## Abstract

Since the fundamental defect in both type 1 and type 2 diabetes is β-cell failure, there is increasing interest in the capacity, if any, for β-cell regeneration. Insights into typical β-cell age and lifespan during normal development and how these are influenced in diabetes is desirable to realistically establish the prospects for β-cell regeneration as means to reverse the deficit in β-cell mass in diabetes. We assessed the mean β-cell age and lifespan by the classical McKendrick-von Foester equation that describes the age-based heterogeneity of β-cells in terms of the time-varying β-cell formation and loss estimated by a β-cell turnover model. This modeling approach was applied to evaluate β-cell lifespan in a rodent model of type 2 diabetes in comparison with nondiabetic controls. When rats were 10 mo old, mean β-cell lifespan was 1 mo vs. 6 mo in rats with type 2 diabetes vs. controls. A shortened β-cell lifespan in a rat model of type 2 diabetes results in a decrease in mean β-cell age and thus contributes to decreased β-cell mass.

- β-cell age
- mathematical model
- diabetes

pancreatic β

-cells play an essential role in determining the amount of insulin that is secreted to maintain blood glucose levels within a narrow range (14). Both type 1 (T1DM) and type 2 diabetes (T2DM) are characterized by a deficit in β-cell mass and increased β-cell apoptosis (1, 3). As a consequence of the progressive β-cell loss and dysfunction underlying both T1DM and T2DM, insulin daily demand eventually exceeds insulin secretion, leading to diabetes (14). Since the fundamental defect in both T1DM and T2DM is β-cell failure, there is increasing interest in the capacity, if any, for β-cell regeneration (4).

In this context, we (12) recently proposed a dynamic model to quantify β-cell turnover. The model describes β-cell number as the balance between β-cell formation and loss. β-Cells are added either by replication of existing β-cells or by other sources of β-cells (OSB), and they are mainly lost through β-cell apoptosis. The model was applied to the HIP rat model of T2DM and its wild-type (WT) counterpart (control). The HIP rat is transgenic for human IAPP on a Sprague-Dawley background. This rat model of T2DM has endoplasmic reticulum stress-induced β-cell apoptosis that leads to a progressive deficit in β-cell number and develops both anatomical as well as functional islet phenotype comparable to that in humans with T2DM (2, 10, 13). The β-cell turnover model revealed that in rats formation and maintenance of adult β-cells largely depend on OSB. Moreover, this source adaptively increases in the HIP rat, implying attempted β-cell regeneration that substantially slows loss of β-cells.

Although this model has provided insights into β-cell dynamics, how β-cell turnover affects mean β-cell age and lifespan still remains unknown. Insight into typical β-cell age and lifespan during normal development and how these are influenced in diabetes is desirable to realistically establish the prospects for β-cell regeneration as means to reverse the deficit in β-cell mass in diabetes. Perl et al. (13) recently applied the radiocarbon dating technique, originally proposed by Spalding et al. (17) for adipocytes, to quantify β-cell turnover in humans. Because of limited material, only three individuals were studied. The available data implied minimal β-cell turnover by replication in humans after age 30 yr, consistent with other studies (7, 11). That approach may not account for β-cells that arise by differentiation from any precursor pool. Also, the approach does not provide a measure of β-cell lifespan.

Under steady-state conditions (i.e., when the number of β-cells remains constant and their formation equals their loss; Fig. 1, *left*), mean β-cell age and lifespan coincide with the reciprocal of per capita β-cell death rate, i.e., the ratio between β-cell loss rate and number. In tracer technique studies, this is assimilable to the notion of mean residence time, i.e., the mean time that a particle, introduced in a compartment, spends in the compartment before irreversibly leaving it (6, 9). In nonsteady state, this simplification is not valid, since β-cell number, formation, and loss are time-varying (Fig. 1, *right*). In this study, we address this problem by using the classical McKendrick-von Foester equation. In statistics, this equation is the gold standard method to describe the dynamics of a population (i.e., birth and death rates) considering the age-based heterogeneity of individuals that constitute the population (5). Spalding et al. (17) applied this approach to the population of fat cells in humans. The mean age of adipocytes was measured by analyzing ^{14}C level in genomic DNA of the cell, and a statistical model for the mean age of fat cells based on the classical McKendrick-von Foester equation was proposed. The mean age of fat cells was expressed in terms of both adipocyte formation and death rates, which were assumed to be constant. By fitting the mean age of adipocytes in this model, the unknown fat cell formation and death rates were estimated. In contrast, we assess the mean β-cell age and lifespan by the classical McKendrick-von Foester equation that describes the age-based heterogeneity of β-cells in terms of the time-varying ingredients of the β-cell turnover model (12): β-cell formation and loss rates.

