VALIDATION OF A POROELASTIC MODEL

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1 VALIDATION OF A POROELASTIC MODEL Tobias Ansali 21/06/2012 Abstract This stuies aim, is to verify from a mathematical point of view the valiity of a moel for the infusion of a ug insie a cancer tissue. Avery important parameter has been optimise in the escription of the phiscal fenomenon the hyaulic conuctivity K. The optimal value has been etermine with Lagrange's approach. The function K was consiere as the optimal value that better represent sperimental ata. The results show how the optimise function has a reasonable tenency from a phisycal point of view an furthermore has a singular tenency to the one obtaine in the moel. The moel In this reasearch we hypothize the soli tumor to have a spherical form.the changes (eformations) that the tumor unergoes are to be consiere innitesimal, for that reason the eformations are governe by Hooke's law (elastic el). We stuy the transfer of the therapeuic agent insie the tumor consiering a porelastic meium carachterize by hyaulic conuctivity k of the tissue (given the relationship between inamic viscosity an permeability of the meium) an by Lame's coecient G an λ. The relationship between the therapeutical agent an bloo vessels is governe by starling's law: L p conuttività vascolare [ S V cm mmhg s ] sup erci e vasc olare p er unità i volume[ cm2 cm 3 ] p e pressione vascolare eettiva[mmhg]. p pressione interstiz iale[mmhg]. Ωcan act both as a well or as a source base on the ierence of pressure insie or outsie the bloo vessel. the tumor is by nature strongly heterogeneous, we only consier it in its raial irection, which is expresse by the hyaulic conuctivity of the tumor K. another stuy has observe that conuctivity k is strongly inuence by the eformation of the tumor. The farmaceutical agent is introuce in the center of the tumor, creating a small raius (a) cavity, which its imensions can be compare to the tip of the neele. The general strength acting on the tumor woul be: T tensor of eective stress T σ pi (2) σ tensor of contact stress p interstizial ui pressure (IFP) consiering that the tension eformation of the tumor is governe by hooke's law, throught the equation of the bon we get: T pi + λ( u)i + 2G[ 1 2 (( u) + ( u)t )] (3) where: Ω L p S V (p e p) (1) where ueformation of soli λ 2ν 1 2ν G 1

2 ove ν is the Poisson's coecient assuming we foun ourselves in stationary conitions, transforming the equation in cilinical coorinates, an an remembering that all variables are hypotetically only expresse in accorance of the raial r coorinate, we obtain: (2G + λ) (u + 2u r ) p (4) To be able to n the istribution of the pressure an eformation we nee another equation. We have to consier the conservation of the mass: q Ω (5) Please note that q has the irection an imention of the velocity, But truthfully is not the actual velocity insie the pores, but it refers to the volumetric carrying for unit area. The secon meaning of55 represent, as alreay sai, the relationship between the vascular net an the pharmaceutical agent uring the evolution of this last one. A ui that evolves insie a pore is escribe by Darcy's law: q K p (6) Taking into consieration the 1 an 5 becomes: ( K p) L p S(p e p) (7) Taking into consieration the 1 r 2 (r2 K p ) L S p V (p e p) (8) Regaring the hyaulic conuctivity of the tumor it has been chosen to consier it as epening from the eformation with a semi-empiric exponential law: u M[α K K 0 e +(1 α) u r ] (9) The equations are close by the bounary conitions. p 0 r R p p inf r a u + 2ν u 1 ν u + 2ν 1 ν r 0 r R u r 0 r a where a an R are respectivly the insie an outsie raiuous after the eformation. Linearization of equations The linearization of the equation is mae through a perturbative analysis of the moel which functions in accorance to a characteristic parameter of the problem. p p p inf pe ; T T 2G+λ ; r r R ; u u r ; K K K 0 ; an the equations become: (u + 2u r ) δ p (10) (r2 K p ) r2 γ 2 (p p e) (11) K u M[α e +(1 α) u r ] (12) having γ 2 Lp V R2 e δ p inf p e 2G+λ Its aroun the aimensional value of δ that the linearization is execute an stopping at the rst orer we have: K 0 S u u 0 + u 1δ K K 0 + K 1 δ p p 0 + p 1δ Please note that we are taking into account the hypothesis of small movements an therefore we can express the exponential as the evelopment in series of mclaurin stopping at the rst orer 2

