Abstract
Background
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.
Objective
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 stretchbased damage criterion for VSMC and evaluate if several commercially available stents are able to resolve vasospasm.
Results
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.
Conclusion
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.
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Change history
23 April 2019
<Emphasis Type="Bold">Correction to:</Emphasis>
<Emphasis Type="Bold">Clin Neuroradiol 2019</Emphasis>
<ExternalRef><RefSource>https://doi.org/10.1007/s00062019007762</RefSource><RefTarget Address="10.1007/s00062019007762" TargetType="DOI"/></ExternalRef>
The original version of this article unfortunately contained a mistake. The Acknowledgements were missing. The correct information is given …
Abbreviations
 \(p_{\mathrm{sys}}\) :

Systolic pressure (value: 16 kPa)
 \(H/R\) :

Ratio of unloaded thickness to unloaded radius (value: 0.2)
 \(\lambda _{z }\) :

Axial stretch (value: 1.3)
 \(k_{E}\) :

Material parameter elastin (value: 93.14 kPa)
 \(k_{M}^{\mathrm{pass}}\) :

Material parameter VSMC (passive) (value: 22.09 kPa)
 \(k_{M}^{\mathrm{act}}\) :

Material parameter VSMC (active) (value: 18.07 kPa)
 \(k_{{C_{M}}}\) :

Material parameter medial collagen (value: 639.5 kPa)
 \(k_{{C_{A}}}\) :

Material parameter for adventitial collagen (value: 5115.6 kPa)
 \(f_{p}\) :

Passive stiffness factor (before vasospasm) (Value: 1)
 \(f_{a}\) :

