Skip to main content
SpringerLink
Log in

Multiscale computational modeling of aortic valve calcification

  • Original Paper
  • Published:
Biomechanics and Modeling in Mechanobiology Aims and scope Submit manuscript

Abstract

Calcific aortic valve disease (CAVD) is a common cardiovascular disease that affects millions of people worldwide. The disease is characterized by the formation of calcium nodules on the aortic valve leaflets, which can lead to stenosis and heart failure if left untreated. The pathogenesis of CAVD is still not well understood, but involves several signaling pathways, including the transforming growth factor beta (TGF\(\beta\)) pathway. In this study, we developed a multiscale computational model for TGF\(\beta\)-stimulated CAVD. The model framework comprises cellular behavior dynamics, subcellular signaling pathways, and tissue-level diffusion fields of pertinent chemical species, where information is shared among different scales. Processes such as endothelial to mesenchymal transition (EndMT), fibrosis, and calcification are incorporated. The results indicate that the majority of myofibroblasts and osteoblast-like cells ultimately die due to lack of nutrients as they become trapped in areas with higher levels of fibrosis or calcification, and they subsequently act as sources for calcium nodules, which contribute to a polydispersed nodule size distribution. Additionally, fibrosis and calcification processes occur more frequently in regions closer to the endothelial layer where the cell activity is higher. Our results provide insights into the mechanisms of CAVD and TGF\(\beta\) signaling and could aid in the development of novel therapeutic approaches for CAVD and other related diseases such as cancer. More broadly, this type of modeling framework can pave the way for unraveling the complexity of biological systems by incorporating several signaling pathways in subcellular models to simulate tissue remodeling in diseases involving cellular mechanobiology.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9
Fig. 10

Similar content being viewed by others

References

  • Abdelhalim MAK (2011) The effects of size and period of administration of gold nanoparticles on rheological parameters of blood plasma of rats over a wide range of shear rates: in vivo. Lipids Health Dis 10:191. https://doi.org/10.1186/1476-511X-10-191

    Article  Google Scholar 

  • Ahamed J, Burg N, Yoshinaga K, Janczak CA, Rifkin DB, Coller BS (2008) In vitro and in vivo evidence for shear-induced activation of latent transforming growth factor-\(\beta 1\). Blood 112:3650–3660. https://doi.org/10.1182/blood-2008-04-151753

  • Aikawa E, Libby P (2017) A rock and a hard place: chiseling away at the multiple mechanisms of aortic stenosis. Am Heart Assoc 135(2)

  • Amindari A, Saltik L, Kirkkopru K, Yacoub M, Yalcin HC (2017) Assessment of calcified aortic valve leaflet deformations and blood flow dynamics using fluid–structure interaction modeling. Inform Med Unlocked 9:191–199

    Google Scholar 

  • Anderson ARA (20056) A hybrid mathematical model of solid tumour invasion: the importance of cell adhesion. Math Med Biol A J IMA 22:163–186. Retrieved from http://academic.oup.com/imammb/article/22/2/163/770979/A-hybrid-mathematical-model-of-solid-tumour, https://doi.org/10.1093/imammb/dqi005

  • Ankeny RF, Thourani VH, Weiss D, Vega JD, Taylor WR, Nerem RM, Jo H (2011) Preferential activation of smad1/5/8 on the fibrosa endothelium in calcified human aortic valves-association with low bmp antagonists and smad6. PLoS ONE 6(6):e20969

    Google Scholar 

  • Arzani A, Masters KS, Mofrad MR (2017) Multiscale systems biology model of calcific aortic valve disease progression. ACS Biomater Sci Eng 3(11):2922–2933

    Google Scholar 

  • Bakhaty AA, Mofrad MR (2015) Coupled simulation of heart valves: applications to clinical practice. Ann Biomed Eng 43:1626–1639

    Google Scholar 

  • Balachandran K, Sucosky P, Yoganathan AP (2011) Hemodynamics and mechanobiology of aortic valve inflammation and calcification. Int J Inflamm 2011

  • Benjamin EJ, Muntner P, Alonso A, Bittencourt MS, Callaway CW, Carson AP (2019) Heart disease and stroke statistics–2019 update: a report from the American heart association. Circulation 139(10):e56–e528

