Multiscale Systems Biology Model of Calcific Aortic Valve Disease

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A multiscale systems biology model of calcific aortic valve disease progression Amirhossein Arzani, Kristyn S Masters, and Mohammad R. K. Mofrad ACS Biomater. Sci. Eng., Just Accepted Manuscript • DOI: 10.1021/acsbiomaterials.7b00174 • Publication Date (Web): 12 Jun 2017 Downloaded from http://pubs.acs.org on June 17, 2017

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ACS Biomaterials Science & Engineering

A multiscale systems biology model of calcific aortic valve disease progression Amirhossein Arzani,† Kristyn S. Masters,‡ and Mohammad R. K. Mofrad∗,† †Department of Bioengineering, University of California Berkeley, Berkeley, CA, USA ‡Department of Biomedical Engineering, University of Wisconsin, Madison, USA E-mail: [email protected]

Abstract Calcific aortic valve disease is a common cause of aortic stenosis, a life threatening condition. In this study a mathematical model is developed to simulate the cascade of mechano-sensitive biochemical events that occur upon damage to the endothelial layer, leading to calcification. The model contains two phases. In the initiation phase, the model accounts for low-density lipoprotein (LDL) penetration into the sub-endothelial space, oxidation of LDL, and monocyte penetration and differentiation to activated macrophages. In the calcification phase, transforming growth factor beta is secreted from macrophages, inducing differentiation of valvular interstitial cells into activated myofibroblasts that can enable calcium deposition. Wall shear stress and mechanical strain are taken into account with simplified models updated based on calcification progression. The model parameters are estimated based on experimental data. Next, a statin therapy simulation is performed to evaluate the effect of lipid lowering therapy on calcification progression, demonstrating an age-dependent effectiveness in statin therapy. A new potential therapy targeting transforming growth factor-β activation is proposed and simulated. The long-term evolution of calcification is compared to

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two sets of published longitudinal clinical data, showing promising agreement. The proposed model can provide clinically valuable data, potentially guiding surgeons in valve replacement decision makings. Keywords: calcification progression, mathematical model, atherosclerosis, inflammation, hemodynamics

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Introduction Calcific aortic valve disese (CAVD) is a life threatening condition where calcific nodules develop in the valve leaflets causing accelerated stiffening of the valve, which can cause incomplete valve opening and closure, leading to aortic stenosis. The etiology of CAVD involves a multi-stage process. 1 It is believed that the initiation phase resembles the inflammatory processes involved in atherosclerosis where the penetration of low-density lipoprotein (LDL) into the sub-endothelial space and the subsequent monocyte infiltration creates an inflammatory environment. While the initiation phase shares some of the pathways involved in atherosclerosis, CAVD and atherosclerosis are distinct, therefore, CAVD and atherosclerosis may not be necessarily concurrent. There are two proposed mechanisms for the second phase of CAVD where the valvular interstitial cell (VIC) plays a key role. 2 In the osteogenic pathway, the VICs undergo osteoblastic differentiation producing bone-like calcification. In the myofibroblastic pathway, the VICs differentiate into myofibroblasts and subsequently apoptose, creating nucleation sites for the deposition of calcium. The second pathway is the focus of the current study. The relative importance of these two pathways in-vivo is unknown. 3 In the myofibroblastic pathway, the cytokines released by the inflammatory cells signal the second phase of calcification. Transforming growth factor-β (TGF-β), particularly TGF-β1, represents a prominent protein involved in aortic valve calcification initiation, 2,4–6 inducing the highest level of calcification among other cytokines and growth factors, in-vitro. 7,8 The interaction of TGF-β with its corresponding receptors on the valvular interstitial cell (VIC) membrane initiates the Smad signaling cascade 9–11 that results in differentiation of VICs to activated myofibroblasts. It has been hypothesized that the nodules formed by myofibroblasts create a safe environment where osteogenic differentiation can occur, leading to calcification. 1,2 It has been well documented that CAVD mostly occurs on the fibrosa, the aortic side of the valve. 12–14 Interestingly, the aortic side possesses a disturbed hemodynamic environment p. 1



