Ensuring Congruency in Multiscale Modeling: Towards Linking Agent Based and Continuum Biomechanical Models of Arterial Adaptation

Abstract

There is a need to develop multiscale models of vascular adaptations to understand tissue-level manifestations of cellular level mechanisms. Continuum-based biomechanical models are well suited for relating blood pressures and flows to stress-mediated changes in geometry and properties, but less so for describing underlying mechanobiological processes. Discrete stochastic agent-based models are well suited for representing biological processes at a cellular level, but not for describing tissue-level mechanical changes. We present here a conceptually new approach to facilitate the coupling of continuum and agent-based models. Because of ubiquitous limitations in both the tissue- and cell-level data from which one derives constitutive relations for continuum models and rule-sets for agent-based models, we suggest that model verification should enforce congruency across scales. That is, multiscale model parameters initially determined from data sets representing different scales should be refined, when possible, to ensure that common outputs are consistent. Potential advantages of this approach are illustrated by comparing simulated aortic responses to a sustained increase in blood pressure predicted by continuum and agent-based models both before and after instituting a genetic algorithm to refine 16 objectively bounded model parameters. We show that congruency-based parameter refinement not only yielded increased consistency across scales, it also yielded predictions that are closer to in vivo observations.

Appendix

 

Symbol (units)	Classification	Definition
$K_{{\sigma_{\theta } }}^{\text{c}}$	CMM: Bounded	Gain parameter governing circumferential-stress mediated rate of production of collagen. See Eq. (1)
$K_{{\sigma_{\theta } }}^{\text{m}}$	CMM: Bounded	Gain parameter governing circumferential-stress mediated rate of production of smooth muscle. See Eq. (1)
$K_{{\tau_{w} }}^{\text{c}}$	CMM: Bounded	Gain parameter governing shear-stress mediated rate of production of collagen. See Eq. (1)
$K_{{\tau_{w} }}^{\text{m}}$	CMM: Bounded	Gain parameter governing shear-stress mediated rate of production of smooth muscle. See Eq. (1)
MMP-10 (pg)	ABM: Bounded	Baseline mass of MMP-1
MMP-1%A (%)	ABM: Bounded	Percent of MMP-1 that is active
C0 (pg)	ABM: Bounded	Baseline mass of collagen
C TGF	ABM: Bounded	Rate parameter governing TBFb mediated rate of production of collagen (pg of collagen per pg of TGFb)
M p	ABM: Bounded	Rate parameter governing PDGF mediated SMC proliferation rate (pg of SMC per pg of PDGF)
M 0 (pg)	ABM: Bounded	Baseline SMC proliferation rate (pg)
M a1	ABM: Bounded	Apoptosis chance for SMC (1)
M a2	ABM: Bounded	Baseline apoptosis chance for SMC (2)
${\text{PDGF}}_{{\sigma_{\theta } }}$ (pg/kPa)	ABM: Bounded	Rate parameter governing hoop stress mediated rate of production of PDGF (pg of PDGF per kPa of stress)
PDGF0 (pg)	ABM: Bounded	Baseline mass of PDGF
${\text{TGF}}\beta_{{\sigma_{\theta } }}$ (pg)	ABM: Bounded	Rate parameter governing hoop stress mediated rate of production of TGFb (pg of TGFb per kPa of stress)
TGFβ0 (pg)	ABM: Bounded	Baseline mass of TGFb
$G_{h}^{\text{e}}$	CMM: Observed	Homeostatic stretch when elastin is deposited
$G_{h}^{\text{c}}$	CMM: Observed	Homeostatic stretch when collagen is deposited
$G_{h}^{\text{m}}$	CMM: Observed	Homeostatic stretch when SMC is deposited
$c^{\text{e}}$ (kPa)	CMM: Calculated	Neo-Hookean material parameter for the stored energy of elastin
