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MODELLING THE GROWTH AND TREATMENT
OF TUMOUR CORDS
A. Bertuzzi1, A. Fasano2, A. Gandolfi1, C. Sinisgalli1
1
2
Istituto di Analisi dei Sistemi ed Informatica del CNR
Viale Manzoni 30, 00185 Rome, Italy
Dipartimento di Matematica “U. Dini”, Università degli Studi di Firenze
Viale Morgagni 67/A, 50134 Florence, Italy
RESENSITISATION AFTER TREATMENT
• After a single dose of radiation or drug administered as a bolus, important
changes will occur in the oxygenation and nutritional status of surviving cells,
as well as in the distribution of cells between the proliferating and quiescent
compartment and among the cell cycle phases.
• These changes are transient, because tumour regrowth tends to restore the
pretreatment status.
• As a consequence, changes in the sensitivity of the cell population to a
successive administration of the agent are expected to occur.
AIM
• We have studied the reoxygenation related to cell death in tumour cords and
its effect on the response to split-dose treatment.
• We did not consider the redistribution among cell cycle phases (resensitisation
and resonance effects).
RADIOTHERAPY
Irradiation induces a lethal damage in a fraction of cells (that become clonogenically
dead). Clonogenically dead cells will die at a later time.
We assume that before irradiation all cells are clonogenically viable. After irradiation:
clonogenically viable cells
live cells
clonogenically dead cells
 Kinetics of damage induction and repair (Curtis, 1986; Hlatky et al., 1994).
Two pathways of lethal damage production: direct action and binary misrepair.
where:
N number of clonogenically
. viable cells, U mean number of DNA double strand
breaks (DSB) per cell, D dose rate, w DSB repair rate, k binary misrepair rate.
(No proliferation of viable cells)
The kinetic model explains the linear-quadratic (LQ) model for the surviving fraction
(Thames, 1985):
Surviving fraction
 Single impulsive dose (D)
where
 Split-dose (D/2+ D/2)
where T is the inter-fraction interval. Note: the survival after a split-dose is
larger than the survival after the undivided dose.
The radiosensitivity parameters a and b depend on the oxygenation level of the cells.
The LQ model does not include the effect of reoxygenation and regrowth.
IDEAL VASCULARIZATION
Krogh’s cylinders and tumour cords
Assumption: vessels move
solidly with the tumour tissue.
r radial distance
r0 vessel radius
rN cord radius
B cord (+necrosis)
outer boundary
MAIN ASSUMPTIONS
• Cylindrical symmetry.
• Cell population variables and concentration of chemicals independent of z.
• Cell velocity u radially directed and independent of z.
• No exchange of matter at the boundary r=B(t).
• Oxygen is the only “nutrient” considered. (r,t) denotes its local concentration.
• Cells die if  falls to a threshold N. Additional cell death induced by treatment can
occur in the viable region. Dead cells are degraded to a fluid waste with rate mN.
• Viable cells are subdivided into proliferating (P) and quiescent (Q). nP and nQ denote
the volume fractions occupied locally by P and Q cells. P cells proliferate with rate .
Transition PQ is governed by the rate (), and transition Q  P by  ().
• The volume fraction of extracellular fluid, nE , is constant.
• All components have the same mass density.
BDFG, Bull. Math. Biol., 2003; BFG, SIAM J. Math. Anal., 2004
MODEL EQUATIONS (impulsive irradiation)
 Cell populations
nP, nQ, n†, nN : local volume fractions of (clonogenically) viable P cells, viable Q
cells, lethally damaged cells and dead cells; XP, XQ mean number of DSBs per P,Q
cell;  (r,t) oxygen concentration.
nQ


