Effective Treatment Planning with Imperfect Models and Uncertain Biological Parameters

Transcription

1 Effective Treatment Planning with Imperfect Models and Uncertain Biological Parameters ob Stewart, Ph.. Associate Professor and Assistant Head of Health Sciences irector, adiological Health Science Program School of Health Sciences Purdue University Presented at 008 S Annual Meeting (Sept( 1-5, 008) Boston, MA Translating Physics and Biology into the Clinic Tuesday September 3, 008 :30 to 4:30 pm School of Health Sciences

2 Slide /5 From whence the biological parameters? Inter-patient heterogeneity Combined treatments LQ model andom and systematic patient setup errors Analysis Poisson TCP Imperfect and highly non-linear models Intra-tumor heterogeneity osimetry Interpretation of clinical outcomes Biological Parameters Tolerance ose (T T 50 ) α, α/β epair rate (τ)( Cell and tissue kinetics Clinical Outcomes LKB-NTCP model Outcomes VA Semenenko and XA Li, Phys. Med. Biol. 53, (008)

3 Slide 3/5 Biologically Guided Treatment Planning? Temporal Alterations in ose elivery (e.g., external beam vs brachytherapy implants) New treatment technologies Altered dose distributions Altered radiation quality Imperfect and highly non-linear models Uncertain Biological Parameters Tumor Control Probability Clinical Outcome (Better?, Same? Worse?) Fraction: Gy (35 fractions) Current Treatment 0.0 α = 0.15 Gy -1, α/β = 3.1 Gy (5% standard deviation) Total Treatment ose (Gy)

4 Slide 4/5 Equivalent ose to a Tumor Target What dose must be delivered to achieve the same level of biological damage as another treatment? eference Treatment TCP ( ) = TCP ( ) Alternate Treatment ( ρvs ) ( ρvs ) exp ( ) = exp ( ) Poisson TCP model ρ = cell density (# cm -3 ) V = tumor volume (cm 3 ) When considering radiation effects in the same patient, ρ and V may be considered treatment independent constants. S ( ) = S ( ) Two biological parameters (ρ and V) eliminated from modeling process For individual patients, iso-tcp = iso-survival survival

5 Slide 5/5 Equivalent ose derived from the LQ eference Treatment = Alternate Treatment S ( ) = S ( ) exp ( α ) exp( βg = α βg ) α and β (or α/β) ) characterize intrinsic radiation sensitivity G is the dose protraction factor Take logarithm, apply quadratic formula and rearrange terms α / β 4Gln S( ) α / β 4G G = 1+ 1 = G αα ( / β) G ( α/ β) α/ β is the total treatment dose needed to achieve same biological effect e as a reference treatment that delivers total dose etermined by the value of α/β and the dose protraction factor for the reference and alternate treatments (G( and G )

6 Slide 6/5 Lea-Catcheside ose Protraction Factor Instantaneous absorbed dose rate (e.g.,( Gy/h) ) at time t t = () ( )exp{ λ( )} Probability per unit time sub-lethal damage (= SB) is rejoined G dt t dt t t t Absorbed dose (Gy) λ = ln τ epair half-time ose d (fraction size) ) delivered during time interval Δt (fraction delivery time) x g = ( e + x 1) / x x λδ t =Δtln / τ Series of n daily fractions g 1 G = n n if Δt << τ (assumes repair complete between fractions) g is always between 0 (large( delivery time) ) and 1 (short( delivery times)

7 Slide 7/5 Effect of Fraction elivery Time Assume the fraction delivery time for the reference treatment is very short compared to the repair half-time (Δt( << τ) G 1 = n eference Treatment G = g n Alternate Treatment x g = ( e + x 1) / x x =Δtln / τ α / β 4G G = G ( α / β) + α / β n( α / β) 4g = g n( α / β) n ( α / β) fraction delivery time d = n Alternate Fraction Size and d = n eference Physical Parameters: n, n, Δt, (small or no uncertainty) Biological Parameters: α/β and τ (large uncertainty)

8 Slide 8/5 Effect of fraction delivery time equired Boost in Fraction Size (%) = 70 Gy (( Gy/fx) α/β = 1.5 Gy τ = 0.5 h Need to boost dose by 3% τ = h equired Boost in Fraction Size (%) = 37.1 Gy (6.18( Gy/fx) α/β = 1.5 Gy τ = 0.5 h τ = h Fraction delivery time (minute) Fraction delivery time (minute) Keep fraction delivery time < minutes to maximize local tumor control Nominal impact on normal tissues as long as dose small compared to α/β

