## Abstract

Maintenance of the endothelial cell (EC) layer of the vessel wall is essential for proper functioning of the vessel and prevention of vascular disorders. Replacement of damaged ECs could occur through division of surrounding ECs. Furthermore, EC progenitor cells (EPCs), derived from the bone marrow and circulating in the bloodstream, can differentiate into ECs. Therefore, these cells might also play a role in maintenance of the endothelial layer in the vascular system. The proliferative potential of both cell types is limited by shortening of telomeric DNA. Accelerated telomere shortening might lead to senescent vascular wall cells and eventually to the inability of the endothelium to maintain a continuous monolayer. The aim of this study was to describe the dynamics of EC damage and repair and telomere shortening by a mathematical model. In the model, ECs were integrated in a two-dimensional structure resembling the endothelium in a large artery. Telomere shortening was described as a stochastic process with oxidative damage as the main cause of attrition. Simulating the model illustrated that increased cellular turnover or elevated levels of oxidative stress could lead to critical telomere shortening and senescence at an age of 65 yr. The model predicted that under those conditions the EC layer could display defects, which could initiate severe vascular wall damage in reality. Furthermore, simulations showed that 5% progenitor cell homing/yr can significantly delay the EC layer defects. This stresses the potential importance of EPC number and function to the maintenance of vascular wall integrity during the human life span.

- atherosclerosis
- computer simulation
- endothelial cells
- progenitor cells

the endothelial cell (EC) layer forms the interface between circulating blood cells and plasma and the rest of the vascular wall. Maintenance of this layer seems critical in preventing coagulation, adhesion, and the intrusion of monocytes, which initiates the formation of plaques (25). Because the layer is under continuous threat of becoming defective, repair mechanisms are continuously active. Replacement of apoptotic or damaged ECs (10, 19) can be established by division of surrounding ECs or by homing of bone marrow-derived EC progenitor cells (EPCs) (6, 12, 13, 24). Cell division is associated with telomere shortening, leading to senescence once telomere length reaches a critical threshold. Intraluminal injury was shown to result in increased EC proliferation and the emergence of senescent ECs (10). In vivo, telomeres were found to shorten more rapidly in ECs located at regions susceptible to atherosclerosis (3, 20). Along with the finding that senescent ECs accumulate on the surface of atherosclerotic plaques (18, 33), these data indicate that repair of the endothelium may be limited by telomere shortening and subsequent senescence of ECs (27).

Telomere shortening is modulated by the rate of cell turnover and oxidative stress (35). Exposure of cultured ECs to homocysteine-induced oxidative stress significantly accelerated the rate of EC senescence and telomere length shortening (37). Scavenging of the level of reactive oxygen species by antioxidants may decelerate telomere attrition, as previously demonstrated in vitro by enrichment of intracellular vitamin C in human ECs (11). Furthermore, telomere length is decreased in arterial sites exposed to high hemodynamic insults (3). Such areas are prone to the development of atherosclerosis and are characterized by turbulent blood flow, disturbed shear stress, and high EC turnover rate (7). Shear stress at physiological levels strongly contributes to maintenance of EC function (36); however, low or extremely high shear induces EC apoptosis (2, 8). Thus several factors that are known to be strongly involved in the initiation and progression of vascular damage also accelerate telomere shortening and may lead to accelerated EC senescence.

There is a growing understanding that, in addition to division of ECs in surrounding tissue, EPCs contribute to repair of dysfunctional endothelium (6, 12, 13, 24). These cells originate from hematopoietic stem cells, which also exhibit telomere shortening (34). Low cell turnover (16) and high telomerase activity (26) in the adult bone marrow result in a lower rate of telomere attrition compared with somatic tissue. Maintenance of the endothelial monolayer by stem cell-derived EPCs may therefore delay senescence in the EC layer. Recently, a strong indication became available that in states associated with cardiovascular risk EPC number and function is diminished (13). Moreover, atherosclerosis in aging apolipoprotein E-deficient mice was inhibited by injection of EPCs from young mice (23). The link among the dynamics of EC layer maintenance, telomere shortening, and senescence is complicated, and the relative importance of EC division and EPC homing in EC layer homeostasis is currently being debated.

