Numerical solution of the bidomain equations

2009 ◽  
Vol 367 (1895) ◽  
pp. 1931-1950 ◽  
Author(s):  
S. Linge ◽  
J. Sundnes ◽  
M. Hanslien ◽  
G.T. Lines ◽  
A. Tveito
Keyword(s):  
Mathematical Models ◽  
Cardiac Electrophysiology ◽  
Mathematical Reasoning ◽  
Full Potential ◽  
Processing Unit ◽  
Bidomain Model ◽  
Central Processing ◽  
Cellular Models ◽  
Bidomain Equations ◽  
The Past

Knowledge of cardiac electrophysiology is efficiently formulated in terms of mathematical models. However, most of these models are very complex and thus defeat direct mathematical reasoning founded on classical and analytical considerations. This is particularly so for the celebrated bidomain model that was developed almost 40 years ago for the concurrent analysis of extra- and intracellular electrical activity. Numerical simulations based on this model represent an indispensable tool for studying electrophysiology. However, complex mathematical models, steep gradients in the solutions and complicated geometries lead to extremely challenging computational problems. The greatest achievement in scientific computing over the past 50 years has been to enable the solving of linear systems of algebraic equations that arise from discretizations of partial differential equations in an optimal manner, i.e. such that the central processing unit (CPU) effort increases linearly with the number of computational nodes. Over the past decade, such optimal methods have been introduced in the simulation of electrophysiology. This development, together with the development of affordable parallel computers, has enabled the solution of the bidomain model combined with accurate cellular models, on geometries resembling a human heart. However, in spite of recent progress, the full potential of modern computational methods has yet to be exploited for the solution of the bidomain model. This paper reviews the development of numerical methods for solving the bidomain model. However, the field is huge and we thus restrict our focus to developments that have been made since the year 2000.

scholarly journals Mathematical analysis and 2-scale convergence of a heterogeneous microscopic bidomain model

2018 ◽  
Vol 28 (05) ◽  
pp. 979-1035 ◽  
Author(s):  
Annabelle Collin ◽  
Sébastien Imperiale
Keyword(s):  
Mathematical Analysis ◽  
Cardiac Electrophysiology ◽  
Convergence Theory ◽  
Bidomain Model ◽  
Periodic Homogenization ◽  
Homogenization Procedure ◽  
Uniform Estimates ◽  
Bidomain Equations ◽  
Homogenization Process

The aim of this paper is to provide a complete mathematical analysis of the periodic homogenization procedure that leads to the macroscopic bidomain model in cardiac electrophysiology. We consider space-dependent and tensorial electric conductivities as well as space-dependent physiological and phenomenological nonlinear ionic models. We provide the nondimensionalization of the bidomain equations and derive uniform estimates of the solutions. The homogenization procedure is done using 2-scale convergence theory which enables us to study the behavior of the nonlinear ionic models in the homogenization process.

scholarly journals Optimal monodomain approximations of the bidomain equations used in cardiac electrophysiology

2014 ◽  
Vol 24 (06) ◽  
pp. 1115-1140 ◽  
Author(s):  
Yves Coudière ◽  
Yves Bourgault ◽  
Myriam Rioux
Keyword(s):  
Boundary Conditions ◽  
Cardiac Electrophysiology ◽  
Propagation Speed ◽  
Neumann Boundary ◽  
Test Cases ◽  
Bidomain Model ◽  
Bidomain Equations ◽  
Monodomain Model ◽  
The Difference ◽  
Computationally Intensive

