Abstract
Diabetic wounds are common in patients with type 2 diabetes; they are ischemic and inflammatory, and difficult to heal without intervention. Hyperbaric oxygen therapy (HBOT) is a standard treatment, but its effectiveness is limited to a subset of the aging population. Senescent fibroblasts, a hallmark of aging, impair wound healing, and senolytic drugs, like quercetin (Q), which target senescent cells, may improve healing. In this study, we developed a mathematical model that defines biological aging through two parameters, 
and 
, that decline with age. These parameters reflect the biological age of an individual, where 
represents fibroblast proliferation and 
represents the production of the angiogenetic protein VEGF. Our model predicts that treatment with only HBOT achieves wound closure, within normal expectable time, for patients with a limited subset pairs of 
, and this subset is increased to a larger subset by combining Q with HBOT. The two subsets of 
are determined explicitly by simulations of the model. To make these results applicable in clinical setting, one will have to relate the aging parameters 
and 
to tangible marks of biological-aging factors.
Subject terms: Computational models, Applied mathematics, Statistics, Differential equations
Introduction
The process of wound healing is divided into four overlapping stages: homeostasis, inflammatory, proliferation, and remodeling. In homeostasis, immediately after injury, platelets from damaged capillaries in the wound bed release PDGF, which stimulates resident fibroblasts1, who are then attracted to the wound and begin to secrete PDGF2. Actually, neutrophils are one of the first cells that are recruited to the site of the wound. Their primary role is to prevent infection by attacking any microbe attempting to invade the body through the open skin wound3. For simplicity, we do not include in our model neutrophils and this very early phase of inflammation. In the inflammatory phase, blood monocytes are attracted to the wound microenvironment, where they differentiate into inflammatory macrophages 
4 that produce inflammatory cytokines, in particular TNF-
5. Fibroblasts secrete TGF-
6, which induces polarization from M1 to anti-inflammatory M2 macrophages7, and M2 macrophages secrete TGF-
5. TNF-
induces polarization from M2 to M1 macrophages8. In the proliferation phase, M2 macrophages and fibroblasts secrete VEGF4,9, and VEGF promotes angiogenesis by stimulating proliferation of endothelial cells and the blood capillary system, thereby increasing the supply of oxygen to the cells in wound microenvironment10 and enabling growth of tissue into the wound. Collagen deposition by fibroblasts, promoted by TGF-
11, enables wound closure and end of the actual healing of the wound, except for wound remodeling. The fourth phase of wound remodeling and scar formation may take many months. Here we focus on the first three phases and on wound closure.
In normal dermal wound healing, the inflammatory phase takes at most one or two weeks, and wound closure takes weeks. Such wounds are called acute wounds. Wounds that do not heal in normal expectable time are called chronic wounds. Chronic wounds are consistently inflamed and may not heal without intervention. Chronic wounds include diabetic foot ulcer, ischemic wounds (arterial insufficiency), and pressure ulcer. Chronic wounds are more common in older rather than younger individuals12,13. Reviews of chronic wounds and potential therapies in adults are given in12–15.
Cellular senescence is a permanent arrest of normal cell cycle, while maintaining cell viability. Senescent cells secrete senescence-associated secretory phenotype (SASP) which include variety of proteins. Cellular senescence is the primary hallmark of aging, but tissue disruption associated with cutaneous wounds also gives rise to senescent cells16. SASP of senesent fibroblasts secrete VEGF17 and IL-618,19.
A review of cellular senescence in skin aging and age related pathologies, including dermal wounds, is given in20. The total number of fibroblasts is reduced by 35% in aged skin (>80 years) while the number of senescent fibroblasts is increased significantly with age21,22. Senescent fibroblasts in aging negatively affect dermal wound healing, and may lead to chronic wounds23. Senescent fibroblasts and macrophaes, exacerbate inflammation by secreting IL-614,19, which, in chronic wounds, enhance M2
M1 polarization18,24. The production of VEGF by macrophages and fibroblasts in wound healing is reduced in aging25 by cellular senescence26, resulting in impaired angiogenesis25,26.
In this paper we consider diabetic wounds: wounds that are more common in patients with diabetes type 2. Such wounds are consistently inflammatory. Indeed the rate of 
correlates with the degree of insulin resistance27. This means that the polarizations M1
M2 are weighted toward M1, or that the normal transition from M1 to M2 is partially blocked.
Diabetic dermal wounds (e.g., diabetic foot ulcer) are ischemic, while the adaptive response to hypoxia is impaired in diabetes due to hyperglycemia28. Diabetic wounds are treated by oxygen therapy, either by topical wound therapy29, or by hyperbaric oxygen therapy30–32 where the patient spends several hours daily in the hyperbaric chamber where the oxygen pressure is three times the pressure in air.
