Cimpa School. Santiago de Cuba. June 13 th, 2016

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1 Cimpa School Santiago de Cuba June 13 th, 2016

2 On mathematical models for Permeabilization of cells in an Electric field

3 Otared Kavian Département de Mathématiques Université de Versailles 45, avenue des Etats Unis Versailles cedex (France)

4 Joint work with colleagues from Inria & université de Bordeaux: Michael Leguèbe Clair Poignard Lisl Weynans Classical Electropermeabilization Modeling at the Cell Scale published in Journal of Mathematical Biology, December 2012, pp. 1 31, DOI: ANR project MEMOVE, including other collaborators Frédéric de Gournay (Toulouse) Lluis Mir & Aude Silve (CNRS, Institut Gustave Roussy, Villejuif) laboratoire Ampère (Ecole Centrale de Lyon)

5 Today s talk On Cells and Membranes What is electropermeabilization? Mathematical questions A molecular dynamics approach A model for the membrane Schwan s model A static model A dynamic model

6 On Cells and Membranes

7 On Cells and Membranes A cell consists of a membrane surrounding the cytoplasm. A typical, and maybe the simplest, example is a vesicle: here the green head + the yellow tail represent a phospholipid

position of the membrane On Cells and Membranes The cell membrane forms a barrier in order to protect the cytoplasm from the outside aggressions filter the ion exchanges between the inside and the

8 position of the membrane On Cells and Membranes The cell membrane forms a barrier in order to protect the cytoplasm from the outside aggressions filter the ion exchanges between the inside and the outside The membrane is a barrier protecting the cytoplasm against intrusions, and filters the ion exchange between the inside and the outside. Composition It is composed of a thin phospholipidic bilayer, with a few transmembrane proteins a thin phospholipidic bilayer a few nanometer thick few transmembrane proteins, ionic channels and gap junctions 5nm Phospholipids Transmembrane proteins 3 / 36

9 On Cells and Membranes A phospholipid has a «fatty» tail, which is hydrophobic, and a hydrophilic head which is an electric dipole. This dipole may rotate around the axis of the tail, to some extent. Relative sizes are as follows: Cell s diameter nm Membrane thickness 5 nm Distance between two cells 100 nm Distance between two phospholipids 1 nm A molecule of water 0.1 nm

10 On Cells and Membranes A typical example of a phospholipid is (image from

11 On Cells and Membranes The schematic scales involved can be depicted as follows: (image from

12 What is electropermeabilization?

13 What is electropermeabilization? File:Pore schematic.svg - Wikipedia, the free encyclopedia It has been observed (R. Stampfli 1958, E. Neumann & K. Rosenheck 1972) that the cell sfile:pore membraneschematic.svg reacts in a special manner to an intense electric field and its permeability changes. From Wikipedia, the free encyclopedia File Before any electric field: no molecule may traverse across the bilayer File history File links 03/04/09 11:39 In particular for a cell membrane this means that no external molecules may enter the cell.

14 What is electropermeabilization? After an electric field (here parallel to the tails) is imposed: some molecules may traverse the bilayer, as if a «pore» had appeared Pore_schematic.svg! (SVG file, nominally 578 " 538 pixels, file size: 4.13 MB) For a cell membrane this means that some external molecules do enter the cell. This is a file from the Wikimedia Commons. The description on its description page there is shown below. Commons is a freely licensed media file repository. You can help. Such a phenomenon may be of interest in order to deliver molecules of drugs or genes into the cell. Summary Description English: Diagram of possible lipid arrangements at the edge of a pore through a lipid bilayer. (above) In the absence of any re-arrangement the pore walls will be hydrophobic since the alkane tails are exposed. (below) Some researchers believe that the lipids at the edge re-orient to line the pore wall with hydrophilic head