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accompanied by complex multi-directional wall shear stress (WSS) patterns, 13,15 which has led to the notion that biomechanical pathways are responsible for this focal susceptibility. 16 The permeability of the valvular endothelium to macromolecules like LDL is spatially heterogeneous, 17 likely due to the spatially varying hemodynamics. The disturbed WSS patterns on the fibrosa oppose the mostly unidirectional WSS patterns on the ventricularis (ventricle side of valve). In addition to the role played by WSS, the mechanical stress acting on VICs and the subsequent strain intensifies calcification. 18 The progression of CAVD is a spatially and temporally multi-scale process. The biochemical pathways involved in the atherosclerosis and calcification phases take place on the biomolecular (nm) and cellular (µm) scales, whereas the hemodynamic forces cover a larger scale (cm). Temporally, the biochemical pathways and notable calcification happens during months/years, whereas the hemodynamics time scale is a cardiac cycle (∼ 1 s). The disparity between the spatial and temporal scales, along with the biological complexity and uncertainty overburdens the task of building a predictive model of CAVD progression. The goal of our study is to take a first step in building a simple model that takes into account the prominent biochemical and biomechanical pathways involved in CAVD. To this end, a simple cell scale systems biology model coupled with organ scale hemodynamics is developed to model the trajectory of the prominent biochemicals and cells involved in CAVD. The initiation phase of the model is built similar to the inflammatory processes in previous atherosclerosis studies. 19,20 The link between the initiation and calcification phase is made through the macrophage and TGF-β pathway and modeled based on available experimental data in the literature. To keep the model simple, the distinction between M1 and M2 macrophages is not considered. The macrophage and TGF-β interaction has been modeled in previous mathematical models of wound healing and inflammation. 21–24 Aortic valve calcification has been previously modeled with preassumed growth laws, 25 however, to the best of our knowledge a predictive model of long-term calcification growth has not been developed.

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Materials and Methods Herein a mathematical model of CAVD progression is developed. Our model is developed based on the myofibroblastic pathway. As suggested above, we assume that the nodules formed in this pathway correlate with calcification. An aging model based on the data reported in the literature is also developed to account for aging. The procedure used in parameter estimation is explained in the Supporting Information. A schematic of the proposed biochemical pathways leading to calcification is depicted in Figure 1. The model consists of an initation phase linked to a calcification phase. The cascade of reactions is initiated by penetration of LDL into the subendothelial space. Conversion of LDL to oxidized LDL (oxLDL) leads to monocyte capturing by the ECs. Differentiation of monocytes to macrophages and formation of foam cells creates an inflammatory environment. These macrophages secrete TGF-β. Once TGF-β is activated, it stimulates VIC differentiation to a procalcific phenotype, inducing nodule formation. These nodules can accompany subsequent calcium deposition. These biochemical events are mediated by mechanical cues exerted on the ECs and VICs. These mechanical cues depend not only on the patient-specific leaflet geometry and blood pressure, but also on the progression of calcification. The leaflets increasingly stiffen as the calcific nodules expand, which affects valve opening/closure and the overall hemodynamics. These pathways are modeled with a set of ordinary differential equations (ODEs) described below. The concentrations reported are spatial averages. The parameters used in the model are either obtained from previous studies or estimated from experimental data reported in the literature (see the Supporting Information). In order to model the regulation of hemodynamics (WSS and strain) by calcification, a simple functional form is assumed where the hemodynamics depend on the amount of calcification within a physiologic range, therefore, providing a simple link between the organ scale hemodynamics and the cell scale biology. The hemodynamic variables in the model correspond to the fibrosa, since calcification is known to primarily occur on the aortic side of the valve. The initiation (inflammation) p. 3



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and the integrity of the ECs. This parameter will be the input to our model. Once LDL penetrates into the leaflets, it can be oxidized:

C˙ oxLDL =

dLDL CLDL | {z }

k C CM | L oxLDL {z }

−

LDL conversion to ox-LDL

,

(2)

ox-LDL reacting with macrophages

where CoxLDL is ox-LDL concentration, CM is macrophage concentration, and kL is the reaction rate. The circulating ox-LDL is ignored in the model, since the major source of LDL oxidation is within the subendothelium. 27 The presence of ox-LDL sends signals to ECs, which promote the expression of adhesion molecules on the ECs that can capture monocytes:

C˙ m =

fm (τ, CoxLDL ) | {z }

d C | m{z m}

−

monocyte capturing by ECs

−

monocyte differentiation to macrophages

m C | d{z m}

,

(3)

monocyte apoptosis

where Cm is monocyte concentration in the subendothelial space, and fm (τ, CoxLDL ) = mr 1+ ττ

CoxLDL Cml , where Cml is the concentration of monocytes on the blood-contacting side

0

of the ECs and mr determines the rate of monocyte influx when WSS is zero. Once the monocytes penetrate into the valve leaflet they can differentiate to macrophages:

C˙ M =

d C | m{z m}

−

monocyte differentiation to macrophages

αkL CoxLDL CM | {z }

,

(4)


where α is a unit converting factor and the apoptosis of macrophages is assumed negligible. Finally, reaction of ox-LDL with macrophages leads to foam cell formation:

C˙ f =

αkL CoxLDL CM {z } |

−


k C | f{z f}

,

(5)

foam cell efferocytosis and apoptosis

where Cf is foam cell concentration and kf is the foam cell efferocytosis and apoptosis rate. Foam cell apoptosis was neglected due to lack of data and its known low degradation rate.

