MODELLING THE GROWTH AND TREATMENT

OF TUMOUR CORDS

A. BertuzziA. Bertuzzi11, A. Fasano, A. Fasano22, A. Gandolfi, A. Gandolfi11, C. Sinisgalli, C. Sinisgalli11

1 Istituto di Analisi dei Sistemi ed Informatica del CNR

Viale Manzoni 30, 00185 Rome, Italy

2 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:

live cellsclonogenically dead cells

clonogenically viable 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, dose rate, DSB repair rate, k binary misrepair rate. (No proliferation of viable cells)

.D

The kinetic model explains the linear-quadratic (LQ) model for the surviving fraction (Thames, 1985):

Single impulsive dose (D)

where

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 and depend on the oxygenation level of the cells.

Surviving fraction

Split-dose (D/2+ D/2)

The LQ model does not include the effect of reoxygenation and regrowth.

IDEAL VASCULARIZATIONIDEAL VASCULARIZATIONKrogh’s cylinders and tumour cordsKrogh’s cylinders and tumour cords

r radial distance

r0 vessel radius

N cord radius

B cord (+necrosis)

outer boundary

Assumption: vessels move solidly with the tumour tissue.

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 N.

• Viable cells are subdivided into proliferating (P) and quiescent (Q). P and Q denote the volume fractions occupied locally by P and Q cells. P cells proliferate with rate . Transition PQ is governed by the rate (), and transition Q P by ().

• The volume fraction of extracellular fluid, E , 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 populationsCell populations

Since we assume P+Q+†+N =* constant, the velocity field u(r,t) is given by

P, Q, †, N : 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.

Q

P

†

N

N

*

N

Direct action of radiation will be represented by the initial conditions.

TreatmentTreatmentA sequence of impulsive irradiations given with dose Di at time ti, i=1,2…n, t1=0.

where

At t = 0-, P(r, 0-) = P0(r), Q(r, 0-) = Q0

(r) and all the other state variables are zero.

(Wouters & Brown, 1997)

Oxygen concentration and viable cord boundaryOxygen concentration and viable cord boundary

non-material interface material interface

Outer boundary:

In the absence of necrosis:

In the presence of surrounding necrosis:

SINGLE-DOSE RESPONSESINGLE-DOSE RESPONSENo necrosisNo necrosis

(Crokart et al., 2005)

SPLIT-DOSE RESPONSEComparing 11 dose D vs. 22 doses D/2 delivered with interval T.

Time course of viable cells

T

0

SPLIT-DOSE RESPONSESPLIT-DOSE RESPONSE

Survival Ratio =Survival Ratio =min min ClVClV22

min min ClVClV11

ClVClV11 volume of clonogenically viable cells after 11 dose DD

ClVClV22 the same quantity in the case of 22 doses DD/2/2

Comparing Comparing DD vs.vs. DD/2+/2+DD/2/2 by Survival Ratio by Survival Ratio

Closed symbols: SR predicted by the model (no necrosis). Apparent radiosensitivity at t = 0: = 0.304 and = 0.054. D = 8Gy.

Open symbols: SR predicted with and independent of and equal to and

0

T

t = 0

Split-dose responseSplit-dose response

Change in inter-vessel distanceChange in inter-vessel distance Change of repair rateChange of repair rate

Till & McCulloch, 1963

B=80m

90m

100m

necrosis

=2

0.5

0.25

ADVANTAGE OF FRACTIONATIONADVANTAGE 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 and 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 INDEXTHERAPEUTIC INDEXSplit-dose treatment

I I ==SSnn

**

SSt t **

SSnn** minimal survival acceptable for normal tissue. For each T, let D* be the

dose producing the survival Sn*

SStt** survival of tumour cells after the dose D*

Tumour cord model

Extended LQ model

Tumour: = 0.5 Gy-1, = 0.05 Gy-2 (left); = 0.6 Gy-1, = 0.075 Gy-2 (right)

Normal tissue: = 0.2 Gy-1, = 0.067 Gy-2

CONCLUSIONS ICONCLUSIONS 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).

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; * 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.

Denoting by the symbol (^) the concentrations on the cell surface, the transport equations will be given in terms of the variables:

cell

b1 b2

BFGS, Springer, 2007

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.

Mass balance of drug, Mass balance of drug, r r ( ( rr0 0 , , NN ))

In the necrotic region, we define:In the necrotic region, we define:

where N is the area of cellular surface per unit volume in N, taken as N /*= N /*.

We have:

Boundary conditions at rN

Boundary condition at r=B

Distribution of free (Distribution of free (blueblue) and bound () and bound (redred) Ab following i.v. bolus) Ab following i.v. bolus- Low convection case -- Low convection case -

low Ab affinityhigh

Distribution of free (Distribution of free (blueblue) and bound () and bound (redred) Ab following i.v. bolus.) Ab following i.v. bolus.- High convection case -- High convection case -

low Ab affinityhigh

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.