The overall aim of the present study was to develop a computational model of the dynamics of damage and repair of the EC layer, including telomere shortening as a consequence of EC turnover and oxidative stress. The model was used to evaluate time-dependent changes in telomere length and senescence. The potential effect of cell geometry was investigated because the cell shape determines the number of adjacent neighbor cells and therefore might influence the dynamics of endothelial maintenance. Moreover, the model was applied to predict at which levels of EC damage and oxidative stress the EC layer will be impaired. Finally, different rates of homing of EPCs were simulated to determine the effect of this repair mechanism on the state of the vessel wall.

## MATERIALS AND METHODS

#### General considerations.

In the model (Fig. 1), ECs were integrated in a two-dimensional structure resembling the endothelium in a large artery. Upon EC damage, either neighbor cells divided or EPCs homed into the endothelium to fill up the discontinuities. Stem cells divided to maintain EPC concentration constant. Division of ECs and stem cells led to telomere shortening. Excessive telomere erosion induced senescence, whereas accumulation of senescent cells resulted in denudation upon EC damage. We investigated the effects of EC geometry, turnover, the level of reactive oxygen species, and homing percentage of EPCs on the rate of EC senescence in the endothelium. An overview of the model parameters, obtained as described below, is given in Table 1.

The initial condition of the simulation was considered to be tissue of a healthy 20-yr-old male subject. The simulations were terminated at the arbitrary age of 65 yr. At this stage, the mean rate of telomere shortening and percentage of EC senescence were calculated. The spatial model structure was investigated to determine whether senescent cells accumulated and the integrity of the endothelium was affected.

#### Spatial model structure.

We incorporated a spatial structure in the model to simulate the repair of damage by neighbor cell division and the formation of discontinuities. Because the endothelium is a monolayer of cells lining the vascular wall, we modeled it as a two-dimensional structure. This structure was generated from a random distribution of cell centers within a square sheet. In the model, the cell positions are described by a matrix **z**: (1) Here, *x* and *y* are the cell center coordinates of the cell with index *i* and *N*_{e} is the number of ECs. This random distribution was evenly spaced by carrying out small displacements until a stable configuration was obtained. These displacements of the cell center positions were calculated based on the distance of the cell with adjacent cells. The new cell positions *k* + 1 are dependent on the old positions *k* and the positions of the surrounding cells, defined by the collection χ: (2) Here, χ is the set of cells within a target distance *d*_{0} from the center of the cell with index *i*: (3) The cell displacements **z**_{i}(*k* + 1) − **z**_{i}(*k*) were modeled as a function of the sum of distances with surrounding cells: (4) where *j* is an index for the neighbors of cell *i* and *R* is a coefficient for the rigidity of the cells (0 < *R* ≤ *d*_{0}). In the model, *N*_{e} cells in an area *A* were simulated. The target distance *d*_{0} was determined based on N_{e} circles within A: (5) To avoid large effects at the four boundaries of the simulation area, periodic boundary conditions were implemented. Fifty iterations were found to be sufficient for the system to obtain an evenly spaced configuration. Cell edges were calculated using the Voronoi algorithm, used to represent epithelial tissue previously (14). The resulting structure was generated with 500 cells in an area of 3.5 × 10^{5} μm^{2} based on a cell radius of 15 μm.

#### Dynamics of endothelial damage and repair.

The model was simulated with time steps of 1 yr. Every simulated year, a certain number of ECs was damaged and replaced subsequently to maintain a continuous monolayer. The computational model incorporated two mechanisms of replacement: division of surrounding cells and homing of EPCs.

