Evolution of spherical aneurysms

Transcription

1 Outlines Evolution of spherical aneurysms Stress-driven remodeling and control laws A. Tatone July 13, 2007

2 Outlines People involved in this project A. Di Carlo a, V. Sansalone b, V. Varano a, A. Tatone c a Università degli Studi Roma Tre, Roma b Université Paris XII Val-de-Marne, France c Università degli Studi dell Aquila, L Aquila

3 Outlines Background references A. Di Carlo, S. Quiligotti, Growth and balance, Mechanics Research Communications, 29, , A. Di Carlo, Surface and bulk growth unified, Mechanics of Material Forces (P. Steinmann & G.A. Maugin, eds.), 53 64, Springer, New York, NY, M. Tringelová, P. Nardinocchi, L. Teresi and A. DiCarlo, The cardiovascular system as a smart system, in Topics on Mathematics for Smart Systems, eds. V. Valente and B. Miara (World Scientific, Singapore, 2007).

4 Outline of Part I Outlines Part I: Aneurysms Part II: Mechanical Model Part III: Numerical Simulations 1 A possible natural history for saccular aneurysms A recent survey in Japan Histology Remarks

5 Outline of Part II Outlines Part I: Aneurysms Part II: Mechanical Model Part III: Numerical Simulations 2 General framework Kinematics Working & balance Energetics & constitutive issues 3 Saccular aneurysms Geometry & kinematics Working & balance Energetics & constitutive issues

6 Outline of Part III Outlines Part I: Aneurysms Part II: Mechanical Model Part III: Numerical Simulations 4 Growth driven by mean hoop stress value a single case different cases compared 5 a first case high hoop resistance higher hoop gain 6 case one

7 A possible natural history for saccular aneurysms Part I Aneurysms 1 A possible natural history for saccular aneurysms A recent survey in Japan Histology Remarks

8 A possible natural history for saccular aneurysms A recent survey in Japan Histology Remarks A possible natural history for saccular aneurysms Intracranial saccular aneurysms are dilatations of the arterial wall. [J.D.Humphrey, Cardiovascular Solid Mechanics, 2001]

9 A possible natural history for saccular aneurysms A recent survey in Japan Histology Remarks A possible natural history for saccular aneurysms An initial insult may cause a local weakening of the wall and thus a mild dilatation. This raises the local stress field above normal values, thus setting into motion a growth and remodeling process that attempts to reduce the stress toward values that are homeostatic for the parent vessel. If degradation and deposition of collagen are well balanced, this could produce a larger, but stable lesion. If degradation exceedes deposition at any time, this could yield a

10 A possible natural history for saccular aneurysms A recent survey in Japan A recent survey in Japan Histology Remarks EBM of Neurosurgical Disease in Japan Case No. Sex Age Location of aneurysm Table 2 Summary of patients with ruptured aneurysms Motive for examination Past history 6 months Size of aneurysm (mm) 12 months 24 months At rupture Treatment 1 F 53 single rt ICA ex. for intracranial hypertension embolization disease (27 months) 2 F 71 multiple AcomA ex. for intracranial hypertension, clipping disease pituitary adenoma (7 months) 3 F 77 multiple lt MCA ex. for anxiety hypertension, heart disease dead (7 months) 4 F 42 multiple lt MCA ex. for anxiety hypertension clipping (18 months) AcomA: anterior communicating artery, ex.: examination, ICA: internal carotid artery, MCA: middle cerebral artery. Fig. 1 Process of growth and rupture of aneurysms. Type 1: aneurysm ruptures within a time span as short as several days to several months after formation, Type 2: aneurysm builds up slowly for a few years after formation and ruptures in this process, Type 3: aneurysm keepsgrowing slowly for many years without rupturing, Type 4: aneurysm grows up to acertain size, probably under 5 mm in diameter, and thereafter remains unchanged. A. Tatone SAH: subarachnoid Evolution hemorrhage. of spherical aneurysms

