25 Image-Guided/Adaptive Radiotherapy Di Yan

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25 Image-Guided/Adaptive Radiotherapy

Di Yan

D. Yan, D.Sc.

Director, Clinical Physics Section, Department of Radiation Oncology, William Beaumont Hospital, Royal Oak, MI 48073- 6769, USA

CONTENTS

25.1 Introduction 321

25.2 Temporal Variation of Patient/Organ Shape and Position 323

25.3 Imaging and Veriﬁ cation 324

25.3.1 Verifi cation with Radiographic Imaging 324 25.3.2 Verifi cation with Fluoroscopic Imaging 325 25.3.3 Verifi cation with CT Imaging 325

25.4 Estimation and Evaluation 327

25.4.1 Parameter Estimation for a Stationary Temporal Variation Process 327

25.4.2 Parameter Estimation for a Non-stationary Temporal Variation Process 328

25.4.3 Estimation of Cumulative Dose 329

25.5 Design of Adaptive Planning and Adjustment 330 25.5.1 Design Objectives 330

25.5.2 Adaptive Planning and Adjustment Parameters 330 25.5.3 Adaptive Planning and Adjustment Parameter and

Schedule: Selection and Modiﬁ cation 330 25.6 Adaptive Planning and Adjustment 332 25.6.1 Indirect Method 332

25.6.2 Direct Method 333

25.7 Adaptive Treatment Protocol 334

25.7.1 Example 1: Image-Guided/Adaptive Radiotherapy for Prostate Cancer 334

25.7.2 Example 2: Image-Guided/Adaptive Radiotherapy for NSCLC 334

25.8 Summary 335 References 335

25.1 Introduction

Adaptive radiotherapy is a treatment technique that can systematically improve its treatment plan in response to patient/organ temporal variations observed during the therapy process. Temporal variations in radiotherapy process can be either patient/organ

geometry or dose-response related. Examples of the former include inter- and intra-treatment variations of patient/organ shape and positions caused by patient setup, beam placement, and patient organ physi- ological motion and deformation. Examples of dose response characteristics include the variations of size and location of tumor hypoxic volume, the appar- ent tumor growth fraction, and normal tissue dam- age/repair kinetics. Furthermore, tumor and normal organ dose response also induce changes in tissue shape and positions.

This chapter focuses on the use of adaptive strategies to manage patient/organ shape and position related temporal variations; however, the con- cepts of adaptive radiotherapy can be extended to a much broader range, including the management in temporal variations of patient/organ biology.

It is generally accepted that temporal variations are the predominant sources of treatment uncertainty in conventional radiation treatment. Numerous imaging studies have demonstrated that substantial temporal variations of patient/organ shape and position could occur during a typical radiotherapy course (Brierley et al. 1994; Davies et al. 1994; Halverson et al. 1991; Marks and Haus 1976; Moerland et al.

1994; Nuyttens et al. 2001; Roeske et al. 1995; Ross et al. 1990). Consequently, the radiation dose delivered to the target and a critical normal organ adja- cent to the target can signiﬁ cantly deviate from that calculated in the pre-treatment planning. This deviation causes a time-dependent, or temporal, variation in the organ-dose distribution, consisting of both dose variation per treatment fraction and cumulative dose variation in each subvolume of the organ, and results in major treatment uncertainties. These uncertainties induce fundamental obstacles to assuring treatment quality and understanding the normal tissue dose response, thereby hindering reliable treatment optimization.

Temporal variations in patient/organ shape and

position during the radiotherapy course can be sepa-

rated into a systematic component and a random

component. The systematic component represents

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a consistent discrepancy between the patient/organ shape and position appearing in pre-treatment simulation/planning and that at treatment delivery;

therefore, it is also called treatment preparation error (Van Herk et al. 2000). The random component represents patient/organ shape and position variations between treatment deliveries; these are also referred to as treatment execution errors. Because there is al- most always some random component in a temporal variation, observations achieved by imaging the patient repeatedly during the treatment course are essential to characterize the variations. The most important function of repeat imaging is to identify the systematic component of a temporal variation, and consequently eliminate its effect in the treatment.

Due to intrinsic temporal variations, targeting in radiotherapy process is, in principle, a four-dimensional (4D) problem, i.e., involving not only space but also time; therefore, it is an adaptive optimal control methodology ideally suited to manage this process.

A general adaptive radiotherapy system consists of ﬁ ve basic components: (a) treatment delivery to de- liver radiation dose to the patient based on a treatment plan; (b) imaging/veriﬁ cation to observe and verify patient/organ temporal variation before, during, and/or after a treatment delivery; (c) estimation/

evaluation to estimate, based on image feedback, the parameters which can characterize the undergoing temporal variation process, and evaluate the corresponding treatment parameters, such as the cumulative dose, biological effective dose, TCP, NTCP, etc.;

(d) design of adaptive planning/adjustment to design and update planning/adjustment parameters, as well as modify imaging, delivery, and adjustment sched-

ules, in response to the estimation and the evaluation;

and (e) adaptive planning/adjustment to perform a 4D conformal or IMRT planning with using the planning parameters speciﬁ ed in the adaptive planning/

adjustment design, and adjust treatment delivery accordingly. A typical adaptive radiotherapy system is illustrated in Fig. 25.1. In the standard textbook of adaptive control (Astrom and Wittenmark 1995), this system is called the self-tuning regulator (STR), indicating a system that can update its planning and control parameters automatically. Patient treatment in this system is initiated by a pre-treatment plan and resides within two feedback loops. The inner loop consists of treatment delivery, imaging/verifi cation, and planning/adjustment, which have been designed primarily to perform online image-guided treatment adjustment. The planning/adjustment parameters are updated and modifi ed most likely offl ine in the outer loop, which is composed of the imaging/verifi cation, parameter estimation/evaluation for a temporal variation process, design of adaptive planning/

adjustment, and adaptive planning/adjustment. In addition, the schedules of adaptive planning/adjustment, treatment delivery, and imaging/verifi cation in the adaptive radiotherapy system are most likely pre-designed and specifi ed in a clinical adaptive treatment protocol; however, these schedules can be modifi ed and updated during the treatment based on new observation and estimation (the dashed lines in Fig. 25.1).