Senolytic drugs (e.g., quercetin, dasatinib, fisetin) are drugs that eliminate senescent cells. Such drugs are currently under study in diseases that are exacerbated in aging patients. In the present paper we consider diabetic wounds treated by combination of hyperbaric oxygen therapy (HBOT) and quercetin (Q). Experimental studies report that quercetin improves wound healing33 and wound closure34.
In this paper, we develop an age-structured mathematical model for diabetic wounds, represented by a system of partial differential equations. We use this model to evaluate the effectiveness of combining Q with HBOT in improving wound closure in diabetic patients.
We conduct simulations to assess the percentage of wound closure in patients of various ages under treatment with HBOT alone, Q alone, and combination of Q and HBOT.
Here “age” is defined by two parameters, 
and 
(
), which are increasing with chronological age. We determine the set of points 
where wound closure in normal expectable time can be achieved by combining Q with HBOT, but not by HBOT alone.
Mathematical model
The mathematical model is based on the network shown in Fig. 1, where the blue connections represent the network for acute wound healing, and the red connections represent the network for chronic wound healing. The blue connections in Fig. 1 exclude senescent cells and diabetes-related variables. In this case, we assume the wound heals in normal expectable time without any treatment. On the other hand, the red connections in Fig. 1 incorporate perturbations to the healing process caused by aging (senescent cells, denoted by 
) and diabetes (denoted by 
and 
), which contribute to the development of chronic wounds. Table 1 lists the variables of the model; densities and concentrations are all in units of g/
.
Fig. 1.
Network of diabetic wound in aging. w=oxygen, E=endothelial cells, F=fibroblast, 
=senescent fibroblast, 
=ECM. 
and 
represent blockades in diabetic wounds.
Table 1.
Variables used in the model. Densities and concentrations are in units of g/
.
| Variables | Descriptions | Variables | Description |
|---|---|---|---|
 |
Density of M1 macrophages |  |
Density of M2 macrophages |
| E | Density of endothelial cells | F | Density of fibroblasts |
 |
Density of senescent fibroblasts |  |
Concentration of ECM |
 |
Concentration of TNF-
|
 |
Concentration of TGF-
|
 |
Concentration of IL-6 | P | Concentration of PDGF |
| V | Concentration of VEGF | w | Concentration of oxygen |
The wound region is assumed to be a cylinder with an axis normal to the skin and, for simplicity, we consider only the two-dimensional circular cross-section (e.g., the base of the cylinder) as in Fig. 2, with time-varying boundary defined by 
. The partially healed tissue is the shell 
, while the shell 
represents normal healthy tissue. For simplicity, as in36, we ignore the thickness for the wound and assume that the “flat” wound is radially symmetric, depending solely on (r, t). Throughout the healing process, the extracellular matrix (ECM) and its density 
experience continuous movement with velocity 
, and that 
is increasing.
Fig. 2.
Geometry around the wound. Wound area is 
, partially healed area is 
, healthy area is 
. This figure was reproduced from35.
To determine the velocity v, we adopt the approach outlined in36, assuming that the partially healed tissue exhibits viscoelastic properties, which we model as a single-phase upper convected Maxwell fluid with pressure depending on its density: elastic over short timescales and viscous over longer ones. Given the slow dynamics of the healing process, we treat it as quasi-static. Letting 
represent the internal isotropic pressure associated with 
in the partially healed medium, we use the following equation for v:
 |
where 
is defined as in36 as follows:
 |
1 |
for some positive parameters 
and 
.
Fibroblasts and M2 macrophages are sensitive to hypoxia due to their dependence on oxidative phosphorylation, whereas M1 macrophages rely on glycolysis for respiration and are not affected by low oxygen levels37,38. Accordingly, we assume that the growth rates of fibroblasts F and M2 macrophages (but not M1 macrophages) are proportional to:
 |
while the apoptotic death rate for these cells increases in proportion to:
 |
where 
represents the oxygen concentration in healthy tissue, 
denotes the oxygen level in extreme hypoxia, and H(s) is defined as 0 if 
and 1 if 
.
We assume that all cells within the partially healed tissue are moving with the same radial advection velocity v, and, in addition, they undergo diffusion. We can then write the dynamics of each species X of cells, in the partially healed tissue, in the following form:
 |
2 |
where 
is the diffusion coefficient, and 
represents the balance of the mass of X by the exchanges indicated in Fig. 1. We use the same structural equation for each cytokine X, but drop the advection velocity, since it is negligible compared to large diffusion coefficients of cytokines. An expression in 
of the form 
describes a process where species Y (e.g., proteins) is absorbed by cells Z at rate 
; 
is called the half-saturation of Y.
