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Field verification tests of the newly developed flow cone tool—In-situ measurements of hydraulic soil properties

  • Received: 20 February 2019 Accepted: 18 July 2019 Published: 26 September 2019
  • Hydraulic soil properties, and in particular the hydraulic conductivity, is important in a number of geotechnical design cases. However, attention is often drawn towards the method of analysis rather than the quality and validity of the hydraulic properties used as input. Intact samples of sand or silt are difficult or impossible to obtain, and prediction of hydraulic soil properties from available in-situ tools represents a challenge. With the aim of quick and reliable in-situ measurements of hydraulic properties of sands and silts, NGI developed a prototype tool referred to as the flow cone. The tool combines the widely used cone penetration test with an add-on pumping system that allows water to flow into the surrounding sediments during cone penetration and stand-still. By measuring flow rates and pore water pressures, the hydraulic properties of the surrounding soil can be estimated. The flow cone prototype was tested at the NGTS sand site (Øysand, Norway) in September 2018. The aim of this paper is to present the measured and interpreted results. The results are evaluated within the context of available data from the sand site including cone penetration tests, in-situ falling head tests, grain size distributions and constant head tests from laboratory. Recommendations for further work and potential applications in engineering practice are discussed.

    Citation: Aleksander S. Gundersen, Pasquale Carotenuto, Tom Lunne, Axel Walta, Per M. Sparrevik. Field verification tests of the newly developed flow cone tool—In-situ measurements of hydraulic soil properties[J]. AIMS Geosciences, 2019, 5(4): 784-803. doi: 10.3934/geosci.2019.4.784

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  • Hydraulic soil properties, and in particular the hydraulic conductivity, is important in a number of geotechnical design cases. However, attention is often drawn towards the method of analysis rather than the quality and validity of the hydraulic properties used as input. Intact samples of sand or silt are difficult or impossible to obtain, and prediction of hydraulic soil properties from available in-situ tools represents a challenge. With the aim of quick and reliable in-situ measurements of hydraulic properties of sands and silts, NGI developed a prototype tool referred to as the flow cone. The tool combines the widely used cone penetration test with an add-on pumping system that allows water to flow into the surrounding sediments during cone penetration and stand-still. By measuring flow rates and pore water pressures, the hydraulic properties of the surrounding soil can be estimated. The flow cone prototype was tested at the NGTS sand site (Øysand, Norway) in September 2018. The aim of this paper is to present the measured and interpreted results. The results are evaluated within the context of available data from the sand site including cone penetration tests, in-situ falling head tests, grain size distributions and constant head tests from laboratory. Recommendations for further work and potential applications in engineering practice are discussed.


    Mammalian cell division patterns are critical to understanding human tumor progression. It has drawn the attention of many researchers, and it has been the topic of intense research for a long time. Several research works utilize age-structured frameworks to investigate the cell cycle. Some examples of age-structured growth models include epidemic [1,2,3], microscopic virus [4,5] and cell population [6,7,8,9] models. The concealed molecular complexity of a tissue, on the other hand, demand a more thorough modeling framework that includes special cellular and molecular interactions.

    Cells that divide in living tissues can be divided into two categories: proliferating and quiescent. The proliferating cells go through different phases in the cell cycle (G1,S,G2,M) while dividing. Quiescent cells, however, do not grow or divide, but rather move to the G0 phase, where they remain until they differentiate or undergo apoptosis. To preserve tissue homeostasis, cells must be capable to transition between proliferative and quiescent phases. The transition between proliferating and quiescent compartments, however, is dependent on signaling molecules called anti-growth or growth factors [10]. In a population of tumor cells, proliferating cells multiply till the tumor becomes active and aggressive. Additionally, to maintain homeostasis, the total number of cells in all sub-populations stays at equilibrium; hence, in a healthy and mutated cell populations, the proliferative compartment is constrained in size. Schematics in Figure 1 depicts multiscale-modeling framework used in this paper.

    Figure 1.  Model schematics. Both healthy and mutated subpopulations of proliferating and quiescent cells are with various transition effects depicted. Healthy proliferating cells can transition to cancer proliferating cells upon mutation with rate m. Microscale (or cell cycle) dynamics with predominating protein states along with their interactions are shown, as indicated by the legend in the bottom right corner. The growth factors gf from the macroscale influences the cell-cycle.

    Mutations in genes that regulate cell growth and division can disturb the finely-tuned equilibrium that governs cell proliferation, which in turn can result in the emergence of cancer. Although other factors like environmental exposures and lifestyle choices may also play a role in cancer development, mutations in genes are a major contributor to this disease. Mutations can be of several types, those that are particularly significant for cancer involve increased potential for proliferation, decreased apoptosis, genetic instability, and reduced tumor suppression [11]. Furthermore, studies have shown that the transformation of a normal cell into a cancerous one usually requires the accumulation of one to ten mutations [11,12]. The mutated cell population also consists of its own progeny including different differentiation stages and quiescent cells.

    This research focuses on the model development of the cell population in all healthy and mutated cell populations. We investigate the coupling dynamics of tissue cell density and cell cycle proteins. Previous studies have used age-structured models to investigate cell populations in quiescent phase [6] only, the proliferating phase [13,14] only, or both phases together [7,15,16,17,18,19]. Despite this, the impact of molecular interactions on the interplay between proliferative and quiescent phases at the subcellular level has yet to be examined. Thus, the primary aim of this paper is to develop a multiscale model utilizing mathematical techniques that can capture the intricacies of a complex system existing at the sub-cellular level.

    The emphasis is laid upon the interaction between the macro- (population dynamics) and micro-scale (cell cycle dynamics) at two distinct levels. The concept of age is used to refer to the time since the last division, as described in previous studies [14,20]. In age-structured models, an additional variable, referred to as "age" (a), is introduced, which has a physiological rather than physical interpretation. The idea of cell age reflects the variability (biological) within a population of dividing cells and is distinct from the physical time variable (t). To model the behavior of cell populations in both the proliferative and quiescent stages at the macroscale, we use partial differential equations (PDEs). For predicting sub-cellular protein interactions related to cell cycle dynamics, we employ ordinary differential equations (ODEs). The two scales are connected via the feedback incorporated in both directions. Within a cycle of cell division (G1, S, G2, M), cells in the early proliferating phase (G1) can move to the quiescent-phase until the restriction point (R).

    It is clear that the restriction point (R) cut the G1 phase into two portions, where before R, the cells become quiescent but once R is passed, cells can no longer avoid division [21,22]. During the quiescent phase, cells do not grow or divide, but they still complete other cellular functions. The bidirectional transition of cells between quiescent and proliferation phases in both healthy and mutated cell populations plays a crucial role in maintaining tissue homeostasis, and it is controlled by the conditions of extracellular environmental [10]. In tumors, the transitional balance between two compartments is disturbed, and cells can grow uncontrollably [23]. Recent experimental results support that cyclin proteins are the most important to regulate any change in the cell-cycle phase [24]. Therefore, we utilize a vital aspect of cell cycle dynamics (i.e., the G1S phase-transition) to predict how the transitional balance between proliferating and quiescent subpopulations evolve, which is essential for maintaining homeostasis.

    Several proteins are involved in regulating the transitions between different phases of the cell cycle at the microscale. The interactions between these proteins have been mathematically modeled and simulated using ODEs by various researchers, as documented in [25,26,27,28,29,30] and related references. However, to keep things simple, we focus only on four proteins: Cyclin DCDK4/6, p21, E2F, and Rb. These proteins are mainly involved in Cyclin D activity and the transition of cells from G1 phase to S phase. Experimental data supports this choice, as Cyclin D has been shown to regulate the transition between the G0 and G1 phases [31,32,33]. In addition, over-expression of Cyclin D leads to cell division commitment in the proliferative phase, while under-expression leads to the quiescent phase. It is essential to note that these molecular-interactions are considered to occur in a rapidly growing cell population rather than a single cell. Furthermore, we assume the average protein concentrations in both quiescent and proliferating cell subpopulations, thus omitting the variability between individual cells. The proteins involved in the cell cycle play a vital role in regulating its progression. Cyclin proteins, which are structural proteins, along with their inhibitors cyclin-dependent kinases (CDK), control the various phases of the cell cycle. Each phase of the cycle is regulated by a specific Cyclin/CDK complex.

    The initiation of a cell cycle is triggered when the microenvironment of a cell receives sufficient growth signals [34], which induce the activity of Cyclin-CDK 4-6 complex. During the G1 phase, Cyclin D is activated only by growth signals, and in case of low or no growth factors, its concentration decreases, and the cell doesn't enter the cycle. Growth factors bind to specific receptors which are situated on the cell's external cytoplasmic membrane, activating intracellular signaling pathways (such as Ras/Map/Rap kinase), eventually which directing to the production of Cyclin D (see [35,36,37], for more details). The active complex of Cyclin D and CDK4/6 is synthesized at maximum. Cyclin complex then triggers the transcription factor E2F's activation by phosphorylating retinoblastoma protein Rb. Consequently, the transcription factor E2F accumulates and triggers some other important cyclins. It is important to note that these interactions occur in a fast growing population of cells rather than in a single cell.

