Modelling Drug Delivery and Tumour Growth

Transcription

1 Modelling Drug Delivery and Tumour Growth Matthew Hubbard and Christian Groh, Pamela Jones, Brian Sleeman, Steven Smye, Christopher Twelves (Leeds) Paul Loadman, Nagaraja Periasamy, Roger Phillips (Bradford) Helen Byrne (Oxford) (Partly funded by Yorkshire Cancer Research) University of Greenwich, 11th December 214

2 PART ONE DRUG DELIVERY (TO TUMOURS)

3 Drug Delivery Tumours contain poorly organised and dysfunctional vasculature. Hence, they are poorly perfused with blood. This difference in microenvironment has a profound effect on tumour response to chemotherapy. A major obstacle to effective chemotherapy is inadequate delivery of drug to cells. Can we model the penetration of drug from blood vessels, through the surrounding layers of cells?

4 Drug Delivery Blood Vessels (red): Drug (blue): Hypoxia (green) Primeau, Rendon, Hedley, Lilge, Tannock (25)

5 Drug Delivery Realistic tumour vasculatures are difficult to obtain. They would also be expensive to simulate. Start by considering a single blood vessel and the surrounding layers of cells. This set-up resembles a tumour cord.

6 Tumour Cords BV = Blood Vessels, N = Necrosis, Glut1 Hypoxia

7 Cell Model k 1 k 2 C 1 C 2 C 3 with C 3 C V 1 V 2 extracellular space intracellular space k 2 Transport of drug across cell membrane (k 1 ) Binding and unbinding of drug in the cell (k 2,k 2 ) Potential saturation of binding sites (C ) Interstitial drug diffusion will link the cells (D)

8 Mathematical Model V 1 dc 1 dt V 2 dc 2 dt V 2 dc 3 dt = ak 1 (C 2 C 1 ) = ak 1 (C 1 C 2 ) V 2 k 2 C 2 (C C 3 ) + V 2 k 2 C 3 = V 2 k 2 C 2 (C C 3 ) V 2 k 2 C 3 Biochemical parameters: k 1 k 2 k 2 C Determined from in vitro experiments. Physical parameters: V 1 V 2 a Determined by observation of tissue.

9 Binding Experiments The in vitro experiments considered the action of the drug (Doxorubicin) on cells (DLD-1) in suspension. The initial extracellular drug concentration (C 1 ) was specified. Measurements were taken at later times of: extracellular drug concentration (C 1 ), intracellular drug concentration (C 2 + C 3 ). All the required parameters are identifiable from the data collected.

10 Parameter Sensitivity In fact, the model is extremely insensitive to the transmembrane transport rate (k 1 ) as long as it is fast enough.

11 Timescales The fit to the data suggests two distinct time-scales, which are confirmed by mathematical analysis and can be linked to the literature. 11 Free extracellular drug, C 1 1 Free intracellular drug, C 2 2 Bound intracellular drug, C Numerical simulation Full asymptotic approximation Fast-scale asymptotic approximation Concentration (µm) 7 Concentration (µm) 6 4 Concentration (µm) Numerical simulation Full asymptotic approximation Fast-scale asymptotic approximation 2 Numerical simulation Full asymptotic approximation Fast-scale asymptotic approximation Time (min) Time (min) Time (min)

12 Radially Symmetric Models A n A 2 A 1 A L L V n V 2 V 1 l V l d d Compartment Model Continuum Model

13 Radially Symmetric Models A compartment model. Each cell layer constitutes 3 compartments. Drug is exchanged between layers and compartments within layers. Effectively a finite volume approach with the mesh size dictated by the biological cell size. A continuum model. Reaction-diffusion partial differential equations. Discretised using a spectral approach. Adaptive time-stepping is used (ode15s in Matlab).