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However, a value of kf = 1.7 × 10−7 1s previously used for macrophages 28 was tested where minimal influence on the results was observed. Table 1 lists all the parameters used in the initiation phase of the model. CLDL and CoxLDL units are in Cf units are in

cell . m3

g m3

whereas the Cm , CM , and

All of the variables have zero initial condition.

Calcification phase The inflammatory environment created by the macrophages and foam cells leads to secretion of TGF-β, which plays a prominent role in calcification. Herein, we assume that the sum of macrophage and foam cell concentrations (CF = Cf + CM ) lead to TGF-β secretion. TGF-β is secreted in its latent form: ˙ = TGF

ft (CF ) | {z }

−

latent TGF-β production

λTGF | {z }

aF CF TGF | {z }

−

TGF-β degradation

,

(6a)

TGF-β activation by macrophages

g 3.3 × 10−7 CF V [ ], ft (CF ) = 4 CF V + 2.84 × 10 s m3

(6b)

where TGF is latent TGF-β concentration, V is the total aortic valve leaflets volume, λ is TGF-β degradation rate, aF determines TGF-β activation rate. and ft (Cf ) models TGF-β secretion by macrophages (see Supporting Information). TGF-β activation is assumed to be induced by the inflammatory environment created by macrophages:

˙ act = TGF

aF CF TGF | {z }

λa TGFact | {z }

−

TGF-β activation by macrophages

,

(7)

TGF-β degradation

where TGFact is active TGF-β concentration and λa is active TGF-β degradation rate. Finally, calcification is modeled using TGFact as an input signal.

˙ = Ca

γk TGFact (1 + h(ǫ)) | c {z }

TGF-β induced calcification promoted with strain

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−

dCa Ca | {z }

Calcification removal


,

(8a)

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Table 1: The parameters used in the inititation phase of the model. Some of the parameters are estimated based on experimental data. The estimation procedure is explained in the Supporting Information. Other parameters are directly taken from previous studies. The degradation/apoptosis rates are obtained from the half-life using d = ln(2) , where t 1 is the t1 2

corresponding half-life. Parameter

Value

dLDL

3 × 10−4

dL

2.4 × 10−5

C¯LDLin

9.44 × 10−5

τ0

1 Pa

kL

1.2 × 10−18

m3 s cell

dm

1.15 × 10−6

1 s

md

2.76 × 10−6

1 s

mr

6.6 × 10−12

m3 gs

Cml

5.5 × 1011

α

5.7 × 106

Mr1

2.83 × 10−11

m3 s mol

Mr2

9.25 × 10−24

m3 s cell

kf

Neglected. Only efficient in early stages.

1 s 1 s g s m3

cell m3

cell g

2

Description

Reference

LDL oxidation

20,29

LDL diffusing out

19,30

LDL influx to intima

estimated from 31

reference WSS

19,20

macrophage reaction with ox-LDL monocyte differentiation to macrophage monocyte apoptosis rate monocyte penetration rate at zero WSS monocyte concentration at the lumen wall M α = Mrr1

20,32

ox-LDL leading to foam cell foam cell formation

20,35

foam cell efferocytosis and degradation rate

36

19,33

34

20

20

20

2

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20,35


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h(ǫ) = 4.435 × 104 exp(6.404ǫ) − 4.435 × 104 ,

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(8b)

where Ca is calcium concentration in Agatston score units, γ is a parameter used to convert calcium concentration from nodules per well unit (reported in in-vitro experiments 18 ) to Agatston score (reported in clinical trials 37,38 ), kc is the rate of calcification in the absence of mechanical strain, and h(ǫ) models calcification enhancement in the presence of mechanical strain ǫ (Supporting Information). The circumferential strain is used in this equation, since the VICs are aligned in the circumferential direction. Table 2 lists all of the parameters used in the calcification phase of the model and parameter estimation is explained in the Supporting Information. All of the above variables have an initial condition of zero. TGF-β units are in

g m3

and Ca units are in Agatston score.