15 Appendix Future directions The electric model of the cell before electroporation Models of the electroporation process he pore creation: an appealing theory What is electropermeabilization? he pore creation: an appealing theory As a matter of fact, one may imagine two kinds of «pores» Hydrophobic pore Hydrophilic pore Figure: Figure: (a) (a) hydrophobic hydrophobic pore pore (b) hydrophilic (b) hydrophilic pore pore However it is not known whether there is creation of «pores», or rather there is a «permeabilization» process under the intense electrical field / 36

16 What is electropermeabilization? This is a very important phenomenon, since through the pore molecules may enter the cell. Based on this, a new treatment of cancer tumors has been set up by several teams around the world (among others Lluis Mir in France, Julie Gehl in Denmark): electrochemotherapy A drug, such as bleomycin, is injected to the patient, locally or intravenous. Within 8 to 12 minutes one creates around the tumor a strong electric field, with several short pulses (from microsecond to nanosecond durations). It is expected that the membranes of the cells in the tumor will be more permeable, and thus the drug and some molecules of bleomycin would enter into such cells

17 What is electropermeabilization? The electroporation of a single vesicule Electroporation in vivo Appendix A rough description The electric model of the cell before electroporation Models of the electroporation process Future directions The process of electropermeabilization, or electroporation, is assumed to be as follows The electroporation process (a) Initial configuration (b) High short pulses (c) Interruption of the pulses 4 / 36

18 What is electropermeabilization? The electroporation process at the cell scale: a rough description The electric model of the cell before electroporation Models of the electroporation process Future directions The electrochemotherapy is treatment developed by Lluis Mir at the I.G.R and which has been validated at the european level. Principle of electrochemotherapy (according to Damian Miklavčič) Figure: Principle of the electrochemotherapy (by D.Miklavcic, Ljubljana). 2 / 45

apparatus Models of the electroporation used

19 What is electropermeabilization? he electroporation process at the cell scale: a rough description The electric model of the cell before electroporation The typical apparatus Models of the electroporation used is as follows process (according to Lluis Mir) Future directions

20 What is electropermeabilization? Electropchemotherapy is much more efficient than the usual chemotherapy alone The treatment of a tumor is local and thus has much less side effects The patient has to undergo a lesser number of treatment Electropermeabilization can be reversible or not: the irreversible phenomenon can be used for the destruction of cells, and it has also other applications (killing germs, improving extraction of juice from fruits and vegetables)

21 What is electropermeabilization? An example of dramatic results may be seen on the chest of this patient (according to Lluis Mir and Julie Gehl) Before treatment After 5 days After 2 weeks After 8 weeks

22 What is electropermeabilization? However there are a few drawbacks As of today deep tumors cannot be treated, although there are several experimentations undertaken around the world Due to the high voltage of the electric field (in the range of 5 25 kv/cm) some patients cannot undergo the treatment (pacemakers, anticoagulant therapy) The process of the electropermeabilization, or electroporation, is not yet well understood, thus the dosage may not be well calibrated

23 Mathematical questions

24 Mathematical questions Understand, via a mathematical model, the electropermeabilization phenomenon for a cell Experiments show that the electropermeabilization is reversible: the permeability of the membrane disappears after a certain time once the electric field is turned off: one would like to estimate the amount of time during which the membrane remains «permeable» Understand how to go from one cell to a cluster of cells which makeup a tumor Set up a model for electropermeabilization so that the electrochemotherapy can be coupled with models in which the growth of tumors can be predicted

25 Mathematical questions Use a mathematical model for the elctropermeabilization in order to optimize the placement of the electrodes Once an affordable model is set up, and the «direct» problem is solved, study the «inverse» problem, that is the determination of characteristics of the cell membrane

26 A molecular dynamics approach

27 A molecular dynamics approach One considers N molecules indexed by j, located at x j (t), having mass m j and charge q j for 1 j N Then one writes the equilibrium of the forces acting on each molecule, assuming that there is an electric field E: = m i d 2 x i dt 2 j i q i q j x i x j 3(x i x j ) + q i E(t)... and one solves numerically these equations...

