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Optimization of Cancer Radiation Treatment Schedules
Jiafen Gong* and Thomas Hillen
Department of Mathematical & Statistical Science, University of Alberta
*[email protected]
1. Abstract
We derive two models to measure cancer cell killing (Total Survival Fraction)
and quantify normal tissue complication (Cumulative Radiation Effect) in nonuniform cancer radiotherapy. Furthermore, we use these two formulae to
optimize realistic cancer radiotherapy treatment schedules.
2. Introduction
3. Model derivation
4. Results
3.1 Total Survival Fraction (TSF)
4.1 Observation 1: Tumor Growth
TSF is a measure for the percentage of cancer cells that survive the radiotherapy.
Denote M i , M i' as cancer cell # before and after i- th radiation, then the cancer
cell # will change as a sequence
Rad ( M  )
Growth ( M  )
Rad ( M  )
M 1 
 M 1' 

M


2
T1
2.1 Cancer



Cancer is the uncontrolled growth and spread of cells. Cancer often invades
surrounding tissue and can metastasize to distant sites.
Cancer caused 30% of the deaths in Canada in 2005. Prostate Cancer is the
most common cancer found in men in Canada.
Surgery, Radiotherapy and Chemotherapy are three major treatment methods.
2.2 Radiotherapy of prostate cancer
Growth ( M  )
Rad ( M  )
'


M


M
N
N
TN 1
Total survival fraction
M N'
TSF (Ti , N , d ) 
, i  1,
M1
N  1.
(1)
Usher (1980) derived a formula for uniform treatment Ti  T . We derive
a formula for non-uniform treatment (various T’s).
 Rad term: for survival fraction, two choices

'
 (d )  e
(a) Linear quadratic (LQ)

M

i   (d ) M i
d / D0 n

 (d )  1  (1  e
)
(b) Multi-target (MT)
Parameter α/ β in (a) is the measure of radio-sensitivity, the higher α/ β, the more
sensitive the cells are. The value for prostate cancer is α/ β≈1 Gy-1. n in (b) is the
number of the cancer cells, D0 is the mean lethal dose per cell.
 d   d 2

Fig 1: Prostate from different views
When we irradiate prostate, then always part of bladder and rectum will be
affected. To minimize complication of the normal tissue, 3-dimension
conformal radiotherapy or intensity modulated beam radiotherapy are used.
Growth term: three choices
1. Exponential growth:
dM (t )
 M
dt
2. Logistic growth:
dM (t )
M
  M (1  )
dt

dM (t )
M
  M log( )
dt

3. Gompertzian growth:
whereλis the growth rate , ө is the carrying capacity and G(Tn )is the solution of
growth equations.
CRE is an empirical measure for radiation damage to normal tissue.
N cd
CREN  b
T
A
2
1
5
72
41
a
1
2
B
2
1
6
70
47
b
1
2
C
3
1
5
72
32
c
1.5
2
D
2.4
1
5
72
40
d
1.2
2
E
4
1
5
72
24
e
2
2
Tab1: Schedules used in prostate cancer literature. Capital letters label standard schedules,
lower case letters correspond to hyper-fractionated schedules.
Treatment schedule is denoted by (T , N , d ) .
where, T: time between each fraction (fr.), N: #. of frs., d: dose per fr.
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f (n)  n  (n  1) .
c
c
1. http://www.prostate-cancer-radiotherapy.org.uk/standard_ebrt.htm.
2. Kirk et al, Clinical Radiation, 1971, 22: 145-155.
3. Ushers J. R., Mathematics Bioscience, 1980, 49:157-189.
4. Zaider M. and ,Minerbo G. N., Phy. Med. Bio., 2000, 45: 279-293.
5. Dawson, A. and Hillen T, Comp. and Math. Meth. In Med., 2006, 7: 121-142.
6. Yurtseven O., Master thesis in University of Alberta, 2006.
1. A higher ranking means lower total survival fraction (TSF).
2. The higher the dose per fraction, the more efficient is the treatment.
3. The relative ranking is identical to the ranking of Yurtseven (2006),
where the tumor control probability is used.
4. The standard schedules removes more cancer cells than the
corresponding hyper-fractionated schedules .
(3)
This formula makes sense because
1. formula (2) and (3) are identical, if Ti  T is uniform.
2. f(n) is a decreasing function as c<1, which is consistent with the
clinical observations that
• less normal tissue is left as radiation goes on.
• there exists repair between each fraction.
Reference
Fig 4: ln(TSF) as function of time for the schedules from Tab 1, with
exponential (left) and logisitic (right) re-growth. As labeled in the graph, each
line corresponds to the TSF for one treatment schedule.
CRE
Damage to normal tissue limits the radiotherapy! Radiation is fractionated so
that normal tissue can have time to recover!
Label
Dose (d Gy)
Total
Days/week
frs./day
Total days
/fraction
dose
(2)
Parameters b, c are depending on the type of radiation.
For example, c=0.65 , b=0.11 for X-ray.
We derive a new formula for non-uniform treatment schedule
d
CRE1  b ,
T1
d
CREn  CREn 1  f (n) b ,
Tn
4.2 Observation 2: Schedule Ranking
We can see from Fig 4 that the overall ranking is
E>e>C>c>D>A>d>B>a>b.
Kirk et al. (1971) consider uniform treatment Ti  T and use
2.3 Treatment Schedules
1. As most radiation is given when tumor is small, Gompertzian growth is
problematic for a study of initial stage tumor radiation treatment since
the growth near 0 is unbounded.
2. Exponential growth gives similar results as logistic growth.


'

M

M

n 1
n G (Tn ),


3.2 Cumulative Radiation Effect (CRE)
Fig 2: Treatment Method. 3-D conformal radiotherapy (left) and
intensity modulated beam radiotherapy (right).
Fig 3: Number of tumor cells as function of radiation times
CRE threshold
fraction
Fig 5: CRE for the ten schedules of Tab 1 as function of the number of fractions
(c=0.65). The left figure is CRE for each schedule, the right figure is the CRE
difference between standard and hyper-fractionated schedule.
5. The hyper-fractionated schedules have lower CRE-value, hence are
less harmful to healthy tissue.
6. Combining both TSF & CRE, schedule a is best which is used in clinic.