Hemodynamics update In order to simulate the long-term progression of calcification, the hemodynamic variables need to be updated according to the change in the mechanical environment. Namely, calcification increases stiffness of the aortic valve leading to incomplete opening and closure of the leaflets (aortic stenosis). The localized increase in stiffness increases strain locally due to a compliance mismatch. Although the strain may decrease globally, it is the localized increase in strain that is hypothesized to contribute to calcification progression. Moreover, the increasing stiffness of the annulus can also increase the circumferential strain. 18 An incomplete opening of the valve generally reduces the WSS on the aortic side of the valve 42 due to more disturbed flow patterns. To simulate the long-term progression of calcification, a multiscale model that accounts for the hemodynamics and the mechano-sensitive biochemical reactions is needed. An accurate modeling of these processes requires computationally expensive patient-specific modeling of the aortic valve. In the present study, we obtain the hemodynamic variables using simplified functions that vary in the anticipated physiological range of the hemodynamic variables.

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Table 2: The parameters used in the calcification phase of the model. Some of the parameters are estimated based on experimental data. The estimation procedure is explained in the Supporting Information. Other parameters are directly taken from previous studies. The degradation/apoptosis rates are obtained from the half-life using d = ln(2) , where t 1 is the t1 2

corresponding half-life. Parameter

Value

Description

Reference

V

27 × 10−8 m3

39

λ

1.28 × 10−4

1 s

λa

5.78 × 10−3

1 s

total aortic valve (tri-leaflet) volume, assuming 0.3 mm thickness and 3 cm2 leaflet area latent TGF-β degradation rate active TGF-β degradation rate

aF

5.2 × 10−15

m3 s cell

kc

−6 4.3×10 3

γ

m nodules/well sg Agatston 0.09 nodules/well

dCa

Neglected

TGF-β activation rate by macrophages calcification rate calcification unit conversion calcification removal rate

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2

40

40

23,41

estimated from 18 estimated No clear mechanism proposed.


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WSS dependence on calcification is modeled using:

τ=

36 [P a] . Ca + 181

(9)

This function is approximated such that when no calcification is present (Ca = 0), τ ≈ 0.2 Pa, which is the spatially averaged time-average WSS reported for a trileaflet aortic valve in a recent study. 43 Moreover, as Ca score approaches 3840, which is the assumed maximum Agatston score (see Supporting Information), τ approaches zero, which is due to an almost complete blocking of the aortic valve. Finally, the WSS and velocity reported in 42 is matched with a clinical study 44 to obtain a third pair of WSS-Ca point to enable data fitting. Circumferential strain dependence on calcification is modeled with:

ǫ=

0.2 Ca + 30 . Ca + 333.33

(10)

This form is assumed such that when no calcification is present (Ca = 0), the physiological value of strain (ǫ = 0.09) is recovered. 45 At a typical pathological strain of ǫ = 0.15 46 a Ca score of 450 was assumed, which is within the range of an established calcification. 37 Furthermore, as Ca becomes very large, ǫ approaches a maximum value of 0.2 reported in previous studies. 47,48 While it might be perceived that in extreme cases ǫ might exceed 0.2, we assume that further increase in strain does not affect calcification. This assumption is similar to previous studies 49 and makes ǫ fall within the range of experimental data used to estimate parameters of the model (see Supporting Information). Therefore, ǫ in our model is the assumed effect of strain on calcification, not necessarily the physical strain. It should be further emphasized that these simple functions are only approximating the complex hemodynamic environment on the aortic side of the valve. Detailed fluid-structure interaction (FSI) simulations are needed to increase the accuracy of these functions.

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Steady state concentrations The proposed system of differential equations, equipped with the approximation equations for the hemodynamic variables, could be solved to obtain the long-term evolution of the 8 state variables. Namely, LDL, ox-LDL, monocytes, macrophages, foam cells, latent and active TGF-β, and calcium. This was done by solving the system of ODEs using Matlab’s ode15s solver, which is suitable for stiff ODEs. The steady state values could be solved analytically, providing insight on the long-term dependence of each variable on different parameters. To make the analytical steady state solution tractable, we neglect the dependence of WSS on Ca and keep τ as a parameter. The steady state values of the state variables could be obtained by letting the time-derivative to be equal to zero. From Eq. 1 we obtain:

CLDL =

CLDLin 1 + ττ0

!