3 until a R R an from 1 to R. p(a) + p(a) (a a) K 1 + M[α u + (1 α)u r ] Now lets move on to express the conitions on the bounaries of points a R an R Rassuming we are moving linearly from point a R For example p(a ) p inf with a a u(a). We can now write our equations in orerδ 0 : c.c. ( u 0 + 2u 0 r ) 0 p 0 ) r 2 γ 2 (p 0 p e) (r 2 K 0 K0 1 + M[α u 0 + (1 α) u 0 r ] p 0 0 r 1 p 0 0 r a R u 0 + 2ν u 0 1 ν r 0 r 1 u 0 + 2ν u 0 1 ν r 0 r a R It can also be emonstrate thatu 0 0 K 0 1. Even p 0 has an analytical answer such as: p 0 p e + A r eγr + B r e γr with A an B known from bounary conitions At the orer δ 1 the equations are: ( u 1 + 2u 1 r ) p 0 c.c. p 0 ) r 2 γ 2 p 1 (r 2 K 0 K1 1 + M[α u 1 + (1 α) u 1 r ] p 1 u p 0 1 r 1 p 1 u p 0 1 r a R u 1 + 2ν u 1 1 ν r 0 r 1 u 1 + 2ν u 1 1 ν r 0 r a R The carrying that goes through the tumor in the non linear moel has value (aimensional): Q 4πr 2 q In the linear case it woul be where Q 4πr 2 (q 0 + δq 1) (13) q 1 K 1 q 0 p 0 p 0 p 1 Optimal K 1 etermination why K 1? It has been ecie to optimize the conuctivity of the hyaulic mean K for essentially two reasons: - The relationship between K an the eformation u is crucial for it's use in the poroelastic theory. It is as a matter of fact the mainly responsible for the interation between ui an pharmaceutical - Both in literature an in our moel it is still not clear how to combine K with the eformation u. This stuy oes not oubt the exponential relationship that goes on between eformation an conuctivity, but wants to verify if the exponential curve that better approximates the experimental values (strictly etermine by a mathematical metho) is qualitatively similar to the one chosen in our moel( etermine with a physical-empiric metho). The experimental ata reporte in the 1were taken from the work of McGuire et al. an for each value of p inf (37,52,69 mmhg)... is calculate Q inf meia (Q I 0.15,Q II 1.4,Q III 0.35). 3

4 - χ 0 (Q I ) π 2 a 4 ( p0 )2 2 p0 + 8Qπa - χ 1 32π 2 p0 ɛr4 + 8Q I πr 2 ɛ - χ 2 32π 2 ɛr 4 ( p0 )2 + 8Q I πr 2 ɛ p0 We can now ene the lagrangiana enition as follows: L(p 1, K 1, α, b, c) J 1 a α ( 2 p p 1 r γ 2 p 1 + ( 2 p 0 r + 2 p 0 )K ( p0 K1 )( )) b (p 1(a) + u 1 (a) p0(a) ) c (p 1 (1) + u 1 (1) p0(1) ) where α, b, c are moltiples of Lagrange. imposing: Figure 1: experimental values Lagrangian approach To make it easier from now on we will omit remembering that all measurements are aimensional From the optimization we have K ott for every value of Q inf. For this reason the size that has to be minimize has been expresse as follows: J (Q(a ) Q I ) 2 + ξ 2 ˆ 1 a K 2 1 (14) The optimization has been one for the linear equations stopping at the 1 orer, furthermore, for reasons tie to the metho of functionality... is expresse as follows Q(a ) ˆ 1 a Q(r)δ D (15) where δ D is a known function elta i Dirac. Taking into account that an remembering that 14 linearization has stoppe on orer 1 the 14 becomes : J χ 0 + where: ˆ 1 a (χ 1 p 1 + χ 2K 1 )δ D + ξ 2 ˆ 1 a K 1 (16) L 0 (17) We obtain the conitions necessary for the resolution of the problem. Therefore we have: L p α 2 α r γ2 α 8πɛ α(a) 0 α(1) 0 b α (a)) c α (1) ((4π p0 r4 +Q I r 2 )δ D ) L K 1 0 K 1 ξ [ 8πɛ ((4π 2 p 0 r 4 + Q I p0 2 r2 )δ D ) + α( 2 p 0 r + 2 p 0 ) p0 2 (α )] L α 0 2 p 1 ( p0 + 2 r K1 )( p 1 γ 2 p 1 + ( 2 p 0 r + 2 p 0 ) 0 L b 0 p 1 (a) u(a) p0(a) L c 0 p 1 (1) u 1 (1) p0(1) please note that: 2 )K Deriving from p 1 we are able to etermine the three multiplicators of lagrange α, b, c - - Deriving in respect to K 1 we obtain K 1,ott 4