The bidomain model is the current most sophisticated model used in cardiac electrophysiology. The monodomain model is a simplification of the bidomain model that is less computationally intensive but only valid under equal conductivity ratio. We propose in this paper optimal monodomain approximations of the bidomain model. We first prove that the error between the bidomain and monodomain solutions is bounded by the error ‖B - A‖ between the bidomain and monodomain conductivity operators. Optimal monodomain approximations are defined by minimizing the distance ‖B - A‖, which reduces for solutions over all ℝd to minimize the Lp norm of the difference between the operator symbols. Similarly, comparing the symbols pointwise amounts to compare the propagation of planar waves in the bidomain and monodomain models. We prove that any monodomain model properly propagates at least d planar waves in ℝd. We next consider and solve the optimal problem in the L∞ and L2 norms, the former providing minimal propagation error uniformly over all directions. The quality of these optimal monodomain approximations is compared among themselves and with other published approximations using two sets of test cases. The first one uses periodic boundary conditions to mimic propagation in ℝd while the second is based on a square domain with common Neumann boundary conditions. For the first test cases, we show that the error on the propagation speed is highly correlated with the error on the symbols. The second test cases show that domain boundaries control propagation directions, with only partial impact from the conductivity operator used.

Perangkat keras komputer

2020 ◽  
Author(s):  
Roudati jannah
Keyword(s):  
Data Storage ◽  
Central Processing Unit ◽  
External Memory ◽  
Storage Device ◽  
Input Device ◽  
Processing Unit ◽  
Central Processing ◽  
Output Device

Perangkat keras komputer adalah bagian dari sistem komputer sebagai perangkat yang dapat diraba, dilihat secara fisik, dan bertindak untuk menjalankan instruksi dari perangkat lunak (software). Perangkat keras komputer juga disebut dengan hardware. Hardware berperan secara menyeluruh terhadap kinerja suatu sistem komputer. Prinsipnya sistem komputer selalu memiliki perangkat keras masukan (input/input device system) – perangkat keras premprosesan (processing/central processing unit) – perangkat keras luaran (output/output device system) – perangkat tambahan yang sifatnya opsional (peripheral) dan tempat penyimpanan data (storage device system/external memory).

CENTRAL PROCESSING UNIT (CPU)

2020 ◽  
Author(s):  
Ika Milia wahyunu Siregar
Keyword(s):  
Processing System ◽  
Central Processing Unit ◽  
Processing Unit ◽  
Central Processing

Perkembangan IT di dunia sangat pesat, mulai dari perkembangan sofware hingga hardware. Teknologi sekarang telah mendominasi sebagian besar di permukaan bumi ini. Karena semakin cepatnya perkembangan Teknologi, kita sebagai pengguna bisa ketinggalan informasi mengenai teknologi baru apabila kita tidak up to date dalam pengetahuan teknologi ini. Hal itu dapat membuat kita mudah tergiur dan tertipu dengan berbagai iklan teknologi tanpa memikirkan sisi negatifnya. Sebagai pengguna dari komputer, kita sebaiknya tahu seputar mengenai komponen-komponen komputer. Komputer adalah serangkaian mesin elektronik yang terdiri dari jutaan komponen yang dapat saling bekerja sama, serta membentuk sebuah sistem kerja yang rapi dan teliti. Sistem ini kemudian digunakan untuk dapat melaksanakan pekerjaan secara otomatis, berdasarkan instruksi (program) yang diberikan kepadanya. Istilah Hardware komputer atau perangkat keras komputer, merupakan benda yang secara fisik dapat dipegang, dipindahkan dan dilihat. Central Processing System/ Central Processing Unit (CPU) adalah salah satu jenis perangkat keras yang berfungsi sebagai tempat untuk pengolahan data atau juga dapat dikatakan sebagai otak dari segala aktivitas pengolahan seperti penghitungan, pengurutan, pencarian, penulisan, pembacaan dan sebagainya.

Perangkat lunak komputer

2020 ◽  
Author(s):  
Intan khadijah simatupang
Keyword(s):  
Data Storage ◽  
Central Processing Unit ◽  
External Memory ◽  
Storage Device ◽  
Input Device ◽  
Processing Unit ◽  
Central Processing ◽  
Output Device

Komputer adalah serangkaian mesin elektronik yang terdiri dari jutaan komponen yang dapat saling bekerja sama, serta membentuk sebuah sistem kerja yang rapi dan teliti. Sistem ini kemudian digunakan untuk dapat melaksanakan pekerjaan secara otomatis, berdasarkan instruksi (program) yang diberikan kepadanya. Istilah Hardware computer atau perangkat keras komputer, merupakan benda yang secara fisik dapat dipegang, dipindahkan dan dilihat. Software komputer atau perangkat lunak komputer merupakan kumpulan instruksi (program/prosedur) untuk dapat melaksanakan pekerjaan secara otomatis dengan cara mengolah atau memproses kumpulan instruksi (data) yang diberikan. Pada prinsipnya sistem komputer selalu memiliki perangkat keras masukan (input/input device system) – perangkat keras pemprosesan (processing/ central processing unit) – perangkat keras keluaran (output/output device system), perangkat tambahan yang sifatnya opsional (peripheral) dan tempat penyimpanan data (Storage device system/external memory).

