to internal organ motion, the actual dose delivered to anorgan volume could be significantly different from the oneprecalculated during treatment planning. Therefore, it couldbe ambiguous to evaluate external-beam treatment based

upon dose–volume relationship from the treatment planningalone.

To incorporate internal organ motion into the treatmentplanning for treatment evaluation and optimization, oneneeds to solve the difficult problem of reconstruction of acumulative dose distribution in a deforming organ. Themost critical step in solving the problem is to track thedisplacement of each volume element in the organ betweenmoments of dose delivery. Unlike the relatively rigid bodymotion of human bony anatomy, physiological actions cancause nonrigid motion or deformation of human tissues.Tracking the displacement of a rigid object is relatively easybecause the distance between any pair of volume elementsin the object remains unchanged before and after the mo-tion. Therefore, a simple linear transformation with the

This work was presented at the ASTRO meeting, Orlando, FL,1997

Reprint requests to: Di Yan, D.Sc., Department of RadiationOncology, William Beaumont Hospital, 3601 W. Thirteen Mile

Rd., Royal Oak, MI 48073-6769.Acknowledgement—This work was supported in part by NCI grant#CA71785.

Accepted for publication 30 December 1998.

Int. J. Radiation Oncology Biol. Phys., Vol. 44, No. 3, pp. 665–675, 1999Copyright © 1999 Elsevier Science Inc.Printed in the USA. All rights reserved

0360-3016/99/$–see front matter

665

parameters of shifting and rotation can be constantly appliedto every volume element in the object to map it from oneposition to the other. By assuming rigid organ motions,some efforts (11–13) have been made to illustrate the cor-responding variation in dose distribution due to internalorgan motion and the possible impact on treatment out-come. However, dose evaluation based upon the assumptionof rigid organ motion is questionable due to the inherentnonrigid or deformed nature of internal organ motion. As aconsequence, conclusions in these studies could be unreal-istic and might lead to erroneous compensation.

Tracking individual elements in a deformable organ re-quires basic information of organ shape before and afterorgan motion. Although human anatomy is not rigid and,therefore, can exhibit high deformability, the shapes beingimaged are strongly structured. Human organ deformationhas been extensively studied in the areas of biomechanicsand biomedical image analysis (14–17). Most of these stud-ies have concentrated on exploring the relationship betweenorgan motion and physiological behavior. For radiationtherapy, it is pertinent to determine the effect of organmotion on the radiation dose delivered to substructures orsubvolumes of an organ throughout the entire course oftreatment. This is important, not only to achieve a betterunderstanding of the clinical dose response of human or-gans, but also to explore further possibility of optimizing theradiation treatment. As a primary study, this paper illus-trates a method to quantify internal organ motion, and amethod to construct the cumulative dose distribution in anorgan of interest during the course of radiation treatment. Tosimplify the notation, the following discussion focuses onintertreatment organ motion alone. However, the conceptscan be extended to intratreatment organ motion.

METHODS AND MATERIALS

In this study, the following notation and assumptions willbe used. First, an organ of interest V (normal organ ortarget) is assumed to consist of numerous volume elements$(v), where v represents a basic element (subvolume orsubstructure) in the organ. The organ volume in the treat-ment process, on the other hand, is indicated as Vi, i 5 0,1,..., N, which represent the instance of the organ volume attreatment planning (i5 0) and at each of the treatmentdeliveries (i5 1,. . ., N) for a total of N fractions. It is alsoassumed that the number of volume elements in each Vi isequal, and the mass of each volume element is identicaleven though the volume of Vi can change. Let Xi 5 (x1, x2,x3)i indicate the spatial position of a volume elementvappeared at treatment planning or a treatment fraction. Thedose calculated in treatment planning and imparted tov in atreatment delivery can then be written as d(X0) and d(Xi).Therefore, the cumulative dose to the volume elementv,calculated during treatment planning, will be D(X0) 5 N .