5 - eriving in repect to α we obtain the irect - eriving in respect tob e c we obtain the initial conitions of the irect Discretization of the metho The equation system generate from the 15 becomes iscretize by an explicit metho at the secon orer Figure 2: rst case an 6000 points Results As state before, we face the problem separately for each of the three values of the infusion range. Given the obvious impossibility to impose a punctual conition on the function it has become necessary the introuction of a gaussiana function such as f C(θ)e θr2 (elta i Dirac approssimato)....which woul have the eect of blocking the information on all of r given ierent weights for every point. In this way, acting onθ We can choose the strength of our control. Obviously accoring to the choices mae on δ D the calculus gri must be moi- e. To make the calculations easier, the stuy will be execute with a δ D the most ample possible t.c. 1 a δ D which oesn't have to move from the unit for more than a variation of 1.5%. Finally, for a particular case we must emonstrate the convergence of the results at the varying of δ D As long as the gri was chosen accurately. In this case stuy will be shown 3 ierent cases with a δ D ever more force. 1 a δ D ; 1 a δ D ; 1 a δ D 3 1. δ D1 : If we know ecie to keep on using a gri of 6000 points even forδ D3 we woul obtain something like this: Figure 3: thir case an 6000 points It is clear that if we wish to keep on using this gri we woul have to raise the number of points on the gri. The following will show a gri of points: 5

6 In the gure below are reporte the values of the solutions not optimize an not linearize for each of the three experiments (Q I, Q II, Q III ) confronte with the values of the optimization at the change of the parameter ξ: Case 1 (Q I ) Figure 4: fourth case an points Whileδ D2 has the following form (gri of 8000) Figure 5: secon case an 8000 points The optimization is obviously strongly inuence by the freeom that is given to the control. The parameter responsible for such choice is ξ,the more it assumes small values the more the control is free to go its own way espite the energy spent to make all this possible. Figure 6: Q for ierent values of ξ Which enlarge in the point of interest shows the valiity of the optimization: 6

7 Figure 7: zoom Figure 9: zoom The black curve represents the non linear solution the others have a value of ξ respectively of 1, 0.1, 0.05, 0.04, The gure beneath, is in reference to the ierent hyaulic conuctivity of the mean K: Beneath are reporte the values taken by J at the varying of the control parameter ξ: Figure 10: J for ierent values of ξ Figure 8: K for ierent values of ξ Which enlarge in the point of interest: In the en, to verify if the chosen gri is sucient the values are reporte for one of the cases calculate above before, with the 6000 point gri an then with a 1000 point gri 7

8 Figure 11: conversetion Figure 13: The problem has now reache perfect conversion. Case 2 Q II We report in the same orer of the rst case the results obtaine. Figure 12: Figure 14: 8

9 Figure 15: Figure 17: case 3 (Q III ) The optimization in the thir case has taken the following values: Figure 16: Figure 18: 9

10 Figure 19: Convergence The graphic shows the convergence of the compute when you increase the ots on the gri: Figure 20: 10

Lecture XII. where Φ is called the potential function. Let us introduce spherical coordinates defined through the relations

Lecture XII. where Φ is called the potential function. Let us introduce spherical coordinates defined through the relations Lecture XII Abstract We introuce the Laplace equation in spherical coorinates an apply the metho of separation of variables to solve it. This will generate three linear orinary secon orer ifferential equations: