A biochemomechanical model of collagen turnover in arterial adaptations to hemodynamic loading

The production, removal, and remodeling of fibrillar collagen is fundamental to mechanical homeostasis in arteries, including dynamic morphological and microstructural changes that occur in response to sustained changes in blood flow and pressure under physiological conditions. These dynamic processes involve complex, coupled biological, chemical, and mechanical mechanisms that are not completely understood. Nevertheless, recent simulations using constrained mixture models with phenomenologically motivated constitutive relations have proven able to predict salient features of the progression of certain vascular adaptations as well as disease processes. Collagen turnover is modeled, in part, via stress-dependent changes in collagen half-life, typically within the range of 10–70 days. By contrast, in this work we introduce a biochemomechanical approach to model the cellular synthesis of procollagen as well as its transition from an intermediate state of assembled microfibrils to mature cross-linked fibers, with mechano-regulated removal. The resulting model can simulate temporal changes in geometry, composition, and stress during early vascular adaptation (weeks to months) for modest changes in blood flow or pressure. It is shown that these simulations capture salient features from data presented in the literature from different animal models.


Introduction
Blood vessels exhibit a remarkable ability to adapt to changes in bio-chemo-mechanical stimuli through evolving vasoactivity and matrix turnover, which can change the luminal diameter, wall thickness, axial stretch, and key mechanical metrics like wall stress and stiffness.The two primary structural constituents within blood vessels are elastic fibers and fibrillar collagens.Elastin endows the wall with an ability to store elastic energy during systole, which can be used to work on the blood during diastole and augment blood flow.Functional elastin is produced primarily before adulthood and under normal conditions has an extremely long half-life, on the order of 50 years.Hence, the normally slow turnover rate of elastin, less than 1% per year (Wagenseil and Mecham 2009), is typically negligible.By contrast, fibrillar collagens endow the arterial wall with stiffness and strength.These fibers are synthesized and degraded continuously, with a normal half-life on the order of 70 days.Their turnover thus contributes both to maintaining arterial integrity over long periods and facilitating adaptations to altered hemodynamic loads.The specific turnover rate can vary with disease and injury, indeed even with changes in mechanical load.Differences between the rates of synthesis and degradation of collagen can lead to fibrosis (as in aging or hypertension) or weakening of the wall (as in dissection or rupture).There is a pressing need, therefore, for computational models that capture possible imbalances in collagen turnover over time (Bishop et al. 1994;Bruno et al. 1998;Chyatte and Lewis 1997;Majamaa and Myllyla 1993;Rodriguez-Feo et al. 2005;Sluijter et al. 2004a).
One of the most important stimulators of procollagen production by intramural cells is the cytokine transforming growth factor (TGF)-β, which also affects smooth muscle cell (SMC) proliferation (Bishop 1998;Suwanabol et al. 2012).Conversely, the most important contributors to collagen removal are matrix metalloproteinases (MMPs), with actual removal depending on balances or imbalances between the activation of latent MMPs and tissue inhibitors of metalloproteinases (TIMPs).Computational models of soft-tissue growth (changes in mass) and remodeling (changes in structure) have included effects of TGF-β and MMPs (Gierig et al. 2021;Sáez et al. 2013), but focused on deposition and degradation of mature collagen fibers.That is, this class of models has not included intracellular synthesis/degradation or the transition to matured collagen fibers.
Over the past two decades, constrained mixture models of arterial homeostasis and pathogenesis have proven capable of describing and predicting salient features of the progression of diverse vascular responses and conditions (Baek et al. 2007a, b;Braeu et al. 2017;Humphrey 2021;Wagenseil 2011).Briefly, these models assume that blood vessels are solid-bounded mixtures wherein individual constituents are incorporated within extant matrix under a pre-stretch and subjected thereafter to vasoactive changes and stressmediated growth and remodeling (G&R).Although these stress-mediated models can capture arterial adaptations and maladaptations over weeks to months or years, the underlying constitutive relations are yet phenomenological (Latorre and Humphrey 2020;Valentín et al. 2009).By contrast, this paper seeks to extend a previous constrained mixture model of short-term vascular adaptations by incorporating knowledge of the intracellular and extracellular biochemical processes of collagen production and removal.
2 Brief review of flow-and pressure-induced adaptations