Equation for ECM (
)
Fibroblasts produce ECM proteins (e.g. collagen) in a process that is enhanced by TGF-
11,35. We assume that the ECM in the partially healed region is also undergoing advection with the velocity v, and write the equation for 
as follows:
 |
3 |
where 
, the carrying capacity of ECM, is larger than 
in Eq. (1) and 
is the degradation rate of 
.
Equations for M1 and M2 macrophages
We assume, as in35, that there is a constant source of M1 macrophages from the blood to the wound, 
, and a constant polarization from M1 to M2 macrophages (at rate 
). According to Fig. 1, PDGF attracts and activates M1 macrophages4, and TGF-
induces polarization from M1 to M2 macrophages7; and IL-618,24 and TNF-
8 promote polarization of M2 to M1 macrophages. Hence 
satisfies the following equation:
 |
4 |
where 
is a source of inactive M1 macrophages in the skin, 
is the death rate, and 
is the chemotactic coefficient of PDGF (P).
M2 macrophages are derived from the exchange M1
M2, their growth rate is proportional to G(w), and their death rate increases due to a hypoxia. Hence,
 |
5 |
Equation for fibroblasts (F) and senescent fibroblasts (
)
Fibroblasts are attracted to PDGF2. As in35, we model their proliferation by a logistic growth model, with both proliferation and death rates being dependent on oxygen level. Fibroblasts can become senescent due to a variety of intracellular and extracellular stressors16. We assume a constant rate of senescent fibroblasts formation. In order to distinguish between old and young patients, we introduce a parameter 
in that regulates the proliferation of senescent cells. Hence,
 |
6 |
 |
7 |
where, 
and 
are constants, and 
. The parameter 
is decreasing with chronological age16.
Equation for endothelial cells (E)
We assume a constant source (
) of endothelial cells.
VEGF chemoattracts endothelial cells and stimulates their proliferation10. Hence E satisfies the following equation:
 |
8 |
where 
is the chemotactic coefficient of V.
Equation for PDGF (P)
PDGF is released by damaged platelets from the wounds39 and fibroblasts2, and PDGF is depleted through chemoattracting F and 
. We assume that the source of damaged platelets from the wound is proportional to the radius R(t) of the wound. Hence PDGF satisfies the following equation:
 |
9 |
where 
is the degradation rate of P and 
is constant in chronic wounds.
Equation for TNF-
(
)
TNF-
is produced by M1 macrophages5, hence
 |
10 |
where 
is the degradation rate of 
.
Equation for TGF-
(
)
TGF-
is produced by fibroblasts and M2 macrophages6, so that
 |
11 |
Equation for IL-6 (
)
IL-6 is secreted by senescent fibroblasts18,19, so that
 |
12 |
Equation for VEGF (V)
VEGF is produced by fibroblasts9, senescent fibroblasts17,40,41 and M2 macrophages4,9. The senescent fibroblasts secrete more VEGF than their presenescent counterparts40,41. However, in diabetic wounds of old people, VEGF production is decreased by both F and 
25. VEGF is also depleted when it combines with receptors on endothelial cells. Hence,
 |
13 |
where A(t) is a decreasing function in t. We take
 |
14 |
where 
and 
are a constants. The parameter 
is decreasing with chronological age.
Equation for oxygen (w)
Hence, w satisfies the following equation: We identify the density of blood vessel by E. Hence oxygen is increased proportionally to E, and is utilized by fibroblasts and M2 macrophages35. Hence,
 |
15 |
Boundary conditions
We take the boundary conditions at 
to be the average serum concentration of the variables in diabetes type 2:
 |
16 |
On the free boundary 
we prescribe the following conditions:
 |
17 |
Note that in Eq. (17) all species, except P, satisfy no-flux condition, while there is an influx of P from the wound, proportional to the size of the wound.
The partially healed area has a fixed boundary 
and a boundary 
that moves in response to the pressure 
. Accordingly, we take
 |
18 |
 |
19 |
We assume that the wound boundary 
decreases with the velocity v of the ECM, that is
 |
20 |
v is expected to have negative values.
Initial conditions
We take 
cm, 
cm, and initial conditions for 
as follows:
 |
21 |
for all other species, where 
is the steady state value as in (16) and 
is the dry weight of tissue.
Since the level of senescent cells depends on the age of the patient, we take
 |
22 |
where 
represents the level of 
for the oldest patient (with 
).
Note that the initial conditions are consistent with the boundary conditions at 
and 
.
Simulations and results
All the computations were done using Python 3.7.3. The parameter values of the model equations are estimated in Section 7 and are listed in the Table 4. We introduce the percent of wound closure (PWC) by the formula
 |
23 |
Table 4.