11 A. Tatone 358 Evolution aneurysms of spherical aneurysms (94.2%) in Type 4. The sum 1 multiple AcomA ex. for intracranial hypertension, A recent survey 4.9 in Japan clip A possible natural history disease for saccular aneurysms pituitary adenoma Histology (7 7 multiple lt MCA ex. for anxiety hypertension, Remarks dea heart disease (7 2 multiple lt MCA ex. for anxiety hypertension clip A recent survey in Japan (1 erior communicating artery, ex.: examination, ICA: internal carotid artery, MCA: middle cerebral artery. s of growth and rupture of aneurysms. Type 1: aneurysm ruptures within a time span as short as several days s after formation, Type 2: aneurysm builds up slowly for a few years after formation and ruptures in this proces sm keeps growing slowly for many years without rupturing, Type 4: aneurysm grows up to a certain size, probab diameter, and thereafter remains unchanged. SAH: subarachnoid hemorrhage.

12 A possible natural history for saccular aneurysms A recent survey in Japan Histology Remarks Remodeling of Saccular Cerebral Artery Aneurysm Wall Is Associated With Rupture: Histological Analysis of 24 Unruptured and 42 Ruptured Cases Juhana Frösen, Anna Piippo, Anders Paetau, Marko Kangasniemi, Mika Niemelä, Juha Hernesniemi and Juha Jääskeläinen Stroke 2004;35; ; originally published online Aug 19, 2004; DOI: /01.STR da Stroke is published by the American Heart Association Greenville Avenue, Dallas, TX Copyright 2004 American Heart Association. All rights reserved. Print ISSN: Online ISSN:

13 A recent survey in Japan Histology Remarks A possible natural history for saccular aneurysms 2290 Stroke October 2004

14 A possible natural history for saccular aneurysms A recent survey in Japan Histology Remarks Before rupture, the wall of saccular cerebral artery aneurysms undergoes morphological changes associated with remodeling of the aneurysm wall. Some of these changes, like SMC [ smooth muscle cell ] proliferation and macrophage infiltration, likely reflect ongoing repair attempts that could be enhanced with pharmacological therapy.

15 A possible natural history for saccular aneurysms A recent survey in Japan Histology Remarks The morphological changes that result from the MH [ myointimal hyperplasia ] and matrix destruction are collectively referred to as remodeling of the vascular wall. Although MH is an adaptation mechanism of arteries to hemodynamic stress, in SAH [ subarachnoid hemorrhage ] patients, for undefined reasons, vascular wall remodeling [ is ] insufficient to prevent SCAA [ saccular cerebral artery aneurysm ] rupture.

16 General framework Saccular aneurysms Part II Mechanical Model 2 General framework Kinematics Working & balance Energetics & constitutive issues 3 Saccular aneurysms Geometry & kinematics Working & balance Energetics & constitutive issues

17 General framework Saccular aneurysms Kinematics Working & balance Energetics & constitutive issues Kinematics: gross and refined motions gross placement body gradient p : B E p b : T b B VE element-wise configuration (prototype) P b : TB b VE,

18 General framework Saccular aneurysms Kinematics Working & balance Energetics & constitutive issues Kinematics: gross and refined motions refined motion (p(τ), P(τ)) : B E (VE VE ) realized velocity ( ṗ(τ), Ṗ(τ)P(τ) 1 ) : B VE (VE VE ). test velocities (v, V ) : B VE (VE VE ) (gross velocity, growth velocity)

19 Working General framework Saccular aneurysms Kinematics Working & balance Energetics & constitutive issues total working ( ) A i V S Dv B + ( b v + A o V ) + t B v B B (integrals taken with respect to prototypal volume and prototypal area) prototypal gradient Dv := ( v)p 1