#### Repair by surrounding cells.

Previously, an in vivo mitotic index of 0.1%/day was determined in ECs (5), i.e., normally 36.5% replications/yr. The simulated endothelium represents adult tissue, and therefore it was assumed that all newly formed cells replaced damaged cells. Under default, reference conditions (e.g., no repair by EPCs; see Table 1), the yearly EC turnover was set at 36.5% and, because time steps of 1 yr were used in the model, all cells had a probability *p*_{d} = 0.365 to be replaced. During a time step of 1 yr, the cells lost due to damage were chosen at random from the EC population. After the damaged cells were determined, random nonsenescent neighbor cells were selected to divide to repair the damage. We used our model to investigate changes in the rate of cell turnover on the rate of telomere shortening and cellular senescence. This was achieved by simulating the model with cell turnover rates of *p*_{d} = 0.2–0.8.

#### Repair by EPC.

In particular simulations, we modeled that every year, a defined number of progenitor cells homed into the two-dimensional structure of ECs. We assumed that equilibrium existed between the rate of EPC homing and the rate of division and mobilization of stem cells in the bone marrow; hence, the stem cell population maintained the number of EPCs at a constant level. Let *k*_{s} be the rate of division of stem cells in the bone marrow, *N*_{e} the number of endothelial cells, and *N*_{s} the number of stem cells. The probability by which the progenitor cells home into the endothelium (*p*_{h}) can then be calculated by: (6) Vaziri et al. (34) determined a value of *k*_{s} = 0.25 stem cell doublings/yr, based on adult bone marrow data. Crosby et al. (6) found that the contribution of EPCs was maximally 1.4% in 4 mo in stable adult endothelial tissue. This can be approached by 4.2% EPC homing/yr. We simulated numbers of stem cells from 0 to 100, which led to homing percentages of 0–5% (*p*_{h} = 0–0.05) via *Eq. 6*. The model assumed that after homing of EPCs, these cells completely differentiated into ECs, i.e., their properties equaled those of ECs. It was assumed that the rate of stem cell divisions per year remained unaltered (*k*_{s} = 0.25).

#### Morphology.

ECs are known to elongate and align as a result of flow (5). To simulate the effect of in vivo blood flow on cell morphology, a different range in *x* and *y* directions of the EC sheet was used (*Eq. 1*). For the in vivo geometry, the contact surface of an EC with its neighbors alongside its longitudinal axis is larger than with the neighbors at the tip ends. Therefore, it could be that the effective number of neighbors per EC is less in the elongated and aligned in vivo geometry than in the circular cell shape that would represent an in vitro shearless culture (or an in vivo situation with reduced shear). Because neighbor cells have an important role in the maintenance of the EC monolayer, it was investigated what would be the effect if differences in morphology would imply different number of neighbors. This hypothesis was implemented by assuming that ECs were only considered to be (effective) neighbors if their contact facet was larger than a certain percentage of the cell's circumference (ε). We simulated values of ε from 0 to 25%.

#### Telomere shortening.

Division of ECs and stem cells leads to loss of telomeric DNA in these cells. The model kept track of the length of 92 telomeres/cell. Because long overhangs of the TTAGGG strand are present at the telomere ends, the mean length of both strands was used in the model.

We assumed that telomeres shortened during cell division only. To model cell division, the parental cell was duplicated, and the telomeres of both cells were then shortened according to the following equation in the discrete time domain: (7) Here, *T*(*k*) is the old length of a telomere in a cell (in kbp), *T*(*k* + *1*) is the new length, and Δ*T* is the telomere loss per cell division. Thefragments Δ*T* lost were modeled as geometric random variables. This is assuming a constant probability of breakage of telomeric DNA during replication. In our model, the mean telomeric reduction was dependent on the level of oxidative stress.

In accordance with previous model predictions (21, 32), we assumed that when at least one of two specific telomeres (so-called proliferation restriction telomeres) in a cell reached a critical telomere length (*T*_{sen}), the cell became senescent and lost the ability to take part in the repair process.

#### Parameters of telomere shortening.

Telomere restriction fragments (TRF) contain a subtelomeric region, which is about 3 kbp (15). In the model, the telomere length without the subtelomeric sequence was described. We assumed an initial mean telomere length of 8 kbp in the model at age 20 yr (1, 20). The individual telomere lengths were drawn from a normal distribution with this mean; for the standard deviation, we used a value of 2 kbp, which is in agreement with the wide variance as observed in TRF measurements of ECs in culture (3, 20, 38). The initial telomere lengths of the stem cells were drawn from a distribution with the same parameters. The critical telomere length *T*_{sen} was assumed to be 2 kbp, after measurements in human fibroblasts (17) and estimated for ECs (21). In our model, stem cells did not become senescent during the human life span.

