Radiation Biophysics - Time, Dose, and Fractionation (TDF)

Time, Dose, and Fractionation (TDF) in Radiobiology

Cell Survival Curves

• A cell survival curve describes the relationship between the surviving fraction of cells and the absorbed dose.
• Surviving fraction is plotted on a logarithmic scale, while dose is plotted on a linear scale.
• The type of radiation influences the shape of the cell survival curve.
• Densely ionizing radiations exhibit a cell survival curve that is almost an exponential function of dose, appearing as a straight line on a log-linear plot.
• Sparsely ionizing radiation curves show an initial slope followed by a shoulder region and then become nearly straight at higher doses.
• As dose increases, survival curves become steeper.
• Cells with a high capacity for repair will have a less steep curve at low doses, resulting in a curvier survival curve.

Linear Quadratic (L-Q) Model

• The linear-quadratic model is often used to describe the cell survival curve: S(D) = e^{-\alpha D - \beta D^2}
  • S(D): Fraction of cells surviving a dose D.
  • \alpha: Constant describing the initial slope of the cell survival curve.
  • \beta: Constant describing the quadratic component of cell killing.
• The ratio \alpha/\beta gives the dose at which the linear and quadratic components of cell killing are equal.
• Single-particle events, where both arms of DNA are damaged simultaneously, are caused primarily by high LET radiation (low energy electrons with high energy photons).
• Linear part of the model: S(D) = e^{-\alpha D}
• Two-particle events involve one arm of DNA damaged as a linear function of dose, D; the probability of damage in an adjacent arm is also a linear function of dose.
• The probability that both arms will be damaged by two different single-particle events is a function of D^2.
• The surviving fraction of cells due to single particle events is given by: S(D) = e^{-\beta D^2}
• Problem with L-Q model: too many unknowns (alpha and beta) for reliable values determined from analysis for clinical data.
• These can be reduced to one parameter by dividing -ln S by \alpha to give the biologically effective dose (BED) equation.
  -ln S = (\alpha D + \beta D^2)
  For n fractions:
  -ln S = n(\alpha d + \beta d^2)
  Hence:
  BED = -\frac{ln S}{\alpha} = nd(1 + \frac{d}{\alpha/\beta})
• Assumptions made for \alpha/\beta:
  • For tumors and acute reactions: \alpha/\beta = 10 Gy
  • For late-reacting normal tissues: \alpha/\beta = 3 Gy
  • For prostate cancer, \alpha/\beta may be as low as 1.5 Gy, and for breast cancer as low as 4 Gy.

The 4 R's of Radiobiology

• The 4 R's are:
  • Repair: favor lower \alpha/\beta or high Dq value (late > early > tumor).
  • Reoxygenation: favor tumor killing.
  • Redistribution (Reassortment): advantage for slow proliferating tissues.
  • Repopulation: tumors and early tissues have advantage.

Acute vs. Late Responding Tissues

• Response to radiation damage is expressed as either acute effect or late effect.
• Acute effects manifest soon after exposure to radiation, characterized by inflammation, oedema, desquamation of epithelia and hematopoietic tissue, and haemorrhage.
• Late effects are delayed and may be caused by absorption of radiation directly in the target tissue or consequential to acute damage in overlying tissue. Examples include fibrosis, atrophy, ulceration, stenosis, or obstruction of the intestine.
• Acute responses occur during or within a short time after RT, such as mucositis, skin reaction, bowel responses, and bone marrow depression. They are more related to total dose and time, less related to fraction size.
• Late responses occur months to years after RT, such as fibrosis of tissue, radiation myelopathy, and renal damage, usually irreversible changes when damage is formed. They are more related to fraction size and total dose, less related to total time.
• The behaviors of tumors are more similar to those of acute responding tissues.

Time Factor

• The time factor can be compensated for acute responding tissue by extra dose.
• Prolonging overall treatment time spares early but not late responding tissues.
• Accelerated repopulation can occur in tumors.