20 Balance principle General framework Saccular aneurysms Kinematics Working & balance Energetics & constitutive issues B ( ) A i V S Dv balance of brute forces + ( b v + A o V ) + t B v = 0 B B Div S + b = 0 on B S n B = t B on B balance of accretive couples A i + A o = 0 on B

21 Energetics General framework Saccular aneurysms Kinematics Working & balance Energetics & constitutive issues free energy Ψ(P)= ψ P (ψ free energy per unit prototypal volume)

22 General framework Saccular aneurysms Kinematics Working & balance Energetics & constitutive issues Constitutive issues: theory and recipes The constitutive theory of inner forces rests on two main pillars, altogether independent of balance: the principle of material indifference to change in observer the dissipation principle Both of them deliver strict selection rules on admissible constitutive recipes for the inner force. None of them applies to the outer force, which has to be regarded as an adjustable control on the motion.

23 General framework Saccular aneurysms Kinematics Working & balance Energetics & constitutive issues Constitutive issues: theory and recipes B ( ) A i V S Dv + ( b v + A o V ) + t B v = 0 B B In our theory of the biomechanics of growth, the outer accretive couple A o plays a primary role, representing the mechanical effects of the biochemical control system, smartly distributed within the body itself: ignoring the chemical degrees of freedom does not make negligible their feedback on mechanics.

24 General framework Saccular aneurysms Kinematics Working & balance Energetics & constitutive issues Material indifference to change in observer Change in observer p(b, τ) = x o (τ) + Q(τ) ( p (b, τ) x o (τ)) P(b, τ) = P(b, τ) ṽ(b) = Q(τ) v(b) + w(τ) + W(τ) ( p (b, τ) x o (τ)) Ṽ(b) = V(b)

25 General framework Saccular aneurysms Kinematics Working & balance Energetics & constitutive issues Material indifference to change in observer The working expended over each body-part on each test velocity (v, V ) by the inner force constitutively related to each refined motion (p, P) should be invariant under all change in observer. The free energy Ψ should be constitutively prescribed in such a way as to be invariant under all change in observer.

26 General framework Saccular aneurysms Kinematics Working & balance Energetics & constitutive issues Material indifference to change in observer brute Cauchy stress T := (det F) 1 S F T = T warp F := Dp = ( p) P 1 (measures how the body gradient of the gross placement differs from the prototypal stance)

27 General framework Saccular aneurysms Kinematics Working & balance Energetics & constitutive issues Material indifference to change in observer If we further assume that the response of the body element at b filters off from (p, P) all information other than p b, p b, P b we obtain the following reduction theorem: there are constitutive mappings Ŝb, Âi b and ψ b such that S(b, τ) = R(b, τ) Ŝb(l b, τ) A i (b, τ) = Âi b (l b, τ) ψ(b, τ) = ψ b (l b, τ) l b := ( U b, P b ) F = R U

28 Dissipation principle General framework Saccular aneurysms Kinematics Working & balance Energetics & constitutive issues ( S (D ṗ) Ai ( ṖP 1 ) ) ( ψ + ψ I ( ṖP 1 ) ) 0 power expended : {working expended by the inner force constitutively related to the motion on the velocity realized along the motion} power dissipated : {power expended along a refined motion} {time derivative of the free energy along that motion} Dissipation principle : the power dissipated should be non-negative, for all body-parts, at all times

29 General framework Saccular aneurysms Time derivative of the free energy Kinematics Working & balance Energetics & constitutive issues ( Ψ(P) = = = P P P ) ψ ω = (ψ ω) P ( ψ ω + ψ ω) ( ψ + ψ I ( ṖP 1 )) ω. (the prototypal-volume form ω evolves in time with ṖP 1 )