Parameters for the model.
| Parameters | Descriptions | Values | References |
|---|---|---|---|
 |
Rate of production of ECM by F | 21.71 
|
est. |
 |
Rate of recruitment of 
|
|
est. |
 |
Rate of  -induced transition 
|
|
est. |
 |
Rate of  -induced transition 
|
|
est. |
 |
Rate of  -induced transition 
|
|
est. |
 |
Rate of transition 
|
|
est. |
 |
Rate of proliferation of F | 0.15 
|
est. |
 |
Rate of transition 
|
0.015 
|
est. |
 |
Rate of maturation of E by EPC |
|
est. |
 |
Recruitment rate of P |
 g/
|
est. |
 |
Rate of production of P by F |
|
est. |
 |
Rate of production of 
|
|
est. |
 |
Rate of production of  by F
|
|
est. |
 |
Rate of production of  by 
|
|
est. |
 |
 -factor of  -production by F
|
0.1 | est. |
 |
Rate of production of V by 
|
3.17 
|
est. |
 |
Rate of production of V by F | 3.17 
|
40,41est. |
 |
Rate of production of V by 
|
31.7 
|
40,41est. |
 |
Rate of production of  by 
|
|
66est. |
 |
Rate of circulation of oxygen |
|
est. |
 |
Rate of Q-enhanced proliferation of F | 0.2 d 
|
est. |
 |
Rate of consumption of oxygen | 0.3  /g 
|
est. |
 |
Factor of internal isotropic medium | 0.19 
|
est. |
 |
Chemotactic coefficient of P | 1000  /g d |
81est. |
 |
Chemotactic coefficient of V | 1000  /g d |
81est. |
 |
Fraction of effect of  on production of  by F
|
0.1 | est. |
 |
Coefficient of maturation of E by V | 1 | est. |
 |
Half-saturation of 
|
 g/
|
70est. |
 |
Half-saturation of 
|
 g/
|
69est. |
 |
Half-saturation of P |
 g/
|
71est. |
 |
Half-saturation of 
|
 g/
|
72est. |
 |
Half-saturation of V |
 g/
|
9est. |
 |
Healthy state of ECM | 0.04 g/
|
61est. |
 |
Threshold of pressure of ECM | 0.008 g/
|
61est. |
 |
Carrying capacity of ECM | 0.044 g/
|
61est. |
 |
Recruitment rate of 
|
 g/
|
est. |
 |
Oxygen concentration |
 g/
|
82,83est. |
 |
Oxygen concentration in extreme hypoxia |
 g/
|
82,83est. |
 |
Steady state in health of 
|
 g/
|
63est. |
 |
Steady state in health of 
|
 g/
|
63est. |
 |
Steady state in health of F |
 g/
|
64,65est. |
 |
Steady state in health of 
|
 g/
|
66est. |
 |
Steady state in health of E |
 g/
|
67,68est. |
 |
Degradation rate of ECM | 0.37 
|
36 |
 |
Degradation rate of PDGF | 33 
|
84est. |
 |
Degradation rate of TNF-
|
216.6 
|
76est. |
 |
Degradation rate of TGF-
|
495.1 
|
85est. |
 |
Degradation rate of VEGF | 16.5 
|
86est. |
 |
Degradation rate of oxygen |
|
87est. |
 |
Death rate of 
|
0.033 
|
78est. |
 |
Death rate of 
|
0.099 
|
78est. |
 |
Death rate of endothelial cells | 0.045 
|
80est. |
 |
Death rate of F | 0.02 
|
88est. |
 |
Death rate of 
|
0.025 
|
est. |
 |
Death rate of 
|
1.073 
|
73est. |
 |
Depletion rate of P by F |
|
est. |
 |
Depletion rate of P by M |
|
est. |
 |
Depletion rate of E by V | 19.8 
|
est. |
 |
Depletion rate of  by w
|
|
est. |
 |
Death rate of F by w |
|
est. |
 |
Rate of Q-enhanced secretion of  by F
|
 d 
|
est. |
 |
Rate of elimination of  by Q |
|
est. |
 |
Diffusion coefficient of 
|
|
89est. |
 |
Diffusion coefficient of 
|
|
89est. |
 |
Diffusion coefficient of E |
|
89est. |
 |
Diffusion coefficient of F |
|
89est. |
 |
Diffusion coefficient of 
|
|
89est. |
 |
Diffusion coefficient of P |
|
90est. |
 |
Diffusion coefficient of 
|
|
91est. |
 |
Diffusion coefficient of 
|
|
91est. |
 |
Diffusion coefficient of V |
|
92est. |
 |
Diffusion coefficient of w | 2 
|
93est. |
 |
Diabetic effect of oxygen flow blockade | 10 | This work |
 |
Diabetic effect of  blockade |
2 | This work |
 |
Diabetic effect of  blockade |
50 d | This work |
Diabetic wounds without senolytics
In the sequel we define “age” by the two parameters, 
and 
, which are decreasing when the chronological age (t) increases. In particular, a person with (
) is older than a person with 
if 
and 
.