Under normal conditions, telomere attrition is around 100 bp/cell division, whereas oxidative stress accelerates telomeric shortening (35). Homocysteine, an agent inducing oxidative stress, augmented telomeric loss in cultured ECs to about 250 bp/cell division (37). In the model, values for the mean of Δ*T* ranging from 50 to 250 bp were simulated. It was assumed that telomere attrition in bone marrow cells was 100 bp/cell division at all times.

#### Statistical analysis.

To calculate the mean telomere length, we averaged the telomere lengths of all cells in a simulation. The mean rate of telomere shortening was obtained by averaging the difference in telomere length between subsequent years over the simulated time span. Per data point, thirty Monte Carlo simulations were carried out to obtain values independent of the number of simulations. Results are presented as means ± SD. Statistical differences were demonstrated by one-way multivariate ANOVA, followed by Scheffé's post hoc test. Differences were found to be significant when *P* values were below 0.05.

## RESULTS

#### General results.

We simulated the model with a mean telomere loss of 100 bp/cell division and a rate of cell turnover of 0.365. In these initial simulations, stem cell homing was deactivated. Under these conditions, 5 ± 2% of the ECs were senescent at the age of 65 yr and the rate of telomere shortening was 79 ± 2 bp/yr. Figure 2 shows the results of a representative simulation. Senescent cells could also be damaged and replaced by division of a viable neighbor cell in the simulations. This causes the fluctuations in the percentage of senescent cells. The proliferative capacity of the ECs in the sheet was sufficient to fully maintain the integrity of the endothelium.

#### EPC homing.

The computational model was used to investigate the effect exerted by small percentages of EPC homing as reported previously (6). The mean rate of telomere shortening per year in ECs decreased to 56 ± 3 bp/yr upon 5% EPC homing. The percentage of senescent cells showed a significant reduction to 2 ± 1% (Fig. 3). The rate of telomere shortening in stem cells was 43 ± 3 bp/yr at all homing percentages when EPC homing was activated.

#### Geometry and number of neighbors.

When all surrounding ECs that have any contact with the center cell are regarded as neighbors that can participate in its repair (ε = 0%), the average number of effective neighbor ECs was 6 ± 0.6. This number decreased to 2.6 ± 1.0 when the contact facet was required to be at least one-fifth of the cell's circumference (ε = 20%) for cells to be considered (effective) neighbors (Fig. 4*A*). When the average number of neighbors was reduced below 4 (ε ≥ 20%) and without EPC homing, at least 4.6 ± 1.0% of the defects in the endothelium could not be repaired at the age of 65 yr (Fig. 4*B*). The endothelial integrity rapidly deteriorated when the number of neighbors decreased toward one. Only in this extreme, physiologically unlikely situation did the effect of cell geometry diminish the repair by surrounding ECs.

#### Alterations in EC damage.

Increasing the probability of cell damage *p*_{d} did not only lead to an augmented rate of telomere shortening but also resulted in an enhanced percentage of senescent cells at the age of 65 yr (Fig. 5*A*). At *p*_{d} = 0.8, the mean rate of telomere shortening per year was 160 ± 5 bp/yr, whereas the percent senescent cells at the age of 65 yr increased to 67 ± 26%. Discontinuities accrued and often occurred when cells were surrounded by senescent cells or other denudated spots upon damage (Fig. 5*B*). At *p*_{d} = 0.8, discontinuities appeared at an average age of 40 yr. Simulations were conducted without EPC homing.

#### Alterations in oxidative stress.

The effect of oxidative stress was investigated by changing the amount of base pairs lost per cell division (Δ*T*). Different rates of telomeric loss were simulated without EPC homing. The rate of telomere shortening and the percentage of senescent cells after 65 yr significantly increased (Fig. 6). Upon a mean telomeric loss of 250 bp/cell division, accumulations of denuded endothelium and senescent cells are present, indicating that endothelial repair was insufficient to maintain a confluent EC layer (Fig. 7*A*). In the worst case condition, the proliferative capacity of the original ECs was only sufficient to fully maintain the integrity of the endothelium up to an age of ∼35 yr.

