## Abstract

Brain aging is a complex process involving many functions of our body and described by the interplay of a sleep pattern and changes in the metabolic waste concentration regulated by the microglial function and the glymphatic system. We review the existing modelling approaches to this topic and derive a novel mathematical model to describe the crosstalk between these components within the conceptual framework of inflammaging. Analysis of the model gives insight into the dynamics of garbage concentration and linked microglial senescence process resulting from a normal or disrupted sleep pattern, hence, explaining an underlying mechanism behind healthy or unhealthy brain aging. The model incorporates accumulation and elimination of garbage, induction of glial activation by garbage, and glial senescence by over-activation, as well as the production of pro-inflammatory molecules by their senescence-associated secretory phenotype (SASP). Assuming that insufficient sleep leads to the increase of garbage concentration and promotes senescence, the model predicts that if the accumulation of senescent glia overcomes an inflammaging threshold, further progression of senescence becomes unstoppable even if a normal sleep pattern is restored. Inverting this process by “rejuvenating the brain” is only possible via a reset of concentration of senescent glia below this threshold. Our model approach enables analysis of space-time dynamics of senescence, and in this way, we show that heterogeneous patterns of inflammation will accelerate the propagation of senescence profile through a network, confirming a negative effect of heterogeneity.

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## Funding

We acknowledge support by the grant of the Ministry of Education and Science of the Russian Federation Agreement No. 074-02-2018-330(1). AZ received MRC grant MR/R02524X/1.

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This article is a contribution to the special issue on Immunosenescence: New Biomedical Perspectives - Guest Editors: Claudio Franceschi, Aurelia Santoro, and Miriam Capri

## Appendices

### Appendix 1: Model detalization

The changes in the concentration of garbage in a certain volume of brain are determined by the balance of its accumulation during neuronal activity on the one hand, and its degradation and clearance by microglial cells and astrocytes on the other. In space-time studies, we neglect the garbage diffusion, assuming it to be much slower than that of SASP molecules (see Eq. 6 below). The dynamics of garbage concentration is then described by the following equation:

where *Q*(*t*) denotes the (time-dependent) garbage production rate, *γ*_{1} is the rate of spontaneous degradation of garbage, and the two remaining terms describe two mechanisms of active garbage elimination: uptake by astrocytes and microglia (term with *γ*_{2}) and the effect of glymphatic system (term with *γ*_{3}).

The rate of garbage uptake by the glial cells depends on the state thereof, which can be active leading to fast cleaning, and normal quiet state associated with slow cleaning. The activation of the healthy glial cells is assumed to be determined by garbage concentration and is described by the function *h*(*R*), which we assume to be a sigmoid function of the following form:

where *x*_{0} denotes the rate of waste clearance by non-activated microglia, *R*_{1} is the activation midpoint, and *η*_{1} is the inverse slope of the activation curve (see Fig. 4a).

The glymphatic system is astrocyte-dependent and is activated during sleep. This daily variability is accounted for by the function *T*(*t*).

The saturation of both garbage elimination systems is described by normalized nonlinear functions *f*_{2,3}(*R*). Given no specific experimental data on either of them, we use a single function obtained from a kinetic description of garbage binding to glia:

where *R*_{0} is the saturation curve midpoint (which is the level of garbage concentration, where the elimination rate is half the maximum).

The conversion of glial cells to the senescent state is assumed to be induced by glial overactivation promoted by garbage accumulation, and also by SASP molecules, which are in turn constantly produced by the senescent cells. The progression of glial senescence is described by the following dynamical equation for the fraction of the senescent glia *G*_{S} (total quantity of glia assumed constant):

where *γ*_{B} is the background (always present) senescence rate, *γ*_{R} and *γ*_{S} control the rates of senescence induced by garbage and SASP, *h*_{R}(*R*) and *h*_{S}(*S*) are corresponding normalized activation functions, which are assumed to be sigmoid functions:

where *R*_{A} and *S*_{A} are senescence activation midpoints for garbage and SASP, *η*_{R} and *η*_{S} determine the widths of the smoothed step transition (see Fig. 4).

The changes of SASP molecule concentration \(\tilde {S}\) are determined by the balance of their production (by senescent astrocytes and microglia), natural degradation, and diffusion, and are described by the equation:

where *σ*_{G} is the rate of SASP production by senescent cells, *σ*_{S} is SASP degradation rate, *D* is the diffusion coefficient, and *Δ* is the Laplacian operator.