30 General framework Saccular aneurysms Kinematics Working & balance Energetics & constitutive issues Constitutive assumptions: free energy and inner force ψ(b, τ)= ψ b ( U b, P b, τ) = φ b (F(b, τ)) The dissipation principle is fulfilled if and only if for each b the mappings Ŝ and Âi satisfy Ŝ = φ + + S, Â i = E + + A together with the reduced dissipation inequality + A(l, τ) ( Ṗ(τ)P(τ) 1 ) + S(l, τ) Ḟ(τ) 0 Eshelby couple extra-energetic responses E := F Ŝ φ I + S, + A

31 Saccular aneurysms General framework Saccular aneurysms Geometry & kinematics Working & balance Energetics & constitutive issues

32 General framework Saccular aneurysms Geometry & kinematics Geometry & kinematics Working & balance Energetics & constitutive issues paragon shape D of B centered at x o E spherical coordinates ( ξ(x), ϑ(x), ϕ(x)) radius of x ξ(x) = x x o gross placement p : D E x x o + ρ ( ξ(x)) e r ( ϑ(x), ϕ(x)) actual radius ρ : [ ξ, ξ + ] R

33 General framework Saccular aneurysms Geometry & kinematics Geometry & kinematics Working & balance Energetics & constitutive issues spherically symmetric vector fields v : D VE radial component of v v : [ ξ, ξ + ] R v(x) = v(ξ) e r (ϑ, ϕ).

34 General framework Saccular aneurysms Geometry & kinematics Geometry & kinematics Working & balance Energetics & constitutive issues spherically symmetric tensor fields L ε : D VE VE L ε (x) = L r (ξ) P r (ϑ, ϕ) + L h (ξ) P h (ϑ, ϕ) orthogonal projectors P r (x) := e r (ϑ, ϕ) e r (ϑ, ϕ) P h (x) := I P r (x)

35 Refined motion General framework Saccular aneurysms Geometry & kinematics Working & balance Energetics & constitutive issues gradient of the gross placement prototype p x = ρ (ξ) P r (ϑ, ϕ) + ρ(ξ) ξ P h (ϑ, ϕ), P(x, τ) = α r (ξ, τ) P r (ϑ, ϕ) + α h (ξ, τ) P h (ϑ, ϕ). warp (Kröner-Lee decomposition) F := ( p) P 1 = λ r P r + λ h P h λ r (ξ, τ) = ρ (ξ, τ) α r (ξ, τ), λ h(ξ, τ) = ρ(ξ, τ) ξα h (ξ, τ).

36 Refined motion General framework Saccular aneurysms Geometry & kinematics Working & balance Energetics & constitutive issues gross velocity and growth velocity ṗ(x, τ) = ρ(ξ, τ) e r (ϑ, ϕ) Ṗ P 1 (x, τ) = α r(ξ, τ) α r (ξ, τ) P r(ϑ, ϕ) + α h(ξ, τ) α h (ξ, τ) P h(ϑ, ϕ)

37 General framework Saccular aneurysms Geometry & kinematics Working & balance Energetics & constitutive issues Dynamics: brute and accretive forces; balance principle working D ( ) A i V S v + o A V + t D v, D D (integrals taken with respect to the paragon volume and paragon area) ξ+ ξ ( ) ( ) A r V r +2 A h V h S r v 2 S h v/ξ 4 π ξ 2 dξ + 4 π ξ 2 ξ t v A := A i + A o = A r P r + A h P h.

38 Balance laws General framework Saccular aneurysms Geometry & kinematics Working & balance Energetics & constitutive issues 2 ( S r (ξ) S h (ξ) ) + ξ S r(ξ) = 0 A r (ξ) = A h (ξ) = 0 S r (ξ ) = t } (ξ < ξ <ξ + )

39 Energetics General framework Saccular aneurysms Geometry & kinematics Working & balance Energetics & constitutive issues Ψ(P)= J ψ P (ψ free energy per unit prototypal volume) J := det(p) = α r α 2 h > 0 (J ψ free energy per unit paragon volume)