d(X0) in the equal fractionation regimen. On the other hand,the actual cumulative dose tov will be $(v) 5 Si 5 1,...,N

d(Xi). It should be cautioned that the dose matrix d(·),

calculated in the treatment planning, could be also quitedifferent from the actual dose in space during a treatmentdelivery due to the changes of tissue’s density, skin contour,or machine output. However, in the following notation, wewill ignore this difference and specifically mention whenthese changes have to be considered. In general, the positionof a volume element at each treatment fraction with respectto the reference coordinate of the therapy machine is de-pendent on both internal organ motion and patient setuperror. Patient setup error during external-beam treatmenthas been quantified elsewhere. Therefore, we will focus onquantifying organ motion alone for the first part of thestudy, and then incorporate both internal organ motion andpatient setup error in the reconstruction of the cumulativedose. As the patient organ moves, each volume elementv inthe organ will move away from the planning position X0.The new position ofv at treatment day i can be representedas Xi 5 X0 1 Ui, where Ui 5 (u1, u2, u3)i is the displace-ment of v from its planning position to the ith treatmentposition. The real dose distribution in an organ of interestcannot be obtained without knowing the displacement ofeach volume element in the organ. Therefore, the followingsection will focus on determining the displacements ofvolume elements in an organ of interest during intertreat-ment organ motion.

Quantification of internal organ motion of specific patientOrgan data preparation.The CT image for treatment

planning and the other acquired immediately before or aftera treatment delivery are needed as primary data set toquantify intertreatment organ motion. First, the two imagesare registered with respect to patient bony anatomy toeliminate patient setup error. Therefore, organ motion in thestudy is quantified within the reference coordinate of patientbony anatomy. In our study, the planning CT image is usedas the reference image, and organ motion at each treatmentfraction is calculated with respect to its planning position.After image registration, a user contours each organ ofinterest on two images separately. The organ volume, V0,manifested on the reference image contains the informationof original shape and position before the motion. Mean-while, the organ volume, Vi, shown on the treatment imagerepresents the final shape and position of the organ at thetreatment. To minimize the contouring artifacts and achievebetter representation of anatomy, a numerical method (cu-bic-spline method) is adopted in the study to smooth thecontours on each transverse CT slice.

Selection of boundary fiducial points.Fiducial points onthe organ boundary need to be specified as primary infor-mation of the boundary condition, to determine the dis-placement for the rest of the volume elements in the organ.The user defines these corresponding points on the organ’sboundary in both CT images. The basic rule for defining theboundary fiducial points is to select them so that the bound-ary segment between any two fiducial points will remain auniform force or stress after the motion achieves an equi-librium or stable state. This implies that one should always

666 I. J. Radiation Oncology● Biology ● Physics Volume 44, Number 3, 1999

select a point for which the mechanical property of adjacenttissue on the boundary changes. For example, selecting apoint where the organ boundary has adjacent structures ofbone and soft tissue. It is worthwhile to mention that theselection of boundary points is one of the most importantprocedures for solving organ motion, regardless rigid ornonrigid body motion. The accuracy of the solution willdepend on this selection. Boundary fiducial points could beeasily determined for a superficial organ (i.e., breast). In thiscase, radiomarkers can be placed repeatedly at the samelocations on patient skin for multiple scanning. On the otherhand, for an internal organ, boundary points can be locatedthrough reference to surrounding anatomy. For example,when liver motion during patient respiration is studied, theboundary points can be determined by referring to thesurrounding anatomies, such as costal cartilage, ribs, rightkidney, gallbladder, inferior vena cava, portal vein, andstomach.

Mesh of organ volume.Volume elements in an organ aregenerated using a volume mesh in the reference image. Themeshing procedure basically divides the entire organ vol-ume into a number of subvolumes or volume elements {v}.The three-dimensional (3D) position of each volume ele-ment in the organ before motion is represented by the

corresponding nodal position of the mesh. The number ofvolume elements used for an organ of interest should bedetermined considering both computation speed and theresolution of the dose-calculation grid, and could be ad-justed by selecting the meshing resolution. It is unnecessaryto select a higher resolution for mesh of organ volume thanfor a grid of dose calculation. In current implementation, themesh is especially constructed to approximately equalizethe volume for all elements (Fig. 1).