The adaptive radiotherapy system shown in Fig. 25.1 has a very rich conﬁ guration. Only a few po- tentials have been investigated thus far in radiotherapy which are outlined in this chapter as the exam-

Clinical Adaptive Treatment Protocol

(Schedules of Imaging, Delivery, & Planning / Adjustment) Pretreatment Plan

Re-schedule Adjustment Schedule Delivery Schedule Imaging Schedule Design of Adaptive

Planning / Adjustment

Estimation/

Evaluation

Adaptive Planning /

Adjustment Treatment Delivery Imaging

Veriﬁ cation Process Parameters

(Temporal Variation Related) (Ofﬂ ine feedback loop) Planning / Adjustment

Parameters

(Online feedback loop)

Fig. 25.1 Adaptive radiotherapy system

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ples of clinical implementation. In this chapter, each key component in the adaptive radiotherapy system is described. In section 25.2, temporal variations of patient/organ shape and position are outlined and classiﬁ ed based on their characteristics. In section 25.3, X-ray imaging and veriﬁ cation techniques are described. Estimation and evaluation of temporal variation-related process parameters are introduced in section 25.4. In section 25.5 design and selection of control parameters for adaptive planning and adjustment are discussed. These parameters are directly used in the 4D planning/adjustment described in section 25.6. Finally, section 25.7 provides two typical adaptive treatment protocols that have been implemented or intends to be implemented in the clinic.

25.2 Temporal Variation of Patient/Organ Shape and Position

Temporal variation of an organ shape and position with respect to the radiation beams can be determined by the position displacement of subvolumes in the organ (V ). For a given patient treatment, patient organ (target or normal structure) can be deﬁ ned as a set of subvolumes or volume elements v, such that V={v}.

The notation v x v

x v x v x v

t

R

t t t

( )

( ) ( ) ( )

=

⎡

⎣

⎢ ⎢

⎢

⎤

⎦

⎥ ⎥

⎥ ∈

1 2 3

3

indicates the three-

dimensional (3D) position vector of a subvolume v at a time instant t; therefore, shape and position variation of the organ of interest during the entire treatment period, T, is speciﬁ ed as

v v v

x v

_t

( ) = x v

_r

( ) + u v

_t

( ), ∀ ∈ v V t ; ∈ T (1) where xv

r

(v)DR

³

is the subvolume position manifested on a pre-treatment CT image for treatment planning, and

u v v

u v u v u v

t

R

t t t

( )

( ) ( ) ( )

=

⎡

⎣

⎢ ⎢

⎢

⎤

⎦

⎥ ⎥

⎥ ∈

1 2 3

3

is the displacement vector of the subvolume at a time instant t.

Denoting T

_i

, i=1, ..., n to be the time interval of dose delivery (<5 min) in each of the number n treatment fractions, then the organ shape and position variation represented by the subvolume displacements during the entire course of treatment delivery

can be modeled as a process of time, or a temporal variation process, as

v U

u v

_t

t T

_i

T v V

i n

( ) | ∈ ⊂ ,

⎧ ⎨

⎩

⎫ ⎬

⎭ ∀ ∈

=1

(2) On the other hand, the organ shape and position variation during each dose delivery can be modeled as

u v v

_t

( ) | t ∈ T

_i

, v V i ; ,..., n

{ } ^{∀ ∈} ⁼¹ ⁽³⁾

It is clear that the processes [Exp. (3)] are subprocesses of the whole treatment process [Exp. (2)], and have been called intra-treatment process. Patient/organ shape and position variations have been clas- siﬁ ed into the inter-treatment variation, deﬁ ned as u v v

_t

( ) = const , ∀ ∈ t T

_i

, and the intra-treatment variation where uv

t

(v) changes within T

_i

; however, both the variations most likely exist simultaneously during the treatment delivery and cannot be easily separated.

Typical example of inter-treatment variation is the daily treatment setup error with respect to patient bony structure. Meanwhile, the typical example of intra-treatment variation is the patient respiration- induced organ motion.

Given an organ subvolume, the displacement sequence, denoted as a set of random vectors in Exps. (2) or (3), can be modeled as a random process within the time domain T of the treatment course or T

_i

of a treatment delivery. Using Eq. (1), subvolume displacement in the random process can be decom- posed (Yan and Lockman 2001) such that

v v v

u v

_t

( ) = µ

_t

( ) v + ξ

_t

( ) , v ∀ ∈ v V t ; ∈ T (4) where µ v

t

( ) v = E u v [ v

t

( ) ] is the mean of the displacement or the mean of the random process, and v

ξ

t

( ) v is the random vector which has a zero mean but same shape of probability distribution of the displacement.

The mean, by deﬁ nition, is the systematic variation, and the standard deviation,

σ ^v

^t

^{( )} ^v ⁼ ^E _⎡⎣ ξ ^v

^t

^{( )} ^v _⎤⎦

²

⁼ ^E [ ^{u v} ^v

^t

^{( )} ⁻ ^vµ

^t

^{( )} ^v ]

²

, is used to characterize the random variation v

ξ

t

( ). v In addition, the mean and the standard deviation have been proved to be the most important factors to inﬂ uence treatment dose distribution; thus, they have been selected as the primary process parameters of temporal variation considered in the design of an adaptive treatment plan.

A temporal variation process can be a stationary

random process if it has a constant mean during

the treatment course, such as µ v

t

( ) v = µ v ( ) , v ∀ ∈ t T ,

and a constant standard deviation, such as

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v v

σ

t

( ) v = σ ( ) , v ∀ ∈ t T . The condition of the constant standard deviation is slightly stronger than the for- mal deﬁ nition of the stationary (wide sense) random process in the textbook (Wong 1983); however, it is more suitable for describing a temporal variation process in radiotherapy.

Followed by the above deﬁ nition, a temporal variation process can be described using the stationary random process if its systematic variation and the standard deviation of the random variation are constants within the entire course of radiotherapy; otherwise, it is a non-stationary random process. Most of temporal variation process of patient/organ shape and position in radiotherapy can be considered as a stationary process. Examples of non-stationary process are most likely dose-response related, such as a process of organ displacement with its mean displacement drifted due to reopening of atelectasis lung, a process affected by organ ﬁ lling that is changed by radiation dose, or a process with a normal organ ad- jacent to a shrinking target.

A subprocess of intra-treatment variation, u v v

_t

( ) | t ∈ T

_i

{ } , can also be classifi ed as the stationary and non-stationary. In this case, example of the stationary process could be related to patient respiration induced organ motion. On the other hand, example of the non-stationary process could be related to an organ- fi lling process such as intra-treatment bladder fi lling.

25.3 Imaging and Veriﬁ cation

Imaging (sampling) patient/organ shape and position frequently during the treatment course is the major means of verifying and characterizing anatomical variation in radiotherapy. Ideally, imaging should be performed with patient setup in treatment position and with a sampling schedule compatible with the frequency of the temporal variation considered. Commonly, imaging schedule in an adaptive radiotherapy is pre-designed in the treatment protocol based on speciﬁ cations required for the estimation and evaluation of temporal variation process parameters, which are further discussed in section 25.4.

Three modes of X-ray imaging have been implemented in the radiotherapy clinic to observe patient anatomy-related temporal variation, which are radiographic, ﬂ uoroscopic, and volumetric CT imaging.

In addition, 4D CT image can also be created (Ford et al. 2003; Sonke et al. 2003). Onboard imaging de- vices with partial or all three modes are commercially

available, which include onboard MV or kV imager, in-room kV imager, CT on rail, tomotherapy unit with onboard MV CT, and onboard cone-beam kV CT.

25.3.1 Veriﬁ cation with Radiographic Imaging

Onboard MV radiographic imaging has been used to verify patient daily setup measured using the position of patient bony anatomy, or position of a region of interest with implanted radio-markers. Normally, the position error is determined using the rigid body registration between a daily treatment radiographic image and a reference radiographic image, most likely a digital reconstructed radiographic (DRR) image created in treatment planning. There have been numerous methods on 2D X-ray to 2D X-ray registration, which have been outlined in a survey paper (Antonie Maintz et al. 1998). Capability of using radiographic image for treatment veriﬁ cation has been extensively studied and is conclusive. It can be applied to determine bony anatomy position as a surrogate to verify patient setup position. In addition, it can also be used to locate implanted radio-markers’ position as a surrogate to verify the position of a region of interest.

Patient/organ position displacement caused by a rigid body motion at the i

^th

treatment delivery has been denoted using a vector of three translational parameters and a 3

_v _v

¥3 matrix with three rotational parameters as

_v _v u v_t( )=^∆_t( )p +R p_t( )⋅

[

x v_r( )−x p_r( ) ,

]

^{∀ ∈}v V; t^∈T_i

^, where v

∆

t

t t t

p ( )

( ) ( ) ( )

=

⎡

⎣

⎢ ⎢

⎢

⎤

⎦

⎥ ⎥

⎥ δ δ δ

1 2 3

is the translational vector with

shift, δ

t j

( )

, along j

^th

axis determined with respect to a reference point p pre-deﬁ ned on the reference image.

R p

_t

( ) = R R R

^{( )}¹ ^{( )}² ^{( )}³

is the rotation matrix with respect to the same reference point and rotation around individual axis, such that

R

t t

( ) ( ) ( )

( ) ( )

( )

cos sin

sin cos

1 1 1

,

1 1

2

1 0 0

0 0

= −

⎛

⎝

⎜ ⎜

⎜

⎞

⎠

⎟ ⎟

θ θ ⎟

θ θ

==

−

⎛

⎝

⎜ ⎜

⎜

⎞

⎠

⎟ ⎟

⎟

=

cos sin

sin cos

( ) ( )

( )

θ θ

t t

R

2 2

3

0 0 1 0

0 and

ccos sin

sin cos

( ) ( )

θ θ

t t

3 3

0 0

0 0 1

⎛ −

⎝

⎜ ⎜

⎜

⎞

⎠

⎟ ⎟

⎟

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representing the subrotation matrix around j

^th

axis by an angle θ

t

( )j

. Since the displacements of all subvolumes in a region of interest are uniquely determined by the translational vector and rotation matrix, only the six parameters, δ

t

θ

j t

j

( ) ( )

; , =1 2 3 , , , are needed to determine patient/organ rigid body motion. Conventionally, the translational vector, v

∆

t

, t = t

1

, ..., t

n

, observed using a portal imaging de- vice before, during, and/or after treatment delivery, have been used to represent the temporal variation of patient setup error, when rotation error in patient setup is insigniﬁ cant.

25.3.2 Veriﬁ cation with Fluoroscopic Imaging

Fluoroscopy has been conventionally used to observe patient respiration-induced organ motion at a treatment simulator to guide target margin design in radiotherapy planning of lung cancer treatment.

Recently, due to the availability of onboard kV imaging, it is being applied to verify intra-treatment organ motion induced by patient respiration (Hugo et al. 2004). This verifi cation has been established by comparing the online portal fl uoroscopy to the digital reconstructed fl uoroscopy (DRF) created using the 4D CT image. Respiration-induced organ motion can be determined by tracking the motion of a land- mark or a radio-marker implanted in or close to the organ of interest. Consequently, the frequency or the density of the motion can be derived by calculating the ratio of an accumulated time, within which the patient respiration-induced displacement is equal to a constant, versus the entire interval of breathing motion measurement (Lujan et al. 1999).

Symbolically, the motion frequency or density function for a point of interest p can be calculated as

c T

≡ ∀

{ u p }

ϕ τ

_τ

τ

( , ) | ( ) ,

p c T

i i

= v ∈

,

where uv

_o

(p), o ŒT

i

is the respiratory displacement of p measured using the ﬂ uoroscopic image within the time interval T

_i

. Figure 25.2 shows a typical time-position curve of patient breathing motion of a point of interest and its corresponding density function.

In clinical practice, both the respiratory motion and its frequency are important for adaptive treatment design and planning to compensate for a patient respiration-induced temporal variation (Liang et al.

2003). For treatment planning purpose, ﬂ uoroscopic image can be obtained in treatment position from either a simulator or an onboard kV imager; how-

ever, onboard fl uoroscopy is preferred for verifying treatment delivery. Positions and frequency of points of interest, specifi cally the mean and the standard deviation of the displacement, measured from an online fl uoroscopy, are compared with those pre-determined from the DRF created in the adaptive planning to verify the treatment quality.

25.3.3 Veriﬁ cation with CT Imaging

Volumetric CT has been the most useful imaging mode in verifying temporal variation of patient anatomy. Using this mode, the treatment dose in organs of interest could be constructed. The treatment plan can be designed in response to changes of patient/organ shape and position during the therapy course; however, due to overwhelming information contained in a 3D and 4D anatomical image, it also brings a great challenge in the applications of volumetric image feedback.

One of the most difﬁ cult tasks in applying volumetric image feedback in adaptive treatment planning is the image-based deformable organ registration. Unlike rigid body registration that has been well developed and discussed everywhere, deformable organ registration is quite immature. Methods

Fig. 25.2 A typical example of patient respiration-induced mo- tion of a subvolume position and its corresponding position density distribution

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of volumetric image-based deformable organ registration have been conventionally classiﬁ ed into two classes (Antonie Maintz et al. 1998): the segmentation-based registration method and the voxel property-based registration method. Segmentation- based registration utilizes the contours or surface of an organ of interest delineated from the reference image to elastically match the organ manifested on the second image (McInerney and Terzopoulos 1996). On the other hand, voxel property-based registration method utilizes mutual information manifested in two images to perform the registration (Pluim et al. 2003). Both registration methods, in principle, share a same problem on the interpretation of the rest of points of interest. Mathematically, this problem can be described as for given conditions {xv

r

(v) | v DV } – the subvolume position of an organ of interest manifested on the reference image and { x v v

_t

( ) = x v v

_r

( ) + u v v v

_t

( ) | ∈∂ ⊂ V V } – the boundary condition of surface points or mutual information, determining { x v v

_t

( ) = v x v

_r

( ) + u v v V v

_t

( ) | ∈ − ∂ V }

– the rest of subvolume positions manifested on the secondary image. Existing methods of interpretation are the ﬁ nite element analysis that determines subvolume position based on the mechanical con-

stitutive equations and tissue elastic properties, and the direct interpretation of using a linear or a spline interpolation. Applications in radiotherapy include using the ﬁ nite element method to perform CT image-based deformable organ registration for organs of interest in the prostate cancer treatment (Yan et al. 1999), the GYN cancer treatment (Christensen et al. 2001) and the liver cancer treatment (Brock et al. 2003). Deformable organ registration followed by volumetric image feedback provides the distribution of organ subvolume displacements (Fig. 25.3), which plays an important role in the adaptive or 4D planning; however, there is no clear answer thus far as to what degree of registration accuracy can be achieved utilizing each interpretation method and what is needed for an adaptive treatment planning.

Two types of sequential CT imaging have been applied in adaptive treatment planning. The ﬁ rst one has a longer elapse (day or days) of imaging (sampling) to primarily measure an inter-treatment temporal variation. Clinical applications of using multiple daily images have been limited to prostate cancer treatment (Yan et al. 2000), colon-rectal cancer treatment (Nuyttens et al. 2002), and head and neck cancer treatment. The second sequential imag-

AP (cm) SI (cm) RL (cm)

Fig. 25.3 A typical example of subvolume displacement distribution for a bladder wall and a rectal wall. The color map from red to blue indicates the range (large to small) of the standard deviation of each subvolume displacement in centimeters

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ing has been aimed to detect organ motion induced by patient respiration. These images, which manifest organs of interest at different breathing phases, can be obtained from a respiratory correlated CT imaging (Ford et al. 2003) or a slow rotating cone-beam CT (Sonke et al. 2003). Clinical application of this measurement has been focused on lung cancer treatment; however, it has no limit to be extended to the other cancer treatment where patient respiration effect is signiﬁ cant, such as liver cancer treatment. In principle, both types of sequential imaging are 4D CT imaging, although the terminology of 4D CT imaging has been speciﬁ cally used to describe the 3D sequential CT images induced by patient respiration.

Commonly, either an onboard or an off-board volumetric imager can be applied to measure patient organ motion for the purpose of offl ine planning modifi cation; however, onboard imaging is a favorable tool for both online and offl ine treatment planning modifi cation and adjustment.

25.4 Estimation and Evaluation

It has been discussed that temporal variation of patient/organ shape and position during the whole treatment course could be modeled as a random process of organ subvolume displacement, denoted as ^{u v} v

^t

^t U ^T

ⁱ

^T ^{v V}

i n

( ) | ∈ ⊂ ,

⎧ ⎨

⎩

⎫ ⎬

⎭ ∀ ∈

=1

, or multiple subprocesses during each treatment delivery, denoted as { ^{u v} ^v

^t

^{( ) |} ^t ^∈ ^T

ⁱ

} ^, ^{∀ ∈} v V ^; ⁱ ⁼¹ ^,..., ⁿ . The former has been primarily considered in the design of ofﬂ ine imaging and planning modiﬁ cation as indicated as the outer loop of the adaptive system in Fig. 25.1;

the latter, on the other hand, has to be considered additionally in the design of online image-guided adjustment – the inner loop of Fig. 25.1. Although process parameter estimation and treatment evaluation are normally performed in the outer feedback loop, they will be utilized to modify and update the planning and adjustment parameters for both ofﬂ ine and online planning modiﬁ cation and adjustment.