In principle, this modeling approach may be applicable to both T1DM and T2DM as long as β-cell formation and loss rates are known. To illustrate a practical application, this approach is applied to the HIP rat model of T2DM and to its WT counterpart to estimate mean β-cell age and lifespan.

## MATERIALS AND METHODS

#### Model.

In an individual of age *t* [time], β-cells differ from each other by their own age, which ranges from zero (for cells formed just at time *t*) to the age of the individual *t* (for cells formed at time zero, when gestation starts). To describe how the age of β-cells, denoted by a [time] distributes between these limits, let us introduce the density function n(*a*,*t*) [number of cells/time], which represents the relative likelihood for a β-cell to have an age equal to *a*, in an individual of age *t*. For example, considering a population of β-cells with a formation rate that linearly increases with the age of the individual while its death rate is constant, the resulting profile for the density n(*a*,*t*) is shown in Fig. 2. Since the formation rate linearly increases with the age *t* of the individual, the number of young β-cells also increases with *t*, as shown in the *top left* corner of the figure.

β-Cell number at time *t*, N(*t*) is the integral of the density function n(*a*,*t*) with respect to β-cell age, i.e.,

The ratio between density n(*a*,*t*) (*Eq. 1*) and β-cell number at time *t*, that is,
*Eq. 2*) represents the mean β-cell age [time] at time *t*, that is,

To calculate this value in practice (*Eq. 3*), a model for the density n(*a*,*t*) [and consequently for the probability density function (*Eq. 2*)] is required. On the following, this model is obtained in terms of β-cell formation and loss estimated by the β-cell turnover model proposed in (12) with a number of assumptions [Data Supplements Part S1 (Supplementary materials are found in the online version of this paper at the Journal website)]. Briefly, the turnover model equation
*t*)/d*t*; number of β-cells/time] as the balance between β-cell formation rate, F(*t*) [number of new formed β-cells per time], either by replication of existing β-cells or by other sources of β-cells, and loss rate, A(*t*) [number of apoptotic β-cells per time], mainly by β-cell apoptosis (i.e., loss rate = rate of β-cell apoptosis). Since this model is based on the homogeneity of β-cells in terms of turnover, the following assumptions are required to describe the age-based heterogeneity of β-cells as a function of the components of the β-cell turnover model (*Eq. 4*).

We first assume that after an event of β-cell replication one daughter cell retains the age of the parent cell and one is considered new. This implies that the density of β-cells of age zero at time *t* [i.e., n(0,*t*)] equals the new β-cell formation F(*t*). This assumption of one new and one aged cell after a cell division is required to permit evaluation of mean β-cell lifespan. Since, following cell division, sorting of established cellular components to the new cells is asymmetric, and by definition ∼50% of the cytoplasmic components are synthesised de novo, this assumption might be reasonably challenged by the alternative assumption that the two resulting cells are in aggregate one-half the age of the original cell. From a practical point of view, since after the period of postnatal expansion β-cell replication is relatively infrequent, this largely philosophical consideration of cell age becomes relatively unimportant, and this will especially be the case when applied to humans.

Second, we assume that all the β-cells have the same probability of dying, i.e., undergoing apoptosis. This implies that the per capita death rate at age *a* at time *t*, i.e., μ(*a*,*t*) [1/time], is independent of *a*, that is, μ(*a*,*t*) = μ(*t*). It is possible to demonstrate that this quantity equals the ratio between the rate of β-cell apoptosis and β-cell number (Data Supplements Part S1), namely,

Third, we hypothesize that time zero (i.e., *t* = 0) corresponds to the start of gestation, so that at *t* = 0 density of aged *a* β-cells, β-cell number, new β-cell formation, and per capita death rate are all zero.