9 Slide 9/5 Equivalent Fractionation Schedules An equivalent treatment dose can be determined by adjusting the physical parameter n n( α / β) 4g = g n( α / β) n ( α / β) x = ( + 1) / = Δ ln / g e x x x t τ eference Treatment ( clinical experience ) = total dose (Gy) n = number fractions Δt << τ ( short fraction delivery time) d = /n r (fraction size) New Treatment n = desired number fractions Uncertainty in primarily arises from uncertainties associated with α/β. g 1 when Δt << τ

10 Slide 10/5 Equivalent Treatments (local( tumor control) α/β = 1.5 Gy α/β = 1.5 Gy 5.5 Fraction Size (Gy) x 3 Gy 0 x 3 Gy (34 x 3.15 Gy/fx fx) 44 x 1.8 Gy Fraction Size (Gy) Use any point along iso-effect line for the reference treatment 0 x 3 Gy 1.5 = 79. Gy 37 x Gy = 60 Gy Number of Fractions, n Number of Fractions, n 1.5 n( α / β) 4g = g n( α / β) n ( α / β)

11 Slide 11/5 Intra-Tumor and Inter-Patient Heterogeneity n( α / β) 4g = g n( α / β) n ( α / β) Formula premised on the idea that α/β (and τ) are the same for the reference and alternate treatments. Inter-Patient Heterogeneity All patients have a different value for α/β (unknown distribution). BUT same value for α/β is appropriate (as( a first approximation) ) for all treatments in the same patient. Intra-Tumor Heterogeneity Ιndividual tumor cells have different values of α/β (unknown distribution). But, again, same for all treatments epresentative CE-CT CT image of a tumor (Courtesy Minsong Cao, IUSM) How sensitive are estimates of to uncertainties in α/β?

12 Slide 1/5 Sensitivity to α/β Fraction Size (Gy) % CI 44 x 1.8 Gy Fraction Size (Gy) x 3 Gy 1.5 = 79. Gy ( clinical( experience ) = 60 Gy ( clinical( experience ) Number of Fractions, n Number of Fractions, n 10,000 values for α/β sampled from a uniform pdf (range 1 to 10 Gy) 1.5

13 Slide 13/5 Non-Uniform ose istributions Apply formula on a voxel-by by-voxel basis to determine biologically equivalent 3 dose distributions. dose to i th voxel i n( α / β) 4gi, i, = g n( α / β) n ( α / β) 1.8 Gy.94 Gy α/β = 1.5 Gy fractions 0 fractions Extrapolation is non-linear and not overly sensitive to value of α/β

14 Slide 14/5 anking and Comparing 3 istributions Same basic approach can be used to convert 3 dose distributions into an equivalent uniform dose (EU) SEU ( ) = Savg Surviving fraction averaged over target volume 1 S ρv exp α βg ρ V ( ) avg i i i i i i i i α / β 4GlnS avg EU = 1+ 1 G αα ( / β) Niemierko, Med Phys 6, 1100 (1999) Niemierko Med Phys 4(1), (1997) geu N 1 a = i N i = 1 1/ a i α / β 4Gln S( i, ) = 1+ 1 G αα ( / β) voxel-by by-voxel corrections for dose rate, fraction size,

15 Slide 15/5 Effects of adiation Quality exp Outcome Low LET = Outcome High LET S ( ) = S ( ) L ( α ) exp( LL βlgll = αll βlgll) H BE L H H Take logarithm, apply quadratic formula and rearrange terms nh( α / β) H 4gHL α L gll = g n ( α / β) α n ( α / β) H H H H L L Formula has explicit corrections for total dose, fraction size, and dose rate effects the effects of radiation quality are implicit in the biological parameters. α H, (α/β) H, (α/β) L, α L τ H and τ L (expect τ L < τ Η )

16 Slide 16/5 Effects of LET on α and α/β α = θσ+ κz F Σ LET zf 0.04 d θ α / β = + zfσ Σ κ θ and κ are biological parameters that are independent of LET up to ~ 100 kev/μm Σ = number of SB Gy - 1 cell - 1 (estimated using Monte Carlo simulations) Isoeffect calculations with two adjustable parameters (θ( and κ) ) instead of four [(α/β)[ H, (α/β) L, α L, and α H ] effects of LET explicit

17 Slide 17/5 Effect of LET on α and α/β (Human Kidney T1 cells) α θ κz F = Σ+ Σ α θ = + z Σ Σ κ / β F α (Gy -1 ) α α/β (Gy) α 0.6 proton LET (kev/μm) θ = κ = H + 4 He He + θ = κ = LET (kev/μm) proton

18 Slide 18/5 elative Biological Effectiveness (BE) Gy Protons BE = H / L Gy 0.01 Gy α/β = 1.5 Gy (κ = ) α/β = 1.3 Gy (κ = ) CSA ange (mm) cell diameters (9 MeV = 1 mm) (0.55 MeV) Kinetic Energy (MeV) Kinetic Energy (MeV) Predicted BE in distal end (last( mm) ) of Bragg peak > 1.1 (compare to generic clinical BE of ~ 1.1 for the SOBP)