Mechanosensitive behaviors
Changes in blood pressure and/or flow under physiological conditions can induce modest alterations in arterial stress (e.g., intramural axial and circumferential stress and luminal wall shear stress), which in turn can change the gene expression profile of the vascular cells (including endothelial cells, SMCs, and fibroblasts) and thereby lead to changes in vascular tone and, if sustained, turnover of extracellular matrix (ECM).Such changes allow the vessel to adapt, which is often homeostatic if the perturbations are modest (Humphrey 2008).For a normal artery in humans, wall shear stress and circumferential stress tend to be maintained at certain levels, ~ 1.5 Pa for the former (Ando and Yamamoto 2013) and ~ 150 kPa for the latter (Fridez et al. 2003).
Stress analyses that include residual stress, basal smooth muscle tone, and nonlinear wall properties suggest that the distribution of axial and circumferential wall stress tend to be nearly uniform across the wall in normalcy (Humphrey 2002).Mean values of key stresses are: where f 1 , P , and Q are the axial force, transmural pressure, and volumetric flow rate; r i and h are the inner radius and thickness of the artery; is the viscosity of the blood; and 1 , 2 are the axial and circumferential intramural stresses, respectively, and w is the wall shear stress that acts on the endothelial surface.
Although orders of magnitude smaller than the intramural stresses, flow-induced wall shear stress is an important mechano-stimulus for the regulation of vasoactive molecules produced by the endothelial cells, which in turn control the vasoactive tone of the SMCs.Increases in flow tend to upregulate endothelial-derived nitric oxide synthase, which catalyzes the synthesis and secretion of the potent vasodilator nitric oxide (NO), which causes SMC relaxation and thus vasodilatation (Hsiai 2008;Lu and Kassab 2011;Moncada et al. 1989).Conversely, decreases in flow tend to upregulate endothelial-derived endothelin-1 (ET-1), which causes SMC contraction under physiological conditions and thus vasoconstriction (Kuchan and Frangos 1993;Morawietz et al. 2000).Both of these changes determine the state in which ECM turns over within the arterial wall.

Arterial collagen synthesis and degradation
Collagen fibrils and then fibers are formed via complex biosynthetic pathways involving intracellular regulation and synthesis plus extracellular post-translational modifications, each with respective time scales (Berg et al. 1980;Trackman 2005).Importantly, SMC production of collagen tends to be attenuated by NO but augmented by ET-1 (Humphrey 2008).In this way, endothelial sensing of sustained changes in flow leads not only to a change in lumen caliber but also a change in collagen turnover within the altered mechanical state (Rudic et al. 1998).That is, if an altered flow is sustained over a long period, G&R can occur in a dilated or constricted configuration and thereby help adapt the wall to the new mechanical stimulus.
Changes in circumferential and axial stress appear to be sensed directly by SMCs, which then change their gene expression profile (Bishop 1998;Chiquet 1999).Amongst the many different changes, increased wall stress leads to an increased SMC production of ECM, particularly collagen (Sasamura et al. 2005).This increase in collagen production is driven by various stress-mediated biomolecules, including TGF-β (Ryan et al. 2003).Hence, if a pressure-induced increase in wall stress persists for a modest period, say days to weeks, the arterial wall will thicken due to increase in collagen production, both in the media and adventitia (Fridez et al. 2003;Matsumoto and (1) Studies also indicate that excess stretch or stress suppresses MMP production and their effectiveness (Maeda et al. 2013), thus reducing collagen degradation.
The active mechanosensitive processes within vascular cells depend in large part on a physical linkage of the ECM to the cytoskeletal structure (Davis and Hill 1999;Liu and Lin 2022).Transmembrane adhesion molecules, that is, integrins, play a central role in this linkage, enabling cells both to mechano-sense and mechano-regulate their ECM in response to changes in mechanical stimuli (Bishop 1998;Chiquet 1999;Lehoux et al. 2006).Although the resulting mechano-transduction signals are complex and control a host of transcriptional changes, we focus on the turnover of fibrillar collagen, and in particular collagen I which constitutes about 60-70% of all vascular collagen (Sasamura et al. 2005).The synthesis, deposition, and degradation of fibrillar collagen represent a complex series of processes, simplified and summarized in Fig. 1.
Following mechanotransduction, associated mRNAs within the nucleus are transported to cytoplasmic ribosomes (Laurent 1987).The appropriate amino acids (over 1000 per chain) are then sequenced in a (G-X-Y) n motif to form an α-chain, where G is glycine and X and Y are often proline or hydroxyproline but any of a number of amino acids.Three α-chains are combined to form procollagen, a triple (super) helix molecule.Prepared in the Golgi apparatus, the procollagen molecules are transported in vesicles that can coalesce at the cell membrane and be secreted in a mechano-regulated fashion.Once in the extracellular space, propeptides are cleaved from the ends and the tropocollagen molecules (or simply collagen, ~ 1.5 nm diameter and 300 nm long with a MW ~ 300,000 daltons) (Jawad and Brown 2011;Nimni 1992) are assembled to form micro-fibrils with a release of water molecules; subsequent Not all of the synthesized procollagen is secreted, however.About 10-50% (Goldberg 2003;McAnulty and Laurent 1987;Schubert et al. 2000), with some estimates of 10-20% (Berg 1986) or 30-40% (Bienkowski et al. 1978), is degraded inside the cell within minutes of synthesis.Such natural intracellular degradation represents a type of quality control, ensuring that secreted molecules contain the precise chemical structure to ensure effective incorporation into structural fibers (Goldberg 2003;Ishida et al. 2009).The secreted procollagen that is converted into tropocollagen (Kao et al. 1979) aggregates within about 20 min (Gelman et al. 1997).There is a general claim that procollagen is converted into collagen around the moment of exocytosis (Berg 1986).Daily, 3-10% of the extractable collagen is degraded (Sodek 1976).
From the remaining extractable collagen, about 1-8% per day is cross-linked and converted into mature cross-linked collagen fibers (Niedermüller et al. 1977), which have a normal half-life from 50 to 100 days (Gineyts et al. 2000;Nissen et al. 1978).According to Nimini (1992), formation of individual α-chains, assembly into a triple helix motif, and packaging and secretion to the extracellular space take on the order of ~ 7, ~ 8, and ~ 20 min, respectively.Not including time for gene transcription, and depending on the specific tissue and collagen type, it takes from about 30 min to 24 h to produce collagen (Marchi and Leblond 1983;Nimni 1992).Although the extracellular cross-linking can occur quickly, the overall degree of cross-linking may change over longer periods (hours to weeks or months) as the fibers 'mature'.For example, after chronic flow reduction in mice, a permanent change in arterial structure may be noticeable as early as 3 days, with changes persisting for weeks or more (Rudic et al. 2000).
Extracellular collagen can be degraded via two different pathways: intracellular or extracellular.The intracellular pathway may dominate under physiological conditions (Everts et al. 1996) and is accomplished via a phagocytotic ingestion by the cells.By contrast, the extracellular pathway appears to dominate in cases of disease, injury, and perhaps even perturbations in mechanical loading; it is affected via a variety of enzymes, including MMPs, serine proteases, and cysteine proteases.MMPs appear to be the major class of enzymes responsible for the degradation of collagen fibers in most cases of arterial G&R (Chase and Newby 2003).