40 Dissipation principle General framework Saccular aneurysms Geometry & kinematics Working & balance Energetics & constitutive issues S ( ṗ) A i ( Ṗ P 1 ) (J ψ) 0 ψ(x, τ) = φ ( λ r (ξ, τ), λ h (ξ, τ); ξ ) S r = J φ,r /α r + + S r A i r = J [ S r α r λ r /J φ ] + + A r S h = J φ,h /α h + + S h A i h = J [ S h α h λ h /J φ ] + + A h reduced dissipation inequality + S r α r λr S h α h λh + A r α r /α r 2 + A h α h /α h 0

41 General framework Saccular aneurysms Constitutive prescriptions Geometry & kinematics Working & balance Energetics & constitutive issues + S r α r λr S h α h λh + A r α r /α r 2 + A h α h /α h 0 + A r = J D r α r /α r, + S r = + S h = 0 D r > 0, D h > 0 + A h = J D h α h /α h

42 General framework Saccular aneurysms Incompressible elasticity Geometry & kinematics Working & balance Energetics & constitutive issues incompressibility det F = λ r λ 2 h = 1 λ r = 1/λ 2 h. reactive inner force S = J π ( 1 α r λ r P r + 1 α h λ h P h ), A = J π I free-energy restriction φ : λ φ ( 1/λ 2, λ )

43 General framework Saccular aneurysms Incompressible elasticity Geometry & kinematics Working & balance Energetics & constitutive issues active and reactive components S r = J ( π (λ h /3) α r λ r S h = J ( π + (λ h /6) φ ) α h λ h A i r = J (T r φ ) D r α r /α r ( A i h = J T h φ ) D h α h /α h Cauchy stress radial and hoop components T = (J det(f)) 1 S P F T r = J 1 S r α r λ r T h =J 1 S h α h λ h

44 Fung strain energy General framework Saccular aneurysms Geometry & kinematics Working & balance Energetics & constitutive issues φ(λ) = (c/δ) exp ( (Γ/2) (λ 2 1) 2), c := 0.88 N/m δ := m Γ := 12.99

45 Evolution laws General framework Saccular aneurysms Geometry & kinematics Working & balance Energetics & constitutive issues D r α r /α r = ( T r φ ) + A o r /J D h α h /α h = ( T h φ ) + A o h /J 2 ( S r S h ) + ξ S r = 0

46 Part III Numerical simulations 4 Growth driven by mean hoop stress value a single case different cases compared 5 a first case high hoop resistance higher hoop gain 6 case one

47 a single case different cases compared Growth driven by mean hoop stress value Outer accretive couple ( A o r = J G r (T m h T ) T r + φ ) A o h (G = J h (T T m h ) T h + φ ) Evolution laws α r /α r = (G r /D r ) (T m h T ) α h /α h = (G h /D h ) (T T m h ) (T m h mean hoop stress, T target hoop stress)

48 a single case different cases compared radius thickness x10 4 G r/d r 2000 G h /D h 5000 D r/d h 1 D r 0.01 D h 0.01 [case-15-k]

49 a single case different cases compared mean hoop stress x10 4 G r/d r 2000 G h /D h 5000 D r/d h 1 D r 0.01 D h 0.01 [case-15-k]

50 a single case different cases compared x10 4 hoop accr. radial accr. G r/d r 2000 G h /D h 5000 D r/d h 1 D r 0.01 D h 0.01 [case-15-k]

51 a single case different cases compared 5 4 x10 4 hoop accr. vel. radial accr. vel x10 4 G r/d r 2000 G h /D h 5000 D r/d h 1 D r 0.01 D h 0.01 [case-15-k]

52 a single case different cases compared Hoop accretion G r/d r 2000 G h /D h 5000 D r/d h 1 D r 0.01 D h 0.01 [case-15-k]

53 a single case different cases compared Radial accretion G r/d r 2000 G h /D h 5000 D r/d h 1 D r 0.01 D h 0.01 [case-15-k]