As has been discussed in section 25.2, the key parameters of a temporal variation process are the systematic variation and the standard deviation of random variation of subvolume displacement [see also Eq. (4)]. These parameters, therefore, have to be estimated for either ofﬂ ine or online mechanisms.

Moreover, the cumulative dose/volume relationship

of organs of interest and the corresponding biological indexes can also be estimated and evaluated. In addition, effects of dose per fraction on a critical normal organ should also be considered in the cumulative dose evaluation when online image guided hypo- fractionation is implemented, because the effect of temporal variation of organ fraction dose can be sig- niﬁ cantly enlarged in a hypo-fractionated treatment (Yan and Lockman 2001).

Multiple imaging (or sampling) performed in the early part of treatment has been the common means to estimate the mean µ

t

v (the systematic variation) and the standard deviation σ

t

v (the characteristic of the random variation) of a temporal variation process. Estimation can be performed once using multiple images obtained early in treatment course, in batches, or continuously. In general, imaging schedule in an adaptive radiotherapy system has been selected in a treatment protocol based on a pre-designed strategy of adaptive treatment. In case of the single offl ine adjustment of patient position during the treatment course, optimal sampling schedule of four to fi ve observations obtained daily in the early treatment course has been suggested (Yan et al. 2000) and proved to be a favorable selection with respect to the criteria of minimal cumulative displacement (Bortfeld et al. 2002); however, so far, there has not been any systematic study to explore the relationship between the imaging schedule and the treatment dose/volume factor. A preliminary study (Birkner et al. 2003) demonstrated that there was only a mar- ginal improvement for prostate cancer IMRT, when offl ine-planning modifi cation is continuously performed compared with a single modifi cation after fi ve measurements. Parameter estimation for a given temporal variation process could be straightforward.

Example of such process is the organ motion induced by patient respiration. In this case, the motion can be characterized using 4D CT imaging and ﬂ uoroscopy before the treatment if the process is stationary; otherwise, the estimation can be performed multiple times during the treatment course.

25.4.1 Parameter Estimation for a Stationary Temporal Variation Process

Without loss of generality, let { u t ^v

^tⁱ

|

ⁱ

∈ ^{T i} ; = ¹ ^{, ...,} ^k }

be the measurements (the sample size k is commonly

small except for the respiratory motion) of subvol-

ume displacement for an organ of interest obtained

from k CT image measurements, or displacement of

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a reference point when radiographic images or ﬂ uoroscopy are used for the measurements. The standard unbiased estimations of the constant mean, µv , and the constant standard deviation, σv, based on the measurements are

ˆ 1 ˆ v 1 v ˆ

v v

µ = ∑

₌ ⁱ

^; σ = ₋ ₁ ∑

₌

⁽ ^u

ⁱ

⁻ ^µ ^v ⁾

1 1

2

k u

t

k

i k

t i

k

.

In addition, the potential residuals between the true and the estimation can also be evaluated based on the standard conﬁ dence interval estimation as

α2, 1

⋅ ≤

− −

µ µ

1 1

v v

_/

v ; v v

,

σ σ

χ σ

α

− ≤ t

−

− ⋅

k

k 2

ˆ 1 ˆ ,

where the factor t

_α_/₂_,_k₋₁

has the t-distribution with k-1 degrees of freedom and χ

¹²−_α, k−¹

has the χ

²

−distribution with k-1 degrees of freedom, and both them have the conﬁ dence 1- α .

The other method of estimating the systematic variation is the Wiener ﬁ ltering (Wiener 1949).

Applying the Wiener ﬁ ltering theory, the optimal estimation of the systematic variation is constructed by minimizing the expectation of the estimation and the truth, such as Min E µ µ v v u

_t

u

_t

( ) | v , ..., v v v

k

, ,

µ σ

⎡⎣ ˆ −

² ¹

Σ Μ ⎤⎦ , with conditions of the k measurements, the standard deviation of the individual systematic variations,

0 0

v

Σ

_µ

µ µ

µ

σ σ

σ

=

⎛

⎝

⎜ ⎜

⎜⎜

⎞

⎠

⎟ ⎟

⎟⎟

1

2

3

0 0

,

and the root-mean-square of the individual standard deviations of the random variations,

0 0

v

Μ

_σ

σ σ

σ

µ µ

µ

=

⎛

⎝

⎜ ⎜

⎜⎜

⎞

⎠

⎟ ⎟

⎟⎟

1

2

3

0 0

.

Consequently, the estimation of the systematic variation is

( )

k k

= ⋅

u c Σ Σ

v v v v v

v v v

µ = ⋅ ,

_µ

⋅ ⋅ +

_µ _σ

=

∑

−

c k

i ^t k

i

1

Μ

1

ˆ .

It has been proved that for a temporal variation process, Σ v

vµ

and v

σv

Μ could have similar values; therefore, the Wiener estimation can be simpliﬁ ed as

v v

µ = + ⋅

∑

=

1 1

1

k u

_t

i k

i

ˆ _.

In addition to the mean and the standard deviation, knowledge of the probability density ϕ ( ) v of each subvolume displacement in a temporal variation process could also be useful; however, except for patient respiratory motion that can be determined

directly using 4D CT and ﬂ uoroscopy (as described in section 25.3.2), the majority of temporal variations can only be practically measured a few times, and using these small numbers of measurements to estimate a probability distribution is most unlikely possible.

Therefore, pre-assumed normal distribution has been applied in the clinic. It has been demonstrated that the actual treatment dose in an organ subvolume is most likely determined by its systematic variation and the standard deviation of its random variation.

The actual shape of the displacement distribution is less important (Yan and Lockman 2001); therefore, the pre-assumption of the normal distribution,

( ˆ )

v N µ v σ v ˆ ( )

ϕ = ( ), v ˆ

²

( ) v , is acceptable in the treatment dose estimation.

Parameter estimation of stationary temporal variation process can be applied for both offl ine and online feedback. Examples of offl ine feedback include using multiple radiographic portal imaging to characterize patient setup variation, multiple CT imaging to characterize internal organ motion, and 4D CT/fl uoroscopy imaging to characterize respiratory organ motion. Application for online feedback is cur- rently limited to characterize intra-treatment organ motion assessed by multiple portal imaging and portal fl uoroscopy. For the online CT image-guided prostate treatment, parameter estimation for intra-treatment variation also depends on patient anatomical conditions. There has been a study (Ghilezan et al.

2003) that showed that the intra-treatment variation of prostate position was primarily controlled by the rectal ﬁ lling conditions.

25.4.2 Parameter Estimation for a Non-stationary Temporal Variation Process Parameters to be estimated in a non-stationary process are similar to those in a stationary process;

however, instead of constants, they can be piecewise constants, such as respiration-induced organ motion during lung cancer treatment, or a continuous function of time, such as bladder-ﬁ lling-induced motion during treatment delivery. It is relatively simple to estimate the process parameters that are piecewise constants. The estimation in each constant period will be performed as same as the one for a stationary process;

however, the estimation for a process with parameter as a continue function will be less straightforward.

The most common method to estimate a function based on ﬁ nite number of samples is the least-squares estimation. With pre-selected orthogonal base of functions φ v t φ

₁

t

₂

φ φ

( ) ( ) ( ) ( )

j j j

m

t

j

t

= ⎡⎣ ⋅⋅⋅ ⎤⎦ , i.e.

(9)

= = φ

i

j i

t t i m

( )

⁻¹

, 1 2 , ,..., , the estimations for both the systematic variation and the standard deviation of the random variation are

ˆ ˆ

µ σ

µ = a ⋅φ ( );t ^{( )}

( )

( ) ( ) ( )

( ) ( )

t j

i j

t j

t j i

k

i

m u

k m

i i

= −

−

=

∑

²

1

∑

1

∑

=

, , , ,

j 1 2 3 where

ˆ

a

u

j

m j

T T

t j

t j k

1 1 ¹

( )

⯗ ⯗ ;

⎡

⎣

⎢⎢

⎢

⎤

⎦

⎥⎥

⎥=

(

⋅

)

^⋅ ^⋅^⎡

⎣

⎢⎢

⎢

⎤

⎦

⎥⎥

⎥

Φ Φ − Φ ΦΦ =

⎡

⎣

⎢⎢

⎢

⎤

⎦

⎥⎥

⎥ v v ⯗ φ

φ

( )

j

j k

t

1

.

As the extension of the Wiener ﬁ lter, the Kalman ﬁ lter has also been applied to estimate the systematic variation for a non-stationary process assuming that the systematic variation is a linear function of time (Yan et al. 1995; Lof et al. 1998; Keller et al.

2004). In addition, similar to the description in the previous section, the probability density of each subvolume displacement can be estimated as ˆ ϕ

_t

( ) v for a respiratory motion or a normal distribution

ˆ ˆ

µ v σ v

( )

t

( ) v N

t

( ), ( )

ϕ ˆ = v

t²

v .

Applications of parameter estimation for a non- stationary process have been few. One study (Ford et al. 2002) attempted to determine the reproduc- ibility of patient breathing-induced organ motion.

It revealed that the mean of patient respiration-induced organ motion could considerably vary during the course of NSCLC due to treatment and patient related factors; therefore, multiple measurements of 4D CT and ﬂ uoroscopy, i.e., once a week, may be necessary to manage adaptive treatment of lung cancer.

Regarding parameter estimation for a continuous function, one study (Heidi et al. 2004) has been performed to estimate bladder expansion and potential standard deviation for the online image-guided bladder cancer treatment, where bladder subvolume position was modeled as a linear function of time.

25.4.3 Estimation of Cumulative Dose

Including temporal variation in cumulative dose estimation can be performed using the knowledge of subvolume displacement distribution. At pres- ent, the construction is performed assuming time invariant spatial dose distribution calculated from the treatment planning. It implies that the dose distribution remains constant spatially regardless the changes of patient anatomy; however, this assumption can only be acceptable if spatial dose variation

induced by the changes of patient body shape and tissue density is insigniﬁ cant, i.e., during the prostate cancer treatment; otherwise, the dose has to be recalculated using each new feedback image. Of course, this can only be performed when CT image feedback is applied.

Cumulative dose for each subvolume in the organ of interest can be evaluated as

or

with considering the biological effect of dose per fraction, where d

_s

is the standard fraction dose 1.8 or 2 Gy. This dose expression is a very general and can be simpliﬁ ed based on the at- tributes of a temporal variation.

For a stationary process with the time invariance dose distribution, the cumulative dose in an organ of interest can be estimated directly utilizing the probability density of subvolume displacement ␸

^{^}

(v) and the planned dose distribution d

_p

as

(5)

if the planned dose per treatment fraction is ﬁ xed.

When an ofﬂ ine planning modiﬁ cation is performed at the k+1 treatment delivery, based on the previous k image measurements, then the cumulative dose can be estimated by considering the treatments which have been delivered (Birkner et al. 2003), such that

.

The estimation can also be performed using the mean and the standard deviation of a temporal variation (Yan and Lockman 2001), such that

(6)

where is the mean dose gradient in the interval , and is the dose curvature at point .

Equation (6) provides a very important structure on the parameter design of adaptive planning and adjustment (discussed in the next section).

Using the estimated dose, radiotherapy dose response parameters, such as the EUD, NTCP, and TCP, can be evaluated using the common methods that have been discussed elsewhere.

(10)

25.5 Design of Adaptive Planning and Adjustment Design of adaptive planning and adjustment contains computation and rules to select planning and adjustment parameters, and to update the schedule of imaging, delivery, and adjustment. Ideally, imaging/veriﬁ cation, estimation/evaluation, and planning/adjustment should be performed with the identical sampling rates, and the planning/adjustment parameters should be selected in such way that the adaptive radiotherapy system can be completely optimized; however, this is most unlikely possible when clinical practice is considered. Only a few pos- sibilities have been investigated and are discussed here.