Biomechanical representation of internal organ motion.As elastic materials, human tissue motion must obey thelaws of mechanics. Geometrical motion of a living organdue to respiration, blood circulation, internal fluid distribu-tion, and surrounding tissue traction can be represented byusing constitutive equations based on mechanical and math-ematical laws that define the common properties of livingtissue and fluid in both static and dynamic states. Thebiomechanical models of human tissues have been dis-cussed in books and review articles (15, 16) and will not bediscussed here in any detail. The mechanical and mathe-matical expressions used in this study (see Appendix forderivation) represent the motion of human tissues such asbreast, liver, rectum, prostate, etc. The motions of internalfluid, blood, and gas are out of the scope of this study.

Fig. 1. Example of volume mesh for (a) hollow organ (one slice), and (b) solid organ (one slice).

667Fractionation dose in deforming organ● D. YAN et al.

Forces or stresses on each volume element make it move.This motion can be expressed applying Newton’s first lawto the volume element with a fixed mass. As long as theforces on opposite surfaces of a volume element are notequal, the element will keep moving until the forces achieveequilibrium. The process of equilibrium is controlled by thelinks of adjacent elements, according to the elastic propertyof organ tissues based on Hooke’s law. Newton’s law andHooke’s law are the fundamental principles used to modelthe organ motion and to construct the corresponding math-ematical expressions in this study (Appendix). In principle,if parameters that are used to characterize the mechanicalproperties of organ tissue are known, the organ motion canbe precisely quantified using the model. However, someparameters cannot be precisely obtained without biome-chanical experiments of human tissue. In the current study,mechanical properties for an organ of interest are approxi-mated using the properties of the soft tissues describedearlier in Reference (18).

Numerical solution of internal organ motion.Mathemat-ical expression of the model (see Eqs. 2–4 in Appendix)contain a group of partial differential equations for eachvolume element. An organ with volume 100 cc can have 109

volume elements if a 1-mm mesh resolution is applied. Itimplies that millions of differential equations need to besimultaneously solved to determine the displacements of allelements in this organ. To make the calculation practical, awell-known numerical method, known as the “finite ele-ment method,” was adopted for this study. The finite ele-ment method solves the mechanical model numerically byapproximating the partial differential equations to a systemof linear equations in a neighborhood at each volume ele-ment. Then, solving them iteratively starting from the organboundary.

Given V0 and Vi, the organ volume shown on the refer-ence image before the motion and the one on the ith treat-ment image after the motion, the numerical solutions in-clude 3D displacement Ui for each elementv in the organvolume. It implies that volume elementv in V0 with positionX0 is registered to the volume element in Vi with theposition Xi 5 X0 1 Ui. Again, these positions are defined inthe reference coordinate of patient’s bony anatomy. In ad-dition, the motion locus of volume elementv from thereference position X0 at treatment planning to the positionXi at the ith treatment can be also calculated to represent theintermediate positions of the motion. Therefore, a group oforgan volumes can be used as the intermediate shapes oforgan volume during the motion between the referencevolume and the volume in the ith treatment. This featurewill be useful to simulate and characterize the organ motionof the individual.

Construction of cumulative dose distribution in an organof interest

With the quantification of intertreatment organ motion ofthe individual, fractionation doses delivered to each elementin an organ of interest can be registered, and meaningfully

accumulated by also considering the patient setup errors toform the actual dose distribution in the organ volume. Thisis obviously important for treatment evaluation, and can beeasily obtained following the quantification of organ motionif patient can be scanned at every treatment delivery. How-ever, daily CT scanning at every treatment delivery may notbe practical in the current clinical setting. Therefore, it willbe necessary to estimate the cumulative dose distribution inan organ of interest by using few daily CT measurementsacquired in the treatment course of the individual. Thisprocedure is also important in the process of adaptive radi-ation therapy (19), where the cumulative dose distributionin organs of interest needs to be estimated during the earlyphase of treatment delivery and, therefore, the treatmentplan of the individual can be reoptimized accordingly. Inthis section, we focus on a method to estimate the cumula-tive dose distribution in an organ of interest based on fewdaily measurements of intertreatment organ motion andpatient setup error during the treatment course of individuals.