    In summary, our contribution is to develop a multiscale model that describes the coupling between two predominant scales and focuses on the challenges related to the disruption of cell transitioning between proliferating and quiescent states, leading to uncontrolled tumor growth. Specifically, we investigate whether Cyclin D complex is one of the important cause in creating a deregulation in cell transitioning between proliferating and quiescent cells.

    The structure of rest of this paper is as follows. In Section 2, we delve into the details of multiscale mathematical modeling of proliferating and quiescent cells with regards to the dynamics of cell cycle. In Section 3, we investigate the existence and uniqueness of non-negative solutions utilizing spectral and semigroup theory. Moving on to Section 4, we start with deriving steady-state solutions and then establish spectral criteria for their local stability. Specifically, we show that if the linearised semigroup's growth bound is negative, the steady-state solution is locally asymptotically stable, while a positive growth bound implies instability of the steady-state solution. Lastly, Sections 5 and 6 offer results discussion and the concluding remarks of this paper, respectively.

    The cell densities of healthy and mutated cells in the proliferative and quiescent compartments are described by nonlinear hyperbolic transport PDEs that relate the cell density distribution to both physiological age a and time t. Specifically, the densities of healthy cells in the proliferative and quiescent phases are expressed as follows:

    tph(a,t)+a(gh(a)ph(a,t))=γh(N)qh(a,t)(τh(a)+αh(a,x1)+μph(a))ph(a,t), (2.1)
    tqh(a,t)=αh(a,x1)ph(a,t)(γh(N)+μqh(a))qh(a,t), (2.2)

    where the rate evolution of a cell cycle is denoted by gh(a) in the equation. The first term on the right side, γh(N)qh(a,t), represents the transition to proliferating from quiescent cells, while the term τh(a)ph(a,t) represents the cell densities for completing cell division in some age of the proliferating phase. Cells which move to the quiescent phase without undergoing division are represented by αh(a,x1)ph(a,t). The loss in the proliferating cells due to apoptosis/necrosis is represented by the death rate μph(a). The inflow from healthy proliferating cells, regulated by the microscale variable of Cyclin complex concentration x1 for each age, is denoted by the first term in Eq (2.2): αh(a,x1)ph(a,t). The next term depicts a loss in the quiescent cells due to either by returning to proliferating phase at the rate γh(N) or by cell death due to apoptosis (or necrosis), as represented by the death rate μqh(a).

    Next, the cell density of mutated cells in mutated proliferating (pc) and quiescent (qc) phases, respectively, is presented:

    tpc(a,t)+a(gc(a)pc(a,t))=γc(N)qc(a,t)(τc(a)+αc(a,x1)+μpc(a))pc(a,t), (2.3)
    tqc(a,t)=αc(a,x1)pc(a,t)(γc(N)+μqc(a))qc(a,t), (2.4)

    where the terms used are similar to those in the case of healthier cells, as shown in Eqs (2.1) and (2.2). The total cell number in both healthy and mutated populations of cells in quiescent and proliferating phases is denoted by N(t) and is defined in Eq (2.5). In the case of quiescent cells, aging does not occur (i.e., the cells stop aging), so the convection term related to physiological age a is absent in Eqs (2.2) and (2.4). The total number of cells, represented by N(t), represents the sum of all cells in the proliferating and quiescent phases throughout all ages, and can be expressed as

    N(t)=a0(ph(a,t)+qh(a,t)+pc(a,t)+qc(a,t))da, (2.5)

    where maximum age of the cells is given by a. The initial conditions are given below:

    ph(a,0)=ph,0(a),qh(a,0)=qh,0(a),pc(a,0)=pc,0(a),qc(a,0)=qc,0(a),a0. (2.6)

    The boundary conditions are given as follows:

    gh(0)ph(0,t)=2(1m)a0τh(a)ph(a,t)da, (2.7)
    gc(0)pc(0,t)=2a0τc(a)pc(a,t)da+2ma0τh(a)ph(a,t)da, (2.8)

    for t>0, where the number 2 shows the two newborn cells initializing in the proliferating phase, and the parameter m represents the mutation rate. Since healthy cell can acquire a mutation only during a division process, therefore, new born mutated cells start will start at age 0.

    The function τi(a) represents the cell number that finish dividing at a particular age in both healthy and mutated proliferating phases. Here, the index i indicates whether the compartment is healthy or cancerous, denoted by h and c, respectively. The function τi(a) is regulated by the age of the cell, denoted by a, and is almost zero until a minimum cell age. Afterward, the function increases until it reaches the age of a.

    τi(a)=ρ1,iaγ1,iργ1,i2,i+aγ1,i, (2.9)

    The maximum proliferation rate is represented by ρ1,i, while ρ2,i is the age t achieve half-maximum. The Hill coefficient is represented by the exponent γ1,i.

    Next, we establish the rate at which cells transition to the quiescent phase from the proliferating compartments, which depends on both the age of the cell (a) and the quantity of the Cyclin DCDK4/6 complex (x1).

    αi(a,x1)=σ1,iσγ2,i2,i(σγ2,i2,i+xγ2,i1)σγ3,i3,i(σγ3,i3,i+aγ3,i). (2.10)

    The function αi(a,x1) depicts the number non-dividing cells due to anti-growth factors. The age-dependence of αi is motivated by the fact that cells transition to the quiescent phase from the proliferating phase only until they reach a specific age that marks a restriction point (R) in the cell cycle (which is also G1S phase transition). However, before the restriction point, the Cyclin complex's concentration x1 must be below a certain value to enable cells to exit the proliferating phase. In Eq (2.10), the Hill coefficients are represented by γ2,i and γ3,i, while σ2,i and σ3,i denote the concentration of the Cyclin DCDK4/6 complex x1 and the age a, respectively. After γ2,i and γ3,i, the rate function α decreases asymptotically to zero, preventing cells from transitioning to the quiescent phase. This implies that at age σ3,i, cells are inevitably committed to entering the proliferation phase. Finally, σ3,i represents the threshold concentration of the Cyclin complex to determine the restriction point R.

    The function γi(N), which determines the number of cells transitioning to the proliferating phase from the quiescent phase, is represented by a Hill function of N that decreases monotonically:

    γi(N)=νiθκiiθκii+Nκi, (2.11)

    where the Hill function are defined as follows: νi specifies the maximum rate at which cells transition to proliferating from quiescent population, when there are no cells, i.e., N=0; κi is the Hill coefficient, and θi represents the proportion of the total cell population that reaches half the maximum value of νi. This implies that the number of quiescent cells transitioning to the proliferative compartment decreases to zero as the cell population increases, illustrating density inhibition.

    Cell growth is controlled by proteins such as cytokines and other factors that regulate proliferation [38]. Cytokines bind to specific receptors, activating signaling pathways [39]. The cytokine signals that regulate cell numbers are reliant on the total population of cells, as demonstrated by various studies [40]. For a detailed explanation of cytokine signal dynamics, refer to [41,42]. Using the quasi-steady-state approximation, we can express the quantity of growth factors (gf) produced by the entire cell population (N) as

    gf=11+ktN. (2.12)

    We restrict ourselves to only four microscale proteins (states of the model) in our cell cycle model, as mentioned earlier, that are sufficient to account for the reversible transitions between the quiescent and proliferating phases.We utilize Michaelis-Menten kinetics to depict the chemical reactions occurring between enzymes and substrates during the cell cycle. The basics of this approach are briefly outlined below. When adequate growth factors are present, Cyclin D protein combines with its catalytic partner CDK4-6, resulting in the formation of Cyclin DCDK4/6 complex. As outlined in [31,43], this complex is responsible for phosphorylating various proteins involved in the cell cycle. These phosphorylated proteins are essential for advancing through the initial growth phase of the cell cycle and crossing the restriction point R. To provide further details, the Cyclin DCDK4/6 complex induces phosphorylation of the retinoblastoma protein Rb, resulting in its deactivation and subsequent release of the transcription factor E2F. This activates several signals that promote cell growth, facilitating the progression of the cell cycle. In contrast, p21 regulates the cell cycle by inhibiting the functions of various CDK proteins. These proteins are described in Table 1.

    Table 1.  Description of the cell states at the microscale.
    Description State
    Cyclin DCDK4/6 x1
    E2F x2
    Rb x3
    p21 x4

     | Show Table
    DownLoad: CSV

    In this study, we examine the behavior of a single cell to represent the dynamics of all cells within a population. Assuming that all cells behave similarly, we utilize a single ordinary differential equation (ODE) model with similar parameters for all cells in the population to describe the underlying cell cycle dynamics on a microscale. We also account for cells with shorter cycles on a macroscale using the function τi(a). Our model considers a specific age a at which the representative cell completes division. The cell cycle dynamics are described by the ODE system presented in [44].

    dx1da=k1s(gfkgf+gf)k14x4x1k1d(x1k1+x1), (2.13)
    dx2da=k21(x2tx2k2+(x2tx2))x1k32x2x3k2dx2, (2.14)
    dx3da=k3sk32x2x3k31(x3k3+x3)x1k3dx3, (2.15)
    dx4da=k4s+k42(k34k34+x3)x2k41(x4k4+x4)x1k4dx4. (2.16)

    A detailed description of all the terms and parameters involved in the model equations (2.13)–(2.16) can be found in [45]. Although we will not delve into the full derivation of these equations here, inquisitive readers can refer to [44] for a comprehensive explanation. To aid in understanding, we have included simulations of the four microscale states mentioned earlier in Figure 2.