In Fig. 3 we show all the model variables and display the percent of wound closure at day 30 in the no drug case, for the oldest patient and a younger patient. The percentage of wound closure after 30 days is 5.1% in the oldest patient, and 6.7% in the younger patient. In both cases, as expected, the pro-inflammatory 
increases and tends to stabilize at a high value, while the anti-inflammatory 
continues to decrease. The oxygen level initially increases but then decreases due to the hypoxia-driven effects of diabetes. The level of senescent fibroblasts, denoted as 
, increases monotonically with age in both the youngest and oldest patients, but the 
level in the youngest patients remains lower than in the oldest patients. In each case, the level of IL-6 follows the trends observed for 
.
Fig. 3.
Simulations of all the model variables in oldest patient (
), and younger patient (
), with no drug. All the variables are in units of g/
.
It is interesting to note that whereas VEGF increases to 
g/
in young patients, in old patients it increases to only 
g/
. F is also smaller in old compared to young patients, and so is 
.
Simulations with drugs
Quercetin
Senolytics, such as quercetin and fisetin, are currently undergoing clinical trials for various diseases like idiopathic pulmonary fibrosis, chronic kidney disease, Alzheimer’s disease, and diabetes42. However, there have been no clinical trials to date exploring their effectiveness in wound healing.
Quercetin, a natural supplement, is known to promote wound healing by enhancing the proliferation and migration of fibroblasts43 and by eliminating senescent cells44. Quercetin also enhances ECM production13,45. In combination with dasatinib, a tyrosine kinase inhibitor that triggers apoptosis in senescent cells46, quercetin has been shown to significantly reduce the burden of senescent cells across multiple tissues in human studies46–48.
Cutaneous wounds were treated with quercetin in mice (in vivo)49,50 and in humans (in vitro)34. In all these cases, subjects given the senolytic drugs exhibited faster wound healing compared to the control groups.
We simulate the senolytic drug quercetin, and denote it by Q. We accordingly update Eqs. (3), (6) and (7) as follows:
 |
24 |
 |
25 |
 |
26 |
where 
, 
and 
are constants.
Quercetin is given in pills daily. At 1 g/d it can be given safely (from damage to the liver) for up to 12 weeks. The half-life of quercetin is 15–28h, so by approximation we take it as a constant: 
and accordingly estimate the parameters 
and 
.
Oxygen therapy
In hyperbaric oxygen therapy (HBOT) chamber, the oxygen pressure is three times the pressure in air, and the patient remains in the chamber for, typically, 2 hours daily. We represent this treatment in our model by including in Eq. (15) a (constant) source of oxygen, as shown in the following modified equation:
 |
27 |
where
 |
and 
is the dose of oxygen administered between 12:00–14:00 every day. We also change the boundary condition for w on 
, increasing 
by 
during the hours between 12:00–14:00.
Treatment with HBOT and quercetin
In Fig. 4, we show the profiles of the wound radius (R) for 50 days, for two patients of extreme age difference (Younger with 
and Older with 
), both with and without drugs HBOT and Q. The profiles for the no-drug case are similar to those in Fig. 3.
Fig. 4.
Diabetic wound treatment with HBOT and quercetin: Simulations of the wound radius with various combinations of hyperbaric oxygen therapy (HBOT) and quercetin (Q) for the youngest patient 
and the oldest patient 
. The percentages represent the percent of wound closure (PWC) of the respective treatments at day 30; we mark day 30 and day 45.
Without treatment, diabetic wounds do not close (PWC in 5.1%–6.7%). When using Q alone, there is only a mild improvement in the PWC (between 7.2%–11.2%) compared to the case with no drug.
For younger patients, HBOT treatment does not result in complete wound closure within the expectable 30 days (PWC=99.6%); but when Q is combined with HBOT, complete wound closure is achieved. Older patients, on the other hand, require additional time for complete wound closure.
We can use the model to depict cases where complete wound closure can be achieved with Q+HBOT therapy, but not with HBOT alone. Table 2 presents the 30-day PWC for 5 patients, listed in order of decreasing age. In all cases, treatment with only Q does not significantly improve wound closure. Treatment with only HBOT significantly improves PWC but does not fully close the wound. Combining Q with HBOT results in complete wound closure, except for the oldest patient (
).
Table 2.
Percent of wound closure (PWC) at day 30 in patients of various ages under treatment with hyperbaric oxygen therapy (HBOT) and quercetin (Q). The daily dose of quercetin is 1.0 g, and 
g/
.