28 Tieleman results. Applied fields at t = 0: 0.5 A molecular dynamics approach For instance here is what one may see, according to D.P. Tieleman 2004

29 A model for the membrane

30 Conjecture A model for the membrane Step 1: Applied voltage (high short pulses). Step 2: Orientation of the hydrophilic heads. Step 3: Destructuration of the membrane. However one may conjecture that as a matter of fact, what happens is something like That is first the dipoles (heads) get oriented and then there is a partial collapse of the phospholipids. There is no «pore», but rather a sharp decrease of the thickness, which is enough for the passage of molecules. 20 / 36

31 A model for the membrane

32 A model for the membrane We denote by n the normal to the mean surface of the membrane, and by h ± the height of the dipole above or below this surface h ± = ±n l ± + max(0, p ± n) Then h := h + + h is the thickness of the membrane. The dipoles p ± tend to rotate when submitted to an electric field. We assume that they satisfy a Landau-Lifschitz-Gilbert equation, namely: p ± = α 1 p ± ( ) v Γ± + G ± α2 p ± p ± t t p ± (0) = c ± G ±

33 A model for the membrane The electric potential v is a harmonic function in the exterior of the cell, as well as in the cell. The boundary conditions are v c σ c n = σ v e e n v e t v c t = α 3 h(t, x) v0 c n on Γ on Γ where v 0 c is a potential obtained by an asypmtotic analysis, making the thickness δ tend to zero. See the domain. Existence and uniqueness of the solution is established by reducing the system to a new system written on Γ (using Steklov-Poincaré operators). Numerical simulations made by Frédéric de Gournay show that there is «permeabilization» in some spots, this behavior is not symmetric, and the membrane returns to its initial position once the electric field is turned off.

34 Schwan s model

35 Schwan s model The cytoplasm is homogeneous and conducting The membrane is homogeneous and very insulating (σ m [10 7, 10 5 ] S/m, while σ copper 10 7 S/m, σ seawater 5 S/m, σ air S/m). Ω m Ω i lh σ i = 1S/m ε i = 10 2 ε 0 σ m 10 5 to 10 7 S/m ε m 10ε 0 h 10 3 to l 2µm

36 Schwan s model The electric potential v satisfies a time dependent quasistatic equation div(ε v) + div(σ v) = 0 in (0, ) Ω t (1) v(0, x) = 0 in Ω. v(t, x) = v imp on (0, ) Ω Or a quasi-static equation in time harmonic regime: setting ε := ε + iσ/ω, then v satisfies (2) { div( ε v) = 0 in Ω v(t, x) = v imp on Ω

37 Schwan s model The different subdomains involved are as follows: Ω m Ω ε e ε i ε m µ e µm Ω i µ i Ω Γ δ Γ Ω e Ω

38 A static model

39 A static model After obtaining an asymptotic model as h 0 we end up with a nonlinear PDE on the domain Ω ε e µ e ε i µ i Ω Ω i Γ Ω e Ω

40 A static model We consider a positive function λ S m (λ), such that for some δ > 0, (7.1) S m (λ) S L for λ < (1 δ)v, S m (λ) S ep for λ > (1 + δ)v (S is for Siemens, the unit for the electric conductance, which is also 1/Ohm) Typically S m (λ) := S L + (S irr S L )β(λ), where β is a function such as (7.2) β(λ) := [1 + tanh(k ep ( λ V rev ))]/2 We assume that the potential V satisfies (see the domain). V = 0 in Ω e Ω i [σ n V] = 0 on Γ (7.3) S m ([V] Γ ) [V] Γ = σ i n V Γ on Γ V = g on Ω. with [V] := V Γ + V Γ.