### Appendix 2: Model reductions

Here, we derive and analyze the reduced formulations of the model, as described in the “Reduced formulations of the mode”. All notations are additionally summarized in Table 2.

### 2.1 Dynamics of garbage concentration

Assuming the rate of spontaneous garbage degradation to be much smaller than that of active garbage cleaning by glial cells, we neglect the term with *γ*_{1} in Eq. 1, transforming the latter into:

The factor in square brackets here describes the dependence of garbage elimination rate on time *t* and concentration of garbage *R*. Generally speaking, it cannot be factorized into a product of separate functions of *t* and *R*. That said, and taking into account the absence of experimental data to quantify all the unknown functions within the bracketed expression, for the sake of model simplicity, we artificially impose the assumption of such factorization, which reduces the equation to the form:

where *C*(*t*) characterizes the time dependence of garbage elimination, and *f*(*R*) the concentration dependence. For the sake of analytical consideration, we use a simplified piecewise linear saturation function:

and in numerical simulations also the smooth function *f*(*R*) = *f*_{K}(*R*) as defined in Eq. 3.

Following this approach, we essentially abstract from quantifying the effect of glial cell activation (given by the function *h*(*R*) in the full model Eq. 1). If taken into account, glial activation would show up as a section with steeper dependence of garbage elimination rate upon garbage concentration somewhere below the saturation level (at \(R\lesssim R_{0}\)). We argue that this would only have a quantitative impact, while our main qualitative results remain mainly determined by saturation effects, which we necessarily take into account. Therefore, we limit ourselves to a paradigmatic model in the form Eq. 8, considering further detalization currently unattainable due to the aforementioned lack of experimental data.

Hereafter, we assume that *R* is measured in the units of *R*_{0}, which implies *R*_{0} = 1.

For further simplification, we recall that the fraction of the senescent glia is typically a small value. A quantitative reference for the number or frequency of senescent microglia/macrophages in the aged brain is currently lacking, due in part to the complexity of defining specific criteria for glial senescence in vivo. In a recent study, the percentage of senescent microglia ranged from 1 to 4% of the total population [30]. This is consistent with senescent cell numbers in other tissues and reflects a significant number of dysfunctional senescent cells. Indeed, very small numbers of transplanted senescent cells are enough to cause lasting physical dysfunction [115, 116]. Hence, we assume *G* = 1 without significant impairment of model precision.

To account for daily variability of garbage production and elimination rates, we assume that *Q*(*t*) and *C*(*t*) take on different constant values during wakefulness and sleep, these values further respectively denoted as *Q*_{w}, *C*_{w}, and *Q*_{s}, *C*_{s}.

Finally, the simplified version of Eq. 8 which we will analyze reads:

where the upper and the lower symbols correspond to wakefulness and sleep.

As soon as the piecewise linear saturation function *f*_{P}(*R*) Eq. 9 is used, Eq. 10 implies accumulation or decay of garbage at a constant rate above the saturation threshold in *R*:

and reduces to a linear equation below the saturation threshold:

Natural assumptions are:

otherwise, according to Eq. 12, a stationary concentration of garbage \(R_{\infty }=Q_{w}/C_{w}<1\) would be attained even without sleep, thus rendering sleep unnecessary, and

otherwise elimination of garbage during sleep would be impossible.

We measure time in days and denote the duration of sleep as *T*_{s}, and that of wakefulness as *T*_{w} = 1 − *T*_{s}. We note that whenever

Eq. 11 leads to infinite accumulation of garbage; thus, there exists a critical sleep duration \(T_{s}^{\text {crit}}\) determined by

If \(T_{s}>T_{s}^{\text {crit}}\) (sufficient sleep), garbage concentration during sleep falls below the saturation level (*R* < 1), and during wakefulness may or may not exceed the saturation level. Established dynamics of garbage concentration in this regime is shown in Fig. 5a. Here, we use the values of parameters *Q*_{w} = 5, *C*_{w} = 3, *Q*_{s} = 0, *C*_{s} = 6, which are justified below, along with *T*_{s} = 1/3, which corresponds to a sleep duration of 8 h.

At the critical sleep duration \(T_{s}=T_{s}^{\text {crit}}\), garbage concentration *R*(*t*) in the stationary regime demonstrates a saw-like profile with minimum at the saturation threshold *R* = 1 and with maximum

Established dynamics of garbage concentration in the critical regime is shown in Fig. 5b with the value \(R=R_{\max \limits }^{\text {crit}}\) marked with a red dashed horizontal line. Here, sleep duration is *T*_{s} = 1/4 (equivalent to 6 h), other parameters same as above.