Distribution of patient-specific organ motion.First, weassume that intertreatment organ motions of individual pa-tient in k days of treatment delivery (In the adaptive process,it will be the first k days.) are measured using multiple dailyCT scanning. Therefore, for an organ of interest, we havek11 volumes {V0, V1, ..., Vk} that manifest on the planningCT image and the k treatment CT images, respectively.Using the method discussed in the last section, all pairs ofvolumes, Vi and Vj, in {V 0, V1, ..., Vk} are registered on theelement by element basis. Meanwhile, a group of interme-diate shapes of volume between the Vi and Vj are alsogenerated to represent possible organ motion during thetreatment course. Therefore, a large sample set, S, of organvolumes of specific patient can be obtained. Each of thecomponents in S represents a possible volume of the organappearing during the course of treatment. For each volumeelementv in the organ, letf(v, X) be a probability densityfunction defined as the frequency of the position X, withinthe reference coordinate of patient’s bony anatomy, occu-pied by the volume elementv throughout the entire courseof dose delivery. This function gives a position distributionof the elementv with respect to patient bony anatomy in thetreatment process, and is generated by random samplingfrom S. It is important to note that this sampling processindexes every volume element to a specific organ volume inthe sample set S.

Distribution of patient-specific setup error.We assumethat treatment setup of individual patients in the first ntreatment delivery are monitored using a portal imagingdevice. Therefore, 3D setup errors {e1, ..., en} quantified asrigid body motion between radiation beam and patient’sbony anatomy can be obtained, whereei represents thepatient setup error in 3D at the ith treatment delivery. It hasbeen shown in a previous study (20) that a Gaussian distri-bution, c(e, m, s), of specific patient can be used to char-acterize setup error of the individual patient, and the meanm and the standard deviations can be predicted using the nmeasurements (on average, n5 6).


Cumulative dose distribution in an organ of interest.With knowledge of the distributions of patient-specific or-gan motion and setup error, the cumulative dose distributionin an organ of interest can now be estimated as the proba-bility expectation of the dose with respect to the distribu-tions. As discussed above, the distribution of patient-spe-cific geometric variation is, in probability, equal to the jointdistribution f(v, X) · c(e, m, s). The joint distributionrepresents the composite effect of internal organ motion andsetup error on the position of the volume elementv (relativeto the treatment beam). It will be predicted after m5Max{k, n} days of measurements, which include k daily CTscanning and n daily portal imaging, during the course oftreatment delivery. First, it assumes that the dose matrixcalculated in treatment planning is not significantly dis-turbed by the new patient geometry in treatment. Then, thecumulative dose$(v) delivered to each elementv in theorgan can be estimated using the expectation of the dosematrix on the joint distribution of geometric position ofv as:

$~v! < $~v, m!

5 *« *X D(T~«! z X) z f~v, X! z c~«, m, s!dX d«, (1)

where T(e) is the linear transformation matrix due to a setuperrore, defined as the displacement between radiation beamand patient’s bony anatomy; X is the position of the elementv relative to patient’s bony anatomy governed by internalorgan motion; T(e).X represents the new position ofv rel-ative to the radiation beam, when both internal organ motionand setup error with joint distributionf(v, X) · c(e, m, s)are considered; and D(T(e).X) is the precalculated dose(total dose) in treatment planning at the position T(e)zX.