1 . dL + dLDL

(11)

Equations 2, 3, and 4 could be solved simultaneously to obtain:

CoxLDL =

(md + dm ) α dLDL CLDLin , mr Cml dm (dL + dLDL )

(12)

Cm =

α dLDL CLDLin , dm (1 + ττ0 ) (dL + dLDL )

(13)

CM =

mr Cm l d m . (md + dm ) (1 + ττ0 ) α kL

(14)

Subsequently, Eq.5 is used to get:

Cf =

α dLDL CLDLin . kf (1 + ττ0 ) (dL + dLDL )

(15)

In order to obtain the steady state values for the calcification phase, we realize that for physiological parameters ft (Cf ) function saturates quickly as Cf approaches its steady state.

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Therefore, this function is replaced by its asymptotic constant A = 3.3 × 10−7 . Equations 6a and 7 give: A

T GF = λ + aF T GFact =

m r Cm l d m (md +dm ) (1+ ττ ) α kL

+

0

α dLDL CLDLin kf (1+ ττ ) (dL +dLDL ) 0

A/λa λ

1+ aF

α dLDL CLDL mr C m dm l + k (1+ τ ) (d +d in ) (md +dm ) (1+ ττ ) α kL L LDL f τ0 0

,

(16)

.

(17)

Finally, Equation 8a gives the following nonlinear algebraic equation:

K Ca = 1 + 4.435 × 104 exp(6.404

0.2 Ca + 30 ) − 4.435 × 104 , Ca + 333.33

(18)

where K is a constant collecting all the parameters:

K=



λa dCa  1 + A γ kc 

λ aF

m r Cm l d m (md +dm ) (1+ ττ ) α kL 0

+

α dLDL CLDLin kf (1+ ττ ) (dL +dLDL ) 0



   .

(19)

Equation 18 can be solved analytically in two limiting cases. For the case when K ≪ 1, the solution occurs for large Ca values, therefore exp(6.404

0.2 Ca+30 ) Ca+333.33

≈ exp(6.404 × 0.2).

Subsequently,

Ca =

1.15 × 105 [Agatston] . K

(20)

For the case when K ≫ 1, the solution becomes Ca ≈ 0 (Ca has to approach 0 as K approaches infinity, since the right hand side of Eq.18 is always bounded). Note that the constant A can be different in this case. The K ≪ 1 limit is representative of the physiological values of the parameters, therefore Equation 20 could be used to estimate the dependence of the long-term calcification on different parameters.

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4

Lipid accumulation grade

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3

2

1

0 0

20

40

60

80

Age [Years]

Figure 2: Lipid accumulation in aortic valve with aging obtained from 100 human patients. This data is used to construct the pathological aging model. Figure reproduced with permission from Ref. 50 Copyright 1965 American Society for Investigative Pathology.

Aortic valve pathological aging model Sell and Scully 50 have reported changes in lipid accumulation in the aortic valve with aging. The associated increase in lipid accumulation with aging is shown in Figure 2. The data are reported in a qualitative “grade of involvement” unit ranging from 0 to 4. The following function is fitted to this data to obtain the dependence of LDL influx on time:

CLDLin = β C¯LDLin

    6.959 t − 130.8 t + 34.8

  0

if t ≥ 18.8

(21)

if t < 18.8 ,

where t is age in years unit and β is a parameter used to simulate different levels of LDL influx. A β = 0.5 value will be used as default, which assumes that a grade of 2 reported in 50 corresponds to CLDLin = C¯LDLin . In general, β will depend on the patient-specific properties of the endothelium and blood LDL concentration. β will be varied as an input to study its effect on the results. The aging model is obtained from patients that have eventually developed aortic valve calcification. 50 Therefore, the aging model represents a pathological aging where the endothelium becomes damaged and exposed to LDL accumulation.