    Google Scholar 

  • Bischoff J (2019) Endothelial-to-mesenchymal transition. Circ Res 124:1163–1165. https://doi.org/10.1161/CIRCRESAHA.119.314813

    Article  Google Scholar 

  • Bosse K, Hans CP, Zhao N, Koenig SN, Huang N, Guggilam A et al (2013) Endothelial nitric oxide signaling regulates Notch1 in aortic valve disease. J Mol Cell Cardiol 60:27–35. https://doi.org/10.1016/j.yjmcc.2013.04.001

    Article  Google Scholar 

  • Bustamante DJ, Basile EJ, Hildreth BM, Browning NW, Jensen SA, Moldovan L, Moldovan NI (2021) Biofabrication of spheroids fusion-based tumor models: computational simulation of glucose effects. Biofabrication 13(3):035010

    Google Scholar 

  • Butcher JT, Tressel S, Johnson T, Turner D, Sorescu G, Jo H, Nerem RM (2006) Transcriptional profiles of valvular and vascular endothelial cells reveal phenotypic differences: influence of shear stress. Arterioscler Thromb Vasc Biol 26(1):69–77

    Google Scholar 

  • Butcher JT, Simmons CA, Warnock JN et al (2008) Mechanobiology of the aortic heart valve. J Heart Valve Dis 17(1):62

    Google Scholar 

  • Butcher JT, Mahler GJ, Hockaday LA (2011) Aortic valve disease and treatment: the need for naturally engineered solutions. Adv Drug Deliv Rev 63(4–5):242–268

    Google Scholar 

  • Cartlidge TR, Bing R, Kwiecinski J, Guzzetti E, Pawade TA, Doris MK et al (2021) Contrast-enhanced computed tomography assessment of aortic stenosis. Heart 107(23):1905–1911

    Google Scholar 

  • Chandra S, Rajamannan NM, Sucosky P (2012) Computational assessment of bicuspid aortic valve wall-shear stress: implications for calcific aortic valve disease. Biomech Model Mechanobiol 11:1085–1096

    Google Scholar 

  • Chen N, Glazier JA, Izaguirre JA, Alber MS (2007) A parallel implementation of the Cellular Potts Model for simulation of cell-based morphogenesis. Comput Phys Commun 176(11–12):670–681

    Google Scholar 

  • Chowkwale MS (2019) In silico multiscale modeling of endothelial cell mechanobiology in a tumor microenvironment

  • Chowkwale M, Mahler GJ, Huang P, Murray BT (2019) A multiscale in silico model of endothelial to mesenchymal transformation in a tumor microenvironment. J Theor Biol 480:229–240. https://doi.org/10.1016/j.jtbi.2019.08.012

    Article  Google Scholar 

  • Chowkwale M, Lindsey ML, Saucerman JJ (2022) Intercellular model predicts mechanisms of inflammation fibrosis coupling after myocardial infarction. J Physiol. https://doi.org/10.1113/JP283346

    Article  Google Scholar 

  • Clark-Greuel JN, Connolly JM, Sorichillo E, Narula NR, Rapoport HS, Mohler ER III, Levy RJ (2007) Transforming growth factor-\(\beta 1\) mechanisms in aortic valve calcification: increased alkaline phosphatase and related events. Ann Thorac Surg 83(3):946–953

  • Collier IE, Legant W, Marmer B, Lubman O, Saffarian S, Wakatsuki T, Goldberg GI (2011) Diffusion of mmps on the surface of collagen fibrils: the mobile cell surface-collagen substratum interface. PLoS ONE. https://doi.org/10.1371/journal.pone.0024029

    Article  Google Scholar 

  • Dayawansa NH, Baratchi S, Peter K (2022) Uncoupling the vicious cycle of mechanical stress and inflammation in calcific aortic valve disease. Front Cardiovasc Med 9:783543

    Google Scholar 

  • Dongre A, Weinberg RA (2019) New insights into the mechanisms of epithelial-mesenchymal transition and implications for cancer. Nat Rev Mol Cell Biol 20:69–84. https://doi.org/10.1038/s41580-018-0080-4

    Article  Google Scholar 

  • Driscoll K, Cruz AD, Butcher JT (2021) Inflammatory and biomechanical drivers of endothelial-interstitial interactions in calcific aortic valve disease. Circ Res 128(9):1344–1370