Patients
|
1 (0.15,0.56) |
2 (0.3,0.67) |
3 (0.6,0.78) |
4 (0.7,0.89) |
5 Youngest (0.92,1) |
|---|---|---|---|---|---|
| No Drug (%) | 5.41 | 5.67 | 6.17 | 6.35 | 6.71 |
| HBOT (%) | 82.29 | 90.26 | 96.12 | 97.48 | 99.55 |
| Q (%) | 7.89 | 8.53 | 9.82 | 10.28 | 11.24 |
| HBOT+Q (%) | 92.56 | 100 | 100 | 100 | 100 |
In Table 3, we consider 5 patients older than those in Table 2, and show their PWC after 45 days of treatment. As in Table 2. Treatment with only Q does not result in significant wound closure. Under HBOT alone, the youngest patient (#10, 
) achieves complete wound closure. The next two older patients (#8 and #9) achieve wound closure only when Q is combined with HBOT. However, the oldest patients (#6 and #7) do not achieve wound closure with HBOT+Q.
Table 3.
Percent of wound closure (PWC) at day 45 in patients of various ages under treatment with hyperbaric oxygen therapy (HBOT) and quercetin (Q). The daily dose of quercetin is 1.0 g, and 
g/
.
Patients
|
6 (Oldest) (0,0) |
7 (0.001,0.11) |
8 (0.003,0.22) |
9 (0.004,0.33) |
10 (0.03,0.44) |
|---|---|---|---|---|---|
| No Drug (%) | 13.66 | 13.72 | 13.79 | 13.86 | 14.11 |
| HBOT (%) | 83.41 | 87.63 | 95.62 | 99.47 | 100 |
| Q (%) | 22.09 | 22.16 | 22.25 | 22.33 | 22.8 |
| HBOT+Q (%) | 87.81 | 92.3 | 100 | 100 | 100 |
Figure 5A is a color map of PWC(30) under treatment with HBOT, and Fig. 5B is a color map under treatment with Q+HBOT, where on the horizontal axis, 
reflects how well the organism performs angiogenesis functions, and on the vertical axis, 
represents the proliferation of senescent cells. The range of (
) was restricted in order to display, more clearly, in each of the figures, the region where PWC(30)
.
Fig. 5.
Effect of age 
on percent wound closure: Simulations of PWC(30) under HBOT (A) and Q+HBOT (B) as the pair 
vary; note the ranges for 
and 
in each plot. The black curves represent the values of 
where PWC(30)
. The black curves represent the pairs 
where PWC(30)=100%.
Based on Fig. 5, Figure 6 shows the region in the 
plane where treatment with HBOT does not achieve wound closure in 30 days, but treatment with Q+HBOT does. This region lies between two equi-PWC curves: The upper curve includes the extreme points (1, 0.97) and (0.986, 1), and the lower once includes the extreme points (0.265, 0) and (0.236, 1).
Fig. 6.
Effect of age 
on percent wound closure: Simulations PWC(30) under HBOT and Q+HBOT as the pair 
vary. The region between the two curves represents the ages 
under which wound closure is achieved under treatment with Q+HBOT, but not under treatment with HBOT alone.
Discussion
Diabetic wounds represent a growing global health crisis, with approximately 15-25% of diabetic patients developing chronic foot ulcers that frequently lead to infections and amputations51. Current treatment modalities include advanced wound dressings, negative pressure therapy, hyperbaric oxygen therapy (HBOT), and emerging biologic therapies, but these approaches often fail to address the underlying pathophysiology of impaired healing in diabetic patients52. The economic burden is staggering, with annual U.S. costs ranging between $9 to $13 billion due to prolonged hospitalizations, frequent debridements, and high recurrence rates; in addition to the cost for management of diabetes mellitus alone53,54. These costs are significantly lower in European systems like England’s NHS and France’s universal healthcare, where standardized treatment protocols, preventive care models, and price controls reduce both complication rates and expenditures55,56. The medical community in the U.S. has proposed several cost-reduction strategies, including value-based care models, early screening programs, and the adoption of senolytic therapies to target cellular senescence – a key contributor to impaired wound healing57. However, significant obstacles remain, including fragmented healthcare delivery, inconsistent insurance coverage, and a lack of reliable biomarkers to predict treatment response.
This paper contributes to addressing these challenges by developing a mathematical model that quantifies biological aging through two key parameters (
for fibroblast proliferation and 
for VEGF production) to predict patient responsiveness to HBOT and senolytic therapy with quercetin (Q). Our simulations demonstrate that while HBOT alone is effective only for a limited subset of patients with specific 
values, combining HBOT with Q expands the treatable population by addressing cellular senescence16. This approach could reduce costs by enabling clinicians to: (1) avoid ineffective HBOT in non-responders, (2) identify patients who would benefit from adjunct senolytic therapy, and (3) potentially shorten treatment duration through targeted interventions. While the model requires clinical validation to correlate 
and 
with measurable biomarkers, it provides a quantitative framework for personalized treatment decisions that could optimize resource allocation in diabetic wound care58.