    Figure 2.  Over time, the microscale proteins involved in the cell cycle undergo changes. The Cyclin DCDK4/6 complex undergoes a complete cycle of activation and degradation. As the Retinoblastoma protein Rb becomes inactivated due to the rise in the Cyclin DCDK4/6 complex, the transcription factor E2F concentration increases. Likewise, protein p21 concentration increases towards the end of the cell cycle to aid in the degradation of the Cyclin complex.

    This section presents the uniqueness of the solution to the initial-boundary value problem (2.1)–(2.7) and (2.13)–(2.16), which we will simplify by using the microscale model for the entire time t, instead only until age a. We introduce Banach spaces, X=L1(0,a)×L1(0,a)×L1(0,a)×L1(0,a) and Y=L1(0,a)×L1(0,a)×L1(0,a)×L1(0,a), with the norm |ϕ|=4i=1|ϕi|1 for ϕ(a)=(ϕ1(a),ϕ2(a),ϕ3(a),ϕ4(a))TX and |φ|=i=14|φi|1 for φ(a)=(φ1(a),φ2(a),φ3(a),φ4(a))TY, where ||1 is the standard norm of L1(0,a). We first treat the initial-boundary value problem of system (2.1)–(2.7) as an abstract Cauchy problem on Banach space X. We assume that gha,ghaa,gca,gcaaL((0,a)×R+), and non-negative death rates, that is, μph()=μqh()0 and μpc()=μqc()0, and are locally integrable on [0,a). The transition rate αi(a,x1)L((0,a)×(0,a)), and τi(a)L1(0,a). We start by defining a linear operator A1 as follows:

    (A1ϕ)(a)=((gh(a)ϕ1(a))a(τh(a)+μph(a))ϕ1(a)μqh(a)ϕ2(a)(gc(a)ϕ3(a))a(τc(a)+μpc(a))ϕ3(a)μqc(a)ϕ4(a)),ϕ(a)=(ϕ1(a),ϕ2(a),ϕ3(a),ϕ4(a))TD(A1).

    The symbol T denotes the transpose of the vector, and the domain D(A1) is given by the following:

    D(A1)={(ϕ1,ϕ2,ϕ3,ϕ4)|ϕi is absolute continuous on [0,a),ϕ(0)=(2(1m)a0τh(a)ϕ1(a)da,0,2a0τc(a)ϕ3(a)da+2ma0τh(a)ϕ1(a)da,0)T}.

    The nonlinear operator F1:X×YX is given by

    (F1(ϕ,φ))(a)=(νhθκhhϕ2(a)θκhh+(Nϕ)κhαh(φ1,a)ϕ1(a)νhθκhhϕ2(a)θκhh+(Nϕ)κh+αh(φ1,a)ϕ1(a)νcθκccϕ3(a)θκcc+(Nϕ)κcαc(φ1,a)ϕ4(a)νcθκccϕ3(a)θκcc+(Nϕ)κc+αc(φ1,a)ϕ4(a)),ϕX,φY,

    where the linear operator N on L1(0,a)×L1(0,a)×L1(0,a)×L1(0,a) is given by

    Nϕ=a0(ϕ1(a)+ϕ2(a)+ϕ3(a)+ϕ4(a))da.

    Consider υ(t)=(ph(,t),qh(,t),pc(,t),qc(,t))TX. We can define the initial-boundary value problem (2.1)–(2.7) as an abstract semilinear initial value problem (IVP) in X, as shown below:

    ddtυ(t)=A1υ(t)+F1(υ(t),v(t)),υ(0)=υ0X, (3.1)

    where υ0(a)=(ph0(a),qh0(a),pc0(a),qc0(a)).

    Next, we define IVP (2.13) as a Cauchy problem on the Banach space Y. Suppose A2 is a linear operator which reads

    (A2φ)(a)=(0k2dφ2(a)k3sk3dφ3(a)k4sk4dφ4(a)),φ(a)=(φ1(a),φ2(a),φ3(a),φ4(a))TD(A2),

    where the domain D(A2) is

    D(A2)={φY|φi is absolute continuous on [0,a),φ(0)=(0,0,0,0)T}.

    We define the nonlinear operator F2:X×YY by

    (F2(ϕ,φ))(a)=(k1s(gf(Nϕ)kgf+gf(Nϕ))k14φ4(a)φ1(a)k1d(φ1(a)k1+φ1(a)),k21(x2tφ2(a)k2+(x2tφ2(a)))φ1(a)k32φ2(a)φ3(a)k32φ2(a)φ3(a)k31(φ3(a)k3+φ3(a))φ1(a)k42(k34k34+φ3(a))φ2(a)k41(φ4(a)k4+φ4(a))φ1(a)),

    where ϕX,φY. Take v(t)=(x1(t),x2(t),x3(t),x4(t))TY. Then (2.13)–(2.16) can be expressed as an abstract semilinear IVP in Y:

    ddtv(t)=A2v(t)+F2(υ(t),v(t)),v(0)=v0Y, (3.2)

    where v0(t)=(x01,x02,x03,x04), we can now establish a joint Cauchy problem for (3.1) and (3.2) as shown below:

    ddt(υv)=(A100A2)(υv)+(F1(υ,v)F2(υ,v)),(υ(0)v(0))=(υ0v0)Z,
    ddtζ(t)=Aζ(t)+F(ζ(t)),ζ(0)=ζ0Z, (3.3)

    where ζ=(υ,v), ζ0=(υ0,v0), A=(A100A2), F=(F1F2) and the Banach space is Z={X,Y}. Assuming that T(t) is a C0-semigroup generated by A for t0, and the operator F is continuously Fréchet differentiable on Z (specifically, both F1 and F2 are Fréchet differentiable on X and Y), a continuous mild solution tζ(t,ζ0) exists and is unique for each ζ0Z on a maximal interval [0,t1) in Z.

    ζ(t,ζ0)=T(t)ζ0+t0T(ts)F(ζ(s,ζ0))ds,t[0,t1), (3.4)

    where t1 can be either + or limtt1|ζ(t,ζ0)|=. Moreover, if ζ0D(A), then ζ(t,ζ0)D(A) for 0t<t1, and the function ζζ(t,ζ0) is continuously differentiable and satisfies (3.3) on [0,t1). This has been established in Proposition 4.16 [46,47].

    Remark 3.1. Let's take ph,max, qh,max, pc,max, qc,max, x1,max, x2,max, x3,max and x4,max to represent the maximum values of the solution variables. If we normalise the governing equations using N(a)=ph(a,t)+qh(a,t)+pc(a,t)+qc(a,t)+x1(a)+x2(a)+x3(a)+x4(a), then an a-priori estimate on these would lead to ph(a,t)+qh(a,t)+pc(a,t)+qc(a,t)+x1(a)+x2(a)+x3(a)+x4(a)=1.

    Lemma 3.1. Let Ω={(ph,qh,pc,qc,x1,x2,x3,x4)Z|ph0,qh0,pc0,qc0,x10,x20,x30,x40} and let Ω0={(ph,qh,pc,qc,x1,x2,x3,x4) Z|0phph,max,0qhqh,max,0pcpc,max,0qcqc,max, 0x1x1,max,0x2x2,max,0x3x3,max,0x4x4,max}. Then after a finite time, the mild solution ζ(t,ζ0) of (3.3), where ζ0Ω, enters a positively invariant set Ω0.

    Proof. To obtain the solution of Eq (2.1), we will begin by utilizing transformations ˜ph(a,t)=gh(a)ph(a,t) and ˜qh(a,t)=gh(a)qh(a,t) for t[0,t1] and a[a0,a). Then, for t(0,t1) and a(a0,a), we have from Eq (2.1)

    ˜ph(a,t)t+gh(a)˜ph(a,t)a=γh(N(t))˜qh(a,t)(τh(a)+αh(x1(a),a)+μph(a))˜ph(a,t), (3.5)

    Next, we apply the parameter transform to remove the term gh(a) and define a new age variable η for both ph and qh, Lemma 3.1 [41]. This yields the expression:

    η˜ph(a(η),t)=dadηa˜ph(a,t)=gh(a)a˜ph(a,t),wheredadη=gh(a).