Clinical Evidence of Accelerated Repopulation

1. Tumor regrowth in post-C/T or surgery.
2. Continuous hyperfractionated accelerated radiation therapy (CHART).

Q & A for Time Factor

• Q1: Will prolonging total treatment time reduce late complications?
• A: (No explicit answer in transcript)
• Q2: Will prolonging total treatment increase the possibility of repopulation?
• A: (No explicit answer in transcript, but implied yes)
• Q3: The total treatment time should be as short as possible, but what are the limitations?
• A: (No explicit answer in transcript)

Fractionation Factor

• Q1: 3 Gy/per fx. x 10 = 2 Gy/per fx. x 15 (????)
• A: Biological effects 30 Gy/10 fx > 30 Gy/15 fx.
• Q2: Does fractionation scheme bring same effects to early- and late responding tissue?
• A: (Refer to next slide - Fractionation protects late responding tissues damage more than acute responding tissues)
• Q3: Does acute responding tissue have the same dose-response curve with the late responding tissues?
• A: No!!!
• Fractionation protects late responding tissues more than acute responding tissues.
• Late responding tissues are more sensitive to changes in fraction size, indicated by a smaller \alpha/\beta ratio.

Calculation of Biological Effective Dose (BED)

• E = \alpha D + \beta D^2
• E = n(\alpha d + \beta d^2)
• E = (nd)(\alpha + \beta d) = (\alpha)(nd)(1 + \frac{d}{\alpha/\beta})
• BED = \frac{E}{\alpha} = (nd)(1 + \frac{d}{\alpha/\beta})
• The \alpha and \beta components of mammalian cell killing are equal at approximately:
  • For early effects: \frac{\alpha}{\beta} = 10 Gy
  • For late effects: \frac{\alpha}{\beta} = 3 Gy
• The parameter \alpha/\beta has the units of dose (Gy) and is a measure of the shape of the survival curve.
• The parameter \alpha defines the initial slope of the survival curve; the larger the value of \alpha, the steeper the initial part of the curve.
• The parameter \beta defines the curvature of the survival curve, and a large value of \beta implies more curvature.
• Thus, a large value of \alpha/\beta implies a steep curve with little curvature (i.e., a small shoulder to the survival curve), and a small value of \alpha/\beta implies a shallow curve with greater curvature (i.e., a large shoulder to the survival curve).
• Higher capacity for repair of radiation damage: late-responding normal tissues (low \alpha/\beta values).
• Lower capacity for repair of radiation damage: early-responding normal tissues and most tumors (high \alpha/\beta values).

Model Calculations and Clinical Protocols

• Example Calculation:
  • 35 F x 2 Gy/7 week vs. 70 F x 1.15 Gy/7 week
  • Assumptions: complete repair between fractions
  • Calculate which schedule is more efficient and better in terms of late effects.
• Concomitant Boost:
  • ([30 F x 1.8 Gy] + {12 F x 1.5 Gy]/6 week
  • Early effects:
    E/\alpha = (nd)(1 + \frac{d}{\alpha/\beta}) = 54(1 + \frac{1.8}{10}) + 18(1 + \frac{1.5}{10}) = 84.4 Gy_{10}
  • Late effects:
    E/\alpha = 54(1 + \frac{1.8}{3}) + 18(1 + \frac{1.5}{3}) = 113.4 Gy_3
• CHART Protocol:
  • 36 F x 1.5 Gy/12 days
  • Early effects including tumor:
    E/\alpha = (nd)(1 + \frac{d}{\alpha/\beta}) = 54(1 + \frac{1.5}{10}) = 62.1 Gy_{10}
  • Late effects:
    E/\alpha = 54(1 + \frac{1.5}{3}) = 81.0 Gy_3

Dose Conversion with \alpha/\beta Ratio

• D1(1 + \frac{D1}{\alpha/\beta}) = D2(1 + \frac{D2}{\alpha/\beta})
• \frac{D1}{D2} = \frac{1 + D2/(\alpha/\beta)}{1 + D1/(\alpha/\beta)} = \frac{(\alpha/\beta + D2)}{(\alpha/\beta + D1)}
• Example: A treatment protocol gives 40 Gy/20 Fx., if we change the fraction to 3 Gy, how should we adjust the total dose for acute responding tissues (\alpha/\beta = 10) or late responding tissue (\alpha/\beta = 3)?
  • For acute responding tissue:
    \frac{40}{D2} = \frac{3 + 10}{2 + 10} \rightarrow D2 = 37 Gy
  • For late responding tissue:
    \frac{40}{D2} = \frac{3 + 3}{2 + 3} \rightarrow D2 = 33 Gy

Summary

• The total treatment time should be shortened to prevent repopulation of the tumor, but the tolerance of acute responding tissue is the limitation.
• The reduction of fraction size is to protect normal late responding tissues, but the fraction number is increased.