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Results The system of ODEs was solved to quantify the long-term progression of calcification. The time evolution of all of the biochemicals and calcification is shown in Figure 3 and 4, for constant LDL influx and aging models, respectively. Different levels of constant LDL influx (CLDLin = ζ C¯LDLin ) are shown in Figure 3 by varying ζ. As anticipated, an increase in ζ increases macrophage and foam cell concentrations, leading to increased calcification. It is observed that LDL, ox-LDL, and monocyte concentrations reach their steady states relatively quickly. This behavior is consistent with previous atherosclerosis models. 20 Ox-LDL, monocyte, and latent TGF-β reach their peak values in an even shorter time-scale compared to LDL. Subsequently, their concentration is lowered due to differentiation/conversion or reaction with other biochemicals. The constant LDL influx model assumes that upon an injury to ECs, a constant LDL influx occurs. Therefore, this model does not predict an accurate time evolution that can represent aging processes. In the aging model, a time dependent LDL influx is approximated based on experimental data. The corresponding results are shown in Figure 4. The horizontal axis in these plots could be representative of the age of a patient that develops CAVD. Similar to the previous figure, an increase in the level of LDL influx, increases calcification. Compared to the constant LDL influx model, the steady state values are approached in a longer time scale, with this time scale being more dependent on the level of LDL influx. The calcification model is validated by comparison to longitudinal clinical data reported in the literature as shown in Figure 5 and 6 for the constant LDL influx and aging models, respectively. The left panel plots calcification progression rate during an average 2.4 years followup as a function of baseline calcification. The clinical data are reported by Owens et al.. 38 Good agreement is observed between the model (particularly the aging model) and clinical data. The right panel plots calcification progression for the default LDL influx level. The red dashed lines and error bars are the clinical data reported by Messika-Zeitoun et al. 37 during 3.8 years followup. To enable a direct comparison, the clinical data are plotted on top p. 14


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of the results from the calcification model, such that the baseline calcification level matches the model. Good agreement is observed where the calcification models mostly fall within the error bars reported in the clinical data.

Potential therapeutic targets Recent clinical trials have shown failure in statin therapy to stop calcification. 51 Specifically, it has been postulated that once calcification is established at a certain level, the calcification process is no longer influenced by LDL deposition. 1 To evaluate this phenomenon, a statin therapy simulation was conducted. In the aging model, LDL influx (CLDLin ) was set to zero after a certain age, which simulated the beginning of an ideal statin therapy with no lipid deposition. The starting age of statin therapy was varied to evaluate its effect. The corresponding calcification progression results are shown in Figure 7a. In accordance with the clinical trials, statin therapy is minimally effective when it is started at a later age in which calcification is sufficiently established. However, the figure shows that statin therapy can become effective if it starts at a sufficiently younger age. Based on our model, we also propose a potential therapy targeting TGF-β activation. TGF-β activation rate (aF ) is reduced to 1% and 0.1% of its original value in Figures 7b and 7c, respectively. It can be observed that calcification growth in our model can be arrested if aF is sufficiently reduced.

Sensitivity analysis A sensitivity analysis is performed to evaluate the sensitivity of the calcification prediction to various parameters in the model. In order to evaluate the sensitivity, the control coefficients are computed as a measure of normalized sensitivity 52

Si =

∆C / Cr , ∆pi / pi

p. 15


(22)



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Discussion Prediction of calcification progression is a challenging problem due to its nonlinear and complex nature, yet being able to estimate when valve replacement is needed is a crucial clinical decision. In this study, we have proposed a predictive mathematical model considering prominent biochemical and biomechanical contributions to calcification. The model links an inflammation model to a calcification model via macrophage-TGF-β interactions. Biomechanical parameters influence the initiation and calcification phases of the model. Good agreements were observed between the results of our model and published longitudinal clinical data. The proposed model can be potentially used to estimate the progression of CAVD. Interestingly, some patients with a notable level of calcification do not reach the end-stage of the disease where aortic valve replacement is needed. It has been reported that an Agatston score exceeding 1274 and 2065 in women and men, respectively, corresponds to severe aortic stenosis with high mortality rate. 53 The proposed model could be used as a predictive tool to estimate the time these thresholds are reached, providing a guideline for surgery decision making. While LDL deposition and atherosclerotic-mimicking pathways are hypothesized to initiate calcification, they seem to play a less crucial role in the final progression phase of calcification. 1 This has provided an explanation for failure of statin therapy for treatment of CAVD. 51 In this study, we performed an ideal statin therapy simulation to investigate its outcome on calcification progression. The main finding was that statin therapy is efficient, if started at a certain age. This age will in general depend on the amount of calcification present, among other likely factors. For example, a healthier endothelial layer and lower blood lipid levels (smaller β) will lead to less calcification depositions, and therefore statin therapy will be required at a later age. However, the effect of lipid lowering therapy diminishes as the amount of calcification present during the start of the therapy is increased. Our model offers a potentially valuable tool to test the effectiveness of statin therapy, by p. 23