    Therefore, from Eq (3.5), it follows that

    ˜ph(a(η),t)t+˜ph(a(η),t)η=γh(N(t))˜qh(a(η),t)(τh(a(η))+αh(x1(a(η)),a(η))+μph(a(η)))˜ph(a(η),t). (3.6)

    To obtain the explicit relation of ˜ph(a(η),t), we will utilize the method of characteristics (MOC). Specifically, we assume that ˜ph(a(η),t) is governed by an ODE along the curve (a(ψ1(y)),ψ2(y))=ψ(y), and therefore, we have

    ˙ψ1(y):=1ψ1(y)=y+c1,˙ψ2(y):=1ψ2(y)=y+c2,z(y):=˜ph(a(ψ1(y)),ψ2(y)),

    where c1,c2R are constants. Then, it follows

    dzdy=d˜ph(a(ψ1(y)),ψ2(y))dy=˜ph(a(ψ1(y)),ψ2(y))ada(ψ1(y))dψ1dψ1(y)dy+˜ph(a(ψ1(y)),ψ2(y))ψ2dψ2(y)dy=γh(N(ψ2(y)))˜qh(a(ψ1(y)),ψ2(y))(τh(a(ψ1(y)))+αh(x1(a(ψ1(y))),a(ψ1(y)))+μph(a(ψ1(y))))˜ph(a(ψ1(y)),ψ2(y))=γh(N(ψ2(y)))˜qh(a(ψ1(y)),ψ2(y))(τh(a(ψ1(y)))+αh(x1(a(ψ1(y))),a(ψ1(y)))+μph(a(ψ1(y))))z(y). (3.7)

    We can now write ˜ph using an ODE (3.7) so that

    ˜ph(a(y+c1),y+c2)=˜ph(a(ψ1(y)),ψ2(y))=z(y)=exp(y0(τh(a(ψ1(ξ)))+αh(x1(a(ψ1(ξ))),a(ψ1(ξ)))+μph(a(ψ1(ξ))))dξ)[y0exp(ζ0(τh(a(ψ1(ξ)))+αh(x1(a(ψ1(ξ))),a(ψ1(ξ)))+μph(a(ψ1(ξ))))dξ)γh(N(ψ2(ζ)))˜qh(a(ψ1(ζ)),ψ2(ζ))dζ+˜ph(a(ψ1(0)),ψ2(0))]=exp(y0(τh(a(ξ+c1))+αh(x1(a(ψ1(ξ+c1))),a(ξ+c1))+μph(a(ξ+c1)))dξ)[y0exp(ζ0(τh(a(ξ+c1))+αh(x1(a(ψ1(ξ+c1))),a(ξ+c1))+μph(a(ξ+c1)))dξ)γh(N(ζ+c2))˜qh(a(ζ+c1),ζ+c2)dζ+˜ph(a(c1),c2)].

    Now, we define the boundary set Γ as [a0,a)×00×[0,t1], which enables us to use the boundary condition to determine ˜ph(a(c1),c2) if a curve (a(ψ1(y)),ψ2(y)) begins in Γ. In order for (a(y+c1),y+c2) to lie on Γ, either c1=0 or c2=0. Therefore, we have the following two scenarios:

    In the first scenario, we can randomly choose c1=0 and c2[0,t1). In this case, we have

    ˜ph(a(y),y+c2)=exp(y0(τh(a(ξ))+αh(x1(a(ξ)),a(ξ))+μph(a(ξ)))dξ)[y0exp(ζ0(τh(a(ξ))+αh(x1(a(ξ)),a(ξ))+μph(a(ξ)))dξ)γh(N(ζ+c2))˜qh(a(ζ),ζ+c2)dζ+˜ph(a(0),c2)].

    The solution in (a(η),t)|t[0,t1],η[0,min(η,t)) can be obtained using the characteristic solution as follows:

    η!=ψ1(y)=y+c1=yy=ηandt!=ψ2(y)=y+c2c2=ty,

    which implies

    ˜ph(a(η),t)=exp(η0(τh(a(ξ))+αh(x1(a(ξ)),a(ξ))+μph(a(ξ)))dξ)[η0exp(ζ0(τh(a(ξ))+αh(x1(a(ξ)),a(ξ))+μph(a(ξ)))dξ)γh(N(ζ+tη))˜qh(a(ζ),ζ+tη)dζ+˜ph(a(0),tη)].

    Using the above equation, we can obtain the expression for gh(a(η))ph(a(η),t) when η<t. By choosing an arbitrary c1[0,η) and setting c2=0, we obtain

    ˜ph(a(y+c1),u)=exp(y0(τh(a(ξ+c1))+αh(x1(a(ξ+c1)),a(ξ+c1))+μph(a(ξ+c1)))dξ)[y0exp(ζ0(τh(a(ξ+c1))+αh(x1(a(ξ+c1)),a(ξ+c1))μph(a(ξ+c1)))dξ)+γh(N(ζ))˜qh(a(ζ+c1),ζ)dζ+˜ph(a(c1),0)].

    Using the characteristic solution, we can obtain a solution in the set (a(η),t)|t[0,t1],η[t,η) as follows:

    η!=ψ1(y)=y+c1c1=ηyandt!=ψ2(y)=y+c2y=t,

    which results into

    ˜ph(a(η),t)=exp(t0(τh(a(ξ+ηt))+αh(x1(a(ξ+ηt)),a(ξ+ηt))+μph(a(ξ+ηt)))dξ)[t0exp(ζ0(τh(a(ξ+ηt))+αh(x1(a(ξ+ηt)),a(ξ+ηt))+μph(a(ξ+ηt)))dξ)γh(N(ζ))˜qh(a(ζ+ηt),ζ)dζ+˜ph(a(ηt),0)].

    Hence, the relation for gh(a(η))ph(a(η),t) is now established for η>t. As a result, the ultimate solution for gh(a(η))ph(a(η),t) can be expressed as

    ˜ph(a(η),t):={exp(η0(τh(a(ξ))+αh(x1(a(ξ)),a(ξ))+μph(a(ξ)))dξ)[h(tη)η0exp(ζ0(τh(a(ξ))+αh(x1(a(ξ)),a(ξ))+μph(a(ξ)))dξ)γh(N(ζ+tη))˜qh(a(ζ),ζ+tη)dζ],ˉa<t,exp(t0(τh(a(ξ+ηt))+αh(x1(a(ξ+ηt)),a(ξ+ηt))+μph(a(ξ+ηt)))dξ)[p0(a(ηt))+t0exp(ζ0(τh(a(ξ+ηt))+αh(x1(a(ξ+ηt)),a(ξ+ηt))+μph(a(ξ+ηt)))dξ)γh(N(ζ))˜qh(a(ζ+ηt),ζ)dζ],ˉat,

    where the boundary condition ˜ph(a(0),tη) is denoted as h(tη). Note that for positive initial data, the above expression is positive and for gh(a)qh(a,t)0.

    We then derive the solution expression from (2.2) as shown below:

    qh(a,t):=exp(t0μqh(a)+γh(N(t))dt){t0exp(ξ0μqh(a)+γh(N(π))dπ)αh(x1(a),a)ph(a,ξ)dξ+qh,0(a)}. (3.8)

    As a direct consequence, we observe that qh(a,t) is non-negative for positive initial data and whenever gh(a)qh(a,t)0. Similarly, we can obtain the solution expression for pc and qc.

    Next, to ensure the positivity of the coupled ODE model (2.13), we express the system of ODEs as follows:

    {dx1da=F1(x1,x4),dx2da=F2(x1,x2,x3),dx3da=F3(x1,x2,x3),dx4da=F4(x1,x2,x3,x4), (3.9)

    where F1, F2, F3 and F4 correspond to the vector fields of the microscale states x1x4. It is worth noting that in (3.9), F1 does not depend on N (i.e., ph, qh, pc and qc), as N changes with time and is a fixed constant at each time step, which determines the growth factors entire age range.

    To ensure that the solutions of all ODEs is positive, it is essential to verify that the vector fields F1,F2,F3,F4 are smoothly differentiable and oriented in a direction that points away from the negative regions in the state space. Starting with the ODE for x1 from (3.9), we set x4=0 in F1(x1,x4) to obtain ˙x1=F1(x1). We can observe that F1(x1)=k1s(gfkgf+gf)k1d(x1k1+x1)>0 for all a>0 when k1s(gfkgf+gf)>k1d(x1k1+x1). This implies that the concentration of x1 consistently rises more than it falls over time, as the sole source of an increase in x1 concentration is from growth factors. Consequently, during periods when growth factors are at their minimum, the concentration of x1 is also at its minimum, meaning the amount of degradation or decrement cannot surpass the activation of the x1 complex. Given that the solution to system (2.13)–(2.16) is unique for each initial condition, as can be observed from (3.3) and (3.4), we can infer that the solution remains in the first quadrant for any x4>0. Therefore, the positivity of the solution for x1 is guaranteed. To obtain an ODE for ˙x2, we assume x1=0 in F2(x1,x2,x3). This results in ˙x2=F2(x2,x3), whose solution takes the form x2(a)=x02e(k32x3(a)k2d)a. Hence, for any positive initial data, x2(a) remains positive for all ages and x3(a) values. Similarly, we substitute x3=0 in F2(x1,x2,x3) to obtain a nonlinear ODE ˙x2=F2(x1,x2) for x2. Although an explicit solution cannot be computed, the phase portrait of (x1,x2) reveals that the solution trajectories move away from the axis separating the positive and negative space for positive initial data. By following a similar procedure, we can establish sufficient conditions for the positivity of solutions for x3(a) and x4(a). Therefore, we conclude that if ζ0Ω, then ζ(t,ζ0)Ω for all t>0.