    Google Scholar 

  • Dutta P, Lincoln J (2018) Calcific aortic valve disease: a developmental biology perspective. Curr Cardiol Rep 20:1–13

    Google Scholar 

  • Fortuna I, Perrone GC, Krug MS, Susin E, Belmonte JM, Thomas GL, de Almeida RM (2020) Compucell3d simulations reproduce mesenchymal cell migration on flat substrates. Biophys J 118(11):2801–2815

    Google Scholar 

  • Franssen LC, Lorenzi T, Burgess AEF, Chaplain MAJ (2019) A mathematical framework for modelling the metastatic spread of cancer. Bull Math Biol 81:1965–2010. https://doi.org/10.1007/s11538-019-00597-x

    Article  MathSciNet  Google Scholar 

  • Garg V, Muth AN, Ransom JF, Schluterman MK, Barnes R, King IN, Srivastava D (2005) Mutations in notch1 cause aortic valve disease. Nature 437(7056):270–274

    Google Scholar 

  • Gaur T, Lengner CJ, Hovhannisyan H, Bhat RA, Bodine PV, Komm BS et al (2005) Canonical wnt signaling promotes osteogenesis by directly stimulating runx2 gene expression. J Biol Chem 280(39):33132–33140

    Google Scholar 

  • Gottesman S (2014) Coordinating bacterial cell division with nutrient availability: a role for glycolysis. mBio. https://doi.org/10.1128/mBio.00935-14

    Article  Google Scholar 

  • Graner F, Glazier JA (1992) Simulation of biological cell sorting using a two-dimensional extended potts model. Phys Rev Lett 69(13):2013

    Google Scholar 

  • Heino J (2007) The collagen family members as cell adhesion proteins. BioEssays 29(10):1001–1010

    Google Scholar 

  • Hjortnaes J, Shapero K, Goettsch C, Hutcheson JD, Keegan J, Kluin J, Aikawa E (2015) Valvular interstitial cells suppress calcification of valvular endothelial cells. Atherosclerosis 242:251–260. https://doi.org/10.1016/j.atherosclerosis.2015.07.008

    Article  Google Scholar 

  • Horn MA, Trafford AW (2016) Aging and the cardiac collagen matrix: Novel mediators of fibrotic remodelling. J Mol Cell Cardiol 93:175–185. https://doi.org/10.1016/j.yjmcc.2015.11.005

    Article  Google Scholar 

  • Hutcheson JD, Chen J, Sewell-Loftin M, Ryzhova LM, Fisher CI, Su YR, Merryman WD (2013) Cadherin-11 regulates cell-cell tension necessary for calcific nodule formation by valvular myofibroblasts. Arterioscler Thromb Vasc Biol 33(1):114–120

    Google Scholar 

  • Jafari Nivlouei S, Soltani M, Shirani E, Salimpour MR, Travasso R, Carvalho J (2022) A multiscale cell-based model of tumor growth for chemotherapy assessment and tumor-targeted therapy through a 3d computational approach. Cell Prolif 55(3):e13187

    Google Scholar 

  • Jian B, Narula N, Li Q-Y, Mohler ER III, Levy RJ (2003) Progression of aortic valve stenosis: Tgf-\(\beta 1\) is present in calcified aortic valve cusps and promotes aortic valve interstitial cell calcification via apoptosis. Ann Thorac Surg 75(2):457–465

  • Kabla AJ (2012) Collective cell migration: leadership, invasion and segregation. J R Soc Interface 9(77):3268–3278

    Google Scholar 

  • Kawamoto M, Matsunami T, Ertl RF, Fukuda Y, Ogawa M, Spurzem JR, Rennard SI (1997) Selective migration of \(\alpha\)-smooth muscle actin-positive myofibroblasts toward fibronectin in the Boyden’s blindwell chamber. Clin Sci 93(4):355–362

  • Keller EF, Segel LA (1971) Model for chemotaxis. J Theor Biol 30(2):225–234

    Google Scholar 

  • Nekolla K, Rehberg M, Vollmar AM, Zahler S (2016) New view on endothelial cell migration. Arterioscler Thromb Vasc Biol 36:2346–2357. https://doi.org/10.1161/ATVBAHA.116.307870