PERANGKAT KERAS ( HARDWARE ) PADA KOMPUTER

2020 ◽  
Author(s):  
Siti Kumala Dewi
Keyword(s):  
Processing System ◽  
Central Processing Unit ◽  
Input Device ◽  
Processing Unit ◽  
Central Processing ◽  
Output Device ◽  
System P ◽  
System 2

Perangkat keras komputer adalah bagian dari sistem komputer sebagai perangkat yang dapat diraba, dilihat secara fisik, dan bertindak untuk menjalankan instruksi dari perangkat lunak (software). Perangkat keras komputer juga disebut dengan hardware. Hardware berperan secara menyeluruh terhadap kinerja suatu sistem komputer. Berdasarkan fungsinya, perangkat keras terbagi menjadi :1.Sistem Perangkat Keras Masukan (Input Device System )2.Sistem Pemrosesan ( Central Processing System/ Central Processing Unit(CPU)3.Sistem Perangkat Keras Keluaran ( Output Device System )4.Sistem Perangkat Keras Tambahan (Peripheral/Accessories Device System)

Integrating Bioinformatics Strategies in Cancer Immunotherapy: Current and Future Perspectives

2020 ◽  
Vol 23 (8) ◽  
pp. 687-698 ◽  
Author(s):  
Houda N. Washah ◽  
Elliasu Y. Salifu ◽  
Opeyemi Soremekun ◽  
Ahmed A. Elrashedy ◽  
Geraldene Munsamy ◽  
...  
Keyword(s):  
Cancer Immunotherapy ◽  
Therapeutic Potential ◽  
Cell Cancer ◽  
Full Potential ◽  
Future Perspectives ◽  
The Past ◽  
Bioinformatics Tools ◽  
Treatment Paradigm ◽  
Immunotherapy Of Cancer ◽  
Significant Attention

For the past few decades, the mechanisms of immune responses to cancer have been exploited extensively and significant attention has been given into utilizing the therapeutic potential of the immune system. Cancer immunotherapy has been established as a promising innovative treatment for many forms of cancer. Immunotherapy has gained its prominence through various strategies, including cancer vaccines, monoclonal antibodies (mAbs), adoptive T cell cancer therapy, and immune checkpoint therapy. However, the full potential of cancer immunotherapy is yet to be attained. Recent studies have identified the use of bioinformatics tools as a viable option to help transform the treatment paradigm of several tumors by providing a therapeutically efficient method of cataloging, predicting and selecting immunotherapeutic targets, which are known bottlenecks in the application of immunotherapy. Herein, we gave an insightful overview of the types of immunotherapy techniques used currently, their mechanisms of action, and discussed some bioinformatics tools and databases applied in the immunotherapy of cancer. This review also provides some future perspectives in the use of bioinformatics tools for immunotherapy.

Mathematical Issues Arising From the Directed Mutation Controversy

Genetics ◽  
2003 ◽  
Vol 164 (1) ◽  
pp. 373-379
Author(s):  
Qi Zheng
Keyword(s):  
Mathematical Reasoning ◽  
Large Body ◽  
Directed Mutation ◽  
The Past ◽  
New Evidence

Abstract During the past 14 years or so a large body of new evidence that supposedly supports the directed mutation hypothesis has accumulated. Interpretation of some of the evidence depends on mathematical reasoning, which can be subtler than it appears at first sight. This article attempts to clarify some of the mathematical issues arising from the directed mutation controversy, thereby offering alternative interpretations of some of the evidence.

Sign in / Sign up

Export Citation Format

Share Document