    The above analysis implies that the local solution ζ(t,ζ0) of (3.3) with initial conditions ζ0D(A)Ω has a well-defined and finite norm. Consequently, we obtain our final result.

    Theorem 3.1. The abstract Cauchy problem (3.3) has a unique global classical solution on Z with respect to the initial data z0ΩD(A).

    As a result of having positive initial data, the IVP (2.1)–(2.4) possesses a singular positive solution.

    This section aims to determine the steady-state solution of the model and to present sufficient conditions for the existence of a positive steady-state. To this end, we specify some notation. Let X be a real or complex Banach space, and let X denote its dual space. We denote the value of FX at ψX as F,ψ. Additionally, we define a cone X+ as a non-zero set that satisfies X+(X+)=0, λX+X+ for λ0, and X++X+X+. Furthermore, we define the dual cone, denoted as X+, as the subset of the dual space.

    The steady-states of the system (2.1)–(2.4) and (2.13)–(2.16) are denoted by ˉph(a), ˉqh(a), ˉpc(a), ˉqc(a) and ˉx1ˉx4. These steady-states must satisfy the following set of time-invariant ordinary differential equations:

    (gh(a)ˉph(a))a=ˉγhˉqh(a)(τh(a)+αh(a,ˉx1)+μph(a))ˉph(a), (4.1)
    0=αh(a,ˉx1)ˉph(a)(ˉγh+μqh(a))ˉqh(a), (4.2)
    (gc(a)ˉpc(a))a=ˉγcˉqc(a)(τc(a)+αc(a,ˉx1)+μpc(a))ˉpc(a), (4.3)
    0=αc(a,ˉx1)ˉpc(a)(ˉγc+μqc(a))ˉqc(a), (4.4)
    ˉph(0)=2(1m)a0τh(a)ˉph(a)da, (4.5)
    ˉpc(0)=2a0τc(a)ˉpc(a)da+2a0τh(a)ˉph(a)da, (4.6)
    dˉx1da=k1s(ˉgfkgf+ˉgf)k14ˉx4ˉx1k1d(ˉx1k1+ˉx1), (4.7)
    dˉx2da=k21(x2tˉx2k2+(x2tˉx2))ˉx1k32ˉx2ˉx3k2dˉx2, (4.8)
    dˉx3da=k3sk32ˉx2ˉx3k31(ˉx3k3+ˉx3)ˉx2k3dˉx3, (4.9)
    dˉx4da=k4s+k42(k34k34+ˉx3)ˉx2k41(ˉx4k4+ˉx4)ˉx1k4dˉx4, (4.10)

    where ˉγi=γi(ˉN), ˉgf=gf(ˉN) and ˉN=a0(ˉph(a)+ˉqh(a)+ˉpc(a)+ˉqc(a))da. Since the cell cycle model's ODEs are age-dependent and the system is in a steady-state due to the input of growth factors, all cell cycle states attain a steady-state. As a result, we can determine the steady-states of the quiescent and proliferating cell populations, represented by ˉph(a), ˉqh(a), ˉpc(a) and ˉqc(a), without explicitly solving the equations of microscale model. Solving the system (4.10) for ˉph, ˉqh, ˉpc and ˉqc allows us to obtain the values of ˉqh and ˉqc:

    ˉqh(a)=αh(a,ˉx1)ˉph(a)ˉγh+μqh(a),ˉqc(a)=αc(a,ˉx1)ˉpc(a)ˉγc+μqc(a), (4.11)

    and substituting the aforementioned expressions for ˉqh and ˉqc into the equations for ˉph and ˉpc, respectively, results in the following expressions:

    d(gh(a)ˉph(a))da+(αh(a,ˉx1)μqh(a)ˉγh+μqh(a)+τh(a)+μph(a))ˉph(a)=0, (4.12)
    d(gc(a)ˉpc(a))da+(αc(a,ˉx1)μqc(a)ˉγc+μqc(a)+τc(a)+μpc(a))ˉpc(a)=0. (4.13)

    Solving Eq (4.12) for ˉph(a) and ˉpc(a), yields steady-state solutions for ˉph(a), ˉqh(a), ˉpc(a) and ˉqc(a) as follows:

    {ˉph(a)=ˉph(0)exp(a01gh(a)(gh(a)+αh(ˉx1,ξ)μqh(ξ)ˉγh+μqh(ξ)+τh(ξ)+μph(ξ))dξ),ˉqh(a)=αh(a,ˉx1)ˉph(0)ˉγh+μqh(a)exp(a01gh(a)(gh(a)+αh(ˉx1,ξ)μq,h(ξ)ˉγh+μqh(ξ)+τh(ξ)+μph(ξ))dξ),ˉpc(a)=ˉpc(0)exp(a01gc(a)(gc(a)+αc(ˉx1,ξ)μqc(ξ)ˉγc+μqc(ξ)+τc(ξ)+μpc(ξ))dξ),ˉqc(a)=αc(a,ˉx1)ˉpc(0)ˉγc+μqc(a)exp(a01gc(a)(gc(a)+αc(ˉx1,ξ)μq,c(ξ)ˉγc+μqc(ξ)+τc(ξ)+μpc(ξ))dξ).

    It is evident that the system described in Eqs (2.1)–(2.4), (2.13) always has a trivial steady-state.

    Our next objective is to obtain the stability criteria for a positive steady-state solution. Suppose ph(a,t)=ˉph, qh(a,t)=ˉqh, pc(a,t)=ˉpc, qc(a,t)=ˉqc, t0 represent equilibrium solutions to the PDE model (2.1)–(2.4) and ph(a,t), qh(a,t), pc(a,t) and qc(a,t) represent the corresponding perturbation terms:

    ph(a,t)=ˉph+ϵph(a,t),qh(a,t)=ˉqh+ϵqh(a,t),pc(a,t)=ˉpc+ϵpc(a,t),qc(a,t)=ˉqc+ϵqc(a,t).

    After substituting the aforementioned expressions into the PDE model (2.1)–(2.4), we obtain

    {ϵtph(a,t)+a(gh(a)(ˉph+ϵph(a,t)))=(νhθκhhθκhh+(ˉN+ϵn(t))κh)(ˉph+ϵqh(a,t))(τh(a)+αh(a,ˉx1)+μph(a))(ˉph+ϵph(a,t)),ϵtqh(a,t)=αh(a,ˉx1)(ˉph+ϵph(a,t))(νhθκhhθκhh+(ˉN+ϵn(t))κh+μqh(a))(ˉph+ϵqh(a,t)),ϵtpc(a,t)+a(gc(a)(ˉpc+ϵpc(a,t)))=(νcθκccθκcc+(ˉN+ϵn(t))κc)(ˉpc+ϵqc(a,t))(τc(a)+αc(a,ˉx1)+μpc(a))(ˉpc+ϵpc(a,t)),ϵtqc(a,t)=αc(a,ˉx1)(ˉpc+ϵpc(a,t))(νcθκhcθκcc+(ˉN+ϵn(t))κc+μqc(a))(ˉpc+ϵqc(a,t)),(ˉph(0)+ϵph(0,t))=2(1m)a0τh(a)(ˉph+ϵph(a,t))da,(ˉpc(0)+ϵpc(0,t))=2a0τc(a)(ˉpc+ϵpc(a,t))da+2ma0τh(a)(ˉph+ϵph(a,t))da,

    where n(t):=a0(ph(a,t)+qh(a,t)+pc(a,t)+qc(a,t))da. Then, take the derivative with respect to epsilon ϵ, leads to

    {tph(a,t)+a(gh(a)ph(a,t))=ϵ(νhθκhhϵθκhh+(ˉN+ϵn(t))κh)qh(a,t)(τh(a)+αh(a,ˉx1)+μph(a))ph(a,t),tqh(a,t)=αh(a,ˉx1)ph(a,t)(ϵ(νhθκhhϵθκhh+(ˉN+ϵn(t))κh)μqh(a))qh(a,t),tpc(a,t)+a(gc(a)pc(a,t))=ϵ(νcθκccϵθκcc+(ˉN+ϵn(t))κc)qc(a,t)(τc(a)+αc(a,ˉx1)+μpc(a))pc(a,t),tqc(a,t)=αc(a,ˉx1)pc(a,t)(ϵ(νcθκccϵθκcc+(ˉN+ϵn(t))κc)μqc(a))qc(a,t),ph(0,t)=2(1m)a0τh(a)ph(a,t)da,pc(0,t)=2a0τc(a)pc(a,t)da+2ma0τh(a)ph(a,t)da,

    which simplifies to

    {tph(a,t)+a(gh(a)ph(a,t))=νhθκhh(θκhh+(ˉN+ϵn(t))κhκhϵn(t)(ˉN+ϵn(t))κh1(θκhh+(ˉN+ϵn(t))κh)2)qh(a,t)(αh(a,ˉx1)+τh(a)+μph(a))ph(a,t),tqh(a,t)=αh(a,ˉx1)ph(a,t)(νhθκhh(θκhh+(ˉN+ϵn(t))κhκhϵn(t)(ˉN+ϵn(t))κh1(θκhh+(ˉN+ϵn(t))κh)2)μqh(a))qh(a,t),tpc(a,t)+a(gc(a)pc(a,t))=νcθκcc(θκcc+(ˉN+ϵn(t))κcκcϵn(t)(ˉN+ϵn(t))κc1(θκcc+(ˉN+ϵn(t))κc)2)qc(a,t)(αc(a,ˉx1)+τc(a)+μpc(a))pc(a,t),tqc(a,t)=αc(a,ˉx1)pc(a,t)(νcθκcc(θκcc+(ˉN+ϵn(t))κcκcϵn(t)(ˉN+ϵn(t))κc1(θκcc+(ˉN+ϵn(t))κc)2)μqc(a))qc(a,t),ph(0,t)=2(1m)a0τh(a)ph(a,t)da,pc(0,t)=2a0τc(a)pc(a,t)da+2ma0τh(a)ph(a,t)da.