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accounting for the actual LDL reduction level during therapy, which can be as high as 40%. 54 We have also demonstrated the usefulness of a potential new therapy that reduces TGF-β activation rate. It should be pointed out that TGF-β plays various roles in the human body, therefore, such therapy should be able to target TGF-β activation locally. A major limitation of this study is the simple phenomenologically assumed functional dependence of hemodynamics on calcification, which is used to connect the organ scale hemodynamics to the cell scale biology. The limits of the hemodynamic parameters used in these functions as well as the general trends are consistent with the data reported in the literature. The dynamics of the valve is influenced by calcification once calcification is established at a certain level . Obtaining a more accurate estimate of the valve dynamics and therefore the hemodynamic parameters requires computationally expensive FSI simulations. The current model is a gross progression model where spatial patterns are not accounted for. Therefore, the uncertainty in the hemodynamics dependence on calcification is further absorbed into the uncertainty due to spatial variations of hemodynamics. To examine this uncertainty, the functional form of WSS and strain functions were perturbed to evaluate their effect on calcification. The model demonstrated very small sensitivity on the functional dependence of WSS, whereas the highest level of sensitivity in the model was with respect to strain. This is consistent with the observation that calcification typically starts at the coaptation area or from the attachment to the center of the leaflet where highest levels of strain are experienced. 55 We have only considered TGF-β secretion by macrophages, while VICs also secrete TGFβ. However, it is estimated that the macrophages are the major source of TGF-β secretion. 28 We have neglected age related changes in leaflet thickness and volume. Moreover, stiffening related to normal aging was not considered. This is anticipated to play a small role in our model, as our model simulates a pathological calcification progression scenario where calcification induced stiffening likely dominates normal aging effects. Hyperphosphatemia and calcium load are other factors that affect CAVD 56 . Moreover, hypertension, a risk factor

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for CAVD, can influence the permeability of the endothelium and the mechanical strain. Fibrosis is another mechanism that can lead to stiffening and aortic stenosis. Recently it has been shown that the major contribution to aortic stenosis in women is fibrosis, whereas calcification is dominant in men. 57 Future work will incorporate hypertension and fibrosis effects. The role of WSS in calcification was modeled through its effect on EC permeability to LDL as well as adhesion and arrest of monocytes by ECs. However, WSS can play other roles in calcification as well. While WSS magnitude influences ECs permeability and adhesion characteristics, WSS vector field topology influences near-wall transport and thereby accumulation patterns of biochemicals and cells near the vessel wall. Recently, the concept of Lagrangian WSS structures has been introduced where attracting WSS structures lead to accumulation and high concentration of biochemicals in localized regions near the vessel wall. 58,59 Interestingly, these structures can only be present on the fibrosa side of the valve leaflet where multidirectional WSS patterns exist. 15 WSS exposure time, a recently proposed measure can quantify the localized regions of high near-wall concentration, 58,59 although its relevance to deformable media remains to be investigated. WSS can also influence the calcification phase of the model. An increase in WSS magnitude reduces VIC differentiation through paracrine signaling, 60 likely through a nitric oxide mechanism. 61,62 However, an increase in WSS magnitude also increases TGF-β expression, 63,64 which can positively influence calcification. Due to these contradictory roles, we have refrained from incorporating WSS in the calcification phase of the model. In this study, calcification was modeled through the myofibroblastic pathway. The calcification phase of the model is built based on the hypothesis that the nodules formed by myofibroblasts eventually lead to calcification. It has been shown that the nodules formed in-vitro are not truly calcific. 65 However, it is hypothesized that these nodules create an environment where true calcification can happen. 1,2 Another potential mechanism is that the specific topography created by nodules can potentially affect cell differentiation. 66 It is not

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known what percentage of these nodules accommodate calcification. Therefore, the calcification predicted by our model should be interpreted as an upper bound to calcification. Future work should incorporate factors such as bone morphogenic protein (BMP), β-Catenin, and receptor activator of nuclear factor κβ ligand RANKL that promote osteogenic differentiation and calcification. 1,67 However, due to the current lack of appropriate mechano-sensitive VIC osteogenic differentiation data in the literature, our model is a first step towards a comprehensive quantification of CAVD progression.

Supporting Information Parameter estimation and calcification units details.

Acknowledgement This work was supported by a generous grant from the American Heart Association.