14 Compartment Model dc (i) 1 δ 1 V i dt dc (i) 2 δ 2 V i dt dc (i) 3 δ 2 V i dt = A i 1 k (C (i 1) 1 C (i) 1 ) + A i k (C (i+1) 1 C (i) 1 ) + a i k 1 (C (i) 2 C (i) 1 ) = a i k 1 (C (i) 1 C (i) 2 ) δ 2 V i k 2 C (i) 2 (C C (i) 3 ) + δ 2 V i k 2 C (i) 3 = δ 2 V i k 2 C (i) 2 (C C (i) 3 ) δ 2 V i k 2 C (i) 3 k = D/d governs transport between layers. δ 1 and δ 2 are volume fractions (δ 1 + δ 2 = 1).

15 Continuum Model δ 1 C 1 t δ 2 C 2 t δ 2 C 3 t = D ( 2 C 1 r r + αk 1 (C 2 C 1 ) C 1 r ) = αk 1 (C 1 C 2 ) δ 2 [k 2 C 2 (C C 3 ) k 2 C 3 ] = δ 2 [k 2 C 2 (C C 3 ) k 2 C 3 ] Both models have the boundary conditions D C 1 r = k v (C v C 1 r=l ) D C 1 r=l r = r=l

16 Cell-Centre Model This is a multidimensional version of the compartment model. Tumour cord Discrete representation

17 Cell-Centre Model dc (i) 1 δ 1 V i dt dc (i) 2 δ 2 V i dt dc (i) 3 δ 2 V i dt = j N i A ij d ij D ( + a (i) k 1 C (i) 2 C (i) 1 ( ) ( C (j) 1 C (i) 1 + j Vi A ij k v C v (j) ) ( ) = a (i) k 1 C (i) 1 C (i) 2 [ ( ) δ 2 V i k 2 C (i) 2 C C (i) 3 = δ 2 V i [ k 2 C (i) 2 ( ) C C (i) 3 ] k 2 C (i) 3 ] k 2 C (i) 3 ) C (i) 1

18 Numerical Experiments To investigate: the differences between the compartment, continuum and cell-centre models; the effect of the pharmacokinetic profile; the influence of binding affinities.

19 Pharmacokinetic Profiles For a bolus injection the concentration of drug in the vessel (c v (t)) is given by: D τ D τ [ A α ( 1 e αt ) + B ( β 1 e βt ) + C ( γ 1 e γt )] t < τ [ A α (eατ 1) e αt + B ( β e βτ 1 ) ] e βt + C γ (eγτ 1) e γt t τ Robert, Illiadis, Hoerni, Cano, Durand, Lagarde (1982) A simplified form is given by c v (t) = Ae αt. Infusion can be simulated by c v (t) = K. The parameters are chosen to give the same area under curve, i.e. c v dt.

20 Bolus Injection.25 Free extracellular drug, C1 (t = 1h).25 Free extracellular drug, C1 (t = 6h).25 Free extracellular drug, C1 (t = 24h).25 Free extracellular drug, C1 (t = 72h).2 Cell-centre Model Continuum Model Compartment Model.2 Cell-centre Model Continuum Model Compartment Model.2 Cell-centre Model Continuum Model Compartment Model.2 Cell-centre Model Continuum Model Compartment Model Concentration (µm).15.1 Concentration (µm).15.1 Concentration (µm).15.1 Concentration (µm) Distance from vessel centre (µm) Distance from vessel centre (µm) Distance from vessel centre (µm) Distance from vessel centre (µm) 1.5 Bound intracellular drug, C3 (t = 1h) 1.5 Bound intracellular drug, C3 (t = 6h) 1.5 Bound intracellular drug, C3 (t = 24h) 1.5 Bound intracellular drug, C3 (t = 72h) Cell-centre Model Cell-centre Model Cell-centre Model Cell-centre Model Continuum Model Continuum Model Continuum Model Continuum Model Compartment Model Compartment Model Compartment Model Compartment Model Concentration (µm).5 Concentration (µm).5 Concentration (µm).5 Concentration (µm) Distance from vessel centre (µm) Distance from vessel centre (µm) Distance from vessel centre (µm) Distance from vessel centre (µm) Drug distribution: extracellular (top), bound intracellular (bottom). Snapshots at times 1h, 6h, 24h, 72h (left to right).