Conclusion
Cellular senescence is a permanent arrest of cell cycle while maintaining viability. Cellular senescence is the hallmark of aging. Senescent cells secrete proteins that have negative effect on tissue regeneration. Diabetic wounds, wounds that develop in individuals with type 2 diabetes, tend to be ischemic due to the inflammation induced by the diabetic condition. And in aging individuals with diabetes type 2, due to also cellular senescence of fibroblasts, diabetic wounds are more likely to become chronic wounds.
Ischemic wounds are commonly treated with oxygen infusion, where the patient spends several hours per day in hyperbaric oxygen therapy chamber (HBOT) under oxygen pressure three times the pressure in air. Senolytic drugs that eliminate senescent cells, such as quercetin (Q), are currently used in experimental studies in chronic wounds. In this paper we simulate treatments with HBOT and Q, separately or in combination, of diabetic wounds in aging population.
To better understand the impact of aging on wound healing, we developed a predictive mathematical model that defines an individual’s biological age using two key parameters: 
and 
. These parameters decrease with chronological age, where 
and 
. The parameter 
reflects age-related changes in the proliferation of senescent cells, while 
represents age-related production of the angiogenetic protein VEGF. Lower values of 
and 
indicate a biologically older individual. Our model can be used to predict the profile of the open-wound radius, with or without treatment, for any biological-age pair 
.
We simulated the threshold of two 
-sets, 
and 
(
): patients in 
achieve wound closure (in expectable time of 30 days) if treated with HBOT, and patients in 
achieve ful wound closure when treated with Q+HBOT. We conclude that for the biologically-old patients in 
, to achieve wound closure they need to be treated with combination of the senolytic drug and HBOT; HBOT alone will not yield wound closure.
The limitation of the model is that it is unable to precisely associate the biological-age parameters 
to any tangible marks that can be determined and measured for each individual. For this reason, the predictions of the model, at present, are highly qualitative.
Future studies should aim to determine the dependence of 
and 
on factors such as lifestyle, general health, and body mass to further personalize treatment strategies. This would significantly improve the predictive accuracy of our model and allow for more individualized care. Ultimately, the integration of these parameters into clinical practice could significantly improve the management of chronic wounds in the aging diabetic population, providing clinicians with a tool to better predict healing outcomes and optimize treatment plans.
Parameters sensitivity analysis
We performed global sensitivity analyses (see Fig. 7). The output was the percent wound closure at day 30 (PWC), and the p-values for all the parameters were less than 
. The computations were done using Latin Hypercube Sampling/Partial Rank Correlation Coefficient (LHS/PRCC) with a Matlab package by59,60, with the parameters listed 
, 
, 
, 
, 
, 
, 
, 
, 
, 
, 
, 
, 
, 
, 
and 
in Table 4.
Fig. 7.
LHS–PRCC sensitivity analysis of radius reduction at day 30. The horizontal axis lists the parameters and the vertical axis represents the PRCC index. The samplings are done with 
of the baselines for the parameters as in Tables 4, except for 
whose range is 
of the baseline.
The ranges for the parameters were between 
of their baselines in Table 4, except 
whose range was between 
of its baseline in Table 4.
Figure 7 shows that 
and 
are the most positive correlated parameters, with 
, 
and 
also showing significant positive correlations. The parameter 
represents the rate of ECM production; as it increases, ECM production rises, leading to a reduction in wound size and an increase in PWC. Similarly, an increase in 
, which denotes the rate of fibroblast proliferation, results in enhanced ECM production and a larger PWC. Additionally, increases in the parameters 
and 
promote the 
polarization, further contributing to increased ECM production and larger PWC. An increase in 
leads to an increase in V, which in turn results in an increase in E. This implies that w also increases, thereby enhancing the wound healing process. Note that the positive correlation of 
is smaller than the positive correlation of 
; this is because 
is 10 times smaller than 
.
The reverse is true for the parameters 
, 
and 
, which are the most negatively correlated parameters. As 
and 
increase, the density of M1 macrophages also increases, which decreases PWC. An increase in 
leads to higher levels of TNF-
, which similarly results in an increase in M1 macrophages, and decrease PWC.
All other parameters have small correlations, and their positive or negative correlation can easily be inferred from the model equations. For instance, increase in any parameter that results in increase 
will improve healing and hence increase PWC, and is positively correlated.
Parameter estimations
We assume that in steady-state, 
, or 
for any protein species, X, where 
is the half-saturation of X.