    In the limit as ϵ approaches zero, we arrive at a linear system of partial differential equations:

    {tph(a,t)+a(gh(a)ph(a,t))=γh(ˉN)qh(a,t))(αh(a,ˉx1)+τh(a)+μph(a))ph(a,t),tqh(a,t)=αh(a,ˉx1)ph(a,t)(μqh(a)+γh(ˉN))qh(a,t),tpc(a,t)+a(gc(a)pc(a,t))=γc(ˉN)qc(a,t))(αc(a,ˉx1)+τc(a)+μpc(a))pc(a,t),tqc(a,t)=αc(a,ˉx1)pc(a,t)(μqc(a)+γc(ˉN))qc(a,t),ph(0,t)=2(1m)a0τh(a)ph(a,t)da,pc(0,t)=2a0τc(a)pc(a,t)da+2ma0τh(a)ph(a,t)da, (4.14)

    where γi(ˉN)=νiθκii/(θκii+ˉNκi), where i={h,c}. Next, we formulate (4.14) as semilinear problem:

    ddtω(t)=Cω(t),ω(0)=ω0X, (4.15)

    where the generator C is defined on the Banach space X as follows:

    (Cϕ)(a)=((a+1gh(a)(τh(a)+αh(a,ˉx1)+μph(a)))gh(a)ϕ1(a)+γh(ˉN)ϕ2(a)αh(a,ˉx1)ϕ1(a)(γh(ˉN)+μqh(a))ϕ2(a)(a+1gc(a)(τc(a)+αc(a,ˉx1)+μpc(a)))gc(a)ϕ1(a)+γc(ˉN)ϕ2(a)αc(a,ˉx1)ϕ1(a)(γc(ˉN)+μqc(a))ϕ2(a)),

    where

    ϕ(a)=(ϕ1(a),ϕ2(a),ϕ3(a),ϕ4(a))TD(C),

    where D(C) is defined below:

    D(C)={(ϕ1,ϕ2)|ϕi is absolute continuous on [0,a),ϕ(0)=(2(1m)a0τh(a)ϕ1(a)da,0,2a0τc(a)ϕ2(a)da+2ma0τh(a)ϕ1(a)da,0)T}.

    Next, we take the resolvent equation for the operator C:

    (λIC)ϕ=ψ,ϕD(C),ψX,λC, (4.16)

    which leads to

    γh(ˉN)ϕ2(a)+a(gh(a)ϕ1(a))+(λ+τh(a)+αh(a,ˉx1)+μph(a))ϕ1(a)=ψ1(a), (4.17)
    (λ+γh(ˉN)+μqh(a))ϕ2(a)αh(a,ˉx1)ϕ1(a)=ψ2(a), (4.18)
    γc(ˉN)ϕ4(a)+a(gc(a)ϕ3(a))+(λ+τc(a)+αc(a,ˉx1)+μpc(a))ϕ3(a)=ψ3(a), (4.19)
    (λ+γc(ˉN)+μqc(a))ϕ4(a)αc(a,ˉx1)ϕ3(a)=ψ4(a), (4.20)

    and

    ϕ1(0)=2(1m)a0τh(a)ϕ1(a)da,ϕ3(0)=2a0τc(a)ϕ3(a)da+2ma0τh(a)ϕ1(a)da.

    By solving (4.18) and (4.20), we get

    ϕ2(a)=ψ2(a)+αh(a,ˉx1)ϕ1(a)λ+γh(ˉN)+μqh(a),ϕ4(a)=ψ4(a)+αc(a,ˉx1)ϕ3(a)λ+γc(ˉN)+μqc(a), (4.21)

    which after substituting in Eqs. (4.17) and (4.19) gives

    ϕ1(a)=exp(a0τh(ξ)+αh(ˉx1,ξ)+λ+μph(ξ)γh(ˉN)αh(ˉx1,ξ)gh(ξ)(λ+γh(ˉN)+μqh(ξ))dξ)[a0exp(ζ0τh(ξ)+αh(ˉx1,ξ)+λ+μph(ξ)γh(ˉN)αh(ˉx1,ξ)gh(ξ)(λ+γh(ˉN)+μqh(ξ))dξ)1gh(ζ){ψ1(ζ)+γh(ˉN)ψ2(ζ)λ+γh(ˉN)+μqh(ζ)}dζ+ϕ1(0)],ϕ3(a)=exp(a0τc(ξ)+αc(ˉx1,ξ)+λ+μpc(ξ)γc(ˉN)αc(ˉx1,ξ)gc(ξ)(λ+γc(ˉN)+μqc(ξ))dξ)[a0exp(ζ0τc(ξ)+αc(ˉx1,ξ)+λ+μpc(ξ)γc(ˉN)αc(ˉx1,ξ)gc(ξ)(λ+γc(ˉN)+μqc(ξ))dξ)1gc(ζ){ψ3(ζ)+γc(ˉN)ψ4(ζ)λ+γc(ˉN)+μqc(ζ)}dζ+ϕ3(0)].

    Substituting ϕ1(a) and ϕ3(a) back in Eq (4.21) yields

    ϕ2(a)=1λ+γh(ˉN)+μqh(a)[exp(a0τh(ξ)+αh(ˉx1,ξ)+λ+μph(ξ)γh(ˉN)αh(ˉx1,ξ)gh(ξ)(λ+γh(ˉN)+μqh(ξ))dξ){a0exp(ζ0τh(ξ)+αh(ˉx1,ξ)+λ+μph(ξ)γh(ˉN)αh(ˉx1,ξ)gh(ξ)(λ+γh(ˉN)+μqh(ξ))dξ)1gh(ζ){ψ1(ζ)+γh(ˉN)ψ2(ζ)λ+γh(ˉN)+μqh(ζ)}dζ+ϕ1(0)}αh(a,ˉx1)+ψ2(a)],ϕ4(a)=1λ+γc(ˉN)+μqc(a)[exp(a0τc(ξ)+αc(ˉx1,ξ)+λ+μpc(ξ)γc(ˉN)αc(ˉx1,ξ)gc(ξ)(λ+γc(ˉN)+μqc(ξ))dξ){a0exp(ζ0τc(ξ)+αc(ˉx1,ξ)+λ+μpc(ξ)γc(ˉN)αc(ˉx1,ξ)gc(ξ)(λ+γc(ˉN)+μqc(ξ))dξ)1gc(ζ){ψ3(ζ)+γc(ˉN)ψ4(ζ)λ+γc(ˉN)+μqc(ζ)}dζ+ϕ3(0)}αc(a,ˉx1)+ψ4(a)].

    Lemma 4.1. The resolvent of operator C is compact and its spectrum, denoted by σ(C), satisfies the condition:

    σ(C)=σP(C)={λC|1σp(Uλ)}. (4.22)

    Here, σP(C) refers to the point spectrum of C, and Uλ is an operator dependent on λ.

    Proof. The expression of ϕ1(a) and ϕ3(a) can be re-written as

    ϕ1(a)=1αh(a,ˉx1){(λ+γh(ˉN)+μqh)(Uh,λψ2)(a)+γh(ˉN)(Uh,λψ1)(a)},ϕ3(a)=1αc(a,ˉx1){(λ+γc(ˉN)+μqc)(Uc,λψ4)(a)+γc(ˉN)(Uc,λψ3)(a)},

    where the linear operator on Banach space, Ui,λ is given as

    (Ui,λψ)(a)=a0Hi,λ(ζ,a)ψ(ζ)dζ,i={h,c}, (4.23)

    where

    Hλ(ζ,a)=αi(a,ˉx1)gi(ζ)(λ+γi(ˉN)+μqi(a))exp(a0τi(ξ)+αi(ˉx1,ξ)+λ+μpi(ξ)γi(ˉN)αi(ˉx1,ξ)g(ξ)(λ+γi(ˉN)+μqi(ξ))dξ)exp(ζ0τi(ξ)+αi(ˉx1,ξ)+λ+μpi(ξ)γi(ˉN)αi(ˉx1,ξ)gi(ξ)(λ+γi(ˉN)+μqi(ξ))dξ). (4.24)

    Similarly, we rewrite ϕ2(a) and ϕ4(a) as

    ϕ2(a)=(Ui,λψ2)(a)+(Vi,λψ1)(a),ϕ4(a)=(Ui,λψ4)(a)+(Vi,λψ3)(a),

    where the linear operator on Banach space, Vi,λ is given as

    (Vi,λψ)(a)=a0Gi,λ(ζ,a)ψ(ζ)dζ,Gi,λ(ζ,a)=1gi(ζ)(λ+γi(ˉN)+μqi(ξ))(γi(ˉN)Hi,λ(ζ,a)+gi(ζ)a).