Under these assumptions, it is possible to express the density of β-cells of age *a* at time *t* in terms of the same variables appearing in the turnover model (*Eq. 4*), namely,
*a* ≤ *t* (Data Supplements Part S2). The exponential term appearing in *Eq. 6* can be interpreted as the survivor function [dimensionless], which describes the probability of a β-cell surviving until age *a* at time *t* as a function of the per capita β-cell death rate (Data Supplements Part 3). Thus, the density n(*a*,*t*) is given by the product of two terms: the fist term is the formation of β-cells at time *t* − *a* [i.e., β-cells that contribute to the density n(*a*,*t*) have age *a* and were born at time *t* − *a*]; the second term selects these β-cells according to the survival function: only those cells of age *a* that survive until time *t* contribute to the density (i.e., those cells that die according to the per capita β-cell death rate μ are not considered). In other words, *Eq. 6* states that the density of β-cells of age *a* at time *t* is constituted by those β-cells born at time *t* − *a* that survive until age *a*. Having estimated the density n(*a*,*t*) allows to calculate the probability density function (*Eq. 2*) and the mean β-cell age (*Eq. 3*) as functions of the ingredients of the β-cell turnover model (12): β-cell formation and loss rates.

As the density function (*Eq. 6*) describes, in terms of number, how β-cells are distributed in an individual of age *t* based on their own age, the survivor function (exponential term in *Eq. 6*) describes how β-cells are distributed in terms of probability to survive (Data Supplements Part S3). Therefore, as β-cell number at time *t* is the integral of the density function n(*a*,*t*) with respect to β-cell age (*Eq. 1*), the integral of the survivor function between zero and *t* is the mean β-cell lifespan [time] at time *t*, i.e.,

In steady state, mean β-cell age and lifespan coincide with the reciprocal of per capita β-cell death rate (Data Supplements Part S4).

#### Data.

The statistical model described by *Eq. 6* is applied to a rodent model of T2DM, the HIP rat, and its WT counterpart (2, 10, 13). Pancreata from 20 WT and 18 HIP rats, at 0.07 (i.e., 2 days), 2, 5, and 10 mo of age (3–5 WT and HIP rats per type and age group) were analyzed, and an estimation of β-cell number, formation, and loss rates was obtained as formerly reported (12, 13). Recalling briefly the previous studies, pancreatic tissues were stained for insulin (peroxidase staining) and hematoxylin to establish fractional β-cell area. The β-cell mass for each animal was consequently obtained multiplying the fractional β-cell area by the pancreatic weight. Then, on the assumption that in aqueous organs, such as pancreas, 1 g of weight equals 1 cm^{3} of volume, in each rat the total number of β-cells was established as the ratio between β-cell mass and β-cell size (measurement of 100 distances was taken in each animal and the mean volume of a β-cell is calculated considering that, on average, cells approximate a sphere). As to β-cell replication and apoptosis, all β-cells per pancreatic section (two sections per animal) were examined for the frequency of β-cell replication as fractional Ki67- and insulin-positive cells. The frequency of β-cell apoptosis for each rat was similarly computed by examination of the TUNEL-positive β-cells. Rates of β-cell replication and apoptosis were derived from frequencies of β-cell replication and apoptosis [fraction of replicated (apoptotic) β-cells] each multiplied by the respective conversion factors estimated in Ref. 16 and by β-cell number. Rate of change in β-cell number and mean smoothed profiles for rates of β-cell replication and apoptosis obtained with a stochastic regularization method (8) were used to estimate the contribution to β-cell formation rate from other sources of β-cells other than β-cell replication, rewriting *Eq. 4* as
*t*) and OSB(*t*) is F(*t*).

#### Statistical analysis.

All data are represented as means ± SE. All the analysis was done by MATLAB software (The Mathworks, Natick, MA).