The integral can be solved numerically as follows. First,randomly taking a sample of organ volume Vi from the setS, and randomly generating a daily setup errorei using thedistributionc(e, m, s). Then, each volume elementv in thesample volume with the location Xi, relative to the patient’sbony anatomy, will be linearly transformed to the positionT(ei)

.Xi in the radiation beam coordinate. Therefore, thecumulative dose to each volume elementv can be estimatedas Si 5 1,...,N d(T(ei)

.Xi) after taking N samples, whered(T(ei)

.Xi) is the single fractional dose precalculated intreatment planning at the position T(ei)

.Xi. A very importantfeature in this estimation is that both single-dose distribu-tion and cumulative-dose distribution in the organ volumecan be constructed. This could be very useful for treatmentevaluation, because the radiobiological response of humanorgans is not only dependent upon the cumulative dose butalso the fractional doses.

The dose matrix, precalculated in treatment planning,may need to be modified or recalculated using the actualpatient’s geometry on a treatment day. It may be the casewhen organ motion causes large changes in internal tissue

density (i.e., during lung treatments) or in skin contour (i.e.,during breast treatments). In these cases, the dose matrixcreated in treatment planning may need to be recalculated ormodified with the updated patient’s geometry.

In summary, the cumulative dose$(v) delivered to thevolume elementv in the organ is estimated as$(v, m) afterm days of measurements, and updated as$(v, m11) after anew measurement becomes available. Therefore, a contin-uous feedback of the estimation of the dose distribution inorgans of interest can be obtained and updated during thecourse of the radiation treatment.

RESULTS

An integrated process to quantify patient-specific organmotion and estimate the cumulative dose distribution in anorgan of interest was implemented. In this process, com-mercial software packages of 3D treatment planning(PINNACLE1) and finite element method (ABAQUS2)were employed. The process (see Fig. 2) has basic functionsdescribed in the last section, and will be continuously im-proved. Performance of the process will be demonstratedusing an example of intertreatment motion of rectal wallduring the course of a prostate treatment.

1 ADAC Laboratories, Milpitas, CA 95035. 2 Hibbitt, Karlsson & Sorensen, Inc., 1080 Main Street, Paw-tucket, RI 02860.

Fig. 2. Flow chart of the integrated process to quantify organmotions and to estimate the cumulative dose distribution in anorgan of interest.


Fig. 3. Overlaid rectal walls after pelvis bone registration: the rectal wall (yellow) on the treatment-planning CT imagevs. the rectal wall (green) on the fifth-treatment CT image. Red dot5 boundary points on the outer rectal wall; pinkcircle 5 boundary points on the inner rectal wall.

Fig. 4. Daily dose volume histograms of the rectal wall.

Fig. 5. (a) Volume mesh of the rectal wall (red) in the treatment-planning CT; (b)-(e) the intermediate shapes (white);and (f) the final shape (white) of the rectal wall in the fifth-treatment CT image.


Daily rectal wall volumes were obtained from a pros-tate treatment where a patient was treated in the supineposition using a standard four-field-box technique tomanage radiation dose of 68.4 Gy (1.8 Gy3 38) to thetreatment isocenter. In addition to a treatment-planningCT, the patient received 14 extra CT scans to measureintertreatment organ motion. These scans were acquiredtwice per week immediately before or after treatmentdelivery. Furthermore, portal images were acquired daily

for each treatment field. The rectal wall on each daily CTimage was contoured following pelvis bone registrationbetween the planning CT image and the daily treatmentCT image. Figure 3 shows overlaid rectal walls delin-eated from the planning image (yellow) and the imageobtained at the fifth treatment CT scanning (green). Dailydose–volume histograms for the rectal wall are plotted inFig. 4 to demonstrate daily dose variation in the rectumduring the course of treatment.

Fig. 6. The displacement maps of volume elements in the rectal wall at the patient’s (a) lateral direction (right to leftin cm unit), and (b) the anterior-posterior direction (posterior to anterior in cm unit).

Fig. 7. The dose distribution and the specific volume elementv (white dot) in the rectal wall at (a) the treatment-planningand (b) the fifth-treatment CT scanning.