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)
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Funding
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 information
Affiliations
Contributions
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.
Corresponding author
Correspondence to P. Bhogal.
Ethics declarations
Conflict of interest
P. Bhogal is codeveloper and copatent 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, codeveloper and copatent 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.
Appendices
Appendix A
The arterial wall is modelled as a nonlinear elastic cylindrical membrane [10] with thickness \(H\) and radius \(R\) in its unloaded configuration. In vivo, it is subject to an internal pressure \(p\) and axial stretch \(\lambda _{Z}\). The governing equation describing the balance of forces for mechanical equilibrium is:
Where the deformed radius \(r\) is given by \(r=\lambda R\) with \(\lambda\) the circumferential tissue stretch, the deformed thickness of the arterial wall is \(h=H/\lambda _{z }\lambda\), and \(\sigma\) denotes the circumferential Cauchy stress. The arterial wall constituents are modelled as a constrained mixture, so that the total (Cauchy) stress \(\sigma\) component for the arterial wall is the sum of stress contributions from the individual constituents, i. e.
where \(\sigma _{E}\), \(\sigma _{CM}\), \(\sigma _{CA}\), \(\sigma _{\text{VSMCp}}\) and \(\sigma _{\text{VSMCa}}\) denote the circumferential Cauchy stress contributions of elastin (E), medial collagen (CM), adventitial collagen (CA), passive and active VSMCs, respectively. We exclude the mechanical role of endothelial cells due to their negligible role in the mechanical behavior of the vessel [30].
We utilize a strain energy function for the collagen that accounts for the recruitment distribution of collagen fibers [13, 16]
where the distribution of recruitment stretches \(\rho \left(\lambda _{{R_{J}}}\right)\) is taken to be a triangular distribution function [12, 14] and the mechanical response of an individual fiber is linear, i. e.
Analytical expressions for the collagen Cauchy stresses [12] can then be obtained via,
The mechanical responses of elastin and passive VSMCs are modelled using neoHookean constitutive models [9] and it follows that
whilst the active response of VSMCs follows [31, 32]
The material parameters \(k_{E},k_{M}^{\mathrm{pass} }, k_{M}^{\mathrm{act}}, k_{C}\) are inferred by prescribing the proportion of load borne by the constituent in the initial homeostatic configuration [33,34,35]. We assume that at systolic pressure the stretch \(\lambda =1.3\), the elastin bears 70% of the load, passive VSMC bears 10% of the load, active VSMC bears 10% of the load medial collagen bears 10% of the load; adventitial collagen is unrecruited so doesn’t bear any load. Note \(f_{p}, f_{a}\) are dimensionless parameters for increasing the passive and active stress responses following vasospasm to maintain mechanical equilibrium.
Natural Reference Configurations
The mechanical responses are defined as functions of the stretches that individual constituents experience, i. e.
where λ denotes the tissue stretch, \(\lambda _{J}\) denotes the stretches of medial and adventitial collagen (J =C_{M},C_{A}) and VSMC (J =S); the recruitment stretches \(\lambda _{{R_{J}}}\) define the onset of load bearing (Fig. A.1).
Initial values for recruitment stretches are obtained by prescribing the attachment stretches of constituents (or VSMC) in the (prevasospasm) loaded configuration, i. e.
For collagen, each (symmetric) triangular distribution of fibers requires specification of maximum/minimum fiber attachment stretches, i. e. \(\lambda _{C_{J}}^{\max }\) and \(\lambda _{C_{J}}^{\min }\), respectively. We model distinct distributions so that medial collagen bears load in homeostatic configuration, while adventitial collagen is modelled as playing a purely protective role, i. e. begins to bear load after systole.
Appendix B
In vasospasm, the contraction of the VSMC reduces the vessel crosssection, removing load from the ECM components as the stretch of these components drops below one. As a result, the VMSC must bear all of the load, albeit at a smaller diameter. We assume that in the time between vasospasm onset and treatment the mechanical contribution from collagen and elastin remains negligible in comparison to the VSMC contributions (despite possible remodeling of collagen fibers). Although the VSMCs must bear the entire transmural load, the relative contributions of the active and passive components may shift in time. The degree and timing of this shift will depend on a number of factors involving the damage to the VSMC during treatment as well as the VSCM remodeling process for the individual patient. In this Appendix, we do not directly model these changes, but rather perform a parametric study to conservatively explore the possible impact that the largest possible variation in relative load bearing would have on predicted stent requirements.
In the parametric study, the relative roles of active and passive VSMC are controlled by varying parameters \(f_{p}\) and \(f_{a}\) (see Discussion for a biological interpretation of these changes). Both parameters are greater than or equal to 1 and are coupled in that their combined influence maintains equilibrium of the vessel. In the main body of the text, a specific case (\(f_{p}=1.2, f_{a}=1.475)\) was considered. Here results are obtained over the entire parameter space. To do so, we first obtained the solutions for the two extreme cases with \(f_{p}\) = 1, and \(f_{a}=1\), respectively. In particular, if the passive response is not affected (\(f_{p}=1\)), then \(f_{a}\) must increase by 75%; conversely, if \(f_{a}=1\), then \(f_{p}\) increases by 55%, Fig. B.1. We then gradually increased \(f_{p}\) from 1 to 1.55 (with \(f_{p}=1.55\) corresponding to \(f_{a}=1)\) by small constant increments and solved the equation of mechanical equilibrium to obtain the relationship between \(f_{a}\) and \(f_{p}\) shown in Fig. B.2.
For each pair of values, we then computed the critical pressure, i. e. the pressure that should be applied to the arterial wall to cross the failure threshold of VSMCs (Fig. B.2, above). Subtracting systolic blood pressure (16 kPa) from the critical pressure, we obtained the additional pressure, namely the amount of pressure an interventional device should provide in order to treat vasospasm (Fig. B.2, below).
The results show that the minimum additional pressure a device should provide is about 5 kPa, while the maximum is about 11 kPa. Therefore, at most, the predictions for required stent pressure would increase to 11 kPa. Hence, the conclusion that pressures needed to treat vasospasm are attainable by stents and are an order of magnitude below the pressure used during balloon angioplasty treatment still holds.
Appendix C
We start from a table of measurements of the chronic outward force (COF) exerted by the stents, expressed in N/mm; this represents the hoop force exerted by the stent divided by its length. Since the hoop force is measured in the circumferential direction, we use the identities in [20] to obtain:
where \(P\) is the pressure exerted by the stent, \(l\) the length of the stent and \(r\) the radius. We illustrate the COFdiameter relationship for the considered stents in Fig. C.1 and the corresponding pressurediameter curves in Fig. C.2.
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Bhogal, P., Pederzani, G., Grytsan, A. et al. The unexplained success of stentplasty vasospasm treatment. Clin Neuroradiol 29, 763–774 (2019) doi:10.1007/s00062019007762
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Keywords
 Vasospasm
 Stent
 Stentplasty
 Mathematical modeling
 Vascular smooth muscle cells