41 A static model In order to write (7.3) as an equation on the boundary Γ we introduce the following Steklov-Poincaré operators: Λ i is defined on H 1/2 (Γ) when Γ is considered to be Γ := Ω i Λ e is defined on H 1/2 (Γ) when Γ is considered to be Γ := Ω e \ Ω (see the domain) Finally Λ 0 is defined as a mapping H 1/2 ( Ω) H 1/2 (Γ) These operators are (pseudo-diffrential operators) of order 1 and are given by: Λ i ( f ) := n i v, with v i Γ i = 0 in Ω i, v i = f on Γ Λ e ( f ) := n e v e Γ +, with v e = 0 in Ω e, v e = f on Γ +, v e = 0 on Ω. Λ 0 ( f ) := n e v, with v e Γ + e = 0 in Ω e, v e = 0 on Γ +, v e = f on Ω.

42 A static model The equation (7.3) above can be written as follows: Let u e := V Γ + and u i := V Γ, then u H := H1/2 (Γ) H 1/2 (Γ) is given by (7.4) { σe Λ e u e + S m (u e u i )(u e u i ) = σ e Λ 0 (g), σ i Λ i u i S m (u e u i )(u e u i ) = 0. Upon setting F(s) := s 0 S m(z)zdz and (7.5) Λu := ( ) σe Λ e u e σ i Λ i u i One shows that the solution of (7.4) is obtained as the minimum of (7.6) J(u) := 1 2 Λu, u + F(u e (τ) u i (τ))dτ. Γ

43 A dynamic model

44 A dynamic model We consider a positive function (t, λ) S m (t, λ) defined by, (8.1) Sm (t, λ) := S L + (S irr S L )X(t, λ). Here, for some X 0 [0, 1] given and β defined in (7.2), X(t, λ) satisfies ( ) X(t, λ) β(λ) X(t, λ) β(λ) X(t, λ) (8.2) = max,, X(0, λ) = X 0 t τ ep We assume that the potential V satisfies (see the domain). V = 0 in (0, T) (Ω e Ω i ) [σ n V] = 0 on Γ [V] (8.3) Γ C m + t S m (t, [V] Γ ) [V] Γ = σ i n V Γ on (0, T) Γ V = g on (0, T) Ω V(0, x) = V 0 (x) in Ω e Ω i. τ res

45 A dynamic model We write the above equation as follows: First we show that Λ e + Λ i is self-adjoint, positive and invertible. Then we set B := I + Λ 1 e Λ i, and v := u i u e := V Γ V Γ + One checks that u i = V Γ = B 1 (v Λ 1 e Λ 0 g) And setting G := Λ i B 1 Λ 1 e Λ 0 g, ϕ := (V 0 ) Γ (V 0 ) Γ +

46 A dynamic model Then v satisfies { Cm t v + Λ i B 1 v + S m (t, v)v = G on (0, T) Γ v(0, ) = ϕ( ) on Γ. Here S m (t, v) = S L + (S ir S L )X(t, v) and X satisfies ( X β(v) X t = max ; β(v) X ) τ ep τ res X(0, v) = X 0 [0, 1]. One proves that these equations have a unique solution.

47 A dynamic model Other phenomena can also be taken into account, upon setting { Sm (t, v) = S L + S 1 X 1 (t, v) + S 2 X 2 (t, v) P m (t, v) = P L + P 1 X 1 (t, v) + P 2 X 2 (t, v) Here P m is the permeability of the membrane, and X := (X 1, X 2 ) satisfies an equation of the type t X = F(v(t, X), X). Some numerical simulations by Michael Leguèbe are obtained, and may be compared quite satisfactorily with the experiments done by Aude Silve.

Dynamical modeling of tissue electroporation

Dynamical modeling of tissue electroporation Damien Voyer, Aude Silve, Lluis M. Mir, Riccardo Scorretti, Clair Poignard To cite this version: Damien Voyer, Aude Silve, Lluis M. Mir, Riccardo Scorretti,