Intertreatment motion of the rectal wall was quantified usingthe method discussed in the previous sections. First, a volumemesh with resolution of 1 mm was created for the rectal wallvolume manifested on the planning CT image (Fig. 5a). Nodalpositions in the mesh were used to represent the initial posi-tions of volume elements. To apply the model (Appendix),parameters of Young’s modulus (tissue elastic modulus),E 50.5 Newton per square meter, and Poisson’s ratio (tissue com-pressible ratio),g 5 0.3, were selected from the literature (18),assuming that the tissue of rectal wall has similar elasticproperty as muscle with small compressibility. The density(g/cm3) of the rectal wall was converted from the correspond-ing CT numbers. On each CT slice, four fiducial points werepicked up on the inner and outer rectal walls (Fig. 3) for theboundary condition. In this example, the boundary points weresimply determined using the most anterior and posterior pointson both inner and outer rectal walls. The mesh of the rectalwall, the contours of rectal wall on the treatment CT image, thepreselected boundary points, and the model parameters werethen imported into the software for finite element methodanalysis. The intermediate shapes and final shape of the rectalwall between the initial and the fifth treatment CT scan areshown in Fig. 5b–f. The displacement map of volume elementsin the rectal wall from the position in treatment-planning imageto the position in a treatment CT image was calculated. Thefinal displacement map for the rectal wall at the fifth treatmentCT scanning was plotted in Fig. 6a for the patient’s lateraldirection, and Fig. 6b for the patient’s anterior-posterior direc-tion. The displacement in the patient’s superior-inferior direc-tion was minimal (within6 0.1cm), and was not plotted. Thecolor map indicates the final displacement of volume elementsin the rectal wall shown in Fig. 3. The red color in Fig. 6arepresents the maximal lateral displacement of volume ele-ments in the rectal wall. This part of the rectal wall moved 2.18cm to patient’s left from its original position in treatmentplanning; meanwhile, the maximal displacement of volumeelements in the right part of the rectal wall (the dark blue) was1.6 cm to the patient’s right. On this treatment day, the rectumwas full of gas in the anterior portion of the rectum and stoolin the posterior portion of the rectum. The thicker rectal wall inthe planning image was stretched to be thinner in the treatment(see Fig. 3). Therefore, a relatively smaller portion of anteriorrectal wall was in the high-dose region (Fig. 7). To illustratethe corresponding dose variation, the daily doses to a specificelement of the rectal wall, which was selected in a region ofhigh dose gradient (see Fig. 7), were tracked in all daily CTimages and plotted in Fig. 8. The daily setup errors measuredfrom the patient treatment were not incorporated in this dosecalculation.

DISCUSSION

A major problem in the evaluation of treatment for whichinternal organ motion exists is lack of the actual dose–volume relationship in organs of interest. Presently, treat-ment evaluations that correlate clinical outcomes to a radio-biological model of dose response, or directly to a dose–

volume relation, operate under the assumption that anidentical distribution of dose in organs of interest, as pre-calculated, would be repeatedly delivered every day in theclinical treatment. However, both the single-fraction doseand the cumulative dose actually delivered to an organcould be quite different from the planned dose if the pa-tient’s geometry varies over the course of treatment deliv-ery. As a consequence, the evaluation could be quite incon-clusive. Using the daily doses to a specific volume elementcalculated in the example (Fig. 8), the linear-quadratic (LQ)model of cellular radiation,E5 Si 5 1,...,N a:di 1 b:di

