the cartoon guide to radiation oncology modelingthe radiation in radiation oncology • high energy...

The Cartoon Guide to Radiation Oncology Modeling

Joe Deasy, PhDDept of Radiation Oncology

Washington University School of Med Alvin J. Siteman Cancer Center

(Kandinsky)

Overview: the typical treatment planning process (1/2)

• Patient image data is acquired (typically fully 3-D computed tomography (CT) scans).

• The physician outlines the tumor and important normal structures on a computer, based on the CT scan.– The Gross Target Volume (GTV) is the volume which

encompasses know macroscopic disease.– The Clinical Target Volume (CTV) exceeds the GTV to

include regions of suspected microscopic disease.– The Planning Target Volume (PTV) includes margins

for anatomical and patient setup uncertainties.

Overview: the typical treatment planning process (2/2)

• Possible treatment dose distributions are simulated on a computer with the prescribed doses as goals. This is typically done by ‘trial and error,’ (but optimization engines in use will be extensively discussed.)

• The ‘best’ plan is chosen in consultation with the physician after ‘negotiation’.

Foundations I: physics

• What is the radiation used in radiation oncology?

• What is “dose”? (energy absorbed per unit mass!)

• What is fluence? • How is treatment delivered?• Intensity Modulated Radiation Therapy

(IMRT)

Foundations II: radiation biology principles

• What is a tumor?• Cell kill• Why does radiation therapy work?• Why does radiation therapy fail? (normal

tissue toxicity!)

Foundations III: modeling for optimization

• Tumor control probability models• Normal tissue complication probability

models• Equivalent uniform dose• Surrogate dose-based goals• Multiple objectives

The radiation in radiation oncology

• High energy collimated electron beams are accelerated using electromagnetic fields, and directed onto high atomic number targets. – Electron energies are typically 10-20 million electron

volts (MeV; 1 eV = energy an electron would gain by accelerating through a 1 Volt potential difference in vacuum).

• The incident electrons are deflected by the nuclear electrostatic attraction. Photons (=‘X rays’) are emitted, preferentially in the forward direction

Generation of high energy photons for radiotherapy

Relative fluence vs. angle

Relative dose in water tank

Filter to flatten the fluence profile

An important control variable: fluence

Fluence is defined as Number of crossing particlesDefined surface area

Fluence can be modulated. This is the basis of intensity modulated radiation therapy (IMRT).

(From Roesch & Attix (1968))

From radiation to biological effect (on one slide!)

• High energy photons, through collisions, set fast electrons in motion which

• Kick atomic electrons off molecules which• Lead to chemical reactions, which• Lead to impaired biological function of DNA,• Or DNA is damaged directly by fast electrons,

which • Leads to cell death.

Fluence vs. dose

Dose is the energy absorbed locally per unit mass. The Gray unit (Gy) is used, defined as 1 joule per kg.

Fluence controlsthe dose but is not equal to the dose.

Fluence at depth (Fluence at depth z=0) exp[ ].z zλ= × −

The depth variation of dose along the central beam axis in a wide x-ray field is nearly exponential

Photon beam energies are “x MV” meaning that the energy of the electrons which generated the beam had about x MeV energy.

Radiation Delivery

• Linear accelerators can direct radiation onto the patient from variable directions.

• Fields (or ‘beams’ or ‘ports’) are simultaneously delivered fluence fields defined by banks of movable ‘fingers’ or leaves of dense material, called ‘multileafcollimators.’

Cross-firing fields: a basic principle of treatment planning

1

11

30

11

1

Schematic of patient cross sectionand three incident unmodulated beams

Intensity modulated radiation therapy (IMRT)

• In IMRT, the fluence from a given direction is made to be nonuniform (i.e., modulated).

• This allows the high-dose region to follow the shape of the target volume.

• Numerical optimization is used to design the fluence modulation.