    Article  Google Scholar 

  • Kick K, Nekolla K, Rehberg M, Vollmar AM, Zahler S (2016) New view on endothelial cell migration: switching modes of migration based on matrix composition. Arterioscler Thromb Vasc Biol 36(12):2346–2357

    Google Scholar 

  • Kumar S, Kapoor A, Desai S, Inamdar MM, Sen S (2016) Proteolytic and non-proteolytic regulation of collective cell invasion: tuning by ECM density and organization. Sci Rep 6:19905. https://doi.org/10.1038/srep19905

    Article  Google Scholar 

  • Kumarswamy R, Volkmann I, Jazbutyte V, Dangwal S, Park D-H, Thum T (2012) Transforming growth factor-\(\beta\)-induced endothelialto-mesenchymal transition is partly mediated by MicroRNA-21. Arterioscler Thromb Vasc Biol 32:361–369. https://doi.org/10.1161/ATVBAHA.111.234286

  • Lee B, Zhou X, Riching K, Eliceiri KW, Keely PJ, Guelcher SA, Jiang Y (2014) A three-dimensional computational model of collagen network mechanics. PLoS ONE 9:e111896. https://doi.org/10.1371/journal.pone.0111896

    Article  Google Scholar 

  • Leopold JA (2012) Cellular mechanisms of aortic valve calcification. Circ Cardiovasc Interv 5(4):605–614

    Google Scholar 

  • Luraghi G, Migliavacca F, Chiastra C, Rossi A, Reimers B, Stefanini GG, Matas JFR (2019) Does clinical data quality affect fluid-structure interaction simulations of patient-specific stenotic aortic valve models? J Biomech 94:202–210

    Google Scholar 

  • Ma X, Zhao D, Yuan P, Li J, Yun Y, Cui Y et al (2020) Endothelialto-mesenchymal transition in calcific aortic valve disease. Acta Cardiol Sin 36(3):183

    Google Scholar 

  • Mahler GJ, Farrar EJ, Butcher JT (2013) Inflammatory cytokines promote mesenchymal transformation in embryonic and adult valve endothelial cells. Arterioscler Thromb Vasc Biol 33:121–130. https://doi.org/10.1161/ATVBAHA.112.300504

    Article  Google Scholar 

  • Maleki H, Shahriari S, Durand LG, Labrosse MR, Kadem L (2014) A metric for the stiffness of calcified aortic valves using a combined computational and experimental approach. Medical Biol Eng Comput 52:1–8

    Google Scholar 

  • Masjedi S, Lei Y, Patel J, Ferdous Z (2017) Sex-related differences in matrix remodeling and early osteogenic markers in aortic valvular interstitial cells. Heart Vessels 32(2):217–228. https://doi.org/10.1007/s00380-016-0909-8

    Article  Google Scholar 

  • Masur SK, Dewal HS, Dinh TT, Erenburg I, Petridou S (1996) Myofibroblasts differentiate from fibroblasts when plated at low density. Proc Natl Acad Sci 93:4219–4223. https://doi.org/10.1073/pnas.93.9.4219

    Article  Google Scholar 

  • Mathieu P, Bouchareb R, Boulanger M-C (2015) Innate and adaptive immunity in calcific aortic valve disease. J Immunol Res 2015

  • Mendoza M, Chen M-H, Huang P, Mahler GJ (2022) Shear and endothelial induced late-stage calcific aortic valve disease-on-achip develops calcium phosphate mineralizations. Lab Chip 22:1374–1385. https://doi.org/10.1039/D1LC00931A

    Article  Google Scholar 

  • ming Meng X, Nikolic-Paterson DJ, Lan HY (2016) TGF-\(\beta\): the master regulator of fibrosis. Nat Rev Nephrol 12:325–338. https://doi.org/10.1038/nrneph.2016.48

  • Mirza A, Ramaswamy S (2022) Importance of non-newtonian computational fluid modeling on severely calcified aortic valve geometries–insights from quasi-steady state simulations. J Biomech Eng 144(11):114501

    Google Scholar 

  • Misfeld M, Sievers H-H (2007) Heart valve macro-and microstructure. Philos Trans Roy Soc B Biol Sci 362(1484):1421–1436

