Stochastic Modeling of Radiation Damage, Repair and Survival of Cells

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Stochastic Modeling of Radiation Damage, Repair
and Survival of Cells

• A.Rangan, Department of Mathematics, IIT Chennai,
  India
• rangan_at_iitm.ac.in

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1.Introduction

• Basic unit of structure
• Nucleus of cell-DNA-regulates growth and
  replication
• Radiation damage of DNA
• Lesion formation repair process-probabilistic
• Low dose experiments

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Biological Process

• Phenomenon of irradiation
• Charged particles loses energy owing to
  collisions
• Causes ionization and excitation
• LET (Kev/µm)
• Low LET lt 10 Kev/µm UV radiation, X, ? rays
• High LET gt10 Kev/µm ionizing radiation

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Cont..

• Radiation damage - cell nucleus
• Lesions base damage, SSB, DSB, dimers,
  crosslinks
• Enzymatic repair mechanism
• PLL survival correct repair
• Lethal, mutant misrepair
• Lethal lesion inability to divide
• Phases of cell cycle
• Quiescent state of cell

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2. A Stochastic for Model Radiation Effects on
Cell Survival

• Lesion formation Poisson process
• Lesion formation and repair birth and death
  process
• Neyman and Puri (1981), Tobias (1985), Janssen
  (1987), Albright (1989), Yang and Swenberg (1991)
• Repair, Misrepair models parameters dose
  rate, lesion induction, repair, misrepair and
  lethality rates

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Cont

• UV radiation induces dimers cross linking
  between adjacent pyramidine bases on the same DNA
  strand
• Not an absolute block for DNA replication
• DNA replication proceeds past dimers not
  considered in literature - leads to non linearity
  in PGF equations

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Cont..

• Shoulder effect concavity of log survival curve
• Holding time
• Two stage process

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3.The Model

• Total dose D for T units at constant rate ?
• (0,T) irradiation interval
• (T, TTh) holding time
• Poisson arrival of primary particles with rate
  A?, where A is cell volume, ?0 for tgtT
• Lesion formation radiation particle induces PLL
  in (t, t?) with probability A? ?o(?)

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Cont..

• Radiation particle induces LL with probability
  A? ?
• Particle has no effect with probability
  A?1-( ) ?o(?)
• Repair misrepair mechanism
• X(t), Y(t), Z(t) No. of PLL, no. of
  transformed lesions,no. of LL

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Conditional probability of correct repair

• Conditional probability of misrepair ßk? o(?)
• Conditional probability of a damaged DNA
  replicating leading to additional lesion
  ?k?o(?)
• Conditional probability of cell death µk?
  o(?)
• Cell death unrelated to radiation d?o(?)

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• X(t), Y(t), Z(t)- cont time vector Markov
  Process
• Define the PGF

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The backward equations are
The solution is given by
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• The number distribution of PLL

• The cell survival probability S(t)

• To conform to experimental protocol D/?T

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• Thus the survival probability exhibits a shoulder

• In our Model the survival probability at the end
  of the holding period is

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4.Comparison with other Models

• Fitted the survival fraction of yeast cells
  saccharomoyces cerevisiae of Frankenberg-Schwager
  (1980)
• Janeson (1987) and Albright (1989) delinked
  repair process from the lesion induction process
  and made a Markov formulation. Our formulation
  lends itself to a two stage process naturally.

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Cont..

• Chadwick and Leenhouts (1988) LQ Model with

Where p is the probability of DSB leading to cell
death

• We expanded S(D) as a power series to obtain

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Cont..

• We can identify k1 and k2 as functions of cell
  death and lesion induction only which compares
  with Chadwick Leenhouts, LQ Models where pa and
  pß have similar interpretations.

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5.Conclusions

• Analyses and obtains explicitly survival
  probabilities of cells subject to UV radiation
• Replication parameter is a novelty
• Log cell survival shows shoulder
• Number distribution of PLL which forms the core
  of cell studies is obtained
• Brief comparison with other existing models
• L-Q approximation for cell survival probability
  as a function of dose is obtained

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Cont..

• Gives sufficiently good fit for the survival data
  of EAT and CHO cells irradiated with different
  doses of UV and X-rays (Illiakis and Nusse
  (1982), Metting et al (1985))
• Versatile enough to incorporate high LET effects
• Incorporation of holding times comes naturally
• Possible to deal with three stage process,
  including cloning period

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Other Models1.Cell repair based on enzyme
kinetics

• Repair process- B D process
• Repair systems which alter or remove PLL get
  saturated at higher doses
• Saturable repair models explain the usual
  database of radio biological phenomenon including
  some not explained by common bio physical models
• Saturable repair models Powers (1962)

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Cont..

• Existence of a pool of chemical compounds
  protect target molecules hydrogen donation to
  free radicals
• With increasing dose pool depleted rendering
  cell more radio sensitive
• Laurie et al (1972), Calkins (1971), Goodhead
  (1985), Braby and Nelson (1981)

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Cont..

• Report experimental work or deterministic models
  based on classical Michalies Menton enzyme
  kinetics
• Process highly stochastic no theoretical
  framework

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2. Threshold Models

• Accumulation of DNA damage in somatic cells
  basic mechanism of aging
• DNA lesions accumulate interfering in
  transcription and replication functional
  repairement
• Threshold value compensative capability of cell
  cell death

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3.Structured Stochastic Models of Cell
Progression through their Cycles

• Cell growth Gompertzs growth curve
• Initial period of rapid growth at exponential
  rate slows down finally ceases
• Retardation of cell growth quiescent state and
  gradual elongation of different phases of cell
  cycle with aging

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• Experiments with holding times and split doses

?1?
?2?
µ2
µ3
?3?
µ1
p?2
?1
q?2
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4. On the Parity of Cells in Tumor Growth

• Parity no. of off springs of individuals in the
  populations
• Cells in culture limited no. of doubling before
  dying
• Build up of abnormal molecular ways that
  accumulate
• Daughter cells from same mother may receive
  different amounts of errors asymmetric
  partition
• Correlation between cells reflected by parity is
  important

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• 5.OPTIMAL STOPPING IN
• CANCER CHEMOTHERAPY

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6.Optimal Drug Scheduling in Cancer Chemotherapy

• Drug toxicity to the host and drug resistance in
  neo plasia major impediments
• Cell cycle phase specific drugs (Anti HIV drug,
  Zidovudine, anti cancer drug Arabinoside)
  detrimental during S phase no effect in other
  phases
• Bring bone marrow depression and other cyto toxic
  effects
• Results of Zvia Agur on population dynamics in
  perturbed environments

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Cont..

• Drug selectivity increased by manipulating
  dosing interval so that resonance is created for
  host cells minimizing their mortality lack of
  resonance for cancer cells
• Effect achieved by interval drug dosing dosing
  interval being an integral multiple of the
  average drug induced intermitotic interval of the
  susceptible host cells
• Simulation, theoretical results using continued
  fractions for optimal no. of doses.