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


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.

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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

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Correspondence to Reuben H. Kraft.

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Appendix A: Experimental data

High-resolution micro-computed tomography (\(\mu \hbox {CT}\)) and magnetic resonance microscopy (MRM) images of embryonic mice serve as experimental data in our analyses. All use of mice was in compliance with animal welfare guidelines approved by the Pennsylvania State University Institutional Animal Care and Use Committees. \(\mu \hbox {CT}\) images with pixel size and slice thickness ranging from 0.0148 to 0.0168 mm were acquired by the Center for Quantitative Imaging at the Pennsylvania State University ( using the HD-600 OMNI-X high-resolution X-ray computed tomography system (Varian Medical Systems, Inc., Lincolnshire, IL). Image data were reconstructed on a \(1024 \times 1024\) pixel grid as a 16-bit tiff but were reduced to 8-bit for image analysis. Isosurfaces were reconstructed to represent all cranial bone at indicated ages based on hydroxyapatite phantoms imaged with the specimens using the software package Avizo 8.1.1 (FEI Company, Inc.). The minimum thresholds used to create the isosurfaces ranged from 70 to \(100\ \hbox {mg}/\hbox {cm}^{3}\) partial density hydroxyapatite. MRM images were acquired by the High Field MRI Facility at the Pennsylvania State University ( The fixed specimens were immersed in 2% Magnevist (Bayer Health Care, Wayne, NJ) phosphor-buffered solution (PBS) for 7–10 days depending upon the embryonic age of the specimen to reduce the T1 and T2 relaxation times. All MRM experiments were conducted on a vertical 14.1 Tesla Varian (Varian Inc., Palo Alto, CA) imaging system with direct drive technology. To prevent drying and to minimize magnetic susceptibility artifacts during scanning, specimens were immersed in fluorinert liquid, FC-43 (3M, St. Paul, MN). A standard imaging experiment with an isotropic resolution of \(80\ \upmu \hbox {m}\) comprised a field of view of \(15.4 \times 14 \times 11\ \hbox {mm}^{3}\) and a matrix size of \(192 \times 132\,(75\%\ {\hbox {partial}}\,{\hbox {Fourier:}}\,176)\ \times 137\). With eight averages and a repetition time of 75 ms (echo time 25 ms), the total scan time was 3 h. MATLAB (The MathWorks, Inc., Natick, MA) was used for image post-processing. By zero-filling all directions by a factor of two, the pixel resolution of a standard imaging experiment was \(40\ \upmu \hbox {m}^3\).

Appendix B: Assumption test

To test our assumptions and examine the effects of each in detail, we compare the simulation results of activator at E17.5 estimated using our computational model with and without each assumption (Fig. 7).

Fig. 7

Distribution of osteoblasts at E17.5 from simulation results using various computational models. a Result using a model with only assumptions A1 and A2, without the assumption pertaining to mechanical effects. Bones grow and form sutures through reaction–diffusion process. b Result using a model with assumptions A1, A2, and only the assumption about the mechanical effect on production of activator (A3a). Locations of primary ossification centers are specified by the mechanical effect on the production of activator. c Result using a model with all assumptions A1, A2, and A3. Relative speed of bone growth is achieved by the mechanical effect on cell differentiation (A3b)

Figure 7a shows the results estimated using the model with only A1 and A2; mechanical effects on molecular expression and cell differentiation are not considered. As predicted, primary ossification centers form and bone grows from them with sutures forming between the bones. The effect of the inhibitor appears to play a role in suture formation. The number and locations of bones do not agree with experimental observations and reveal a disorganized pattern of bone formation. Figure 7b shows the results estimated using the model with only A1, A2, and A3a, excluding the assumption of the mechanical effect on cell differentiation. Six bones and associated sutures form in locations similar to our experimental observations as predicted. Consequently, the mechanical effect on the production of activator appears to play a role in specifying the number and location of the primary ossification centers. However, limiting the computation to these assumptions results in a similar growing speed across all bones so that their final volumes are similar, a result that does not match experimental observations. Figure 7c shows the result estimated using the model with all assumptions: A1, A2, A3a, and A3b. The results from the coupled model show that apical growth of the frontal and parietal bones is delayed, as observed in experimental animals. Moreover, growth of the interparietal bone is constrained superoinferiorly relative to the growth of the frontal and parietal bones, modeling what is observed in experimental data. The model reveals that localized restriction of growth of the interparietal is due to reduced accumulated volumetric strain apical to the bone (Fig. 4b). These results reveal the importance of local strain in determining the relative growing speed and final shape of each cranial vault bone.

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Lee, C., Richtsmeier, J.T. & Kraft, R.H. A coupled reaction–diffusion–strain model predicts cranial vault formation in development and disease. Biomech Model Mechanobiol 18, 1197–1211 (2019) doi:10.1007/s10237-019-01139-z

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