n†
nP
m


N
nN

*
Since we assume nP+nQ+n†+nN =n * constant, the velocity field u(r,t) is given by
Direct action of radiation will be represented by the initial conditions.
mN
 Treatment
A sequence of impulsive irradiations given with dose Di at time ti, i=1,2…n, t1=0.
where
(Wouters & Brown, 1997)
At t = 0-, nP(r, 0-) = nP0(r), nQ(r, 0-) = nQ0(r) and all the other state variables are zero.
 Oxygen concentration and viable cord boundary
 In the absence of necrosis:
Outer boundary:
 In the presence of surrounding necrosis:
non-material interface
material interface
SINGLE-DOSE RESPONSE
No necrosis
(Crokart et al., 2005)
SPLIT-DOSE RESPONSE
Comparing 1 dose D vs. 2 doses D/2 delivered with interval T.
Time course of viable cells
0
T
SPLIT-DOSE RESPONSE
Comparing D vs. D/2+D/2 by Survival Ratio
t= 0
0
T
ClV1 volume of clonogenically
viable cells after 1 dose D
ClV2 the same quantity in the
case of 2 doses D/2
Survival Ratio =
min ClV2
min ClV1
Closed symbols: SR predicted by the model (no necrosis). Apparent radiosensitivity at t = 0:
a = 0.304 and b = 0.054. D = 8Gy.
Open symbols: SR predicted with a and b independent of  and equal to a and b.
Split-dose response
Change in inter-vessel distance
Change of repair rate
B=80mm
w=2
90mm
0.5
0.25
100mm
necrosis
Till & McCulloch, 1963
ADVANTAGE OF FRACTIONATION
• Dose splitting increases the survival of irradiated cells by sparing the
normal tissue, but reducing the tumour cell killing.
• Since normal tissues are characterized by different a and b values,
conditions can be found in which the sparing of normal tissue exceeds the
reduction of tumour cells killing.
• Therefore, higher total doses can be delivered, obtaining an ultimate gain in
tumour mass reduction.
• The effect of reoxygenation on radiosensitivity is likely to be greater in the
tumour than in the normal tissue, owing to the lower mean oxygenation
level in tumours. This fact further enhances the therapeutic advantage of
dose fractionation.
• A more accurate assessment of reoxygenation dynamics might help in
optimizing this advantage.
THERAPEUTIC INDEX
Split-dose treatment
Extended LQ model
Sn*
I =
St *
Tumour cord model
Sn* minimal survival acceptable for normal tissue. For each T, let D* be the
dose producing the survival Sn*
St* survival of tumour cells after the dose D*
Normal tissue: a = 0.2 Gy-1, b = 0.067 Gy-2
Tumour: a = 0.5 Gy-1, b = 0.05 Gy-2 (left); aM = 0.6 Gy-1, bM = 0.075 Gy-2 (right)
CONCLUSIONS I
• The present analysis confirms that the reoxygenation occurring after the
first radiation dose can substantially affect the efficacy of a successive
dose.
• The extent of cell resensitisation appears to be related in a complex way
to the parameters characterizing the vasculature as well as to the intrinsic
radiosensitivity of cells.
• A more complex model would be needed to describe the phase-specific
effects of radiation and the dynamics of cell redistribution among the
cell-cycle phases. Moreover, the transient changes in perfusion might be
included.
Other applications of the cord model
• A similar approach has been followed to analyse the split-dose response
to a cycle-specific cytotoxic drug, highlighting the role of the possible
cell recruitment from quiescence to proliferation (BFGS, J. Theor. Biol.,
2007).
• To better investigate the transport of drugs, the extracellular fluid motion
and the interstitial pressure have been included in the cord model (BFGS,
Math. Biosci. Engng., 2005; BFG, Math. Mod. Meth. Appl. Sci., 2005).
• The diffusive and convective transport of monoclonal antibodies and the
binding to cell membrane antigens have been modelled (BFGS, Springer,
2007).
ANTIBODY TRANSPORT IN A STEADY-STATE CORD
Monoclonal antibodies specific to tumour antigens and conjugated to radio-nuclides
have been proposed in cancer therapy. We consider the transport and binding of inert
bivalent antibodies (Ab) in a cord at steady state (surrounded by necrosis).
Denoting by the symbol (^) the concentrations on the cell surface, the transport
equations will be given in terms of the variables:
where b1(r,t) and b2(r,t) are the concentrations (referred to the extracellular volume) of
monovalently and bivalently bound Abs; S is the total antigen (Ag) concentration; a*
is the area of cellular surface per unit volume. The concentration of extracellular free
Ab will be denoted by c(r,t).
Free and bound Ag in extracellular fluid
are disregarded.
BFGS, Springer, 2007
b1
b2
cell

Mass balance of drug, r ( r0 , rN )
with
Boundary conditions at r = r0:
where Pe is given by
and D is the interstitial diffusivity, P the vessel permeability, f the retardation factor,
f the filtration reflection coefficient and cb(t) the Ab concentration in blood.

In the necrotic region, we define:
where aN is the area of cellular surface per unit volume in N, taken as aN /a*= nN /n*.
We have:
Boundary conditions at r=rN
Boundary condition at r=B
Distribution of free (blue) and bound (red) Ab following i.v. bolus
- Low convection case high
low Ab affinity
Distribution of free (blue) and bound (red) Ab following i.v. bolus.
- High convection case high
low Ab affinity
CONCLUSIONS II
• The proposed model, that includes the interstitial fluid motion and the
dynamics of necrotic region, provides an improved description of the transport
within the cord of agents of high molecular weight, such as monoclonal
antibodies. The model extends the work by Fujimori et al. (1989).
• The simulation results evidenced that the convective transport may
significantly contribute to raise the bound Ab concentration at the cord
periphery.
• In the case of low convection, the high binding to cell surface antigens results
in a “barrier” to Ab penetration, and generates a more heterogeneous Ab
distribution.