    Let Λ=λC,|1σ(Uλ). For λCΛ, the operators Ui,λ and Vi,λ are compact operators from X to L1(0,a), implying that ϕ1(a) and ϕ3(a) are represented by compact operators, and similarly, ϕ2(a) and ϕ4(a) are also represented by compact operators. As a result, the operator C has a compact resolvent, which confirms that its spectrum σ(C) constitutes only isolated eigenvalues, i.e., σ(C)=σP(C) (see Theorem 6.29 on page 187 in [48]). Hence, CΛρ(C), where ρ(C) is the resolvent of operator C. Therefore, σP(C)=σ(C)Λ. Since Uλ is a compact operator, we have σ(Uλ)0=σP(Uλ)0. If λΛ, there exists an eigenfunction ψλ such that Uλψλ=ψλ. It is easy to see that (ϕ1(a),ϕ2(a),ϕ3(a),ϕ4(a))T provides an eigenvector of C for the eigenvalue λ. Thus, we have ΛσP(C), and we can conclude that (4.22) is satisfied.

    Lemma 4.2. Consider the operator C which generates C0-semigroup for t0. Then, T(t) is eventually norm continuous (ENC), and we have

    ω0(C)=s(C)=supReλ,|λσ(C), (4.25)

    where s(C) represents the spectral bound of the operator C, and ω0(C) denotes the growth bound of the semigroup T(t).

    Proof. To begin, we express the bounded operator C as

    Cϕ=(kh(a)γh(ˉN)00αh(a,ˉx1)γh(ˉN)μqh(a)0000kc(a)γc(ˉN)00αc(a,ˉx1)γc(ˉN)μqc(a))(ϕ1(a)ϕ2(a)ϕ3(a)ϕ4(a)),

    for ϕX, kh(a)=gh(a)aτh(a)αh(a,ˉx1)μph(a) and kc(a)=gc(a)aτc(a)αc(a,ˉx1)μpc(a). To establish the compactness of C, our strategy is to demonstrate that for any bounded sequence (ϕn)nN in X, the sequence (Cϕn)nN contains a subsequence that converges uniformly. To accomplish this, we invoke the Arzelà-Ascoli Theorem, which requires us to verify that (Cϕn)nN is uniformly bounded and uniformly equicontinuous. To prove boundedness, since we assume (ϕn)nN to be bounded, we get

    Cϕn1Cϕn1CsupnNϕn1,

    which determines that (Cϕn)nN is also bounded. For uniform equicontinuity, we consider

    R|(Cϕ)(a+h)(Cϕ)(a)|da=R|C(a+h)C(a)||ϕ(a)|daR|(kh(a+h)γh(ˉN)00αh(a+h,ˉx1)γh(ˉN)μqh(a+h)0000kc(a+h)γc(ˉN)00αc(a+h,ˉx1)γc(ˉN)μqc(a+h))(kh(a)γh(ˉN)00αh(a,ˉx1)γh(ˉN)μqh(a)0000kc(a)γc(ˉN)00αc(a,ˉx1)γc(ˉN)μqc(a))||ϕ1(a)ϕ2(a)ϕ3(a)ϕ4(a)|da=R|kh(a+h)kh(a)000αh(a+h,ˉx1)αh(a,ˉx1)μqh(ah)μqh(a)0000kc(a+h)kc(a)000αc(a+h,ˉx1)αc(a,ˉx1)μqc(a+h)μqc(a)||ϕ1(a)ϕ2(a)ϕ3(a)ϕ4(a)|daϕR|kh(a+h)kh(a)000αh(a+h,ˉx1)αh(a,ˉx1)μqh(ah)μqh(a)0000kc(a+h)kc(a)000αc(a+h,ˉx1)αc(a,ˉx1)μqc(a+h)μqc(a)|da.

    Hence, we have shown that (Cϕn)nN is equicontinuous, and by the Arzelà-Ascoli Theorem, we can conclude that there exists a uniformly convergent subsequence of (Cϕn)nN. Hence, C is compact, which in turn implies that T is an ENC semigroup. Since the spectral mapping theorem can be applied to ENC semigroups, we have the spectral determined growth condition given by ω0(C)=s(C). Thus, we obtain (4.25).

    The local exponential asymptotic stability of the steady-state solution ω=0 of (4.15) is established when ω0(C)<0. Specifically, there exist constants ϵ>0, M1, and γ<0 such that if xX and |x|ϵ, then the solution ω(t,x) of (4.15) exists globally and satisfies |ω(t,x)|Mexp(γt)|x| for all t>0. In order to examine the stability of steady states, it is necessary to identify the dominant singular point within the set Λ, which corresponds to the element with the highest real value. By utilizing (4.22) and (4.25), we can then determine the growth bound of the semigroup T.

    Lemma 4.3. For any λR, the operator Ui,λ is nonsupporting with respect to X+ and

    limλ+r(Ui,λ)=0 (4.26)

    holds.

    Proof. By Eqs. (4.23) and (4.24), we can conclude that the operator Ui,λ,λR is strictly positive. To prove that Uλ,λR is non-supporting, we can easily demonstrate the inequality

    Ui,λψfi,λ,ψc,c=1X+,ψX+, (4.27)

    where the linear function fi,λ, is

    fi,λ,ψ=a0[si(ζ)gi(ζ)(λ+γi(ˉN)+μqi(a))exp(a0τi(ξ)+αi(ˉx1,ξ)+λ+μpi(ξ)γi(ˉN)αi(ˉx1,ξ)g(ξ)(λ+γi(ˉN)+μqi(ξ))dξ)exp(ζ0τi(ξ)+αi(ˉx1,ξ)+λ+μpi(ξ)γi(ˉN)αi(ˉx1,ξ)gi(ξ)(λ+γi(ˉN)+μqi(ξ))dξ)]ψ(ζ)dζ. (4.28)

    Thereby, it leads us to Un+1i,λψfi,λ,ψfi,λ,cnc,n. which holds for all ψX+, where fi,λ is strictly positive and the constant function c=1 is a quasi-interior point of L1(0,a). This implies that F,Uni,λ>0 for every pair ψX+0, FX+0, and therefore Ui,λ,,λR is non-supporting. We then use inequality (4.27) and take the duality pairing with the eigenfunctional Fi,λ of Ui,λ corresponding to r(Ui,λ), yielding

    r(Ui,λ)Fλ,ψFλ,efi,λ,ψ.

    Assuming ψ=c, we obtain the inequality:

    r(Ui,λ)fi,λ,c,

    where

    fi,λ,c=a0si(ζ)gi(ζ)(λ+γi(ˉN)+μqi(ζ))exp(a0τi(ξ)+αi(ˉx1,ξ)+λ+μpi(ξ)γi(ˉN)αi(ˉx1,ξ)g(ξ)(λ+γi(ˉN)+μqi(ξ))dξ)exp(ζ0τi(ξ)+αi(ˉx1,ξ)+λ+μpi(ξ)γi(ˉN)αi(ˉx1,ξ)gi(ξ)(λ+γi(ˉN)+μqi(ξ))dξ)dζ. (4.29)

    It follows that

    fλ,cϵa01gi(ζ)(λ+γi(ˉN)+μqi(ζ))exp(a0τi(ξ)+αi(ˉx1,ξ)+λ+μpi(ξ)γi(ˉN)αi(ˉx1,ξ)g(ξ)(λ+γi(ˉN)+μqi(ξ))dξ)exp(ζ0τi(ξ)+αi(ˉx1,ξ)+λ+μpi(ξ)γi(ˉN)αi(ˉx1,ξ)gi(ξ)(λ+γi(ˉN)+μqi(ξ))dξ)dζ. (4.30)

    By using the positivity of γi(ˉN),μpi,μqi,αi and τi, we conclude the following:

    limλ+r(Ui,λ)=0.

    Hence proved. The previous lemma implies that the function λr(Ui,λ) is decreasing for all λR. Moreover, if there exists a λR such that r(Ui,λ)=1, then it follows that λΛ, as r(Ui,λ)σP(Ui,λ). Combining this with the monotonicity property of r(Ui,λ) and inequality (4.26), we obtain the following result.

    Lemma 4.4. There exists a unique λ0RΛ such that r(Ui,λ0)=1, and λ0>0 if r(U0)>1; λ0=0 if r(U0)=1; λ0<0 if r(U0)<1.

    We will demonstrate that λ0 is a dominant singular point, utilizing Theorem 6.13 in [49].