2, wastested. Including also the dose in treatment planning, the 15daily doses shown in Fig. 8 were used to represent the dailydoses delivered to the volume element in the treatment. Byassuminga 5 0.1 Gy-1 anda/b 5 3 Gy for late effectivetissues (21), the E value was calculated separately usingeither equal daily dose to simulate treatment planning, ornonequal daily dose to simulate the actual treatment. In caseof equal daily dose, a single dose from the 15 was repeat-edly applied in the LQ model for all fractions (N5 38).Therefore, it simulates a possible result when the dosedistribution in the rectal wall calculated from treatmentplanning alone was used for the treatment evaluation. Onthe other hand, for the nonequal daily dose, the 38 fractiondoses were randomly picked up from the 15 daily doses. Ifwe assume that the volume element represents a functionalsubunit (FSU) of the rectum that consists of k5 103

clonogenic cells, then probability of killing the FSU will bePFSU 5 (1 2 exp(-E))k (22). It has values from 0.3% to94.4% for the simulated plan, and 36.5% for the actualheterogeneous daily doses (Fig. 9). Although the aboveexample might not be representative of all prostate treat-ment, it does indicate that daily variation in the dose deliv-ery to the rectum can result in large uncertainty in thecalculation of the radiobiological response. It follows thatsuch treatment variation might result in the large variabilityof the observed treatment outcome. The inability to accountfor the effects of dose variation contributes to the substantialuncertainty in our present knowledge of TCP and NTCP.But with the new wealth of information about patient’sgeometric variations, it may be possible to resolve these

Fig. 8. The daily doses (the first one is the planned dose) to thespecific volume elementv.


uncertainties and establish a better foundation to character-ize treatment outcome. The method introduced in this paperrepresents the beginning of an attempt to do so.

Use of online measurements of patient-specific variationas feedback for treatment plan reoptimization forms thefoundation of adaptive radiation therapy (19, 20). In theadaptive treatment process, daily variations of the individ-ual due to internal organ motion and treatment setup aremodeled as two stochastic processes. These stochastic pro-cesses are characterized by the distributions that are pre-dicted with confidence from daily measurements in the earlyphase of the treatment course. The distributions of patient-specific organ motion and setup error are then used toestimate the cumulative dose distribution in organs of in-terest so that the treatment plan can be adaptively reopti-mized. The distribution of patient-specific setup error hasbeen proven to be predictable from the early days of portalimaging (20). Therefore, cumulative dose distribution in thepatient relative to rigid bony anatomy can be estimated if

internal organ motion can be neglected. However, predict-ing the distribution of patient-specific geometric variationbecomes much more difficult when internal organ motion ispresent. The main reason for this difficulty is due to thenonrigid nature of organ motion. With the current model,nonrigid body motion of an internal organ can now bequantified so that the volume elements in the organ areregistered between any pair of organ volumes before, dur-ing, and after motion. Therefore, the prediction turns out tobe practical. To predict the distribution of patient-specificorgan motion, we proposed to utilize the organ volume inthe daily CT scans acquired during the treatment process, aswell as the intermediate shapes of the organ during themotion that are generated using the model. Performance ofthis prediction will be tested in further study.

To maximize the performance of biomechanical modelsin modeling internal organ motion, one needs to know thetissue’s mechanical properties of elasticity and/or compress-ibility. Organ tissues that have relatively low elastic mod-ulus can easily change shape, resulting in a nonrigid bodymotion during normal physiological action. On the otherhand, some tissues have a relatively high elastic modulus.For these organs, the motion produced during normal phys-iological action may be approximated using rigid bodymotion. It is important to recognize that these motions arealso characterized in a biomechanical model and, therefore,a single model can be used to quantify both rigid andnonrigid organ motions. One of the major procedures inquantifying internal organ motion is to determine the me-chanical properties of human tissues. For most tissues, theseproperties may also have some level of heterogeneity (16).Measuring mechanical properties of human tissues is costly,but can be done in most biomechanical laboratories. Studyof internal organ motion should also focus on this subject toachieve a meaningful precision on the quantification oforgan deformation.

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Fig. 9. Probability of FSU killing for the volume elementv, whichwas evaluated based on (a) the dose in a single CT image (indi-cated by a dot) or (b) the doses in 15 daily CT images (indicatedby the dashed line).


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APPENDIX

Steps of calculating displacement of volume elements ina moving organ with the biomechanical model will beoutlined here. To simplify the following description, thesymbols, (ai) and (bij), represent a 33 1 vector, (a1, a2,a3), and a 93 1 vector, (b11, b12, b13, b21, b22, b23, b31,b32, b33), respectively.

Distances between two volume elements in an organ ofinterest at treatment planning and at a specific day of treat-ment delivery can be used to indicate organ deformation.First, consider two adjacent volume elementsv andv’ in theorgan which are located at X5 (xi) and X 1 dX 5 (xi 1dxi) and have distance\dX\ in the reference CT image.Meanwhile, let Xn 5 (xi)n and Xn 1 dXn 5 (xi 1 dxi)n

represent the locations of the same elements in the nthtreatment CT image, which have distance\dXn\. If \dXni 5\dX\ for all pairs of volume elements in the organ, the

motion is a rigid body motion alone. Otherwise, the organmotion is nonrigid body motion or deformation. Behavior oforgan deformation depends on the strain of elastic tissuethat can be measured as the difference between the squaresof the distances of adjacent elements,\dXn\

2 2 \dX\2,before and after the motion. It is easy to deriveidXni2 2idXi2 5 dXT [Un/XT 1 Un/X 1 Un/XT.Un/X] dX,by considering Xn 5 X 1 Un, where Un 5 (ui)n is the 3Ddisplacement of volume elementv from the position X tothe position Xn. The 33 3 matrix [Un/XT 1 Un/X 1Un/XT.Un/X] is the main parameter to measure thestrain or tissue elasticity, which would be equal to zero forevery volume element in a rigid object during the motion(i.e., a metal block moves under a normal force). To deter-mine the displacement Un 5 (ui)n , the strain-stress relationin continuum mechanics needs to be applied to an organ thathas elastic, pseudoelastic, or viscoelastic properties.

When exterior force or stress on an organ surface varies,unbalanced stresss 5 (sij) on the surface of each volumeelement in the organ (Fig. 10) will make the element moveuntil a new equilibrium of the stress is achieved accordingto Newton’s law of motion. The equations of motion, calledthe Eulerian equation, are represented by partial differentialequations of the stress ratio,

( i51,2,3 s ij / xi 1 r z gj 5 r z aj, for j 5 1, 2, 3,(2)

wherer is the density of the material;gj are components ofgravitational field; andaj are components of mass acceler-ation. Stress on a volume element in Eq. 2 changes duringmotion, which is governed by the strain of neighborhoodtissues. In continuum mechanics, a constitutive equation isused to link the straine 5 (eij) to the stresss 5 (sij)according to Hooke’s law. In this study, we assumed that

Fig. 10. Indication of stress on a volume element.


the organs, such as liver, heart, kidney, prostate, rectum,bladder, etc., can be approximated as materials with elasticand isotropic properties. With these assumptions, a standardrepresentation of strain called the Green’s strain tensor andthe constitutive equation with theLame’s constants can beapplied as follows,

eij 5 @uj /xi 1 ui /xj 1 (k 5 1,2,3~uk /xi !~uk /xj !#/2, (3)

sij 5 l z ~(k 5 1,2,3ekk! z dij 1 2m z eij, for i; j 5 1, 2, 3, (4)

where dij is the Kronecker delta, withdij 5 1 if i 5 j,otherwise equals 0. The twoLame’s constants,l andm, canbe expressed using Young’s modulusE and Poisson’s ratio

g, wherel 5 E . g /[(1 1 g)(1 - 2g)] and m 5 E / [2(1 1g)]. The Young’s modulus and Poisson’s ratio representelasticity and compressibility of organ tissue, respectively.Their values for some specific soft tissues can be found inthe literature (18).

Concurrently solving the equations of motion (2), theequations of strain expression (3) and the constitutive equa-tions (4) with respect to boundary conditions, the displace-ment Un for each volume element in an organ of interest canbe determined. Appropriate boundary conditions includespecific points and/or surface traction on the organ bound-ary. It is worth mentioning that commercial software pack-ages of mechanical models and finite element methods areavailable. The users can customize them for their ownneeds.


A model to accumulate fractionated dose in a deforming organ

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