We simulated elevated cell damage (*p*_{d} = 0.8) and increased telomere shortening as a consequence of increased oxidative stress (Δ*T* = 250 bp/cell division) simultaneously with 5% EPC homing/yr (Fig. 8, *A* and *B*). In both situations, EPC homing returned the rate of telomere shortening to control levels and significantly reduced the number of senescent cells at age 65 yr. It was shown that 5% EPC homing could significantly reduce the part of the endothelium that could not be repaired at the age of 65 yr under potentially unfavorable in vivo morphology (ε = 20%; Fig. 8*C*). Additionally, high rates of cell damage and severe oxidative stress did not lead to discontinuities upon 5% EPC homing (Fig. 7*B*).

## DISCUSSION

The in vitro relation among telomere shortening, senescence, and aging-related phenotypes in primary human cells is well known and can be experimentally studied relatively easily. Until recently, little attention has been paid on the potential impact of vascular cell senescence in vivo on age-related vascular disorders (19). Telomere shortening is likely to be involved in cellular senescence of ECs and subsequent injury of the endothelial layer, although conclusive experimental data are scarce. In the present study, we integrated published experimental data in a stochastic, spatiotemporal model of endothelial maintenance. In the reference situation (*p*_{d} = 0.365, Δ*T* = 100, no EPC homing), the model predicted a rate of telomere shortening of 79 ± 2 bp/yr. Chang and Harley (3) found a telomere attrition rate of 102 bp/yr in human ECs from iliac arteries. Okuda et al. (20) and Aviv et al. (1) determined lower values of 28 bp/yr in the intima of the distal segment of the abdominal aorta and 18 bp/yr using measurements in different segments of the abdominal aorta, respectively. The model yielded similar low rates if small percentages of EPC homing were included in combination with a 50% lower turnover rate or 50% less oxidative damage per division. Our model predicts that loss of telomeric function in ECs subjected to increased damage or oxidative stress causes accumulation of senescent cells at a simulated age of 65 yr. Even more, high levels of oxidative stress cause localized loss of integrity of the endothelial monolayer as a result of limited repair by senescent cells at that time. Importantly, the model predicts that these negative effects can be overcome by homing of EPCs. Homing of stem cell-derived ECs significantly improved the potential replicative capacity of the ECs in the monolayer and delayed EC senescence. Simulation of homing of EPCs also maintained a continuous EC layer.

Disturbed flow increases rates of EC apoptosis in vitro (2, 8) and EC turnover in vivo (4). Induction of apoptosis in vivo could lead to endothelial denudation (9). A number of stimuli affecting endothelial apoptosis have been shown to influence the number of circulating apoptotic ECs. Furthermore, almost all risk factors for atherosclerosis increase the number of circulating endothelial cells in vivo (30). Implementing an elevated turnover in the model showed that EC senescence was significantly increased. The present simulations strengthen the hypothesis that at sites in the cardiovascular system, where blood flow is turbulent, elevated cell turnover increases the rate of telomere shortening, leads to cellular senescence, and increases the risk of vascular disorders.

Telomeric DNA is known to be specifically vulnerable to reactive oxygen species (35). We simulated oxidative stress by increasing the mean rate of telomere shortening per cell division. Our model predicted significantly increased percentages of senescent cells under elevated oxidative stress: the higher variability in the telomeric loss under oxidative stress may prematurely lead to critically short telomeres, invoking cell cycle arrest. These findings support the idea that telomere shortening in vivo is linked to disease states, such as atherosclerosis, in which increased levels of oxidative stress could play a causative role (35).