21 Spatial Variation.2 Free extracellular drug, C 1.2 Free extracellular drug, C 1.2 Free extracellular drug, C t = 1 h t = 6 h t = 24 h t = 72 h t = 1 h t = 6 h t = 24 h t = 72 h t = 1 h t = 6 h t = 24 h t = 72 h Concentration (µm) Concentration (µm) Concentration (µm) Distance from vessel centre (µm) Distance from vessel centre (µm) Distance from vessel centre (µm) 2 Bound intracellular drug, C 3 2 Bound intracellular drug, C 3 2 Bound intracellular drug, C t = 1 h 1.8 t = 1 h 1.8 t = 1 h 1.6 t = 6 h 1.6 t = 6 h 1.6 t = 6 h 1.4 t = 24 h t = 72 h 1.4 t = 24 h t = 72 h 1.4 t = 24 h t = 72 h Concentration (µm) Concentration (µm) Concentration (µm) Distance from vessel centre (µm) Distance from vessel centre (µm) Distance from vessel centre (µm) Drug distribution: extracellular (top), bound intracellular (bottom); for bolus (left), exponential (middle), infusion (right).

22 Temporal Variation.12 Free extracellular drug, C1.12 Free extracellular drug, C1.12 Free extracellular drug, C1 r 26 µm r 26 µm r 26 µm.1 r 18 µm.1 r 18 µm.1 r 18 µm r 19 µm r 19 µm r 19 µm Concentration (µm).6.4 Concentration (µm).6.4 Concentration (µm) Time (h) Time (h) Time (h) 2 Bound intracellular drug, C3 2 Bound intracellular drug, C3 2 Bound intracellular drug, C Concentration (µm) Concentration (µm) Concentration (µm) r 26 µm r 18 µm.4 r 26 µm r 18 µm.4 r 26 µm r 18 µm.2 r 19 µm.2 r 19 µm.2 r 19 µm Time (h) Time (h) Time (h) Drug evolution: extracellular (top), bound intracellular (bottom); for bolus (left), exponential (middle), infusion (right).

23 Exposure 45 Bound intracellular drug, C 3 (t = 24h) 9 Bound intracellular drug, C 3 (t = 72h) 4 tri-exponential mono-exponential 8 35 uniform Exposure (µm h) 25 2 Exposure (µm h) tri-exponential mono-exponential uniform Distance from vessel centre (µm) Distance from vessel centre (µm) Exposure to bound intracellular drug, C 3 dt, after 24h (left) and 72h (right).

24 Varying Binding Affinity 25 Bound intracellular drug, C 3 (t = 72h) Exposure (µm h) k 2 =.4 M 1 s 1 k 2 = 4 M 1 s 1 k 2 = 4 M 1 s Distance from vessel centre (µm) Exposure to bound intracellular drug, C 3 dt, after 72h, for three different binding affinities.

25 Preliminary Observations The two discrete models and the discretised continuum model produce very similar results. The pharmacokinetic profile has less effect on the exposure than expected at long times. The exposure is sensitive to the binding rates. The model is extremely simple. The experimental measurements contain the largest source of variation.

26 The Next Steps Convective transport, validated by transwell experiments and possibly artificial tumour cords. More (parameterised?) processes and cell types. Cell cycle (mitosis, apoptosis, necrosis). Cell response (and a tumour growth model). Drug clearance mechanisms, irreversible binding. With cell movement, realistic vasculature and three dimensions the scientific computing contribution becomes more significant.