Biochemomechanical model
We first present a new chemical kinetics model for collagen production, assembly, and removal in Sect.3.1 and then summarize the existing constrained mixture model in Sect.3.2.The new biochemical model is coupled with an existing constrained mixture model to form a biochemomechanical model of vascular adaptation in Sec.3.3.Finally, simulations are presented in Sect.3.4.

A Simplified biochemical model of collagen turnover
Although the chemical steps of collagen synthesis are now understood and detailed kinetics models could be linked directly to vascular G&R using parameter estimation, it is still difficult to estimate specific reaction rates for each step.Hence, rather than developing a detailed reaction model, with many unknown parameters, we simplify the reactions from intracellular procollagen production to mature collagen assembly via two steps that include four net parameters.
As shown in Fig. 1, the extracellular collagen is classified into an intermediate state, which represents assembled microfibrils, and a final state, which represents fibrillar collagen with mature cross-links.Similar to prior work (Niedermüller et al. 1977), we consider first-order reactions, with an initial rate of supply m p (mol/m 3 sec) scaled by ∈ [0, 1] (fraction of procollagen that is released to the extracellular space), a conversion from intermediate to final parameterized by rate β, and degradation in each of the two extracellular states parameterized by rates 1 and 2 , respectively.The governing equations are thus: where C I and C F are molar concentrations (mol/m 3 ) of inter- mediate and cross-linked matured collagen.Values of , , 1 , and 2 with their corresponding references are in Table 1.Measurement of collagen turnover without considering reutilization of isotopic precursors used to label the collagen can result in longer than actual turnover times (Laurent (2)  (1978) 1987; Sodek and Ferrrier 1988); hence, actual rates may be greater than the values indicated in Table 1.The mean age of the mature collagen is thus given by a mean = 1∕ 2 = 100 days, with 2 constant ( 0.01 day −1 ).
Although degradation of mature fibers depends on additional variables, such as the number of cross-links of the fibers (both enzymatic and non-enzymatic, which typically change with aging), we only use the current state of stress and concentrations of enzymes (mainly MMPs) to model degradation.It appears, for example, that increasing the concentration of MMPs accelerates the rate of degradation while perturbing stress can alter the conformation of the collagen fibers and influence MMP efficacy (Ruberti and Hallab 2005).We also assume that only the cross-linked mature collagen contributes to mechanical load bearing (i.e., strain energy function) in the biomechanical model.