Although still under dispute, homing of EPCs into the damaged vessel wall has been implied to be involved in the maintenance of the endothelial layer (12, 24). The model simulated small percentages of EPC homing per year compared with values found during neovascularization, conforming to previously published findings (6). Our model sheds light on the possible role of EPC homing in stable adult tissue by demonstrating that the simulated low rates of EPC homing could make a significant contribution to the replicative capacity of the endothelium. These results support the recent findings that augmented endothelial injury in the absence of sufficient circulating progenitor cells may enhance the progression of cardiovascular disease (13). Because of lack of data, the different levels of EPC homing were assumed to be constant during the simulated life span. This assumption is probably not valid because, on the one hand, it was shown that in the first decades of increased turnover or elevated oxidative stress EPC homing is not required to repopulate the monolayer. On the other hand, in view of overall aging and age-related diseases, the number of circulating EPCs is likely to decrease with age and in correlation with cardiovascular risk factors (13). If data on changes in EPC levels with age become available, these can readily be incorporated in the model. The same holds for the parameters describing the probability of EC damage and the telomere loss per cell division.

A previous model study (22) on the different mechanisms of telomere shortening predicted an increase in the rate of telomere shortening and senescence under conditions of elevated oxidative stress. This supports the integration of oxidative stress as one of the key factors in our model of vascular wall ageing. The notion that telomere shortening may play a crucial role in age-related diseases was recently endorsed by a mathematical model of telomere shortening in white blood cells (28). To our knowledge, there is only one other computational model in the literature investigating the contribution of EPCs to angiogenesis (31). This model predicted that EPCs have an appreciable impact during the early stages of tumor growth. The novelty of our model study is the link between progenitor cell homing and telomere attrition in the EC layer. Additionally, the limited availability of adjacent cells to divide upon damage of ECs was considered by incorporation of a two-dimensional structure. This structure allowed us to visualize accumulations of noncycling cells and the formation of discontinuities. The spatial structure was also applied to investigate the potential effect of differences in EC geometry in arteries with different levels of shear. It was shown that the cell morphology can affect the number of adjacent cells, but it is highly unlikely that this number is reduced to a level that could deteriorate endothelial maintenance. Mathematical models like the present study have proven to be powerful tools to analyze and organize experimental data. Moreover, they can be used for direction of future research by testing and formulating hypotheses.

The model in its present form focuses on shortening of telomeres as a cause of impaired vessel wall maintenance. Another key step in the formation of atherosclerotic lesions is the accumulation of low-density lipoproteins in the vascular wall, previously modeled by Stangeby and Ethier (29). It has to be noted that the predicted defects are unlikely to be observed in vivo as relatively large areas of denuded endothelium. For a long time of the human life span, the repair systems will be superior to the damage processes and once repair becomes inferior a permanently damaged endothelium would initiate inflammatory responses, whereas severe endothelial erosion could result in acute thrombosis (9). Senescent ECs were found to accumulate on the surface of atherosclerotic plaques (19, 33) and telomeres in ECs located at regions susceptible to atherosclerosis shorten more rapidly (4, 21). Whether telomere exhaustion is a real cause remains to be experimentally confirmed. We and others are currently searching for early markers of cellular senescence, and we hope this will deliver the tools to study the predicted pattern of senescent cells in vivo. The present study does not pretend that telomere-dependent senescence of ECs is the only cause of vascular disease. It rather predicts that senescence in vivo induced by telomere shortening could contribute to pathology of the vascular wall.

In summary, we present the first computer simulation of the dynamics of the EC layer during a life span that incorporates telomere shortening and homing of EPCs. The results suggest that under normal conditions, endothelial monolayer integrity can be maintained by replication of neighboring cells. The model predicts that homing of EPCs will protect the layer from becoming discontinuous in case high cell turnover or increased oxidative stress takes place. The presented computational model may facilitate extrapolation of the consequences of experimental data to longer time periods.

## GRANTS

This research was financially supported by Dutch Ministry of Economic Affairs Senter Grant TSGE1028 and Unilever Research and Development (Vlaardingen, The Netherlands). B. Braam is a fellow of the Royal Dutch Academy of Arts and Sciences.

## Acknowledgments

Two anonymous referees are acknowledged for comments.

## Footnotes

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- Copyright © 2004 by the American Physiological Society