Biomechanics and Modeling in Mechanobiology

, Volume 18, Issue 4, pp 1197–1211 | Cite as

A coupled reaction–diffusion–strain model predicts cranial vault formation in development and disease

  • Chanyoung Lee
  • Joan T. Richtsmeier
  • Reuben H. KraftEmail author
Original Paper


How cells utilize instructions provided by genes and integrate mechanical forces generated by tissue growth to produce morphology is a fundamental question of biology. Dermal bones of the vertebrate cranial vault are formed through the direct differentiation of mesenchymal cells on the neural surface into osteoblasts through intramembranous ossification. Here we join a self-organizing Turing mechanism, computational biomechanics, and experimental data to produce a 3D representative model of the growing cerebral surface, cranial vault bones, and sutures. We show how changes in single parameters regulating signaling during osteoblast differentiation and bone formation may explain cranial vault shape variation in craniofacial disorders. A key result is that toggling a parameter in our model results in closure of a cranial vault suture, an event that occurred during evolution of the cranial vault and that occurs in craniofacial disorders. Our approach provides an initial and important step toward integrating biomechanics into the genotype phenotype map to explain the production of variation in head morphology by developmental mechanisms.


Computational morphogenesis Finite volume method Intramembranous ossification Skull growth and evolution Craniosynostosis Brain Mouse model 



Computations for this research were performed on the Pennsylvania State University’s Institute for CyberScience Advanced CyberInfrastructure (ICS-ACI). We acknowledge Matthew Dolack for checking data on github. This work was supported in part through instrumentation funded by a National Science Foundation Grant OCI0821527, a Burroughs-Wellcome Fund 2013 Collaborative Research Travel Grant, Pennsylvania Department of Health using Tobacco Cure Funds, and by the National Institutes of Health Grants R01DE022988 and P01HD078233. The content is solely the responsibility of the authors and does not necessarily represent the official views of the National Institutes of Health. Source code of the reaction–diffusion–strain model and an example case are available at

Supplementary material

Supplementary material 1 (mp4 2986 KB)


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Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  • Chanyoung Lee
    • 1
  • Joan T. Richtsmeier
    • 2
  • Reuben H. Kraft
    • 3
    • 4
    • 5
    Email author
  1. 1.Department of Mechanical EngineeringPennsylvania State UniversityUniversity ParkUSA
  2. 2.Department of AnthropologyPennsylvania State UniversityUniversity ParkUSA
  3. 3.Department of Mechanical and Nuclear EngineeringPennsylvania State UniversityUniversity ParkUSA
  4. 4.Department of Biomedical EngineeringPennsylvania State UniversityUniversity ParkUSA
  5. 5.Institute for CybersciencePennsylvania State UniversityUniversity ParkUSA

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