If \(T_{s}<T_{s}^{\text {crit}}\) (insufficient sleep), then garbage concentration inevitably rises above the saturation level (*R* > 1) and accumulates infinitely (while the deficiency of sleep is present) with average rate:

Dynamics of garbage accumulation in this regime is shown in Fig. 5c with the average trend \(R(t) \sim \langle \dot R \rangle \cdot t\) shown with a red dashed line. Here, sleep duration is *T*_{s} = 5/24 (5 h).

In order to fit the model parameters to reality, we start with the assumption that garbage production during sleep is negligible compared with that during wakefulness (*Q*_{s} ≪ *Q*_{w}), so further we let *Q*_{s} = 0 for simplicity. The remaining model parameters *Q*_{w}, *C*_{w}, *C*_{s} can be quantified using the following considerations for a healthy human.

During sleep, the dynamics of garbage in the unsaturated regime (when *R* < 1) is described by the linear Eq. 12, which reduces to:

and implies exponential decay of garbage concentration to zero:

with characteristic time scale:

Then, sleep periods greatly exceeding *τ*_{s} (for definiteness, *T*_{s} > 2*τ*_{s}) are excessive in the sense that further sleeping does not improve garbage elimination significantly. Estimating this sufficient sleep duration as 8 h, we get *τ*_{s} equal to 4 h, or in the units of days *τ*_{s} ≈ 1/6, which produces an estimate *C*_{s} ≈ 6.

In order to estimate the two remaining parameters *Q*_{w} and *C*_{w}, we generally need two additional biologically relevant quantitative model outcomes to correlate with reality. For one of them, we use the critical sleep duration \(T_{s}^{\text {crit}}\), for which from Eq. 16 we get:

Estimating \(T_{s}^{\text {crit}}\approx 0.25\) (6 h), and using the above estimate *C*_{s} ≈ 6, we get *Q*_{w} − *C*_{w} ≈ 2. Due to scarcity of quantitative experimental data on the dynamics of garbage concentration in the brain, we further employ the observation of [5, 114] that the rate of garbage elimination during sleep is roughly twice than that during wakefulness, which finally yields *C*_{w} ≈ *C*_{s}/2 ≈ 3, *Q*_{w} ≈ *C*_{w} + 2 ≈ 5. This set of parameters was used to produce the profiles of *R*(*t*) in Fig. 5 and is used hereinafter, unless stated otherwise.

When the piecewise linear saturation function Eq. 9 in the model is replaced by a more realistic smooth function Eq. 3, with parameter values unchanged, the quantitative dynamics *R*(*t*) changes, but the qualitative behavior remains. This can be seen in Fig. 6, where panel (a) corresponds to normal sleep duration (8 h) and panel (b) shows the accumulation of garbage in case of sleep deficiency (5 h of sleep). Remarkably, the expression for average trend Eq. 18 still produces a good estimate for garbage accumulation rate in the saturation regime (shown with the red dashed line in the figure).

### 2.2 Dynamics of glial senescence

The sigmoid activation functions *h*_{R}(*R*) and *h*_{S}(*S*) Eq. 5 from dynamical equation for the fraction of senescent glia Eq. 4 for the sake of analysis can be written in the step form:

where *H*(⋅) is the Heaviside step function.

The balance equation for the unnormalized SASP concentration Eq. 6 allows for arbitrary rescaling of its variable of state, which allows eliminating one of its parameters. We make such rescaling by introducing a normalized SASP concentration variable *S* according to:

so that the unitary concentration *S* = 1 is now defined as the (actually unattainable) stationary concentration of SASP, which would be achieved if all the glia were senescent (*G*_{S} = 1). In this notation, the balance Eq. 6 transforms into

where the parameter *σ*_{S} determines the characteristic time scale *τ*_{S} of SASP concentration equilibration

### 2.3 Local dynamics of garbage and senescence

To characterize the joint dynamics of glial senescence and SASP, we first consider their local dynamics, which implies *D**Δ**S* = 0 in Eq. 25. This corresponds to the absence of diffusion *D* = 0, or to the spatially uniform case with *Δ**S* = 0.