    Google Scholar 

  • Nguyen Edalgo YT, Zornes AL, Ford Versypt AN (2019) A hybrid discrete-continuous model of metastatic cancer cell migration through a remodeling extracellular matrix. AIChE J 65(9):e16671

    Google Scholar 

  • Nkomo VT, Gardin JM, Skelton TN, Gottdiener JS, Scott CG, Enriquez-Sarano M (2006) Burden of valvular heart diseases: a population-based study. The Lancet 368(9540):1005–1011

    Google Scholar 

  • O’Brien F, Harley B, Yannas I, Gibson L (2005) The effect of pore size on cell adhesion in collagengag scaffolds. Biomaterials 26:433–441. https://doi.org/10.1016/j.biomaterials.2004.02.052

    Article  Google Scholar 

  • O’Brien J, Lyons T, Monks J, Lucia MS, Wilson RS, Hines L, Schedin P (2010) Alternatively activated macrophages and collagen remodeling characterize the postpartum involuting mammary gland across species. Am J Pathol 176(3):1241–1255

    Google Scholar 

  • Osman L, Yacoub MH, Latif N, Amrani M, Chester AH (2006) Role of human valve interstitial cells in valve calcification and their response to atorvastatin. Circulation 114:547–552. https://doi.org/10.1161/CIRCULATIONAHA.105.001115

    Article  Google Scholar 

  • Phan SH (2008) Biology of fibroblasts and myofibroblasts. Proc Am Thorac Soc 5(3):334–337

    Google Scholar 

  • Pho M, Lee W, Watt DR, Laschinger C, Simmons CA, McCulloch CA (2008) Cofilin is a marker of myofibroblast differentiation in cells from porcine aortic cardiac valves. Am J Physiol Heart Circ Physiol 294:H1767–H1778. https://doi.org/10.1152/ajpheart.01305.2007

    Article  Google Scholar 

  • Piek A, de Boer RA, Silljé HHW (2016) The fibrosis-cell death axis in heart failure. Heart Fail Rev 21:199–211. https://doi.org/10.1007/s10741-016-9536-9

    Article  Google Scholar 

  • Rajamannan NM, Evans FJ, Aikawa E, Grande-Allen KJ, Demer LL, Heistad DD et al (2011) Calcific aortic valve disease: not simply a degenerative process a review and agenda for research from the national heart and lung and blood institute aortic stenosis working group. Circulation 124(16):1783

    Google Scholar 

  • Ramis-Conde I, Drasdo D, Anderson AR, Chaplain MA (2008) Modeling the influence of the E-cadherin-\(\beta\)-catenin pathway in cancer cell invasion: a multiscale approach. Biophys J 95(1):155–165

  • Rush MN (2018) Chemically modified monolayer surfaces influence valvular interstitial cell attachment and differentiation for heart valve tissue engineering (Unpublished doctoral dissertation)

  • Sadrabadi MS, Eskandari M, Feigenbaum HP, Arzani A (2021a) Local and global growth and remodeling in calcific aortic valve disease and aging. J Biomech 128:110773

    Google Scholar 

  • Sadrabadi MS, Hedayat M, Borazjani I, Arzani A (2021b) Fluid-structure coupled biotransport processes in aortic valve disease. J Biomech 117:110239

    Google Scholar 

  • Sarper M, Cortes E, Lieberthal TJ, del Río Hernández A (2016) Atra modulates mechanical activation of tgf-\(\beta\) by pancreatic stellate cells. Sci Rep 6(1):27639

  • Sánchez-Duffhues G, de Vinuesa AG, ten Dijke P (2018) Endothelial-to-mesenchymal transition in cardiovascular diseases: developmental signaling pathways gone awry. Dev Dyn 247:492–508. https://doi.org/10.1002/dvdy.24589

    Article  Google Scholar 

  • Son KJ, Gheibi P, Stybayeva G, Rahimian A, Revzin A (2017) Detecting cell-secreted growth factors in microfluidic devices using bead-based biosensors. Microsyst Nanoeng 3(1):17025. https://doi.org/10.1038/micronano.2017.25