Decomposition into pencil beams or beamlets

Fluence of i’th beamlet, denoted bi

Radiation source

Port or ‘beam’ of 8 beamlets

A beamlet contributes dose mainly along its path, but also contributessome dose to other “voxels” (tissue sub-volumes).

(From: Chui et al., Medical Physics(2001) 28:2441-2449.)

Fluence map example (a map of the bi’s)

Optimization of beamlet fluence weights results in a ‘fluence map’ for each treatment head position

(From: Kung and Chen, Medical Physics (2000) 27:1617-1622.)

Beam’s Eye View of target volume

First delivered field“segment”

Second segment.

An IMRT fluence map is constructed from a superposition of open static fields or “segments”

Radiation biology principles: what is a tumor?

• Tumors are masses of malignant and normal cells (20-50%), typically 108 cells or more.

• All tumor cells must be killed, either directly or indirectly (e.g., nutrient starvation) during therapy in order to local tumor control to be achieved. “Local control” is the usual goal of radiation therapy.

• Local control does not necessarily translate to survival (e.g., there may already be distant metastatic cells which are viable though clinically undetected).

Radiation biology principles: cell kill

• Treatments are typically delivered in 2 Gy“fractions”, daily, for several consecutive weeks (e.g., 60 Gy in 30 fractions in 6 weeks).

• About 2 Gy will kill half the tumor cells, for any given fraction. That is, a surviving cell is thought to be about as viable as an unirradiated cell.

• Hence:• Normal tissue cells recover better from

fractionated radiation than tumor cells, for reasons which are still incompletely known.

Probability of cell survival = exp(- d).α

Prescription implications

• For local control we need to give all tumor cells a high tumoricidal dose (typically 70-80 Gy).

• However, tumors regress (shrink) during therapy (even though tumor cells are also dividing), and it is difficult to predict the impact of a small cold spot on the edge of a tumor.

Normal tissue toxicity

• There are many undesirable effects from either– too great a dose to some normal tissue or– too large an irradiated volume or– radiation to a particularly sensitive structure.

• For example:– Chronic rectal bleeding due to high doses for

prostate cancer treatment– Reduced lung oxygenation capacity from lung

irradiation for lung cancer treatment.– There are many other undesirable endpoints…

Modeling for optimization: radiobiological models

• Tumor control probability (TCP). The probability of local control given the planned dose distribution.

• Normal Tissue Complication Probability (NTCP). The probability of some defined undesirable effect on the patient due to the irradiation.

TCP and NTCP as a function of dose

Holthusen (1936)

The mathematics of curing a tumor: a simplified TCP model

N

no. clonogens

=

TCP = (1-prob. of survival)(1 - exp(- dose))

(1- N exp(- dose)) exp(- N exp(- dose))

αα

α;;

is the radiosensitivity parameter and is cell-line dependent.

α

A TCP model for nonuniform dose distributions

M Voxels

i=1

M Voxels

i=1

VCP, (VCP = voxel control prob.)

= (N M )

TCP =

exp(- exp(- voxel_dose))α

∏

∏

So a low control probability for any voxel means a low overall probability of control.

(Goitein, Webb and Nahum)

TCP model caveats

• Tumor regression during therapy is common, except for slow growing disease (e.g., breast or prostate cancer). This makes direct application of mechanistic models problematic.

• Inter-patient heterogeneity in tumor cell radiosensitivity, numbers of clonogens, and rate of clonogen reproduction makes models less predictive for a particular patient than they otherwise could be.

NTCP: serial vs. parallel endpoints

• Some tissue endpoints are thought to be ‘serial,’ in that failure of any part of the tissue in question leads to a complication. For example: spinal cord radiation injury.

• Parallel tissue endpoints are those for which the tissue may functionally fail, but there is a ‘functional reserve’ which allows a certain volume fraction of the tissue to lose function before a clinical unacceptable endpoint is reached. An example is parotid salivary irradiation.

IMRT treatment planning: it’s all in the objective function

• Typically, the overall objective function will be some linearly weighted combination of objective functions which embody desirable properties concerning the dose distribution.

