The method of fundamental solutions for the direct elastography problem in the human retina. (English) Zbl 1454.65182
Alves, Carlos (ed.) et al., Advances in Trefftz methods and their applications. Selected papers based on the presentations at the 9th conference on Trefftz methods and 5th conference on method of fundamental solutions, Lisbon, Portugal, July 29–31, 2019. Cham: Springer. SEMA SIMAI Springer Ser. 23, 87-101 (2020).
Solving systems of partial differential equations (PDEs) has attracted a special interest by many mathematicians and engineers due to the applicability of these equations in various modelling scenarios and scientific phenomena. There are various recent studies on solving these equations using analytical and numerical methods such as the double Laplace transform method [the reviewer et al., “New approximate-analytical solutions for the nonlinear fractional Schrödinger equation with second-order spatio-temporal dispersion via double Laplace transform method”, Preprint, arXiv:2010.10977; “Novel methods for solving the conformable wave equations”, J. New Theory 2020, No. 31, 56–85 (2020)], the differential transform method [loc. cit.], the method of fictitious domains and homotopy [I. P. Gavrilyuk and V. L. Makarov, Ukr. Math. J. 72, No. 2, 211–231 (2020; Zbl 1453.65434); translation from Ukr. Mat. Zh. 72, No. 2, 191–208 (2020)], and the method of semigroups [N. Cusimano et al., ESAIM, Math. Model. Numer. Anal. 54, No. 3, 751–774 (2020; Zbl 1452.35237)]. We also refer the reader to other related studies in [Q. Du, Nonlocal modeling, analysis, and computation. Philadelphia, PA: Society for Industrial and Applied Mathematics (SIAM) (2019; Zbl 1423.00007); F. Camilli and A. Goffi, NoDEA, Nonlinear Differ. Equ. Appl. 27, No. 2, Paper No. 22, 37 p. (2020; Zbl 1452.35234); K. Ryszewska, J. Math. Anal. Appl. 483, No. 2, Article ID 123654, 17 p. (2020; Zbl 1436.35323); A. Ghanmi and Z. Zhang, Bull. Korean Math. Soc. 56, No. 5, 1297–1314 (2019; Zbl 1432.34012); H. Dong and D. Kim, J. Funct. Anal. 278, No. 3, Article ID 108338, 66 p. (2020; Zbl 1427.35316); S. D. Taliaferro, J. Math. Pures Appl. (9) 133, 287–328 (2020; Zbl 1437.35697); K. Hidano and C. Wang, Sel. Math., New Ser. 25, No. 1, Paper No. 2, 28 p. (2019; Zbl 1428.35662); B. Zhu and B. Han, Mediterr. J. Math. 17, No. 4, Paper No. 113, 12 p. (2020; Zbl 1452.35248); M. Musso et al., Math. Ann. 375, No. 1–2, 361–424 (2019; Zbl 1452.35244); L. C. F. Ferreira et al., Bull. Sci. Math. 153, 86–117 (2019; Zbl 1433.35185)].
In this paper, the authors have provided a numerical study on a proposed mathematical model based on the human eye. This model discusses the mechanical waves propagation and induced displacements in the human retina. This proposed model is useful for elastography which is a well-known medical imaging modality for mapping the elastic properties and stiffness of soft tissue. The studied model problem in this paper is based on the need for solving a system of PDEs subject to the boundary and acoustic-elastic coupling transmission conditions for each layer where a narrow cylindrical layered domain is assumed, and the interface boundaries in the eye are considered plane within the cylinder in this model.
The authors have applied a technique known as the fundamental method of solutions which is a method for solving partial differential equations where the fundamental solution is used as a basis function. An ansatz is assumed for each layer; therefore, a linear system is needed to be solved to approximate the time-harmonic acoustic waves propagation for various eye media and for the elastic excitation in the human retina. Then, the authors have provided numerical experiments to validate their obtained results where simulation results have been successfully provided in this paper for the obtained approximations of the acoustic and elastic fields using the proposed technique, the fundamental method of solutions. As mentioned by the authors of this research work, there is a drawback of using the proposed method of solution where there is no significant improvement in the accuracy by increasing the number of points after each stage. However, further research studies and investigations are needed to overcome this challenge. In conclusion, the presented study in this research work is very important for providing a basis for more further future research studies on this interesting topic or other related research topics; therefore, I would like to recommend all interested researchers to read this interesting research work and work on investigating more related research problems.
For the entire collection see [Zbl 1445.65001].
In this paper, the authors have provided a numerical study on a proposed mathematical model based on the human eye. This model discusses the mechanical waves propagation and induced displacements in the human retina. This proposed model is useful for elastography which is a well-known medical imaging modality for mapping the elastic properties and stiffness of soft tissue. The studied model problem in this paper is based on the need for solving a system of PDEs subject to the boundary and acoustic-elastic coupling transmission conditions for each layer where a narrow cylindrical layered domain is assumed, and the interface boundaries in the eye are considered plane within the cylinder in this model.
The authors have applied a technique known as the fundamental method of solutions which is a method for solving partial differential equations where the fundamental solution is used as a basis function. An ansatz is assumed for each layer; therefore, a linear system is needed to be solved to approximate the time-harmonic acoustic waves propagation for various eye media and for the elastic excitation in the human retina. Then, the authors have provided numerical experiments to validate their obtained results where simulation results have been successfully provided in this paper for the obtained approximations of the acoustic and elastic fields using the proposed technique, the fundamental method of solutions. As mentioned by the authors of this research work, there is a drawback of using the proposed method of solution where there is no significant improvement in the accuracy by increasing the number of points after each stage. However, further research studies and investigations are needed to overcome this challenge. In conclusion, the presented study in this research work is very important for providing a basis for more further future research studies on this interesting topic or other related research topics; therefore, I would like to recommend all interested researchers to read this interesting research work and work on investigating more related research problems.
For the entire collection see [Zbl 1445.65001].
Reviewer: Mohammed Kaabar (Gelugor)
MSC:
| 65N80 | Fundamental solutions, Green’s function methods, etc. for boundary value problems involving PDEs |
| 74L15 | Biomechanical solid mechanics |
| 74B05 | Classical linear elasticity |
| 76Q05 | Hydro- and aero-acoustics |
| 76Z05 | Physiological flows |
| 78A40 | Waves and radiation in optics and electromagnetic theory |
| 92C10 | Biomechanics |
| 92C55 | Biomedical imaging and signal processing |
| 35J05 | Laplace operator, Helmholtz equation (reduced wave equation), Poisson equation |
| 35Q74 | PDEs in connection with mechanics of deformable solids |
Keywords:
elastography; method of fundamental solutions; coupled acoustic wave propagation; elastic displacement modelCitations:
Zbl 1423.00007; Zbl 1436.35323; Zbl 1432.34012; Zbl 1427.35316; Zbl 1437.35697; Zbl 1428.35662; Zbl 1433.35185; Zbl 1453.65434; Zbl 1452.35237; Zbl 1452.35234; Zbl 1452.35248; Zbl 1452.35244References:
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