Densities/concentrations in steady state
Estimate for
and
. The ECM density is 3–4% of the dry weight of tissue61; we accordingly take
 |
We also take
 |
and
 |
Estimate for 
.
The concentration of oxygen in tissue is given by the following formula (in text, section Materials and Methods of62):
 |
where 
M/mmHg is the oxygen pressure in arterial blood and (from Table 3 in62) 
is the oxygen solubility in the tissue (
). Hence
 |
F and 
cells undergo increased apoptosis as the oxygen concentration (in g/
) decreases from 
to 
. We assume that the rate of apoptosis is porportional to D(w) and take
 |
Estimates for steady 
(of 
) and 
(of 
).
There are 2–4
macrophages per 
in mid-dermis63. We take the number of macrophages to be 
cells/
and the mass of a cell to be 
g. We assume that 
in the steady state of mid-epidermis, and take
 |
and
 |
Estimate for 
.
There are 2100–4100 fibroblasts/
in mid-dermis64. The volume of a fibroblast cell is 
65(Fig. 1B) and accordingly we take its mass to be 
g. Assuming that there are 3000 fibroblasts cells in 
, we get the steady-state of F to be
 |
Estimate for 
Wounds are hard to heal when the accumulation of senescent fibroblasts exceeds 15% threshold66. For diabetic wounds we take
 |
Estimate for 
.
The number of human corneal endothelial cells is 3000 cells/
, or 
cells/
67,68. We assume that the number of endothelial cells in mid-dermis is 
cells/
. Taking the mass of one cell to be 
g, we get the steady-state of E:
 |
Estimate for 
The level of IL-6 in normal human skin is approximately 205.1 pg/g69(Tab. 1).
The level of IL-6 in human skin wounds is increasing after wound initiation69(Fig. 4). We take the average level of IL-6 around the wound to be
 |
Estimate for 
In humans, the level of TGF-
decreases from its high value within the first hours post wound initiation to reach an average of 100 pg/
70(Fig. 2).
Since diabetic wounds do not heal without intervention, we take
 |
Estimate for 
In mice, the level of PDGF in the wound was measured to be approximately 6 ng/mg at day 3 post wound initiation and 1.5 ng/mg at day 7 post wound initiation71(Fig. 2c).
We take in chronic wounds,
 |
Estimate for 
We assume that the average level of VEGF after wound initiation is similar to that of murine skin 5 to 7 days after wound initiation9(Fig. 7) and take
 |
Estimate for 
In murine skin, the level of TNF-
increases from its control level (60 pg/ml) to 120 pg/ml after wound initiation72(Figs. 1,2). We take, in diabetic wounds,
 |
Death/degradation rates
The death or degradation rate 
of a species X is linked to its half-life 
by the formula:
 |
Estimate for
. The half-life of IL-6 is 15.5 hours73, or equivalently 0.646 days. Hence,
 |
Estimate for
. The half-life of PDGF is 30 minutes74. We take 
days, so that
 |
Estimate for
. The half-life of VEGF is 60 minutes75. Hence 
days, and
 |
Estimate for
. The half-life of TNF-
is 4.6 minutes76. Hence 
days, and
 |
Estimate for
. The half-life of TGF-
is approximately 2 minutes77. Hence 
days, and
 |
Estimate for
. The half-life of tissue M1 macrophages is three weeks78, or, equivalently, 21 days. Hence,
 |
Estimate for
. The half-life of tissue M2 macrophages is 7 days78, so that
 |
Estimate for
and
. The death rate of fibroblast is approximately 
per second79, hence,
 |
We take
 |
Estimate for
. The half-life of cardiac endothelial cells is 2.2 weeks80, or, equivalently, 15.4 days. Using the same half-life for dermal epithelial cells, we get
 |
Chemotactic coefficients
We take
 |
which is in the range given in81 for 
.
Estimates by equations
There are still many parameters that need to be estimated, including all the production coefficients. To do that we shall take, in each equation, its steady state, that is, make the right-hand side equal to zero, drop chemotactic terms, and replace each species by its steady state as estimated above.
- Equation (9): We take the “average” value of R(t) in diabetic wounds control case to be
cm, and assume that 
, so that
Next, in steady-state,
. We assume that 
for some 
. Taking 
, we get 
. Hence, 
Acknowledgements
Research reported in this publication was supported by the National Institute Of General Medical Sciences of the National Institutes of Health under Award Number R16GM154782. The content is solely the responsibility of the authors and does not necessarily represent the official views of the National Institutes of Health.
Data availability
All data generated or analyzed during this study are included in this published article.
Footnotes
Publisher’s note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
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Data Availability Statement
All data generated or analyzed during this study are included in this published article.