optimization of nonuniformly fractionated radiotherapy ... · melissa r. gaddy (ncsu) nonuniform...

Optimization of Nonuniformly Fractionated RadiotherapyTreatments

Melissa R. Gaddy1, Sercan Yıldız2, Jan Unkelbach3, David Papp1

1. NC State University, 2. UNC Chapel Hill, 3. University Hospital Zurich

May 2, 2017

Melissa R. Gaddy (NCSU) Nonuniform Fractionation in Radiotherapy May 2, 2017 1 / 31

Outline

1 Optimization in Radiotherapy

2 The Fractionation Problem

3 The SDP Relaxation

4 Future Directions


Optimization in Radiotherapy

Outline




4 Future Directions



External-Beam Radiotherapy

Main objective: Irradiate the tumor while avoiding the organs andsurrounding tissue




Main objective: Irradiate the tumor while avoiding the organs andsurrounding tissue




Linear accelerator Multi-leaf collimator

https://www.varian.com/oncology/products/treatment-delivery/clinac-ix-system, accessed: 2017-11-04.http://newsroom.varian.com/imagegallery?mode=gallery&cat=2473, accessed: 2017-18-04.



Intensity Modulated Radiotherapy (IMRT)

Deliver multiple unmodulated beamsfrom the same angle to superimposetheir effect

Reemsten and Alber. Handbook of Optimization in Medicine, Pardalosand Romeijn (eds.), Springer. 2009.



Intensity Modulated Radiotherapy (IMRT)




Reemsten and Alber. Handbook of Optimization inMedicine, Pardalos and Romeijn (eds.), Springer.2009.

Discretize patient into a 3-D grid ofvoxels

Decision variables:

x : vector of beamlet weightsd : vector of doses delivered to eachvoxel of the patient

Dose-influence matrix D relatesbeamlet weights to dose absorbed bythe patient Dx = d




Dose-based IMRT model:

minx ,d

∑i∈I

wiFi (d)

s.t. Dx = d

x ≥ 0

Each Fi is a piecewise quadratic penalty function to penalize underdose,overdose, or mean dose.

E.g. Fi (d) =∑v∈V

(dv − dpresv )2+ to penalize overdose


The Fractionation Problem

Outline




4 Future Directions



What is Fractionation?

Total treatment is delivered over a series of days or weeks (eachtreatment day is called a fraction)

Currently in the clinic, the same treatment is delivered each day.

Want to evaluate potential improvement of delivering different doseseach day

Uniform Nonuniform

• Same treatment every day • Different treatment each day• Beamlet weights: x , doses: d • Beamlet weights: xt , doses: dt

(t = 1, . . . ,N)• Convex optimization problem • Nonconvex optimization problem• Based on physical dose d • Based on biologically effective

dose b



Biologically Effective Dose (BED) Model

Incorporate the effect of fractionation with biologically effective dose:

b =N∑t=1

(dt +

d2t

(α/β)

)

dt is dose absorbed in fraction t, (t = 1, . . . ,N)

α and β are tissue-specific parameters



Nonuniform Model

Dose-based IMRT model:

minx ,d

∑i∈I

wiFi (d)

s.t. Dx = d

x ≥ 0

(Nonconvex) Nonuniformlyfractionated IMRT model:

minx ,d ,b

∑i∈I

wiFi (b)

s.t. bv =N∑t=1

(dvt + d2vt

(α/β)v) ∀v

Dxt = dt t = 1, . . . ,N

xt ≥ 0 t = 1, . . . ,N



Evaluating Plan Quality

How can we fairly evaluate the benefit of nonuniform fractionation?

Two challenges:

Cannot fairly compare a dose-based plan with a BED-based plan

Compute a BED-based uniform reference plan

Cannot interpret the objective function

Prioritize a single clinical objective



Uniform Model

Assuming x1 = · · · = xN , we rewrite the model to solve for a uniformreference plan.

minx ,d ,b

∑i∈I

wiFi (b)

s.t. bv = Ndv

(1 +

dv(α/β)v

)∀ voxels v

Dx = d

x ≥ 0.

Despite having the same quadratic equality constraints, this is convex.



Constrained Nonuniform Model

Prioritize the clinical objective F1

Let b∗ be the BED obtained in the uniform reference plan.

minx ,d ,b

F1(b)

s.t. Fi (b) ≤ Fi (b∗) i 6= 1

bv =N∑t=1

(dvt + d2vt

(α/β)v) ∀ voxels v

Dxt = dt t = 1, . . . ,N

xt ≥ 0 t = 1, . . . ,N



Experimental Setup

Five liver cases with various geometries (2-D slice)

Five-fraction treatments with 21 equispaced beams

Clinical objectives:

Minimize the mean BED in the healthy liver tissueLower and upper BED bounds on all voxels

