Clinical Neuroradiology

, Volume 29, Issue 4, pp 763–774 | Cite as

The unexplained success of stentplasty vasospasm treatment

Insights using Mechanistic Mathematical Modeling
  • P. BhogalEmail author
  • G. Pederzani
  • A. Grytsan
  • Y. Loh
  • P. A. Brouwer
  • T. Andersson
  • Namrata Gundiah
  • Anne M. Robertson
  • Paul N. Watton
  • Michael Söderman
Original Article



Cerebral vasospasm (CVS) following subarachnoid hemorrhage occurs in up to 70% of patients. Recently, stents have been used to successfully treat CVS. This implies that the force required to expand spastic vessels and resolve vasospasm is lower than previously thought.


We develop a mechanistic model of the spastic arterial wall to provide insight into CVS and predict the forces required to treat it.

Material and Methods

The arterial wall is modelled as a cylindrical membrane using a constrained mixture theory that accounts for the mechanical roles of elastin, collagen and vascular smooth muscle cells (VSMC). We model the pressure diameter curve prior to CVS and predict how it changes following CVS. We propose a stretch-based damage criterion for VSMC and evaluate if several commercially available stents are able to resolve vasospasm.


The model predicts that dilatation of VSMCs beyond a threshold of mechanical failure is sufficient to resolve CVS without damage to the underlying extracellular matrix. Consistent with recent clinical observations, our model predicts that existing stents have the potential to provide sufficient outward force to successfully treat CVS and that success will be dependent on an appropriate match between stent and vessel.


Mathematical models of CVS can provide insights into biological mechanisms and explore treatment approaches. Improved understanding of the underlying mechanistic processes governing CVS and its mechanical treatment may assist in the development of dedicated stents.


Vasospasm Stent Stentplasty Mathematical modeling Vascular smooth muscle cells 



Systolic pressure (value: 16 kPa)


Ratio of unloaded thickness to unloaded radius (value: 0.2)

\(\lambda _{z }\)

Axial stretch (value: 1.3)


Material parameter elastin (value: 93.14 kPa)


Material parameter VSMC (passive) (value: 22.09 kPa)


Material parameter VSMC (active) (value: 18.07 kPa)


Material parameter medial collagen (value: 639.5 kPa)


Material parameter for adventitial collagen (value: 5115.6 kPa)


Passive stiffness factor (before vasospasm) (Value: 1)


Active stiffness factor (before vasospasm) (value: 1)

\(\lambda _{M}^{\mathrm{mean}}\)

Cell stretch at which VSMC active response is maximal (value: 1.1)

\(\lambda _{M}^{\max }\)

Maximum cell stretch at which VSMC’s active response is zero (value: 1.8)

\(\lambda _{{M_{AT}}}\)

VSMC attachment stretch (value: 1.15)

\(\lambda _{C_{M}}^{\max }\)

Maximum medial collagen fiber attachment stretch (value: 1.07)

\(\lambda _{C_{M}}^{\min }\)

Minimum medial collagen fiber attachment stretch (value: 1.0)

\(\lambda _{C_{A}}^{\max }\)

Maximum adventitial collagen fiber attachment stretch (value: 1.0)

\(\lambda _{C_{A}}^{\min }\)

Minimum adventitial collagen fiber attachment stretch (value: 0.8)



G. Pederzani is a PhD student of SofTMech, an EPSRC center for soft tissue mechanics and is funded by a PhD scholarship provided by the Department of Computer Science, University of Sheffield. P. Watton acknowledges partial support towards this work from SofTMech, UK EPSRC (EP/N014642/1).

Author Contribution

P. Bhogal developed the theory. N. Gundiah advised on cell mechanobiology. G. Pederzani, A. Grytsan, A.M. Robertson and P.N. Watton developed the mathematical model. All authors contributed to the editing and reviewing of the paper. M. Söderman is guarantor.

Compliance with ethical guidelines

Conflict of interest

P. Bhogal is co-developer and co-patent holder of the Lumenate Stent and consults for Phenox. G. Pederzani, A Grytsan, N. Gundiah, A.M. Robertson & P.N. Watton have no conflicts of interest. Y. Loh is a consultant for Balt, Neurvana and Medtronic. P. Brouwer is a consultant for Stryker, Medtronic and Cerenovus/Neuravi. T. Andersson is a consultant for Stryker, Covidien, Neuravi, Rapid Medical. M. Söderman is a consultant for Blockade, Neuravi, co-developer and co-patent holder of the Lumenate Stent.

Ethical standards

For this article no studies with human participants or animals were performed by any of the authors. All studies performed were in accordance with the ethical standards indicated in each case. For this type of study formal consent is not required.


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Copyright information

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

Authors and Affiliations

  1. 1.Department of Interventional NeuroradiologyThe Royal London HospitalLondonUK
  2. 2.Department of Computer ScienceUniversity of SheffieldSheffieldUK
  3. 3.Insigneo Institute for in silico MedicineUniversity of SheffieldSheffieldUK
  4. 4.Uniformed Services UniversityUniversity of CaliforniaLos AngelesUSA
  5. 5.Swedish Neuroscience InstituteWashingtonUSA
  6. 6.The Karolinska University HospitalStockholmSweden
  7. 7.AZ GroeningeKortrijkBelgium
  8. 8.Department of Mechanical EngineeringIndian Institute of ScienceBangaloreIndia
  9. 9.Department of Mechanical Engineering and Materials ScienceUniversity of PittsburghPittsburghUSA

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