Conventional Treatment

• 1.8-2 Gy/day
• 5 days/week
• Total dose is usually higher than 6.000 cGy for gross solid tumor.

Hypofractionation

• Dose > 2 Gy/per fraction.
• Palliative:
  • Short term: > 2 Gy/day, usually 3 Gy/day. Total dose is lower than curative dose. Eg. 3 Gy x 10 Fx.
• Curative
  • For organs with parallel functional structure such as lung and liver. If the dose can be conformed to tumor, hypofractionation is considered. Eg. Proton treatment in hepatoma and lung cancer.
  • Stereotactic Radiosurgery: single dose
  • Some Europe countries used large fraction size, their results were also acceptable in some literature. However, large fraction size is less used in the USA for curative attempt.

Hyperfractionation

• Decrease of fraction size – < 1.8 Gy/fraction, usually 1.15-1.6 Gy.
• Increase fraction number. – Usually treat more than 1 fraction/per day.
• Purpose: Spare normal tissue (late complication tissue) when using smaller fraction size.
  • If treated with a similar dose as conventional treatment, complications are expected to decrease.
  • If treated with the increased total dose, tumor control probability is expected to increase but with the same possibility of late complications.

Accelerated Fractionation

• Same fraction size, 1.8 – 2 Gy, same fraction number.
• Shorten total treatment time. –Eg. bid (twice per day), or > 5 fractions/per week
• Purpose: Reduce tumor repopulation during RT.
• Limitation: Severe acute reaction.

Limitations

• “Consequential” late damage.
  • Late damage developed out of the very severe acute effects.
  • Clinical example: skin reactions, G-I reactions
• Incomplete repair between fractions
  • Especially for CNS.

Mixed Type

• CHART (Continuous Hyperfractionated, Accelerated Radiotherapy)
  • 150 cGy/fx. 3 fx./day, W1-7, total dose = 54 Gy.
  • Total treatment time: 12 days.
  • Results:
    • Increased acute toxicity.
    • Late complications: not increased except spinal cord.
    • Similar or slightly better local control for different types of tumor.
  • Implication: Reduce repopulation
• ARCON
  • Accelerated hyperfractionated radiation therapy while breathing carbogen and with the addition of nicotinamide
  • Accelerated to overcome proliferation.
  • Hyperfractionated to spare normal tissues.
  • Carbogen breathing to overcome chronic hypoxia.
  • Nicotinamide to overcome acute hypoxia.

Single-Dose vs. Fractionated Dose

• Threshold dose in radiobiological effects
• Apoptosis of tumor and endothelial cell death
• Cytokine gene expression
• 1 Gy in lung, 7 Gy in brain. “Target-Switching” model (dose-dependent “target”) Nature Medicine, 2005: 11:477
• Accumulated vs. single dose
• Interaction between previous events and subsequent radiation
  • Enhancement? Adaptation?
• Other radiobiological factors: Repopulation, reassortment, re-oxygenation, repair

Tumor Control Probability (TCP)

• Assumption

  • 1 cm3 = 109 tumor cells
  • Survival fraction of 2 Gy = 0.5 = ½
  • No repopulation during treatment, each dose has the same killing effect

• Calculation

  • 2 Gy x 2 = ½ x ½
  • 2 Gy x 3 = ½ x ½ x ½ ……….. 2 Gy x 10 = (1/2)10 = 1/1024 = (10)-3
  • 2 Gy x 30 = (1/2)30 = (10)-9
  • 2 Gy x 31 = (1/2)31 = (10)-9 x ½

• Calculation

  • 1 cm3 = 109 tumor cells
  • After
    • 28 treatment (2 Gy x 28) = 4 cells
    • 29 treatment (2 Gy x 29) = 2 cells
    • 30 treatment (2 Gy x 30) = 1 cell
    • 31 treatment (2 Gy x 31) = 0.5 cell
    • 32 treatment (2 Gy x 32) = 0.25 cell

• Poisson distribution
  Pn = \frac{e^{-a} a^n}{n!}

  • (Pn = the probability of finding n survival cells)
  • (a = expected number)

• Eg: n =1, a=1

  P1 = \frac{e^{-1} 1^1}{1!} = 37\%

  • (P1 =the probability of finding 1 survival cells)
  • (1 = expected number)