19 Slide 19/5 Isoeffect Calculations A Summary Easy-to to-use method to guide the selection of equivalent fractionation schedules and 3 dose distributions Corrections for fraction delivery time ( dose rate effects ), fraction size, radiation quality, repopulation effects, oxygen effects, Small number of biological parameters (α/β most critical one) Potential to use Monte Carlo NA damage simulations to account for oxygen and LET effects with adjust parameters (θ and κ) instead of (α/β) H, (α/β)( L, α L, α H n( α / β) 4g = g n( α / β) n ( α / β) Need approximate value for (α/β) and τ (to compute g) α = θσ+ κz F Σ α θ β = + z Σ Σ κ / F

20 Slide 0/5 Inter-patient and Intra-tumor tumor Heterogeniety Small changes in a fractionation schedule quite reasonable despite uncertainties in biological parameters Fraction Size (Gy) % CI 44 x 1.8 Gy Fraction Size (Gy) x 3 Gy 1.5 = 79. Gy ( clinical( experience ) = 60 Gy ( clinical( experience ) Number of Fractions, n Number of Fractions, n 1.5

21 Slide 1/5 Treatment Individualization? Outcome eference Patient = Outcome i th (specific)) Patient exp S ( ) = S ( ) ( α ) exp( βg = αii βig i i ) i i Take logarithm, apply quadratic formula and rearrange terms ( α / β) i 4G i α G = G ( α / β) α ( α / β) i i i Analysis of clinical outcomes for patient population: (α/β), α, τ (G ) Use, for example, predictive assays or functional imaging to estimate patient-specific parameters (α/β) i, α i, and/or τ i (G i ) αi {measured quantity} α ( α / β ) i {measured quantity} ( / ) α β

22 Slide /5 Hypothetical Patient Groups An analysis of clinical outcomes suggests that the overall tumor response for a patient population can be characterized by α 1 = 0.15 Gy ( α / β ) = 3 Gy A dose limiting normal tissue (α/β = 4 Gy) can tolerate 15 Gy over 0 fractions. Now imagine that a new technique allows us to separate the patient population into two tumor response groups α1 0.4 α = ± Group 1: Group : ( α / β ) 1 ( α / β) = 0 ± 4 α Group : α = ± ( α / β ) ( α / β) = 0.6 ± 0.3 α 1 = 0.3 ± 0.06 Gy ( α / β ) 1 = 60± 1 Gy 1 α = 0.1 ± 0.03 Gy ( α / β ) = 1.8± 0.9 Gy 1

23 Slide 3/5 Potential for ose Escalation (Groups( 1 and ) Tumor Group 1 (α = 0.3 Gy -1, α/β = 60 Gy) Opportunity for dose escalation Group (α = 0.3 Gy -1, α/β = 60 Gy) Fraction Size (Gy) x 3 Gy Tolerance ose (T) (α/β = 4 Gy) Isoeffect T = 4 T4 Fraction Size (Gy) x 3 Gy T = 4 T x 0.75 Gy 0 x 0.75 Gy Number of Fractions, n Number of Fractions, n Might increase number of fractions for group 1 (n( > 0) Very desirable to decrease number of fractions for group (n( < 0)

24 Slide 4/5 Concluding emarks Isoeffect calculations are the preferred way to tackle many treatment-related related activities Assess, design and compare dose escalation studies Studies from multiple clinics Optimal number of fractions (fraction size and total dose) Assess and correct for deviations from a planned treatment Patient setup errors and organ motion Missed treatment days or interrupted treatments Leverage existing clinical experience with low LET radiation when n introducing new techniques and dose delivery technologies Proton therapy, new brachytherapy sources and procedures Combined treatments (e.g., brachytherapy with an IMT boost) Compare and rank 3 dose distributions (EU) Alterations in the temporal pattern of radiation delivery (fraction delivery time, shortened overall treatment time) Aid in the analysis of clinical outcomes Aid in the implementation of individualized treatments esearch still needed to develop and, especially,, validate models and methods but the future looks promising!

25 Slide 5/5 Thank You Translating Physics and Biology into the Clinic Tuesday September 3, 008 :30 to 4:30 pm X. Allen Li, Medical College of Wisconsin epartment of adiation Oncology Joseph O. easy, Washington University School of Medicine, epartment of adiation Oncology Yue Cao, University of Michigan Health System, epartment of adiation Oncology obert Jeraj, University of Wisconsin Medical School, epartment of Medical Physics ob Stewart, Purdue University, School of Health Sciences