25.5.1 Design Objectives

Objectives in the design of adaptive planning/adjustment are commonly specifi ed in an adaptive treatment protocol. The objectives are (a) to improve treatment accuracy by reducing the systematic variation, (b) to reduce the treated volume and improve dose distribution by reducing the systematic variation and compensating for patient specifi c random variation, (c) to reduce the treated volume and improve dose distribution by reducing the both systematic and random variations, and (d) to additionally improve treatment effi cacy by alternating daily dose per fraction and number of fractions.

Clearly, an objective has to be selected based on expected treatment goals and available technolo- gies. The fi rst two can be implemented using an offl ine feedback technique. Conversely, online image guided adjustment or planning modifi cation has to be implemented to achieve the objectives (c) or (d).

Most of ofﬂ ine techniques have implemented the re- planning and adjustment once during the treatment course, except for the case when a large residual ap- peared in the estimation. On the other hand, most of online techniques have aimed to adjust patient treatment position only by moving the couch and/

or beam aperture; therefore, it is also important to implement a hybrid technique, where ofﬂ ine planning is performed to modify the ongoing treatment plan in certain time intervals (e.g., weekly) during online daily adjustment process.

25.5.2 Adaptive Planning and Adjustment Parameters Given organs of interest, the target and surrounding critical normal structures, the aim of an adaptive treatment planning is to design and modify treatment dose distribution in response to the temporal variations observed in the previous treatments.

Considering the dose expressed in Eq. (6), four factors play the key roles on treatment quality and can be considered in the adaptive treatment planning and adjustment design; these are two patient/organ- geometry related factors, the systematic variation µv and the standard deviation of the random variation σv for each subvolume in the organs of interest, and two patient dose-distribution-related factors, the dose gradient ¢d

_t

and the dose curvature ∂

∂

2 2

d x

v

t

at each spatial point in the region of interest. Theoretically, any treatment planning and adjustment parameter, which can control these factors, can be selected to modify and improve the treatment.

Planning and adjustment parameters can be divided into two classes: one contains patient-positioning parameters, such as couch position and rotation, beam angle, and collimator angle, which can be applied to reduce both the systematic and random variations {µv , σv}; however, these parameters can only adjust variations induced by rigid body motion and improve position accuracy and precision, but have lim- its to manage variations induced by organ deformation and cannot improve treatment plan qualities;

the other contains dose-modifying parameters, such as target margin, beam aperture, beam weight, and beamlet intensities. These parameters are typically used to adjust dose distribution, thus modifying

∇ ∂

∂

⎧⎨

⎩

⎫⎬

⎭

d d

t

x ,

t

2

v

2

in the region of interest to improve ongoing treatment qualities. In addition, prescription dose, dose per fraction, and number of fractions have also been used as parameters for adaptive planning (Yan 2000).

25.5.3 Adaptive Planning and Adjustment Parameter and Schedule: Selection and Modiﬁ cation

In an ideal adaptive radiotherapy system, design of

planning and adjustment should have a function of

automatically selecting on going planning/adjust-

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ment parameters and schedules of imaging, delivery, and adjustment; thus, a new treatment plan can be calculated by including the observed temporal variations and estimation, optimized using the selected parameters and executed with the new schedules. In principle, a set of pre-speciﬁ ed rules and control laws could be used in the design, which match the parameters of temporal variation process to the parameters and schedules of adaptive planning and adjustment.

Basic rules can be created utilizing the discrepancies between the ideal treatment under the ideal condition (i.e., no temporal variation occurs) and the

“actual” treatment that includes the temporal variations. The discrepancies can be either the organ volume/dose discrepancy, or the discrepancies of EUD, TCP, and/or NTCP determined from the planned dose, { ^{D v} ^{( )} ^| ^{v V} ∈

ⁱ

^, ⁱ = ¹ ^,..., ^l } , in organs of interest,

V

_i

, calculated without considering temporal variations vs those determined from the estimated dose distribution { ^{D v} ˆ ^{( ) |} v V ∈

ⁱ

^, ⁱ = ¹ ^,..., ^l } constructed from a treatment plan created using pre-speciﬁ ed planning/adjustment parameters and including the estimation of temporal variations. A set of pre- deﬁ ned tolerances { ^{δ δ δ}

_V

^,

_D

^,

^EUD

^, ^δ

^TCP

^, ^δ

^NTCP

} ^is

then used to test whether or not the discrepancies,

∆ ≤ δ ≤ δ ∆

V ( D

_D

)

_V

, EUD ≤δ

_EUD

, ∆ TCP ≤δ

_TCP

, and/or

∆ NTCP ≤δ

_NTCP

, hold within the predefi ned ranges. In addition, these tolerances can also be utilized to evaluate and rank the potential treatment quality with respect to different groups of planning/adjustment parameters and adjustment methodology (offl ine or online). Depended on the variation type (rigid or non-rigid) and the objectives of planning/adjustment, the planning and adjustment parameters could be selected as (a) couch position/rotation, beam angle, and/or collimator angle (online or offl ine position adjustment for a rigid body motion), (b) target margin, beam aperture, beam weight (intensities), and/or prescription dose (online or offl ine planning modifi cation), and (c) beamlet intensity, prescription dose, and/or dose per fraction plus number of fractions (online planning modifi cation). Contrarily, a subset of patients, who have insignifi cant temporal variation, can also be identifi ed; therefore, no re-planning and adjustment are necessary for this subset.

There have been limited studies on utilizing control laws to automatically modify the planning and adjustment parameters. A decision rule (Bel et al.