Biomechanical model
The continuum theory of constrained mixtures is well suited for describing the behavior of arteries (Ateshian and Humphrey 2012;Humphrey and Rajagopal 2002), though it is currently not possible to capture the chemical or mechanical contributions of all of the ECM components (there are on the order of 100 different proteins, glycoproteins, and glycosaminoglycans within the arterial wall).Hence, we focus on the three primary structurally significant constituents: elastin, collagen fibers, and SMCs (Bank et al. 1996;Figueroa et al. 2009).The actual complexity of arterial mechanobiology, in view of the large number of constituents that could be tracked, necessitates judicious choices that capture salient responses of the mixture by following only the key constituents.Thus, let material properties at each place x(t) in the mixture configuration t (B) be modeled by assuming that, in a homogenized sense, multiple constituents co-exist within local neighborhoods and at each time t.
We denote constituents SMCs, collagen fibers (using a 4-fiber family model with angle α), and elastin via i ∈ [m, k, e] , whereby we assume that SMCs are oriented in the circumferential direction, collagen has four families in axial, circumferential, and symmetric diagonal directions ( k = 1, 2, 3, 4), and elastin is an amorphous material.We also assume the same mechanical properties for multiple families k of locally parallel collagen fibers.In this biomechanical model, the time variable ∈ [0, s] is the past time at which the newly deposited cross-linked collagen began to carry stress relative to the current G&R time t .Multiple configura- tions, R (B) and i n( ) , and their transformations from one to other configurations have been defined previously (Baek et al. 2007a, b;Valentín et al. 2009) (Fig. 2).Let the deformation of the ith constituent that was produced at time be described by the linear transformation F i n( ) (t): where F at time t or describes mixture-level deformations and G i h captures individual (homeostatic) pre-stretches at which constituent i is deposited.
For modeling deformations of an idealized artery during G&R, we assume an axisymmetric thin wall with F(t) is given by diag 1 (t), 2 (t) , where 1 (t) = l(t)∕l h , 2 (t) = r(t)∕r h , with l axial length and r the mean radius.F i n( ) , which can be written as diag( i n,1 , i n,2 ) , transforms vectors that belong to the tangent space at x n( ) ∈ n( ) (B) to the tangent space at x ∈ t (B) for the i th constituent that was produced at time ≤ t .Again, following prior work (Baek et al. 2007a, b;Valentín et al. 2009), the strain energy is given as where m i R ( ) > 0 is the rate at which new structural constitu- ents i are produced at time , with a i max the maximum age of constituent i .In this paper, any quantity that is denoted by (⋅) R is expressed relative to a fixed reference configura- tion.The q i (t, ) ∈ [0, 1] is the fraction of constituent i that is produced at time and survives to time t .It is inspired by population dynamics (Thieme 2003) and can be expressed as Functional forms for m i R ( ) and i 2 (s) will be derived in the next section.Finally, Ψ i is the strain energy of a constitu- ent i per unit mass, which can be computed via Because newly produced material could have a different material symmetry than extant material, the material (4) (5) where J = 1 2 denoting the membrane stress due to vascu- lar smooth muscle tone by T act , the total Cauchy membrane stress of a vasoactive vessel is: where T pass is the passive Cauchy membrane stress and T act is computed, by slightly modified constitutive relations from previous work (Baek et al. 2007a, b;Rachev and Hayashi 1999), to improve the shape of wall shear stress dependent active tone: where S basal is the basal vasoactive tone, M m R is the total mass of SMCs, k ton is a scaling constant, and wh is the homeo- static wall shear stress.Moreover, M and 0 are the stretches at which the contraction is maximum and zero, respectively, and act 2 is an active stretch that is computed from (Baek et al. 2007a, b): where K act is constant.
Finally, the normal Cauchy stress equations in Eq. ( 1) can be re-written as:

Biochemomechanical model
In Sects.3.1 and 3.2, we separately presented the biochemical and biomechanical models.To integrate these models into a single biochemomechanical model, we need a common computational configuration.Let R (B) be a fixed reference configuration (at reference position X) that is mapped into in vivo configurations (e.g., t (B) ) at different times during G&R.The use of fixed-reference configuration allows us to use a material description when modeling (15) is used for a computational reference as an initial geometry in homeostatic conditions.The natural configurations, i n( ) are stress-free configurations of constituent i at time .b Deformation of an arterial segment from the idealized thin wall in a homeostatic condition to the deformed wall at t both the kinetics of mass turnover and changes in strain energy without considering volume or area changes.
Let the synthesis rate of intracellular procollagen m pR (t) [Fig. 1, Eq. ( 2)] be given by a stress-dependent scalar function depending on time t.
where h is the homeostatic normal stress and K and K w are gain-type parameters; m k pR (t) is the rate of procollagen mass production and M c R (t) is collagen mass.The ratio M c R (t)∕M c R (0) comes from an assumption that the total cell numbers is increased proportionally during G&R and that each cell produces procollagen (Baek et al. 2005).
After synthesis via m k pR (t) , a change in intermediate col- lagen, C k IR , is given by a differential equation, In order to utilize the reaction equations from intermediate to mature collagen, we consider the production of matured collagen, meaning conversion from the intermediate collagen, which is soluble, to cross-linked collagen fibers, k, where the mass production rate m k R ( ) at time is given by where MW c is the molecular weight of collagen.Written in this way, information from the 'reaction kinetics' can be incorporated directly within the mixture formulation, thus providing additional guidance on reasonable constitutive relations for constituent production and removal.The mass of matured cross-linked collagen fiber family k (and the total mass of collagen, c ) per unit reference area is given by Scalar measures of wall stress and thickness in the collagenous fiber families are given by: where T c = ∑ k T k , y k is unit vector in the direction of colla- gen family k, and f is the mass fraction of fluid in the vessel (70%).Although the relative percentages and mechanical properties of elastin and collagen change with age (Seyedsalehi et al. 2015), we consider adaptations that occur over (17) , much shorter time scales.Hence, we take the baseline state as homeostatic.
The rate of degradation i 2 (t) can be given as a consti- tutive function of stress.It is also related to the relative tension in collagen fiber k , as, for example (Baek et al. 2007a, b): We assume that the tissue has been in a homeostatic state for a long time prior to t = 0. Thus, the condition for a steady state for k th collagen family is [from Eqs. ( 2), ( 3) and ( 19), and observing that It is noteworthy that if a step-change in sustained flow or blood pressure occurs, the above conditions Eqs. (24-26) do not hold during the transition; however, if the sustained flow or pressure is long enough, these steady-state conditions return to a new homeostatic state.
Similarly, SMCs mass will be computed from, where m m R (t, ) is the turnover rate of SMCs and M m R (t) is the total mass of SMCs at time t .The rate of loss of smooth muscle cells (apoptosis) is assumed to be constant, . The scalar measure of the stress is set to: In summary, the presented biochemomechanical model is an extension of an existing biomechanical model to include a new biochemical kinetic model, and differences between them are summarized in Table 2.