54 a single case different cases compared Hoop stress G r/d r 2000 G h /D h 5000 D r/d h 1 D r 0.01 D h 0.01 [case-15-k]

55 a single case different cases compared Hoop stress last G r/d r 2000 G h /D h 5000 D r/d h 1 D r 0.01 D h 0.01 [case-15-k]

56 a single case different cases compared mean hoop stress x10 4 G r/d r 100 G h /D h 1000 D r/d h 10 D r 0.1 D h 0.01 [case-19-k]

57 a single case different cases compared mean hoop stress x10 4 G r/d r 1000 G h /D h 2500 D r/d h 1 D r 0.01 D h 0.01 [case-20-k]

58 a single case different cases compared mean hoop stress x10 4 G r/d r 2000 G h /D h 5000 D r/d h 1 D r 0.01 D h 0.01 [case-15-k]

59 a single case different cases compared mean hoop stress x10 4 G r/d r 100 G h /D h 1000 D r/d h 10 D r 0.1 D h 0.01 [case-19-k]

60 a single case different cases compared mean hoop stress x10 4 G r/d r 100 G h /D h 100 D r/d h 0.1 D r 0.01 D h 0.1 [case-17-k]

61 a single case different cases compared mean hoop stress x10 4 G r/d r 1000 G h /D h 100 D r/d h 0.1 D r 0.01 D h 0.1 [case-18-k]

62 a single case different cases compared mean hoop stress x10 4 G r/d r G h /D h 100 D r/d h 0.1 D r 0.01 D h 0.1 [case-16-k]

63 a first case high hoop resistance higher hoop gain Outer accretive couple ( A o r = J G r (T h T ) T r + φ ) A o h (G = J h (T T h ) T h + φ ) Evolution laws α r /α r = (G r /D r ) (T h T ) α h /α h = (G h /D h ) (T T h ) (T m h mean hoop stress, T target hoop stress)

64 a first case high hoop resistance higher hoop gain radius thickness x10 4 G r/d r 2000 G h /D h 80 D r/d h D r 0.01 D h 10 [case-15e-kp]

65 a first case high hoop resistance higher hoop gain mean hoop stress x10 4 G r/d r 2000 G h /D h 80 D r/d h D r 0.01 D h 10 [case-15e-kp]

66 a first case high hoop resistance higher hoop gain 80 hoop accr radial accr x10 4 G r/d r 2000 G h /D h 80 D r/d h D r 0.01 D h 10 [case-15e-kp]

67 a first case high hoop resistance higher hoop gain 2 x10 3 hoop accr. vel. 1.5 radial accr. vel x10 4 G r/d r 2000 G h /D h 80 D r/d h D r 0.01 D h 10 [case-15e-kp]

68 a first case high hoop resistance higher hoop gain Hoop accretion G r/d r 2000 G h /D h 80 D r/d h D r 0.01 D h 10 [case-15e-kp]

69 a first case high hoop resistance higher hoop gain Radial accretion G r/d r 2000 G h /D h 80 D r/d h D r 0.01 D h 10 [case-15e-kp]

70 a first case high hoop resistance higher hoop gain Hoop stress G r/d r 2000 G h /D h 80 D r/d h D r 0.01 D h 10 [case-15e-kp]

71 a first case high hoop resistance higher hoop gain Hoop stress last G r/d r 2000 G h /D h 80 D r/d h D r 0.01 D h 10 [case-15e-kp]

72 a first case high hoop resistance higher hoop gain radius thickness x10 4 G r/d r 2000 G h /D h 5 D r/d h D r 0.01 D h 10 [case-15b-kp]

73 a first case high hoop resistance higher hoop gain mean hoop stress x10 4 G r/d r 2000 G h /D h 5 D r/d h D r 0.01 D h 10 [case-15b-kp]

74 a first case high hoop resistance higher hoop gain 1.6 hoop accr radial accr x10 4 G r/d r 2000 G h /D h 5 D r/d h D r 0.01 D h 10 [case-15b-kp]