1993) has been proposed and applied for the ofﬂ ine adjustment of systematic variation induced by daily patient setup. This decision rule is constructed by assuming the statistical knowledge of patient setup variation, and automatically schedules the setup ad-

justment based on the estimated systematic variation, and pre-designed “action levels.” The other method to control the offl ine planning and adjustment has been “no action level” but including estimated residuals in the target margin design, and primarily single modifi cation after four to fi ve consecutive observations (Yan et al. 2000; Birkner et al. 2003). An early investigation (Lof et al. 1998) on the adaptive planning has modeled the cumulative dose and beamlet intensities (control parameters) recursively using a linear system, and created a quadratic objective from the prescribed doses and the estimated doses. Based on the optimal control theory (Bryson and Ho 1975), the intensity fl uence adjustment therefore follows a standard linear feedback law – a linear function of the dose discrepancy in organs of interest. Intuitively, the beamlet intensities in the treatment should be ad- justed proportionally to the estimated dose discrepancy. Similar methodology has been also proposed for adaptive optimization using the tomotherapy delivery machine (Wu et al. 2002). In addition to beam intensity fl uence, a control law has also been proposed to manipulate the prescription dose per treatment fraction and the total number of treatment fractions in an online image-guided process (Yan 2000). This control law utilizes temporal variation of dose/volume of critical normal organs to select the most effective dose of the fraction and the total number of fractions.

Most problems in adaptive radiotherapy are easily described but hardly solved. Compared with a direct 4D inverse planning after k number of observations, control laws derived from an ideal system model commonly provide only limited roles in the clinical implementation. For the clinical practice, most temporal variations of patient/organ shape and position can be described using stationary random processes, and therefore the control mechanism is straightforward. Applying one or few planning modifi cations, the systematic variation can be maximally eliminated and thus patient treatment can be signifi cantly im- proved in an offl ine adjustment process. Residuals are commonly inverse proportional to the frequency of the verifi cation, estimation, and adjustment. These residuals could be signifi cantly large to diminish the anticipated gain of adaptive treatment for certain patients; therefore, a decision rule should be applied to modify the schedule of imaging and adjustment if these patients are identifi ed during the treatment course.

Sampling rates of imaging/veriﬁ cation, estima-

tion/evaluation, and adaptive planning/adjustment

should be scheduled to match the rate of the aimed

(12)

temporal variations. Mismatch results in signiﬁ cant downgrading of the expected treatment quality;

therefore, before selecting objectives in an adaptive treatment design, speciﬁ cally for an online adjustment process, one should ensure that appropriate sampling rates of imaging/veriﬁ cation, estimation/

evaluation, and adaptive planning/adjustment could be implemented.

25.6 Adaptive Planning and Adjustment

Adaptive planning and adjustment are implemented with the pre-design parameters. The adaptive planning is often performed including the temporal variations in the planning dose calculation; therefore, it has also been called “4D treatment planning.” There have been two methods to perform a 4D planning.

The ﬁ rst (indirect method) does not directly include the temporal variation in the planning dose calculation. Instead, it constructs the PTV and margins of organs at risk based on the characteristics of patient speciﬁ c temporal variations and a generic planned dose distribution, and then performs a conventional conformal or inverse planning accordingly. The second (direct method) performs treatment planning by directly including the temporal variations in the dose calculation as has been discussed in section 25.4.3.

The adaptive treatment plan designed with the expected dose distribution can best compensate for the temporal variations. Consequently, pre-designed target margin is either unnecessary or used only to compensate for the residuals of the estimation.

25.6.1 Indirect Method

Planning technique in the indirect method is primarily the same as the conventional one except for the defi nition of the planning target volume and the margins of organs at risk. For a rigid body motion without signifi cant rotation, the patient specifi c target margin in each direction j after k observations can be constructed by considering the residuals of the estimations of the systematic and random variations (Yan et al. 2000), such that

1 1

m c t

k c k

j

j ,k

j

k j ( )

/

( )

,

( ) =

−

⋅ + ⋅ − ⋅

( )

− −

2 1 2

σ 1

χ σ

α

ˆ ˆ

α

,

where t

_α_/₂_,k₋₁

and χ

¹2−_α^,^k−¹

have been deﬁ ned in section 25.4.1. The factor c

_j

is determined by ensuring that the potential dose reduction in the target with the corresponding margin is less than a pre-deﬁ ned dose tolerance δ , such that

ˆ ⋅ ≤

⋅ϕ ξ )

(

( )

)

− +

= ⋅

( )

ξ δ

∆D c n

d x

d x m c

j

p r

j

p r

j j j

j

( ) j

( ) (

( )

( ) ( )

( )

−

⎡

⎣

⎢⎢

⎤

⎦

⎥⎥ ξ

( )

−∞

∞

∫

^d ^j

^,

where d

_p

( x

_r^{( )}^j

) is the planning dose around the CTV edge x

_r^{( )}^j

on the j axis. ˆ ϕ ξ ) (

^{( )}^j

is the estimated probability distribution with the mean t

, k

k

j /

( ) 2 −1

⋅σ

α

ˆ

(the residual of the systematic variation) and the standard deviation

1 1

k

− ⋅

j

− −

1 χ

2

σ

α^,

ˆ

( )

.

It is clear that the calculation of dose discrepancy here is approximated assuming the spatial invariance of planning dose distribution. In addition, this evaluation can also be approximated using the dose gradient and curvature around the CTV edge as indicated by Eq. (6), such that

τ π

j

2 1

; (

σ( )

≈ ⋅

∆D c n t

k d x

d x

j

,k p

j

j p

j

( )

[ , ] ( )

( )

/

( )

( ) ( )

⋅

⋅∂

∂ + ⋅∂

∂

−

σ

²

π

2 ²² ₂ ₁

⎛

⎝

⎜⎜

⎞

⎠

⎟⎟

= − = − + ₋⋅

τ x_r^{( )}^j m^{( )}^j(c_j)π x_r^{( )}^j m^{( )}^j c) t_/_,k σ

δ

j k

≤ ˆ

ˆ

ˆ α

α

.

This method can be further extended to construct CTV- to-PTV margin that compensates for much broad type of variations including organ deformation. Let ∂ CTV represent the boundary of CTV, then the 3D margin can be constructed using the vector normal to target surface at each boundary point v ∈ ∂ CTV , such that

25 Image-Guided/Adaptive Radiotherapy Di Yan