27 PART TWO TUMOUR GROWTH

28 Vascular Tumour Characteristics Rapid cell proliferation Imbalance between mitosis and apoptosis Angiogenesis and co-option of vessels Can be spherical or irregular shapes Necrotic regions in tumour core and in avascular areas

29 A Continuum Multiphase Model Based on Breward, Byrne and Lewis (22,23) Phases: cells, extracellular material, blood vessels Incompressible, viscous, fluids with drag between phases Cell-cell interactions included in stress tensors Includes mitosis, apoptosis/necrosis, angiogenesis, vessel occlusion A diffusible nutrient Tumour interacts with external tissue

30 Mass Balance The tissue consists of N p interacting phases, whose volume fractions θ i are governed by mass conservation θ i t + (θ i u i ) = q i i = 1,...,N p. u i are phase velocities q i allow conversion between phases Densities assumed to be constant and equal There are no voids, i.e. N p i=1 θ i = 1 and N p i=1 q i =.

31 A Four-Phase Model Four phases are chosen, to represent healthy cells (α 1 ) tumour cells (α 2 ) extracellular material (β) blood vessels (γ) More could be added, e.g. different tumour cell phenotypes, mature/immature vessels. The q i are chosen to ensure θ i [, 1].

32 Mass Sources For the cell phases: q αj ( ) c = k 1,j α j β c p + c }{{} cell birth ( ) cc1 + c k 2,j α j c c2 + c }{{} cell death c/(cp + c) c 2 Birth rate is half maximum at c = c p k are rate constants c are concentration thresholds (cc1 + c)/(cc2 + c) Necrosis rate is half maximum at c = c c2 Baseline apoptosis rate c

33 Mass Sources For the blood vessel phase: q γ = k 3 γ H(α 1 p α1 + α 2 p α2 p crit,ǫ 3 ) }{{} occlusion ( )( ) β c + k 4 (α 1 + α 2 )γ ε + β (c a + c) }{{ 2 } angiogenesis where H(θ,ǫ) is a smooth switch. For the remaining material: (1 + tanh(p/ǫ3))/2 c/(ca + c) p = p crit p = θ 1 p 1 + θ 2 p q β = q α1 q α2 q γ 1.5 Rate is maximum at c = c a c

34 Momentum Balance Assuming low Reynolds number flow, (θ i σ i ) + F i = i = 1,...,N p. σ i are the phase stress tensors. F i are momentum sources and sinks. Note that mass conservation implies N p i=1 (θ i u i ) =.

35 Constitutive Laws For viscous Newtonian fluids σ i = p i I + µ i ( u i + u T i ) + λ i ( u i )I i = 1,...,N p, though λ i = 2 3 µ i is assumed (local thermodynamic equilibrium). Momentum sources include pressure effects and interphase drag (with coefficients d ij ): F i = p i I θ i + j i d ij θ i θ j ( u j u i ) i = 1,...,N p.

36 Pressure Relations Pressure due to the blood vessel phase is assumed independent of the local flow, i.e. p γ = p γ. Cells act like a fluid with additional cell-cell interactions, i.e. where p α1 = p α2 = p β + Σ α Σ α = max ( ) Λ(α α ) (1 α), 2 with α = α 1 + α 2 and α a natural equilibrium density. max(,(θ θ )/(1 θ) 2 ) θ = θ θ

37 Diffusible Species There are N d additional species whose concentrations c j are governed by D j 2 c j + q j = j = 1,...,N d. They are assumed to evolve much more rapidly than the tissue. They do not contribute to the volume. They provide non-local influence.

38 Diffusion Sources A single species representing nutrient is used here, for which q c = k 5 γ (c v c) }{{} replenishment k 6,1 α 1 c k 6,2 α 2 c }{{} baseline consumption ( ) ( ) c c k 7,1 α 1 β k 7,2 α 2 β c p + c c p + c }{{} consumption due to cell birth It would be simple to include others, e.g. VEGF, ph, glucose.

39 The Whole System θ i t + (θ i u i ) = q i N p i=1 θ i = 1 (θ i [µ i ( u i + u T i ) + λ i ( u i )I]) + j i d ij θ i θ j ( u j u i ) θ i (pi I) = N p (θ i u i ) = i=1 D j 2 c j + q j =

40 Boundary Conditions Mass balance: specify phase volume fractions at inflow. Momentum balance: u β = somewhere on the boundary; zero stress elsewhere. Diffusible species: c n = (zero flux).