75 a first case high hoop resistance higher hoop gain 5 4 x10 4 hoop accr. vel. radial accr. vel x10 4 G r/d r 2000 G h /D h 5 D r/d h D r 0.01 D h 10 [case-15b-kp]

76 a first case high hoop resistance higher hoop gain Hoop accretion G r/d r 2000 G h /D h 5 D r/d h D r 0.01 D h 10 [case-15b-kp]

77 a first case high hoop resistance higher hoop gain Radial accretion G r/d r 2000 G h /D h 5 D r/d h D r 0.01 D h 10 [case-15b-kp]

78 a first case high hoop resistance higher hoop gain Hoop stress G r/d r 2000 G h /D h 5 D r/d h D r 0.01 D h 10 [case-15b-kp]

79 a first case high hoop resistance higher hoop gain Hoop stress G r/d r 2000 G h /D h 5 D r/d h D r 0.01 D h 10 [case-15b-kp]

80 a first case high hoop resistance higher hoop gain radius thickness x10 4 G r/d r 2000 G h /D h 50 D r/d h D r 0.01 D h 10 [case-15c-kp]

81 a first case high hoop resistance higher hoop gain mean hoop stress x10 4 G r/d r 2000 G h /D h 50 D r/d h D r 0.01 D h 10 [case-15c-kp]

82 a first case high hoop resistance higher hoop gain hoop accr. radial accr x10 4 G r/d r 2000 G h /D h 50 D r/d h D r 0.01 D h 10 [case-15c-kp]

83 a first case high hoop resistance higher hoop gain 1 x hoop accr. vel. radial accr. vel x10 4 G r/d r 2000 G h /D h 50 D r/d h D r 0.01 D h 10 [case-15c-kp]

84 a first case high hoop resistance higher hoop gain Hoop accretion G r/d r 2000 G h /D h 50 D r/d h D r 0.01 D h 10 [case-15c-kp]

85 a first case high hoop resistance higher hoop gain Radial accretion G r/d r 2000 G h /D h 50 D r/d h D r 0.01 D h 10 [case-15c-kp]

86 a first case high hoop resistance higher hoop gain Hoop stress G r/d r 2000 G h /D h 50 D r/d h D r 0.01 D h 10 [case-15c-kp]

87 a first case high hoop resistance higher hoop gain Hoop stress G r/d r 2000 G h /D h 50 D r/d h D r 0.01 D h 10 [case-15c-kp]

88 case one Evolution laws D r α r /α r = ( T r φ ) + A o r /J D h α h /α h = ( T h φ ) + A o h /J Outer accretive couple ( A o r = J G r (T h T ) T r + φ ) A o h ( T = J h + φ ) α r /α r = (G r /D r ) (T h T ) α h /α h = 0

89 case one radius thickness x10 4 G r/d r 2000 D r 0.01 D h 10 [case-15b-brx]

90 case one mean hoop stress hoop stress (in) hoop stress (out) x10 4 G r/d r 2000 D r 0.01 D h 10 [case-15b-brx]

91 case one hoop accr. (in) hoop accr. (out) radial accr. (in) radial accr. (out) x10 4 G r/d r 2000 D r 0.01 D h 10 [case-15b-brx]

92 case one 5 4 x10 4 hoop accr. vel. radial accr. vel x10 4 G r/d r 2000 D r 0.01 D h 10 [case-15b-brx]

93 case one Hoop accretion G r/d r 2000 D r 0.01 D h 10 [case-15b-brx]

94 case one Radial accretion G r/d r 2000 D r 0.01 D h 10 [case-15b-brx]

95 case one Hoop stress G r/d r 2000 D r 0.01 D h 10 [case-15b-brx]

96 case one Hoop stress last G r/d r 2000 D r 0.01 D h 10 [case-15b-brx]