## RESULTS

Mean β-cell number (*top*), new β-cell formation (*middle*), and per capita β-cell death rate (*bottom*) in 20 WT and 18 HIP rats, aged 2 wk-10 mo, from prior studies (10, 11), are shown in Fig. 3. HIP rats showed a reduced β-cell number (10^{7} β-cells) by 2 mo of age and an increased per capita β-cell death rate (1/mo) with respect to WT rats. The rate of new β-cell formation in WT rats fell from a high (2.4 × 10^{7} β-cells/mo) at 2 wk of age to a nadir at 4 mo of age, after which it again increased progressively, presumably reflecting the need for increased β-cell mass with age-related insulin resistance. In contrast, in the HIP rat, the rate of new β-cell formation (10^{7} β-cells/mo) was relatively constant, perhaps reflecting attempted regeneration in the setting of a β-cell deficit and shortened β-cell lifespan (vide infra).

The new β-cell formation and the per capita β-cell death rates were applied to *Eq. 6* to calculate the density function n(*a*,*t*), shown in Fig. 4. While the HIP rats (Fig. 4, *bottom*) always show a high density of young β-cells, in the population of β-cells in WT rats the results are more heterogeneous, with a high density of new born β-cells at birth and around 10 mo of rat age.

The mean β-cell age (Fig. 5) and lifespan (Fig. 6) were then calculated from the β-cell turnover data according to *Eqs. 3* and *7*. From Fig. 5, it is evident that mean β-cell age (∼9 days) was comparable in WT and HIP rats at 2 wk of rat age. Thereafter, mean β-cell age increased to 3.5 mo in 10-mo-old WT rats. In contrast, in HIP rats, mean β-cell age reached a maximum of 1.2 mo by 4 mo of rat age and then declined to 0.8 mo by 10 mo of rat age.

The mean β-cell lifespan (Fig. 6) showed a pattern similar to mean β-cell age (Fig. 5). At 2 wk of age, mean β-cell lifespan was comparable in WT and HIP rats: 0.7 mo. In WT rats, the mean β-cell lifespan gradually increased to 4.9 mo by 10 mo of rat age. In contrast, in HIP rats, the mean β-cell lifespan increased to a peak of 1.3 mo by 3.5 mo of rat age, declining gradually thereafter to 0.8 mo by 10 mo of rat age.

## DISCUSSION

To address the general problem of estimating mean β-cell age and lifespan, we proposed the classical McKendrick-von Foester equation used in population dynamics (5). The equation *Eq. 6* statistically described the age-based heterogeneity of β-cell population as a function of β-cell formation (from replication of existing β-cells and from other sources) and β-cell loss (by apoptosis), estimated from the β-cell turnover model recently proposed in (12). Mean β-cell age and lifespan were then calculated from the model based on statistical considerations (*Eqs. 3* and *7*).

There is as yet no effective means to measure β-cell mass in vivo and no immediate prospect of being able to evaluate β-cell turnover and lifespan in vivo. Therefore, insight into the mechanisms subserving the deficits in β-cell mass in T1DM and T2DM are restricted to cross-sectional sampling of pancreas at a single time point. This approach has provided useful insights into the extent of the deficit of β-cell mass and some of the potential mechanisms; it does not readily lend itself to evaluation of the kinetics of β-cell turnover. Recent studies have provided some new tools that permit the present statistics-based approach to evaluate β-cell lifespan and age. Directly measured β-cell turnover by time lapse studies compared with subsequent cross-sectional histological evaluation of the same tissue provided conversion factors for the frequency of β-cell replication and apoptosis to the respective rates (16). These were used to establish a model for β-cell turnover (12), which in turn permitted the present evaluation. Application of the McKendrick-von Foester equation to the β-cell turnover data thus computed provided insights into the mean age and lifespan of β-cells with aging in health and in the HIP rat model of T2DM. In WT rats by 10 mo of rat age, the mean β-cell lifespan was ∼6 mo. In contrast, in the HIP rat model of T2DM the mean β-cell lifespan remained much lower than in WT rats at ∼1 mo (Figs. 5 and 6). In essence, this illustrates the mechanism subserving the deficit in β-cells in this model of T2DM and likely in humans with T2DM, specifically that, despite ongoing β-cell formation, β-cell lifespan is shortened. As with all novel analyses, there are constraints and limitations that should be evaluated.