• Currently, planning systems do not allow prioritization of objectives, although this is highly desirable.

IMRT Treatment planning objective functions (1/10)

• Dosimetric– Minimum dose– Maximum dose – Mean dose– Quadratic deviations in dose– Dose-volume contraints--no more than “x” % of an

outlined structure can exceed “y” dose.• Outcome related

– Equivalent uniform dose (EUD)– TCP--tumor control probability – Minimum cell survival: Ns, the expected number of

clonogens– NTCP--normal tissue control probability


• Dosimetric• Minimum dose

• appropriate as a limit on minimum allowed target dose

• But may not be achievable


• Dosimetric• Maximum dose

• Appropriate to limit tumor hot spots• Or normal tissue doses


• Dosimetric• Mean dose

• Minimizing mean dose could be appropriate for reducing, for example, lung or parotid damage.


• Dosimetric• Quadratic deviations in dose

• Often used as an objective function for tumor dose distributions due to its nice mathematical properties

• But does not adequately penalize cold spots


• Dosimetric• Dose-volume contraints--no more than “x” % of an outlined

structure can exceed “y” dose.• Pro: simple dosimetric interpretation• Pro: can be approximately related to conventional delivery• Con: difficult to get appropriate dose-volume constraints

relating to tissue tolerance from the literature• Con: the biological effect of two dose distributions can vary a

great deal for two dose distributions which obey, say, no more than 50% volume may receive over 20 Gy.

• Compare: [20 Gy, 50 Gy] vs. [20 Gy, 25 Gy]• Multiple dose-volume constraints will typically be

required.


• Equivalent uniform dose (EUD)• Poisson EUD (Niemierko) is biologically based and

is essentially the same as cell survival (discussed later).

• Generalized EUD (Niemierko) is the generalized mean of the dose distribution. Applies to normal tissues or tumors.

• If the GEUD has been determined from 3-D data, it may be a powerful tool for ranking treatment plans.

• However, GEUD for tumors may not penalize a cold spot as heavily as TCP would.

• For tumors: GEUD would be maximized.• For normal tissue endpoints: GEUD would be

minimized or made a maximum value constraint.

Generalized Equivalent Uniform Dose

1/

1

1

GEUD( ; ) ,

GEUD min. dose GEUD max. dose

1 GEUD mean dose0 GEUD geometric mean dose

aNaiN

i

a d

aaaa

=

=

→ −∞ ⇒ →→ ∞ ⇒ →= ⇒ == ⇒ =

∑dv

Known as the “Generalized Mean” in Abramowitz & Stegun, Handbook of Mathematical Equations, Eq.

3.1.14.


• Outcome related• TCP--tumor control probability

• Pro: it’s what we are trying to do!• Pro: significantly penalizes cold-spots• Pro: significantly rewards tumor subvolume boosts • Con: typically doesn’t account for difference

between GTV and CTV and PTV• Con: has nearly a zero slope over much of the

curve, so an optimization routine can lose track of where it is on the curve.

• Con: tumor regression is typical, lessening the effect of cold spots on the edge of dose-distributions and making TCP predictions problematic except for slow growing disease.


• Outcome related• Minimum cell survival: Ns, the expected number of

clonogens• Pro: it’s what we are trying to do!• Pro: significantly penalizes cold-spots• Pro: significantly rewards tumor subvolume boosts• Pro: log(cell survival) is numerically well-behaved

for optimization algorithms• Con: typically doesn’t account for difference

between the GTV, CTV, and PTV.• Again, the original EUD gives the same dose

distribution rankings


• Outcome related• NTCP--normal tissue control probability

• Pro: it is what we are trying to avoid!• Con: validated & highly predictive models are

scarce (nonexistent?)• Con: like TCP, difficult to use in computer

optimization due to sigmoidal shape• Con: we would like to reduce damage well below

NTCP thresholds when possible. • Again, generalized EUD, where fitted to 3-D

clinical data, may be more powerful and more reliable.

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