Simulations
It has been shown previously that 2D (i.e., membrane) models can capture many salient features of arterial G&R (Gleason et al. 2004;Han et al. 2022;Kim et al. 2020;Zhou et al. 2015); hence, we use the same approach here.For the simulation, we assume an idealized circular-cylindrical vessel that is uniform along the axial direction without axial extension during the G&R, in which the mean radius r is the only independent variable with respect to time t during the simulation.Dimensions and corresponding stresses change when the pressure, P , or volumetric flow rate, Q , are perturbed from homeostatic levels.Depending on the duration (Δt) and magnitude of the perturbations ( P or Q ), the vascular dimensions (diameter D(t) , and wall thickness h(t) ) and stresses evolve in time (Fig. 3).
Parameters used in the simulation are listed in Table 3.Briefly, dimensions (r oh , h h, h ), parameters for the strain energy ( 3 ) and smooth muscle tone ( M , 0 , S basal ) are determined by optimization and curve fitting of both passive and active mechanical behavior of a mouse carotid artery data (Gleason et al. 2008), where the homeostatic mass fractions ( e 0 , m 0 , k 0 , f ), stresses ( h , h ), and pressure ( P h ) are assumed constant based on the references in Table 3.The 'fminsearch' function in Matlab is used for the nonlinear regression (Mathwork, MA), and a two-step fitting method was employed in optimization (Seyedsalehi et al. 2015).First, best-fit values of passive and active parameters were obtained using a penalty method based on pre-stretch values ( G c h , G m h , G e 1 , G e 2 ) that were obtained from previous lit- erature.Second, as the cost function's value approaches its minimum, we eliminate the penalty terms and perform the optimization with all of the fitting parameters to determine the lowest cost function.
Figure 4 shows optimized results for intramural pressure vs. outer diameter.Then, assuming that blood pressure and volumetric flow are independent input variables, the artery is assumed to be in the homeostatic state at and before t = 0 ; this state is then perturbed by either changes in the pressure or flow from homeostatic values.We then compute changes in the other arterial parameters as time progresses.Using Eqs. ( 10) and ( 11), mean blood pressure and circumferential membrane stress for a vasoactive vessel are calculated from the equilibrium equation in the circumferential direction:  where pressure ( P ) is assumed to be known.After each time step ( t + Δt ), we determine the radius using (29) Newton-Raphson iteration, where the solution procedure is shown in Fig. 5.

Stress-mediated adaptation
As noted earlier, when the blood flow rate is reduced over a long period, the vessel tends to adapt.We can investigate effects of changes in stress on these adaptation by varying the stress sensitivity parameters: K c and K m (see Eqs. 17 and 27).Many experimental studies have shown that wall shear stress tends to return close to targeted homeostatic values in response to modest changes in flow and similarly for wall stress (Hu et al. 2007a, b;Humphrey 2002).Hence, given information on the flow perturbation (for example, a 30% reduction in volume flow rate at a constant pressure, as shown in Fig. 6), then, using Eq. ( 1), we can predict the final dimension of the artery, i.e., lim = 1 , with P = P h and Q = Q h , yields , which for our case results in  for = 1 and = 0.5 .Our simulation captures this ideal prediction to different degrees depending on the value of stress sensitivity parameters.When mass production was assumed to be independent of nor mal stress (i.e.,K c = K m = 0 ), the internal radius adapted slowly and reduced beyond the expected ideal value (Fig. 6a) indicative of a weak luminal radius regulation.The corresponding shear stress (Fig. 6c) also passed beyond its homeostatic level and continued to increase.Without normal stress-regulation, wall thickness increased excessively (Fig. 6b).Thus, the circumferential stress similarly did not achieve its homeostatic value, as seen in Fig. 6d, but instead continued to fall as the simulation progressed.By contrast, when the strength of the mechano-regulation increased (i.e., increasing values of K c and K m ), the inter- nal radius and shear stress returned to their corresponding homeostatic levels quicker.Conversely, higher values of K c and K m resulted in a large overshoot in thickness change and similarly a large undershoot in hoop stress change.
When the pressure increased 50% above its homeostatic value (at a constant volumetric flow rate), the simulation suggested that internal radius, wall shear stress, and hoop stress returned close to the homeostatic level with reasonable values of the stress-mediated gain parameters (Fig. 7a, c, d).