41 The Whole System The unknowns are θ i phase volume fractions u i phase velocities p i phase pressures c j diffusible species concentrations Using unstructured triangular meshes in two space dimensions, evolve the phase volume fractions in time then update the other variables.

42 Numerical Approximation Phase volume fractions Hyperbolic conservation laws with sources/sinks Important to remain within [, 1] Cell-centre finite volume scheme on refined mesh Upwind with MUSCL slope limiting Forward Euler time-stepping

43 Numerical Approximation Phase velocities and pressures A form of multicomponent Stokes flow Stable finite elements quadratic velocities, linear pressures System is linear but not (quite) symmetric MUMPS (without the MP) Care is needed when θ i =

44 Numerical Approximation Nutrient concentration Standard reaction-diffusion system Galerkin finite elements with linear elements on refined mesh Newton iteration combined with MUMPS Allows nonlinear reaction terms Jacobian approximated numerically

45 Test Case Circular domain, radius 16 Triangular mesh: 2349 nodes, 4539 elements Gives DoFs in FEM solve Free boundaries: zero normal stress except β Seed healthy tissue in centre with α 2 =.5 cos 2 (πr/2) for r 1 otherwise (animations)

46 Phase Volume Fractions Tumour cells: θ2 at t = 3 Healthy cells: θ1 at t = 3 Cell pressure: p1,p2 at t = Extracellular material: θ4 at t = 3 Blood vessels: θ3 at t = 3 2 ECM pressure: p4 at t = p. 46/5

47 Phase Fluxes Healthy cells: θ 1 u 1 at t = 3 Tumour cells: θ 2 u 2 at t = 3 Extracellular material: θ 4 u 4 at t = 3

48 Varying Diffusion Healthy cells: θ1 at t = 225 Healthy cells: θ1 at t = 525 Healthy cells: θ1 at t = Tumour cells: θ2 at t = Tumour cells: θ2 at t = 8 Tumour cells: θ2 at t = p. 48/5

49 Varying Diffusion Extracellular material: θ4 at t = 225 Extracellular material: θ4 at t = 525 Extracellular material: θ4 at t = Blood vessels: θ3 at t = 225 Blood vessels: θ3 at t = 525 Blood vessels: θ3 at t = p. 49/5

50 Parameter Dependence Rmax(t) Rmax(t) Rmax(t) 6 3 D c =.1 D c =.1 D c =1. D c = Λ=.1 Λ=.1 Λ=.1 Λ=1. Λ= µ=1. µ=1. µ=1. µ= Time, t Time, t Time, t Nutrient diffusion rate (left), cell tension (middle), phase viscosity (right).

51 Front Irregularity Tumour cells: θ 2 at t = Tumour cells: θ 2 at t = Variable phase viscosities: low (left), high (right).

52 Model Behaviour The proliferating rim/necrotic core structure is reproduced. Shapes can be almost circular (in 2D) or irregular. The regularity is most sensitive to the phase viscosities and the cell tension constant. Tumour growth is approximately linear. The growth rate is increased by increasing nutrient diffusion rates, cell birth rates or cell tension, or decreasing phase viscosities or cell death rates.

53 Issues Speed and memory Iterative solvers, preconditioning, multigrid Better efficiency needed for 3D, more phases, etc. Physical/biological realism Measurable model parameters Numerical artefacts, resolution issues Adaptive approaches?

54 Current Developments Multiscale asymptotic analysis is allowing us to scale the effects of the capillary-level drug delivery model up to the scale of a full tumour. The large-scale drug delivery model will be linked with a tumour response model. The response model will investigate the combined effects of chemotherapy and radiotherapy. The ultimate aim is to devise optimal, personalised, treatment strategies.