As described in the introduction, under steady-state conditions, mean β-cell age and lifespan are estimated as the reciprocal of β-cell loss per unit of β-cells, i.e., the reciprocal of per capita β-cell death rate μ. To quantify the bias introduced by approximating mean β-cell age and lifespan with the reciprocal of μ in nonsteady state, the three profiles were compared (Fig. 7). The reciprocal of μ generally leads to bad estimates of mean β-cell age and lifespan when steady-state conditions were not verified. This was evident in WT rats, where the error introduced was 216% for mean β-cell age and 117% for mean β-cell lifespan (error evaluated as the percentage relative difference between the areas under the curves). In HIP rats, the bias was less marked (45% for mean β-cell age, 28% for mean β-cell lifespan), since the steady-state conditions were well approximated (Fig. 3), and after 8 mo of rat age the three profiles mostly coincided. This analysis also revealed the difference between mean β-cell age and lifespan, which is expected since the first describes how old on average β-cells are in a individual and results in a function of both β-cell formation and loss (*Eq. 3*), while mean β-cell lifespan is only a function of β-cell loss (*Eq. 7*), since it expresses the life expectancy of β-cells in an individual.

There are also limitations described in detail previously in relation to the model for measurement of β-cell turnover, including the potential confounding effects of heterogeneity in the behavior of β-cells with regard to turnover (12) as well as the certainty and variability of the conversion factors to convert frequency of β-cell replication and apoptosis to the corresponding rates (12, 16). Other limitations derived from the assumptions of the statistical model (*Eq. 6*); that is, after an event of β-cell replication, one daughter cell retains the age of the parent cell and one is considered new; all β-cells have the same probability to die; *t* = 0 corresponds to the start of gestation (Data Supplements Part S1). These hypotheses let us calculate mean β-cell age and lifespan from the available components of the β-cell turnover model, and their impact was determined by comparing the profile of β-cell number N(*t*) with the area under the curve of the density n(*a*,*t*), according to *Eq. 1*. The comparison revealed similar profiles in both WT and HIP rats (Fig. 8), and the percentage relative differences between the areas under the two curves were less than 9%, ensuring that model assumptions did not introduce a relevant bias into results. Alternative assumptions, e.g., the use of parametric profiles for the per capita β-cell death rate, based on the alternative hypothesis that the older a β-cell the higher the probability that it will die, do not improve upon these results.

The model offers new insights into β-cell dynamics as much as it defines the underlying mechanism subserving the β-cell deficit in a rat model of T2DM in which the islet phenotype closely recapitulates that of the islet in humans with T2DM (2, 10, 13). The value in the insights thus obtained is that it provides a framework within which β-cell regeneration might be contemplated as an approach to the therapy of diabetes. For example, it is apparent from these data that therapeutic strategies targeted to permit a more normal lifespan of β-cells would be more effective in restoring β-cell mass in the HIP rat than strategies to enhance the rate of β-cell formation. Moreover, since β-cells with a shortened lifespan are presumably damaged, the impact of the β-cell defect can be predicted to be enhanced by a β-cell deficit in the residual cells. Given the implausibility of measuring β-cell lifespan in humans in health or disease in the foreseeable future, the present approach offers an opportunity to approach this problem in human pancreatic tissue. From such studies it will then become possible to estimate the time course that would be required to reverse the defect in β-cells present in T1DM or T2DM through β-cell regeneration.

In summary, the classical McKendrick-von Foester equation used in population dynamics (5) was proposed to estimate mean β-cell age and lifespan. The modeling approach, applied to the HIP rat model of T2DM and its WT counterpart, showed a decreased mean β-cell age and lifespan in the first as a consequence of increased death rate.

## GRANTS

This study was partially supported by the National Institutes of Health (National Institute of Diabetes and Digestive and Kidney Diseases DK-077967), the Juvenile Diabetes Research Foundation, the Larry L. Hillblom Foundation, and Ministero dell'Istruzione, dell'Università e della Ricerca.

## DISCLOSURES

No conflicts of interest are reported by the authors.

## ACKNOWLEDGMENTS

We thank our colleagues at the Department of Information Engineering, University of Padova, and at the Larry Hillblom Islet Research Center, UCLA, for their useful comments.

Current address of E. Manesso: Division of Computational Biology and Biological Physics, Department of Theoretical Physics, Lund University, Lund, Sweden.

- Copyright © 2011 the American Physiological Society