Parametric study of collagen biochemical kinetics
Three parameters, φ, β, and 1 , are associated with collagen biochemical kinetics within intracellular space and before maturation.Sensitivity to the values of these parameters was analyzed for the case of changes of intraluminal pressure (a 50% step-increased pressure) and volume flow rate (a 30% step-decreased flow rate and a 5% step-increased flow rate).
The parametric sensitivity showed vascular adaptation to be insensitive to changes in φ for altered pressure and flow (Supplementary Document S1).Meanwhile, intermediate collagen removal rate 1 and cross-linking collagen rate β are directly related to temporal dynamics associated with the transition from intermediated to cross-linked collagen.As it should have, thickness increased 50% above its original level in response to a 50% increase of pressure (Fig. 7b), though sensitive to the values that defined the strength of The two timescales in the flowchart represent the real time progression (t) and the timescale that tracks amount of mature collagen (τ) with respect to maximum collagen turnover time (a max ) mechano-regulation and collagen turnover.As the combination of β and 1 was increased from 0.001 to 0.3, the thickness approached to its preferred value faster.Luminal radius and shear stress were relatively insensitive to collagen turnover when pressure was increased and flow rate remained constant.We also studied the sensitivity of luminal changes to a wide range of volumetric flow rates with mechano-regulation included.When the volumetric flow rate increased by 10% or 30% (over a relatively short period), internal radius increased by about 1.8% and 2.1%, respectively; conversely, if flow decreased by the same magnitude, internal radius decreased by about 4.5% and 8.2%, respectively (supplementary Fig. S1).During early changes (e.g., within one day in Fig. S1), we see that initial luminal control depends largely on the vasodilatory versus vasocontractility capacity.This  and K Clearly, mechano-regulation is needed for a homeostatic response finding is qualitatively consistent with previous reports, including those for cerebral arteries (Valentín et al. 2009).Figure 8 shows a case for a 30% increase in flow over 300 days.An ideal adaptation, where wall shear stress returns to its hemostatic level, should yield a final value of the internal radius of 8a shows a result consistent with an ideal adaptation but with a decreasing settling time for increases in collagen turnover constants (starting from about three months for 1 + = 0.001 to about a month for 1 + = 0.3 ).The peak values for internal radius shift to the left and decrease in magnitude as turnover constants increase.Although the final simulated thickness nearly equals the homeostatic value (Fig. 8b), the transient undershoots are very sensitive to the collagen turnover constants.As collagen turnover increases, the magnitude of the undershoot reduces and shifts toward the left.Figure 8c, d shows corresponding variations in the stresses for different collagen turnover constants.Overall, the different panels in Fig. 8 show that, as collagen turnover increases, the adaptation of the artery to a flow perturbation is faster.

Intrinsic changes in smooth muscle tone
Intrinsic changes in smooth muscle tone can also play important roles over different periods.Inspired by prior studies (Valentín et al. 2009), we simulated variations in active stress, act = T act h m , for different circumferential stretches on different days, as G&R progresses (Fig. 9).For a 30% decrease in volumetric flow rate, changes in muscle tone are relatively large over a period of days (Fig. 9a).As the circumferential stretch (λ 2 ) increases from 0.65, the active stress increases and the peak active value is attained at stretches of 2.1, 2.0, 1.9, and 1.8 for muscle mass fractions Fig. 8 Simulations for a 30% step increase in volumetric flow rate: a internal radius, b arterial thickness, c wall shear stress and d) circumferential stress, with 1 = 0.0, β = 0.001 (dotted-dashed); 1 = 0.0 , β = 0.05 (dotted); 1 = 0.05 , β = 0.05 (dashed); similar to a prior study (Butler et al. 2011) and the magnitude of active stress is comparable to another study (Price et al. 1983).