Assuming the time scale *τ*_{S} of SASP dynamics to be much smaller (faster) than that of glial senescence, we can replace *S* in Eq. 4 by its quasi-steady-state approximation from Eq. 25, which is

Additionally, in order to focus on garbage-induced senescence, we neglect the background (garbage-independent) senescence by taking *γ*_{B} = 0. This way Eq. 4 is transformed into

Model Eq. 4 implies that the quantity of senescent glia never decreases. In the version Eq. 28 with step functions Eq. 23 taken for *h*_{R}(⋅) and *h*_{S}(⋅) senescence at best does not progress (*G*_{S} = *c**o**n**s**t*) while *R* < *R*_{A} and *G*_{S} < *S*_{A}, and increases otherwise.

In other words, while senescence remains below its threshold (*G*_{S} < *S*_{A}), the accumulation of senescent glial cells is conditioned by garbage and occurs only when the latter exceeds its respective threshold (*R* ≥ *R*_{A}).

As soon as senescence due to its course-of-life accumulation exceeds its threshold (*G*_{S} ≥ *S*_{A}), the right-hand side of Eq. 28 becomes positive regardless of the garbage level, which implies the monotonous accumulation of glial senescence due to a positive feedback via SASP even in the absence of further induction by garbage.

In this view, *R* = *R*_{A} is the “garbaging” threshold; its temporary overshoot by garbage concentration leads to an increase in the quantity of senescent glia, which however stabilizes (does not accumulate any more) once the garbage overshoot ends. The garbage increase, in turn, may be caused by a temporary deficiency or deprivation of sleep, as it follows from the garbage balance Eq. 8, according to the results of the Section 2.1. The height and the duration of a garbage concentration peak induced by a particular episode of sleep restriction depend increasingly upon the severity and the duration of sleep deficiency.

We illustrate this by simulating the joint dynamics of garbage concentration and glial senescence according to the Eq. 8 and Eq. 28 with smooth nonlinearities Eq. 3, Eq. 5. Parameters of garbage dynamics Eq. 8 are as in Section 2.1, and for senescence dynamics Eq. 28, due to the lack of real data, we pick indicative quantities *γ*_{R} = *γ*_{S} = 10^{− 4}, along with activation function parameters in Eq. 5*R*_{A} = 15, *η*_{R} = 1, *S*_{A} = 0.1, *η*_{S} = 0.005. In Fig. 7a, we show the simulation result on a time interval of 350 days. Most of the time, the sleep duration is 8 h (*T*_{s} = 1/3), except for two episodes of sleep restriction (*T*_{s} = 5/24 or 5 h) lasting for 10 and 20 days, which correspond to the first two peaks in the garbage dynamics (the upper panel in Fig. 7a), and two episodes of complete sleep deprivation (*T*_{s} = 0) lasting for 3 and 7 days, which correspond to the latter two peaks in the garbage dynamics. Indeed, we observe more pronounced garbage peaks as the duration of sleep deprivation or restriction is increased (the 2nd and the 4th peaks, compared with the 1st and the 3rd). Expectedly, complete sleep deprivation (the 3rd and the 4th peaks) produces a steeper increase of garbage than sleep deficiency (the 1st and the 2nd peaks). We have chosen the garbaging threshold value *R*_{A} = 15 so that only the more severe episodes of sleep restriction (20 days of sleep deficiency or 7 days of complete sleep deprivation) lead to the accumulation of glial senescence (the lower panel in Fig. 7a).

To demonstrate the effect of an age-related decrease of garbage elimination rate upon the dynamics of garbage and glial senescence, we performed a similar simulation with garbage elimination rates *C*_{s} = 5.7, *C*_{w} = 2.85, which are cut down by 5% compared with the previous simulation, all other conditions unchanged. The result is shown in Fig. 7b, showing higher garbage peaks and longer recovery times; moreover, all four episodes of sleep deficiency now produce glial senescence. The resultant increment of glial senescence over the same simulated time interval is now about 3 times greater than in the previous simulation (cf. Fig. 7a).

In turn, *G*_{S} = *S*_{A} is the “inflammaging” threshold. When senescent glia accumulates beyond this threshold (due to accumulation of the above mentioned “garbaging” overshoots during the life course), further progression of senescence becomes monotonous due to self-induction via SASP and does not stop until the end of life. This is illustrated by the simulation result shown in Fig. 7c, where all conditions are the same as in Fig. 7b, except for starting from a higher value of the senescence variable *G*_{S}(*t* = 0) = 0.085 and extending the simulated time interval (the relevant part is up to *t* = 500 days; the remaining part of the graph is discussed below). Here, the senescence variable, once reaching the inflammaging threshold *S*_{A} = 0.1, continues to grow even in the absence of garbaging peaks.

