A Biologically-Based Model for Low-Dose Extrapolation of Cancer Risk

from Ionizing Radiation

Doug Crawford-Brown

School of Public Health

Director, Carolina Environmental Program

What’s our task? Extrapolate downwards in dose and dose-rate

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WLM

tum

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inci

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ce

Having trouble finding the right functional form? No problem. We

have in vitro studies to show us that.

0.00E+00

5.00E-04

1.00E-03

1.50E-03

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3.50E-03

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Dose (Gy)

TF

/S

Cells also die from radiation, so we need to account for that

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1.2

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Dose (Gy)

S(D

)

Just use these to create a phenomenological model

PTSC(D) = αD + βD2

S(D) = e-kD

PT(D) = (αD + βD2) x e-kD

So what’s the big deal? Just fit it!

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in vitro Kd “Fitted” Kd

Why does it not work??

• Model mis-formulation even at lower level of biological organization

• New processes appear at the new level of biological organization (emergent properties)

• Processes disappear at the new level of biological organization

• Incorrect equations governing processes• Parameter values differ at the new level of

biological organization

Why does it not work (continued)??

• Dose distributions different at the new level of biological organization

• Computational problems somewhere• Anatomy, physiology and/or morphometry differ

at the new level of biological organization• Errors in the data provided (exposures,

transformation frequency, probability of cancer, etc)

Then let’s get a generic modeling framework

Exposure conditions

Environmental conditions

Deposition and clearance

Dose distribution

Dose-response

Probability of effect

The environmental, exposure and dosimetry conditions

• In vitro doses are uniform as given by the authors, and at the dose-rates provided

• Rat exposures are from Battelle and Monchaux et al studies, under the conditions indicated by the authors

• Human exposures are from the uranium miner studies in Canada

• Rat and human dosimetry models using Weibel bifurcating morphology

• Uses mean bronchial dose in TB region, or dose distributions throughout the TB region and depth in the epithelium

The multi-stage nature of cancer

Initiation

Promotion

Progression

Cell Death

The state vector model

State 0

State 1S

State 6

State 1NS

State 2 State 3

State 4State 5

k23

k34

P45k56

kRS

kRNS

kRNS

kRS

ks

kNS

kNS

kS

State 7k67

k54

)()()( 111 tNtNtN nss

333432233 NkNkNkNk

dt

dNdmi

)()()()()()( 43210 tNtNtNtNtNtNT

)(

)()( 4

4 tN

tNtf

T

3

The Mathematical Development of the SVMLet Let NNii(t) (t) be the number of cell in State be the number of cell in State ii at any time at any time tt::

• Vector Vector represents the state of the represents the state of the cellular community wherecellular community where

• The total cells in all states is denoted: The total cells in all states is denoted:

• Transformation frequency is calculated by:Transformation frequency is calculated by:

• Six Differential equations describe the movement of cells through Six Differential equations describe the movement of cells through statesstates

Example: Example:

)](),(),(),(),([ 43210 tNtNtNtNtN

And now for some parameter values: chromosomal aberrations

Dose (Gy)

0.0 0.4 0.8 1.2 1.6 2.0

To

tal C

hro

mo

som

e A

ber

rati

on

s p

er C

ell

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1

2

3

4

Rate constants for repair rates and transformation rate constants.

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3.50E-03

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Dose (Gy)

TF

/S

Inactivation rate constants

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1.2

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Dose (Gy)

S(D

)

Then for promotion: removal of contact inhibition

I

D

DD

DD

DShowing: Complete removal of

cell-cell contact

inhibition

6 6)(1)(6

cipcipcip

ci

tptpp

F

So, does this work for x-rays? The in-vitro data on transformation

Pooled data from many experiments for the transformation rate for single () and split (O) doses of

X-rays (Miller et al. 1979)

Model fit to in vitro data

Sensitivity to Pci value

Low dose behavior (no adaptive response)

A

Dose (Gy)

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0Tra

ns

form

atio

n F

req

uen

cy

per

Su

rviv

ing

Ce

ll (x

105)

0

2

4

6

8

10

0.000 0.005 0.0100

1

2

3

4

5

Low dose behavior (with adaptive response)

B

Dose (Gy)

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0Tra

nsf

orm

atio

n F

req

ue

ncy

pe

r S

urv

ivin

g C

ell

(x1

05)

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4

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8

10

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1

2

3

But does it work for in vivo

exposures to high LET radiation with

very inhomogeneous

patterns of irradiation?

Helpful scientific picture from EPA web site

The rat data (Battelle in circles and Monchaux et al in triangles)

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Mean Dose (Gy)

P(D

)

So, does this work for rats??

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Mean Dose (Gy)

P(D

)

Well, not so much……..

With dose variability

PC(D) = ∫ PDF(D) * (αD + βD2) * e-kD dD

Incorporating dose variability

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Mean Dose (Gy)

P(D

)

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Mean Dose (Gy)

P(D

)

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0.45

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Mean Dose (Gy)

P(D

)

GSD = 1, 5, 10

Empirically: lognormal with GSD = 8

Deterministic or stochastic?

State 0

State 1S

State 6

State 1NS

State 2 State 3

State 4State 5

k23

k34

P45k56

kRS

kRNS

kRNS

kRS

ks

kNS

kNS

kS

State 7k67

k54

Deterministic or stochastic?

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Mean Dose (Gy)

P(D

)

Back to the issue of differentiation, Rd/s in the kinetics model

Changes in Rd/s

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P(D

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1, 2, 4

Fits to mining data

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With depth-dose information Without depth-dose information

Inverting the dose-rate effect

________________________________________________________________ Exposure (WLM) Exposure rate (WLM/yr) Lung cancer risk ________________________________________________________________ 2.7* 0.007 200 10 0.006

0.27* 0.030 20 10 0.035

________________________________________________________________ *based on an exposure time of 73 years

Conclusions (continued)

• Good fit to the in vitro data, even at low doses if adaptive response is included (IF you believe the low-dose data!)

• Reasonable fit to rat and human data at low to moderate doses, but only with dose variability folded in

• Best fit with Rd/s included to account for differentiation pattern in vivo

Conclusions

• Under-predicts human epidemiological data at higher levels of exposure

• Under-predicts rat data at higher levels of exposure, especially for Battelle data (not as bad for the Monchaux et al data)

Why did it not work??

• Model mis-formulation even at lower level of biological organization: compensating errors that only became evident at higher levels of biological organization

• New processes appear at the new level of biological organization: clusters of transformed cells needed to escape removal by the immune system

• Processes disappear at the new level of biological organization: cell lines too close to immortalization to be valid at higher levels

• Incorrect equations governing processes: dose-response model assumes independence of steps

• Parameter values differ at the new level of biological organization: not true for cell-killing, but may be true for repair processes

Why does it not work (continued)??

• Dose distributions different at the new level of biological organization: we account for the distributions, but we don’t know the locations of stem cells

• Computational problems somewhere: what exactly are you suggesting here (but perhaps a problem of numerical solutions under stiff conditions)???

• Anatomy, physiology and/or morphometry differ at the new level of biological organization: we think we are accounting for this

• Errors in the data provided: well, not all mistakes are introduced by theoreticians