Discussion
Prior constrained mixture G&R models have described and predicted vascular changes in response to altered blood pressure and flow rate over long periods (more than 6 months) (e.g., Baek et al. 2007a, b;Valentín et al. 2009), yet such models assume that changes in loading or stress stimuli directly affect tissue-level adaptations while ignoring the short time duration between applied stress stimuli and intracellular/extracellular biochemical processes.In contrast to these phenomenological models of stress-mediated G&R (see Table 2 for a comparison), the new biochemomechanical model couples the previous macroscopic mass turnover rate with a biochemical turnover process for collagen: from rates of synthesis of pro-collagen to cross-linking collagen fibers while accounting for rates of removal of intermediate and mature collagen within the extracellular space.This biochemical rate model not only allows us to capture a delay in short-term vascular adaptations over weeks and months, it also provides a more mechanistic constrained mixture model that tracks temporal changes in geometry and stress during both short-and long-term adaptations to changes in hemodynamics.
Many arterial adaptations tend to be intuitive-increases in flow drive increases in luminal caliber whereas increases in pressure drive increases in wall thickness.Nevertheless, complex couplings among flow-induced wall shear stressand pressure-induced intramural stress-mediated G&R as well as differential rates of synthesis and degradation renders it difficult to intuit reasons for an overall (mal)adaptation.A computational model can be helpful in this regard.Overall, this new biochemomechanical model successfully predicted salient aspects of canonical arterial G&R, including adaptations over periods from weeks to a few months.One can, for example, assess differences in increased mass fractions of collagen and smooth muscle by studying parametrically the relative contributions of K c or K m to G&R (Eqs.17, 27).
Similarly, one can study potential effects of differential degradation rates for intermediate and mature collagen or conversions of intermediate into mature collagen [by varying β or 1 in Eq. ( 2)], noting that early and long-term mechanisms may differ (Fig. 7).
The simulation results of Figs. 6 and 8 show temporal predictions of vascular adaptation for two cases: a stepdecrease versus a step-increase in blood flow, respectively.For decreases in flow the luminal area decreases while wall thickness increases over the first few months but then returns toward the homeostatic level within 3 months.Conversely, for step increases in blood flow, the simulation predicts an increased luminal area, a reduction in wall thickness, and associated increases in shear and normal stresses over the first few months.The latter two return toward the homeostatic level within 4 months, however.Differences in response to decreases and increases in flow stem primarily from the differential effects of NO and ET-1 on collagen synthesis.By contrast, when pressure increased (Fig. 7), the simulation showed an increase in wall thickness with a preserved luminal area, consistent with most previously reported results (Fridez et al. 2003;Valentín et al. 2009).
An increased thickness can help restore wall stress toward normal, but it can also increase the structural stiffness that influences the hemodynamics, which may not be favorable (Anderson 2006;Humphrey et al. 2016).Simulated parametric studies should be checked against experimental data whenever possible; in cases where data are not available, such models can guide the experimental plan.Figure 10a compares simulated results for evolving diameter against data from carotid ligation studies.The measured volumetric flow rate over 21 days was used as input to the simulation, which yielded results comparable to those reported experimentally (Sluijter et al. 2004b).Because there was a lack of pressure measurements over the reported period, we assumed that pressure remained constant.The ideal luminal change (dotted line) is smaller than the experimental report, and our simulation predicts values that are between the two sets of values after 21 days.
Deviations between the reported experimental values and our simulations could be due to additional effects, including inflammation, that are not included in the current model as well as unknown differences across species (e.g., rabbits versus mice).Comprehensive data are clearly needed to better inform all models, including the present one.We also used a variable pressure (Fig. 10b, c) as an input over 56 days.The simulation is similar to the reported findings.Even though blood pressure increased more than 65%, the internal diameter stayed almost constant, slightly below the homeostatic level consistent with the reported experiment (Fridez et al. 2003).Compared to the prior data (Fridez et al. 2003), however, our model overestimated the chronic thickness of the vascular wall, which may have resulted from an excess generation of collagen or smooth muscle mass.Assuming the volumetric flow rate was constant, our simulation matches the ideal adaptation after about a month.More experimental data are needed, however, to verify the true source of the deviation.When pressure was increased by 50%, the circumferential stress reached its homeostatic level within 14-16 days, and its chronic level (which is slightly lower than the homeostatic value) within 45 days.This result is consistent with prior findings (Hu et al. 2007a, b), wherein circumferential stress returned to the homeostatic level as early as 2 weeks after aortic coarctation to induce hypertension.
Although biochemical processes of intracellular and extracellular collagen turnover are well-studied and ranges of collagen cross-linking and removal rates have been reported (summarized in Table 1), the associated ranges are wide and differences in rates likely vary due to multiple physiological (e.g., age) and pathological (e.g., hypertension) differences, hence complicating predictions of temporal changes during vascular adaptation, with or without pharmaceutical intervention.In our biochemomechanical model, φ represents the fraction of procollagen that is released to the extracellular space, while the other fraction (1 − φ) represents intracellular degradation of procollagen, which may associate with autophagy.Autophagy is a natural degradation process that removes unnecessary or dysfunctional components of a cell through a lysosomal-dependent regulation.Among other factors, autophagy removes misfolded procollagen and basal autophagy is an essential in vivo process for mediating proper vascular function (Ishida et al. 2009).In the present biochemomechanical model, however, simulated arterial remodeling in response to altered pressure or flow was insensitive to changes in φ (shown in Supplementary Document S1).A possible reason for this is that the model assumes for normal collagen kinetics that an increased or decreased value of φ is automatically adjusted for by the initial rate of supply m p .Autophagy may be impaired, however, upon the onset of vascular diseases if stimulated by different stressors (e.g., reactive oxygen species or hypoxia) (De Meyer et al. 2015).Recent studies suggest that an age-dependent decline in autophagy accelerates vascular aging, and its impairment or imbalance promotes pathological aging and its sequelae (Aman et al. 2021;Sun et al. 2021).
The parameter captures the rate of conversion of intermediate to mature collagen, which is in large part due to enzymatic cross-linking of collagen, often driven by lysyl oxidase.For example, inhibitors of lysyl oxidase reduced arterial stiffness in a mouse model of hypertension (Eberson et al. 2015).Herein, appeared to be the most sensitive parameter in our simulations of pressure-and flow-induced adaptations.We simply performed a sensitivity study, however, and did not attempt to associate specific values of with particular vascular conditions.It is known, for example, that the stability of intermediate cross-linked collagen depends on age and genetics (Bailey and Shimokomaki 1971;Eyre and Glimcher 1972;Robins et al. 1973), hence specific values of as well as 1 should be able to be related to specific cases.Regardless, the simulations showed that a reduced ( 1 + ) associated with delayed vascular adapta- tions for altered pressure or flow (recall Fig. 8).There is also a need to elucidate how intracellular procollagen synthesis/ degradation and cross-linking kinetics influence changes in homeostatic set-point values and gain parameters for mechanical homeostasis under various physiological and pathological conditions (Humphrey 2021).Importantly, changes in intracellular degradation can affect total metabolic energy consumption in the body (Deretic and Kroemer 2022;Rabinowitz and White 2010), which may be connected to altering homeostatic set-point values and gain parameters for mechanical homeostasis.
Some of the deviations of our results from reported experiments could also be ascribed to the need to model better the many complex, multifactorial cellular and molecular mechanisms, as, for example, in hypertension (Touyz 2000).We also did not include mechanical damage to elastin, which could happen with marked increases in either flow or pressure (Chow et al. 2013).We also did not account for possible viscoelastic effects, noting that viscoelastic stress relaxation and G&R stress recovery could appear similar experimentally.Given the time scales involved, however, we feel that the short-term dynamics (e.g., cardiac pulsatility) resulting from viscoelasticity should be negligible.We also used a 2D (membrane) rather than 3D model of the wall, which has been shown previously to capture salient changes in geometry and structural stiffness but not details such as residual stress (Karšaj and Humphrey 2012).Given that residual stress related opening angle data were not available in the experimental reports we used, this was not a key issue although 3D models could certainly offer greater insight.Finally, the biochemical model was simplified to two steps to model a binary status of the collagen and we neglected possible stress mediated dynamics of the proteolytic enzymes.Again, additional data will be needed to increase model sophistication.We submit, however, that the present model both recovers prior simulation results based on phenomenological descriptors and enables additional information on collagen turnover to be incorporated once available.A critical next major challenge will be to link such changes in turnover to specific stress-mediated changes in gene expression.