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List of Tables 1

The parameters used in the inititation phase of the model. Some of the parameters are estimated based on experimental data. The estimation procedure is explained in the Supporting Information. Other parameters are directly taken from previous studies. The degradation/apoptosis rates are obtained from the half-life using d =

ln(2) t1 2

2

, where t 1 is the corresponding half-life. . . . . . . . . 2

7

The parameters used in the calcification phase of the model. Some of the parameters are estimated based on experimental data. The estimation procedure is explained in the Supporting Information. Other parameters are directly taken from previous studies. The degradation/apoptosis rates are obtained from the half-life using d =

ln(2) t1

, where t 1 is the corresponding

2

2

half-life. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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List of Figures 1

A schematic of the cascade of biochemical reactions leading to calcification.

2

Lipid accumulation in aortic valve with aging obtained from 100 human pa-

4

tients. This data is used to construct the pathological aging model. Figure reproduced with permission from Ref. 50 Copyright 1965 American Society for Investigative Pathology. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

13

Biochemicals concentration and calcification evolution in 90 years. LDL influx (CLDLin ) is taken constant and various levels of CLDLin = ζ C¯LDLin are simulated. Calcification units is Agatston, monocyte, macrophage, and foam cells units are in

4

cell , m3

and all the other units are in

g . m3

. . . . . . . . . . . .

16

Biochemicals concentration and calcification evolution in 90 years. LDL influx (CLDLin ) is varied according to the aging model. Various levels of β in the aging model are simulated. Calcification units is Agatston, monocyte, macrophage, and foam cells units are in

5

cell , m3

and all the other units are in

g . m3

17

Comparison of the model to clinical data reported in the literature. Constant LDL influx with various levels CLDLin = ζ C¯LDLin are shown in this figure. (a) Rate of calcification progression during 2.4 years followup versus the natural logarithm of baseline calcification (Agatston units). The thick green line is the Lowess fit to the full patient cohort, and the blue circles are the mean rate of progression reproduced with permission from Owens et al. 38 Copyright 2010 Elsevier. The root mean square error between the simulation and the Lowess fit is 7.1 for ζ = 0.5. (b) Calcification progression based on baseline calcification during 3.8 years followup plotted with red dashed lines and error bars reproduced with permission from Messika-Zeitoun et al. 37 Copyright 2007 American Heart Association. To enable comparison, the Messika-Zeitoun et al. data are plotted on top of the calcification simulation (black line) such that the baseline calcification matches the calcification evolution simulation. p. 36


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6

Comparison of the model to clinical data reported in the literature. The aging LDL influx model with various levels are shown in this figure. (a) Rate of calcification progression during 2.4 years followup versus the natural logarithm of baseline calcification (Agatston units). The thick green line is the Lowess fit to the full patient cohort, and the blue circles are the mean rate of progression reproduced with permission from Owens et al. 38 Copyright 2010 Elsevier. The root mean square error between the simulation and the Lowess fit is 4.7 for β = 0.25. (b) Calcification progression based on baseline calcification during 3.8 years followup plotted with red dashed lines and error bars reproduced with permission from Messika-Zeitoun et al. 37 Copyright 2007 American Heart Association. To enable comparison, the Messika-Zeitoun et al. data are plotted on top of the calcification simulation (black line) such that the baseline calcification matches the calcification evolution simulation. . . . . . . . . . .

7

19

Calcification progression in the aging model (β = 0.5) with different therapies commencing at a certain age shown in the figure. (a) Statin therapy. In the statin therapy simulation it is assumed that LDL influx (CLDLin ) becomes zero after the age when statin therapy begins. (b) TGF-β activation (aF = 0.01aF ). The TGF-β activation rate (aF ) is reduced to 1% of its original value after the therapy is started. (c) TGF-β activation (aF = 0.001aF ). The TGF-β activation rate (aF ) is reduced to 0.1% of its original value after the therapy is started.

8

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

20

Temporal evolution of control coefficients corresponding to calcification (Ca) in the constant LDL influx model. Various parameters in the initiation phase of the model are perturbed to asses the sensitivity of the model with respect to these parameters. “WSS function” sensitivity represents the factor in Ca (unity by default) used in the WSS-Ca equation (Eq. 9). . . . . . . . . . . .

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Temporal evolution of control coefficients corresponding to calcification (Ca) in the constant LDL influx model. Various parameters in the calcification phase of the model are perturbed to asses the sensitivity of the model with respect to these parameters. “TGF-β production rate” sensitivity represents the peak value in the TGF-β production equation (Eq. 6b). “Maximum strain” sensitivity represents the peak strain in the ǫ-Ca equation (Eq. 10). “Ca enhancement by strain” sensitivity represents the 4.435 × 104 constant in the h(ǫ) equation (Eq. 8b).

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Multiscale Systems Biology Model of Calcific Aortic Valve Disease

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