This feature of our model implies that inflammaging, once set in, never stops, even if the garbage elimination rate recovers to normal. More precisely, our model does not incorporate any mechanisms for inflammaging to stop. This follows from the assumptions that inflammaging is driven by SASP, which in turn is produced by senescent cells, which remain in this state perpetually.

A hypothetic possibility to extinguish inflammaging would require a mechanism to eliminate senescent cells from the brain faster than they build up. Wong [117] and Clarke with colleagues [22] discuss the replacement of aged microglia with young microglia as the “rejuvenative” therapy. These microglial “replacements” may promote the removal of accumulated garbage, and the slowdown of the cellular senescence process, thereby improving cognitive function in aging. Prospects for the role of techniques for clearance of senescent macrophages in prolonging healthspan are actively discussed in the scientific community now [36]. We simulate such recovery of senescent glia by artificially resetting the variable *G*_{S} to a sub-inflammaging value *G*_{S} = 0.085, as shown in Fig. 7c at *t* = 500. The concentration of senescent cells is now reset, self-induction and progression of senescence are stopped, and future behavior will again depend on whether sleep pattern and garbage elimination are normal or not.

### 2.4 Space-time dynamics of senescence (“propagaging”)

The model in the form of ordinary differential equations considered in Section 2.3, where diffusion is dropped and quasi-steady-state approximation is used to express SASP concentration, applies to the description of the dynamics of glial senescence in the brain taken as a whole, on large time scales up to the lifetime, or in local parts of the brain, while this local senescence does not propagate in space.

In order to describe the progression of senescence with spatial detalization, we revert to the full model consisting of the equations for glial senescence Eq. 4 and SASP balance Eq. 25 including the diffusion term. We focus on propagating solutions where glial senescence propagates itself through the tissue in the wake of diffusing SASP (a phenomenon referred to as propagation of inflammaging [40]). In this regard, we take an initial spatial profile of senescence at the onset of propagation as given, abstracting from its backstory and taking into account the only mechanism of senescence, namely that activated by SASP (*γ*_{S}≠ 0), while assuming *γ*_{B} = 0, *γ*_{R} = 0 in Eq. 4. Although we recognize activation by garbage as the root cause initiating senescence, we consider it as part of the mentioned “backstory,” which is adequately described by local dynamics as in Section 2.3, and hence we omit it in the model of inflammaging propagation.

Finally, we formulate the mathematical model as a system of simultaneous Eqs. 4 and 25 with the assumptions above taken into account:

We simulate a two-dimensional system with spatial inhomogeneity introduced into the initial profile of the senescence variable *G*_{S}, as shown in panel (a) of Fig. 8. To initiate inflammaging propagation, we place a patch with a sufficiently high senescence value in the center of the system. A typical snapshot of the senescence variable in color encoding taken at *t* = 25 is presented in panel (b) of Fig. 8.

We find that the spatial inhomogeneity of the initial senescence background may lead to faster propagation of the senescence front than it would be in case of homogeneous initial senescence with the same average. To quantify this statement, we calculate the ratio *𝜖* of the tissue volume with high level of senescence (where the condition *G*_{S} > *S*_{A} is fulfilled) to the total volume of the tissue. We compute *𝜖* over time in settings with homogeneous and inhomogeneous initial senescence backgrounds with the same average. The resulting plots are shown in Fig. 9 confirming faster propagation of senescence in the inhomogeneous case.

This phenomenon can be explained by quicker propagation of the front in areas with greater initial senescence, which allows the inflammaging to cover quickly larger distance, even though leaving behind some non-inflamed “holes” in places where initial senescence was low. These holes eventually get taken over by the inflammaging propagation process without hindering the fast propagation of the foremost parts of the front.

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Gordleeva, S., Kanakov, O., Ivanchenko, M. *et al.* Brain aging and garbage cleaning.
*Semin Immunopathol* **42, **647–665 (2020). https://doi.org/10.1007/s00281-020-00816-x

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### Keywords

- Inflammaging
- Aging
- Waste solutes
- Glymphatic system
- Sleep
- Microglia
- Cell
- Senescence