Computed globally optimal uniform reference plan

Computed locally optimal nonuniform plan



Locally Optimal Solution of Nonuniform FractionatedModel

High single-fraction dose to subregions of the tumor

Consistent, low dose to healthy tissue



Locally Optimal Solutions of Nonuniform Model

These are qualitatively about the same, but each partitions the tumordifferently



Results

The nonuniformly fractionated plans achieve a 13% - 35% reduction inmean liver BED

Case Description Mean liver BEDin uniformreference plan

Mean liver BEDin nonuniformplan

Mean liver BEDreduction

1 Large central lesion 84.54 75.87 12.75%2 Small lesion 26.14 19.47 34.26%3 Two small lesions 59.54 50.24 18.51%4 Lesion abutting chest wall 47.51 38.65 22.92%5 Lesion abutting GI tract 88.67 77.38 14.59%



Comparing Uniform and Nonuniform Plans

Uniform reference plan Nonuniform DEQ5

Difference


The SDP Relaxation

Outline




4 Future Directions


The SDP Relaxation

Bound on the Maximum Achievable Benefit

Recall the constrained nonuniform fractionation model:

minx ,d ,b

F1(b)

s.t. Fi (b) ≤ Fi (b∗) i 6= 1

bv =N∑t=1

(dvt + d2vt

(α/β)v) ∀ voxels v

Dxt = dt t = 1, . . . ,N

xt ≥ 0 t = 1, . . . ,N

where Fi is of the form∑v∈V

(bpresv − bv )2+ or similar


The SDP Relaxation


Reformulate the model into a QCQP using auxiliary variables

Introduce Xt = xtxTt and write the quadratic inequalities as linear

inequalities in xt and Xt .E.g. Incorporating Dxt = dt , the first constraint

bv =N∑t=1

(dvt + d2vt

(α/β)v)

becomes

bv =N∑t=1

⟨[1 xTtxt Xt

],

[0 eTv D

2DT ev

21

(α/β)vDT eve

Tv D

]⟩


The SDP Relaxation


Replace Xt = xtxTt with the convex relaxation

[1 xTtxt Xt

]< 0.

Add the componentwise inequality Xt ≥ 0 to tighten the bound.

Since the formulation is convex and symmetric in the fractions, wecan assume x1 = · · · = xN and X1 = · · · = XN .


The SDP Relaxation

SDP Relaxation for Constrained Nonuniform Modelmin

x,X ,p,q,rr1

s.t.∑v∈Vi

p2iv ≤ Fi (b∗) ∀i ∈ I+

∑v∈Vi

q2iv ≤ Fi (b∗) ∀i ∈ I−

ri ≤ Fi (b∗) ∀i ∈ Im, i 6= 1

piv ≥ N

⟨[1 xT

x X

],

[− bhiiv

N

eTv D

2DT ev

2Cv

]⟩∀v ∈ Vi , ∀i ∈ I+

qiv ≥ −N⟨[

1 xT

x X

],

[− bloiv

N

eTv D

2DT ev

2Cv

]⟩∀v ∈ Vi , ∀i ∈ I−

ri ≥N

|Vi |∑v∈Vi

⟨[1 xT

x X

],

[−mhi

iN

eTv D

2DT ev

2Cv

]⟩∀i ∈ Im

piv ≥ 0, qiv ≥ 0, ri ≥ 0 ∀i , v[1 xT

x X

]< 0, x ≥ 0, X ≥ 0.


The SDP Relaxation

SDP Lower Bounds

Solved the SDP for the five liver cases

Obtained a lower bound m∗ on the mean BED to healthy liver tissue

Compared the reduction achieved by the nonuniform plans to thebound on the maximum possible reduction

gap closed =munif −mnonunif

munif −m∗


The SDP Relaxation

Results

The nonuniform plans closed 78%-96% of the gap

Case Description Mean liver BEDin uniformreference plan

Mean liver BEDin nonuniformplan

SDPLowerBound

Gapclosed

1 Large central lesion 84.54 75.87 73.38 77.69%2 Small lesion 26.14 19.47 18.58 88.23%3 Two small lesions 59.54 50.24 48.03 80.80%4 Lesion abutting chest wall 47.51 38.65 37.65 89.86%5 Lesion abutting GI tract 88.67 77.38 77.02 96.91%


Future Directions

Ideas for Future Research

Develop a method to solve the large-scale SDP for athree-dimensional patient

A variable for each beamlet (hundreds)A constraint for each voxel (tens of millions)

Find even better local solutions, or tighten the lower bound, or both

Incorporate uncertainty into the model


Future Directions

Conclusion

Locally optimal nonuniformly fractionated treatment plans

Maintained treatment effectiveness in the tumor

Reduced Biologically Effective Dose (BED) in healthy liver tissue by13-35%

Closed 78-96% of the bound on maximum achievable benefit(solutions are near the globally optimal treatment plan)


optimization of nonuniformly fractionated radiotherapy ... · melissa r. gaddy (ncsu) nonuniform...

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