    Lemma 4.5. If there exists a λΛ,λλ0, then Reλ<λ0.

    Proof. Let λΛ and Ui,λψ=ψ. Then |ψ|(a)=|ψ(a)|, and we have |Ui,λψ|=|ψ|. Therefore, we obtain Ui,Reλψψ. By taking the duality pairing with FReλX+, we get r(Ui,Reλ)FReλ,|ψ|FReλ,|ψ|. We have r(Ui,Reλ)1, as FReλ is strictly positive. Since r(Ui,λ),λR is a declining function, we conclude that Reλλ0. Suppose Reλ=λ0. Then Ui,λ0|ψ|=|ψ|. If we assume Ui,λ0|ψ|>|ψ|, then taking the duality pairing with the eigenfunctional F0 corresponding to r(Ui,λ0)=1 results in F0,|ψ|>F0,|ψ|, which is a contradiction. Therefore, we have Ui,λ0|ψ|=|ψ|, and we can deduce that |ψ|=cψ0, where constant c is assumed to be 1, and ψ0 is the eigenfunction relating to r(Ui,λ0)=1. Hence, we have ψ(a)=ψ0(a)exp(iv(a)) for a real-valued function v(a). Substituting this into Ui,λ0ψ0=|Ui,λψ| yields

    αi(a,ˉx1)gi(a)(λ0+γi(ˉN)+μqi(a))a0exp(ζaτi(ξ)+αi(ˉx1,ξ)+λ0+μpi(ξ)γi(ˉN)αi(ˉx1,ξ)λ0+γi(ˉN)+μqi(ξ)dξ)ψ0(ζ)dζ=|αi(a,ˉx1)gi(a)(λ0+iImλ+γi(ˉN)+μqi(a))a0exp(ζaτi(ξ)+αi(ˉx1,ξ)+λ0+iImλ+μpi(ξ)γ(iˉN)αi(ˉx1,ξ)λ0+iImλ+γi(ˉN)+μqi(ξ)dξ)exp(iv(ζ))ψ0(ζ)dζ|.

    Lemma 6.12 [49] implies that Imλ+v(ζ) equals a constant Θ. Using the fact that Ui,λψ=ψ, we obtain the equation exp(iΘ)Ui,λ0ψλ0=ψλ0exp(iv(ζ)). This equation shows that if Θ=v(ζ), then Imλ=0. Therefore, the proof is complete.

    Theorem 4.1. The equilibrium state (ˉph(a),ˉqh(a),ˉpc(a),ˉqc(a))T, for (2.1)–(2.4), is locally asymptotically stable if r(U0)<1 and locally unstable if r(U0)>1.

    Proof. Lemmas 4.4 and 4.5 suggests that supReλ:1σP(Ui,λ)=λ0. This implies that if r(U0)<1, then s(C)=supReλ:1σP(Ui,λ)<0. Conversely, if r(U0)>1, then s(C)=supReλ:1σP(Ui,λ)>0. Therefore, the proof is complete.

    In this section, we show simulations of the model to investigate the evolution of healthy and mutated sub-populations of quiescent and proliferative cells in relation to the cell cycle dynamics. These simulations are performed in the Matlab and we have used finite volume method with discretization central upwind scheme. Table 2 displays the model parameters that were used in the simulations. A maximum cell age of 50 is assumed, and the time step and spatial step size are set to Δt=0.02 and Δa=0.5, respectively. Additionally, we use unit speed, such that gh(a)=gc(a)=1.

    Table 2.  Parameters used in the simulations.
    Para. Description Healthy Mutated Unit
    m Mutation rate 0.2 - day1
    νi Maximum transition rate from quiescent to proliferation phase 0.6 [50] 0.6 day1
    θi Total cell population beyond which Γ is zero 0.095×106 [50] 0.095×106 -
    κi Hill coefficient 1 [50] 1 -
    ρ1,i Maximal effect of Cyclin DCDK4/6 complex on the division of cell 0.7 0.7 -
    ρ2,i Value of Cyclin DCDK4/6 complex to achieve half maximum effect 0.35 0.35 -
    γ1,i Hill coefficient 8 8 -
    σ1,i Maximum rate of switching cells from proliferating to quiescent phase 0.01 0.01 -
    σ2,i Switching Cyclin DCDK4/6 complex value, after that α is close to zero 0.5[45] 0.45
    σ3,i Switching age value beyond which α is close to zero 14 15 h
    γ2,i Hill coefficient 7 7 -
    γ3,i Hill coefficient 7 7 -
    kt Rate constant which measures the effect of total population on growth factors 1.80×109 [51] 1.80×109 -

     | Show Table
    DownLoad: CSV

    Steady-state dynamics of healthy and mutated cell populations:

    This case study aims to study the steady-state dynamics of both healthy and mutated cell populations, where the death rates are set to μph=μqh=0.0014 and μpc=μqc=0.0014. Additionally, γ(N)=6.8964×106 and ρ1=1.0, while the mutation rate is set to m=0.1. The initial conditions for the healthy cell populations are defined as normal distributions ph(a,0)=qh(a,0)=k02πσ2exp((aμ)22σ2), where k0=106, μ=2, and σ2=200. Similar initial distributions are used for the mutated cell populations, but with k0=102. A normal distribution is used because it provides a good approximation of the cell distribution within a population by incorporating heterogeneity with respect to cell age in a population.

    Figure 3 illustrates the number density distribution of different cell populations, namely healthy (a) proliferating, (b) quiescent, and mutated (c) proliferating, (d) quiescent, as they evolve and eventually reach a steady-state. Here, the cell age is measured in time and cell density is measured as cells per cubic millimeter. The mutation rate is set to m=0.1. Over time, the population of mutated proliferating and quiescent cells gradually declines as they grow faster and occupy the tissues completely. However, the total population of cells (consisting of both healthy and mutated proliferating and quiescent cells) denoted by N(t) increases rapidly, as shown in Figure 4(a), before eventually reaching a steady-state. Furthermore, Figure 4(b) demonstrates that the growth factors, which are influenced by the cell population N(t), initially increase due to the low cell count and subsequently decrease until reaching an equilibrium. Finally, Figure 4(c) portrays the transition rate of cells, denoted by γ(N), from quiescent phase to proliferating phase.

    Figure 3.  Cell number density distribution of different cell populations in macroscale.
    Figure 4.  Dynamics of the combined cell population, growth factors, and transition function γ: (a) the total cell population N(t) reaches a steady-state, (b) the growth factors gf decrease as the cell population increases, and (c) both gamma functions, γh and γc, decrease as the total cell population reaches a steady-state.

    Exponential growth of mutated cell populations:

    For this case study, we selected a mutation rate of m=0.2 and used death rates of μph=μqh=0.0014 and μpc=μqc=0.0010. Additionally, we altered several other parameters, including ν1,c=0.045, ρ1,c=1.0, ρ2,c=30, σ1,c=0.040 and σ2,c=0.45. Figure 5 shows the cell density distribution of healthy and mutated proliferating and quiescent cells, respectively. Both healthy subpopulations exhibit the trends of reaching a trivial steady-state with time. However, the mutated cell populations pc(a,t) and qc(a,t) increase exponentially, mimicking cancerous behavior. Figure 6 illustrate the total number of cells, growth factors, and the switching function from quiescent to proliferating phase, γi(N), respectively. The total cell population, which comprises healthy and mutated proliferating and quiescent cell populations, exhibits an exponential increase in cell number over time. The growth factors are initially maximum due to the low cell count and gradually decline until reaching extremely low levels. Finally, the transition functions γh and γc also decline as the cell population grows.

    Figure 5.  Cell number density distribution of different cell populations in macroscale.
    Figure 6.  Dynamics of entire cell population, growth factors, and γ transitions: (a) The total cell population N(t) exhibits exponential growth. (b) As the cell population increases, the growth factors gf decrease. (c) Both gamma functions, γh and γc, decrease as the total cell population increases.

    The proposed model has some limitations that should be considered. Firstly, exclusion of cell heterogeneity, that is a essential aspect to account for cellular noise. Additionally, the feedback model that is comprised of growth factors is fairly simple straight forward, and to characterize the activation of the Cyclin D complex, all signaling pathways should be considered. Moreover, at the microscale, it would have been necessary to model cell cycle dynamics separately for healthy and mutated cell populations to investigate their respective compartments' more heterogeneous behavior. Moreover, while the Cyclin-CDKCDK4/6 are crucial for the G1 to S phase transition, the model overlooks the other restriction point that detects DNA damage in the S phase.

    This reserch presents non-linear, multiscale modeling of physiologically-structured healthy and mutated quiescent and proliferating cells in relation to cell cycle dynamics. We modeled reversible transitioning from quiescent to proliferating cells and vice versa. We checked the wellposedness of the model, derive non-trivial equilibrium solutions and find spectral criteria for local stability. We also performed numerical simulations to study the impact of Cyclin DCDK4/6 complex on the transition between two sub-populations. Furthermore, we predict that the Cyclin' complex plays an important role in the reversible transition, and any abnormality in this transition can result in cancer.

    All authors declare no conflicts of interest in this paper.



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