    Article  Google Scholar 

  • Steitz SA, Speer MY, McKee MD, Liaw L, Almeida M, Yang H, Giachelli CM (2002) Osteopontin inhibits mineral deposition and promotes regression of ectopic calcification. Am J Pathol 161(6):2035–2046. https://doi.org/10.1016/S0002-9440(10)64482-3

    Article  Google Scholar 

  • Stephens EH, Saltarrelli JG, Baggett LS, Nandi I, Kuo JJ, Davis AR, Grande-Allen KJ (2011) Differential proteoglycan and hyaluronan distribution in calcified aortic valves. Cardiovasc Pathol 20(6):334–342

    Google Scholar 

  • Sucosky P, Balachandran K, Elhammali A, Jo H, Yoganathan AP (2009) Altered shear stress stimulates upregulation of endothelial vcam-1 and icam-1 in a bmp-4-and tgf-\(\beta\)1-dependent pathway. Arterioscler Thromb Vasc Biol 29(2):254–260

  • Swat MH, Thomas GL, Belmonte JM, Shirinifard A, Hmeljak D, Glazier JA (2012) Multi-scale modeling of tissues using compucell3d. In: Methods in cell biology, vol 110. Elsevier, pp 325–366

  • Taylor PM (2007) Biological matrices and bionanotechnology. Philos Trans Roy Soc B Biol Sci 362(1484):1313–1320

    Google Scholar 

  • Thampatty BP, Wang JH-C (2007) A new approach to study fibroblast migration. Cell Motil Cytoskelet 64(1):1–5

    Google Scholar 

  • Wakefield LM, Winokur TS, Hollands RS, Christopherson K, Levinson AD, Sporn MB (1990) Recombinant latent transforming growth factor beta 1 has a longer plasma half-life in rats than active transforming growth factor beta 1, and a different tissue distribution. J Clin Investig 86:1976–1984. https://doi.org/10.1172/JCI114932

    Article  Google Scholar 

  • Weinberg EJ, Kaazempur Mofrad MR (2007) Transient, three-dimensional, multiscale simulations of the human aortic valve. Cardiovasc Eng 7:140–155

    Google Scholar 

  • Welsh C, Xu J, Smith L, König M, Choi K, Sauro HM (2023) libRoadRunner 2.0: a high performance SBML simulation and analysis library. Bioinformatics. https://doi.org/10.1093/bioinformatics/btac770

    Article  Google Scholar 

  • Yutzey KE, Demer LL, Body SC, Huggins GS, Towler DA, Giachelli CM et al (2014) Calcific aortic valve disease: a consensus summary from the alliance of investigators on calcific aortic valve disease. Arterioscler Thromb Vasc Biol 34(11):2387–2393

    Google Scholar 

Download references

Acknowledgements

This work was funded by the National Science Foundation under Grant CMMI 1919438.

Author information

Authors and Affiliations

  1. Department of Mechanical Engineering, Binghamton University, Binghamton, NY, 13902, USA

    Javid Azimi-Boulali, Bruce T. Murray & Peter Huang

  2. Department of Biomedical Engineering, Binghamton University, Binghamton, NY, 13902, USA

    Gretchen J. Mahler

Authors
  1. Javid Azimi-Boulali
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Gretchen J. Mahler
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Bruce T. Murray
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Peter Huang
    View author publications

    You can also search for this author in PubMed Google Scholar

Contributions

JA-B, GJM, BTM, and PH contributed to the study conception and outlined the paper. JA-B performed all of the simulations and made figures. JA-B wrote the first draft of the manuscript and GJM, BTM, and PH commented on the previous versions of the manuscript. The final manuscript was read and approved by JA-B, GJM, BTM, and PH.

Corresponding author

Correspondence to Peter Huang.

Ethics declarations

Conflict of interest

The authors declare that they have no competing or financial interests.

Additional information

Publisher’s Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Supplementary Information

Below is the link to the electronic supplementary material.

Supplementary file 1 (pdf 33410 KB)

Rights and permissions

Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Azimi-Boulali, J., Mahler, G.J., Murray, B.T. et al. Multiscale computational modeling of aortic valve calcification. Biomech Model Mechanobiol 23, 581–599 (2024). https://doi.org/10.1007/s10237-023-01793-4

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10237-023-01793-4

Keywords

Search

Navigation