Fig. 1
Fig.1Schematic illustration of the production and removal of mature collagen.On the right-side, collagen production is simplified into three stages: newly synthesized procollagen molecules in the intracel- parameters within Ψ i can change with time.The newly pro- duced material depends on the stress and the history of the deformation.The constitutive relation for the Cauchy membrane stress (force per deformed length) is:

Fig. 2 a
Fig.2a Important configurations and their mappings: The reference configuration R (B) is used for a computational reference as an initial geometry in homeostatic conditions.The natural configurations, i n( ) are stress-free configurations of constituent i at time .b Deformation of an arterial segment from the idealized thin wall in a homeostatic condition to the deformed wall at t

Fig. 3
Fig. 3 An idealized model for growth and remodeling of an artery in response to perturbed hemodynamics

Fig. 4
Fig. 4 Intramural pressure (mmHg) versus outer diameter (µm) for a mouse carotid artery (Gleason et al. 2008).The passive and active parameters for arterial model were estimated from passive (* * *) and active (o o o) experimental data.The fit to data with the model is shown by the solid (passive) and dashed (active) curves

Fig. 5
Fig.5Flow chart indicating the solution procedure for different variables, given hemodynamically induced stress as input.The two timescales in the flowchart represent the real time progression (t) and the timescale that tracks amount of mature collagen (τ) with respect to maximum collagen turnover time (a max )

Fig. 6
Fig. 6 Predicted changes in dimension and stress in response to a sustained 30% step decrease in volumetric flow rate as a function of wall stress-sensitivity: a internal radius, b arterial thickness, c wall shear stress, and d circumferential stress.Results are shown for different vales of key mechano-regulation parameters: K i = 0 (solid), K i = 2 (dashed),

Fig. 9
Fig. 9 Active stress (kPa) vs normalized muscle fiber stretch at different days for a 30% decreased volume flowrate and b 50% increased pressure( K c = K m = 3 , K k 1 = K k 2 = 0.01 , K c

Fig. 10 a=
Fig. 10 a Simulated changes in internal diameter (solid line): ideal adaptation (diamond-dotted) plus experimental data (circles) reported by Sluijter et al.(2004b); b simulated changes in internal diameter (solid line) and experimental data (circles) reported by Fridez

Table 1
Ranges of biochemical parameters estimated from the literature

Table 2
Comparison between the previous biomechanical G&R models and the new biochemomechanical model