$c_{1}^{\text{c}}$ (kPa)	CMM: Calculated	Fung-type exponential material parameter for the stored energy of collagen (1)
$c_{1}^{\text{m}}$ (kPa)	CMM: Calculated	Fung-type exponential material parameter for the stored energy of SMC (1)
$c_{2}^{\text{c}}$	CMM: Observed	Fung-type exponential material parameter for the stored energy of collagen (2)
$c_{2}^{\text{m}}$	CMM: Observed	Fung-type exponential material parameter for the stored energy of SMC (2)
$T_{\text{M}}$ (kPa)	CMM: Observed	Maximum stress generated by SMC
$\lambda_{\text{M}}$	CMM: Observed	Circumferential stretch where active stress is maximum
$\lambda_{0}$	CMM: Observed	Circumferential stretch where active stress is zero
$C_{\text{B}}$	CMM: Observed	Material parameter for the ratio of constrictor concentration to dilator concentration
$C_{\text{S}}$	CMM: Observed	Scaling parameter for shear stress induced change in constrictor concentration scaling factor
$\phi^{\text{c}} \left( 0 \right)$	CMM: Observed	Initial mass fraction of collagen in the arterial wall
$\phi^{\text{e}} \left( 0 \right)$	CMM: Observed	Initial mass fraction of elastin in the arterial wall
$\phi^{\text{m}} \left( 0 \right)$	CMM: Observed	Initial mass fraction of SMC in the arterial wall
$K_{h}^{\text{m}}$ (days−1)	CMM: Observed	Half-life of SMC
$K_{h}^{\text{c}}$ (days−1)	CMM: Observed	Half-life of collagen
NO0 (pg)	ABM: Observed	Baseline mass of NO
NO τw	ABM: Observed	Rate parameter governing shear stress mediated rate of production of NO (pg of NO per kPa of stress)
$\delta^{\text{PDGF}}$ (pg)	ABM: Observed	Parameters of the sigmoid function:
$\alpha^{\text{PDGF}}$		${\text{PDGF}} = M\left( {\delta + \alpha \left( {1 - e^{{ - kx^{n} }} } \right)} \right)$
$k^{\text{PDGF}}$ (kPa−1)
$n^{\text{PDGF}}$
$M^{\text{PDGF}}$		Maximum rate of PDGF production
$\delta^{{{\text{ET{-}}}1}}$ (pg)	ABM: Observed	Parameters of the sigmoid function:
$\alpha^{{{\text{ET{-}}}1}}$		${\text{ET}}1 = M\left( {\delta + \alpha \left( {1 - e^{{ - kx^{n} }} } \right)} \right)$
$k^{{{\text{ET{-}}}1}}$ (kPa−1)
$n^{{{\text{ET{-}}}1}}$
$M^{{{\text{ET{-}}}1}}$		Maximum rate of ET-1 production
$\delta^{{{\text{MMP{-}}}2}}$ (pg)	ABM: Observed	Parameters of the sigmoid function:
$\alpha^{{{\text{MMP{-}}}2}}$		${\text{MMP}}2 = A\left( {M\left( {\delta + \alpha \left( {1 - e^{{ - kx^{n} }} } \right)} \right)} \right)$
$k^{{{\text{MMP{-}}}2}}$ (kPa−1)
$n^{{{\text{MMP{-}}}2}}$
$M^{{{\text{MMP{-}}}2}}$		Maximum rate of MMP-2 production
$A^{{{\text{MMP{-}}}2}}$ (%)		Percent of MMP-2 Active
$D^{{{\text{MMP{-}}}2}}$	ABM: Observed	Degradation rate of MMP-2
$\delta^{{{\text{MMP{-}}}9}}$ (pg)	ABM: Observed	Parameters of the sigmoid function:
$\delta^{\text{MMP{-}9}}$		${\text{MMP}}9 = A\left( {M\left( {\delta + \alpha \left( {1 - e^{{ - kx^{n} }} } \right)} \right)} \right)$
$k^{{{\text{MMP{-}}}2}}$ (kPa−1)
$n^{{{\text{MMP{-}}}9}}$
$M^{{{\text{MMP{-}}}9}}$		Maximum rate of MMP-9 production
$A^{{{\text{MMP{-}}}9}}$ (%)		Percent of MMP-9 active
$D^{{{\text{MMP{-}}}9}}$	ABM: Observed	Degradation rate of MMP-9
$D^{{{\text{MMP{-}}}1}}$	ABM: Observed	Degradation rate of MMP-1

1. Note: Parameters within the ABM or CMM are classified as observed, bounded, or calculated. Observed parameters were either cited elsewhere or obtained via direct measurements or fits to experimental data. Bounded parameters are less well known, but are expected to fall within specified range. Lastly, calculated parameters include those that are needed to satisfy equilibrium under homeostatic conditions.