TEMPLATE DESIGN © Optimization of Cancer Radiation Treatment Schedules Jiafen Gong* and Thomas Hillen Department of Mathematical & Statistical Science, University of Alberta 1. Abstract3. Model derivation 2. Introduction 2.1 Cancer 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. Damage to normal tissue limits the radiotherapy! Radiation is fractionated so that normal tissue can have time to recover! Treatment schedule is denoted by. where, T: time between each fraction (fr.), N: #. of frs., d: dose per fr. Tab1: Schedules used in prostate cancer literature. Capital letters label standard schedules, lower case letters correspond to hyper-fractionated schedules. 2.2 Radiotherapy of prostate cancer LabelDose (d Gy) /fraction Days/weekfrs./day Total dose Total days AaAa BbBb CcCc DdDd EeEe Treatment Schedules Surgery, Radiotherapy and Chemotherapy are three major treatment methods.  Cancer caused 30% of the deaths in Canada in Prostate Cancer is the most common cancer found in men in Canada.   Cancer is the uncontrolled growth and spread of cells. Cancer often invades surrounding tissue and can metastasize to distant sites. 3.1 Total Survival Fraction (TSF) is a measure for the percentage of cancer cells that survive the radiotherapy. TSF as cancer cell # before and after i- th radiation, then the cancer Denote Total survival fraction Rad term: for survival fraction, two choices (a) Linear quadratic (LQ) (b) Multi-target (MT) Growth term: three choices Exponential growth: 1. Logistic growth: 2. Gompertzian growth: 3.   whereλis the growth rate, ө is the carrying capacity and is the solution of growth equations. is an empirical measure for radiation damage to normal tissue. CRE Parameters b, c are depending on the type of radiation. We derive a new formula for non-uniform treatment schedule This formula makes sense because less normal tissue is left as radiation goes on. there exists repair between each fraction. 2. f(n) is a decreasing function as c<1, which is consistent with the clinical observations that (2) (1) 3.2 Cumulative Radiation Effect (CRE) 4. Results 4.1 Observation 1: Tumor Growth 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. 4.2 Observation 2: Schedule Ranking Fig 1: Prostate from different views Fig 2: Treatment Method. 3-D conformal radiotherapy (left) and intensity modulated beam radiotherapy (right). 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. Fig 3: Number of tumor cells as function of radiation times 1. A higher ranking means lower total survival fraction (TSF). 5. The hyper-fractionated schedules have lower CRE-value, hence are less harmful to healthy tissue. 3. The relative ranking is identical to the ranking of Yurtseven (2006), where the tumor control probability is used. cell # will change as a sequence Usher (1980) derived a formula for uniform treatment. We derive a formula for non-uniform treatment (various T’s). c=0.65, b=0.11 for X-ray. For example, 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. Reference 4. Zaider M. and,Minerbo G. N., Phy. Med. Bio., 2000, 45: Dawson, A. and Hillen T, Comp. and Math. Meth. In Med., 2006, 7: Yurtseven O., Master thesis in University of Alberta, Ushers J. R., Mathematics Bioscience, 1980, 49: Kirk et al, Clinical Radiation, 1971, 22: We derive two models to measure cancer cell killing (Total Survival Fraction) and quantify normal tissue complication (Cumulative Radiation Effect) in non- uniform cancer radiotherapy. Furthermore, we use these two formulae to optimize realistic cancer radiotherapy treatment schedules. (3) Kirk et al. (1971) consider uniform treatment and use 1. formula (2) and (3) are identical, if is uniform. 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, D 0 is the mean lethal dose per cell. We can see from Fig 4 that the overall ranking is E>e>C>c>D>A>d>B>a>b. 4. The standard schedules removes more cancer cells than the corresponding hyper-fractionated schedules. 2. The higher the dose per fraction, the more efficient is the treatment. 6. Combining both TSF & CRE, schedule a is best which is used in clinic. CRE CRE threshold fraction