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Mathematical analysis of the Cancitis model and the role of inflammation in blood cancer progression

  • Recently, a tight coupling has been observed between inflammation and blood cancer such as the Myeloproliferative Neoplasms (MPNs). A mechanism based six-dimensional model-the Cancitis model-describing the progression of blood cancer coupled to the inflammatory system is analyzed. An analytical investigation provides criteria for the existence of physiological steady states, trivial, hematopoietic, malignant and co-existing steady states. The classification of steady states is explicitly done in terms of the inflammatory stimuli. Several parameters are crucial in determining the attracting steady state(s). In particular, increasing inflammatory stimuli may transform a healthy state into a malignant state under certain circumstances. In contrast for the co-existing steady state, increasing inflammatory stimuli may reduce the malignant cell burden. The model provides an overview of the possible dynamics which may inform clinical practice such as whether to use inflammatory inhibitors during treatment.

    Citation: Zamra Sajid, Morten Andersen, Johnny T. Ottesen. Mathematical analysis of the Cancitis model and the role of inflammation in blood cancer progression[J]. Mathematical Biosciences and Engineering, 2019, 16(6): 8268-8289. doi: 10.3934/mbe.2019418

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  • Recently, a tight coupling has been observed between inflammation and blood cancer such as the Myeloproliferative Neoplasms (MPNs). A mechanism based six-dimensional model-the Cancitis model-describing the progression of blood cancer coupled to the inflammatory system is analyzed. An analytical investigation provides criteria for the existence of physiological steady states, trivial, hematopoietic, malignant and co-existing steady states. The classification of steady states is explicitly done in terms of the inflammatory stimuli. Several parameters are crucial in determining the attracting steady state(s). In particular, increasing inflammatory stimuli may transform a healthy state into a malignant state under certain circumstances. In contrast for the co-existing steady state, increasing inflammatory stimuli may reduce the malignant cell burden. The model provides an overview of the possible dynamics which may inform clinical practice such as whether to use inflammatory inhibitors during treatment.


    Myeloproliferative Neoplasms (MPNs) is a group of hematopoietic stem cell disorders, including essential thrombocytosis (ET), polycythemia vera (PV) and primary myelofibrosis (PMF) [1,2]. The pathogenesis of these neoplasms is yet to be fully discovered. For patients with MPNs, the mutation JAK2V617F is found present in the most cases of ET (50%) and in 95% of the cases with PV and PMF ultimately leading to acute myeloid leukemia (AML) [3,4]. This suggests a biological continuum where the diseases evolve from early cancers (ET and PV) into the advanced myelofibrosis stage, with an increasing load of JAK2V617F mutations from a low burden at ET and PV to a high load [2,5]. MPNs imply an increased risk for the development of other cancers [1,4].

    Recent research supports that MPNs can be regarded as chronic inflammatory diseases and MPNs has been described as a "human inflammation model", which leads to premature atherosclerosis, clonal evolution and an increased risk of second cancers. [2,3,6]. This is based on evidence from clinical observations, experiments and molecular studies [3].

    Several insightful theoretical studies have been published on control dynamics of biological networks. Mathematical models have been proposed [7,8,9] describing the control networks for regulation of stem cell lineage.Mathematical modelling of cancer is useful for understanding of cancer initiation, progression [10,11], to confirm or dismiss biological/medical hypotheses, and to study effects of single or multi modality treatments in silico. The mathematical model presented in [12] shows that successful therapy may eliminate tumour stem cells. A five-dimensional model given in [13] includes active and quiescent stem cells, progenitor cells, mature cells and one immune compartment describing chronic myelogenous leukemia.In [14] a mathematical model of cancer stem cell dynamics is proposed and the different scenarios of cancer initiation and possible treatments strategies have been discussed. The mathematical model given in [15] is useful for investigating the impact of cytokine-dependence of acute myeloid leukemic cells. In addition, the model allows distinguishing between cytokine-dependent and cytokine-independent acute myeloid leukemia (AML) and both scenarios are supported by patient data.

    However, only a few mathematical models of MPNs exist. Some work includes investigation of the origin of myeloid malignancies with MPNs as a particular example [16]. In [17], a two dimensional model of MPNs is investigated without including the immune response dynamically. The Cancitis model including chronic inflammation as the trigger and driver of MPNs was proposed in [5]. In [5], T-cells are not explicitly considered whereas, in [18], the effect of these cells has been included specifically. The analysis of a two dimensional mathematical model [18] is used to discuss in silico effect of existing and novel treatments. The model presented here is identical to the model presented in [5] except for the simpler functional form of the stem cells niche interaction used here and in [18].

    In the present paper we conduct a thorough mathematical investigation of the Cancitis model and explore the intricate coupling between inflammation and MPNs. We address the following questions which have not been systematically investigated previously: Which steady states of the system are feasible and which trajectories are attracted to the steady states? How do the number and stability of the steady states change when varying the parameters, in particular, the exogenous inflammatory stimuli, self-renewal and death rates of stem cells, and inhibitory strength of the stem cell niche interaction? Which set of clustered parameters control the dynamics of the system? Does the analysis suggest correlated parameters? The bio-medical applications of the model analysis are discussed, e.g. how the inflammation influences the transition between healthy and diseased states. In addition, the analysis predicts effects of ongoing and potential combination therapies.

    The Cancitis model stated in [5] is illustrated in Figure 1, with the system of differential equations shown in system (2.1). In this section the details of the model and the reasoning behind it is presented.

    Figure 1.  The boxes illustrate the compartments of the Cancitis model. The arrows represent the rates of the flows between and out of these compartments. Red stipulated arrows represent the effect of inflammation which is stimulated by exogenous inflammatory stimuli, I. Green stipulated lines represent the bone marrow niches interaction with a 'crowding' competition between HSC and MSC. Stem cells (HSC and MSC) may self-renew, die or differentiate, while mature cells die after a while (MMC, HMC). Dead cells (a) are engulfed by the immune cells (s), that stimulate production of stem cells, increase risk of mutation and increase the removal of dead cells (For more details, see main text).

    The model describes the proliferation of hematopoietic stem cells (HSC) into hematopoietic mature cells (HMC) and likewise malignant stem cells (MSC) into malignant mature cells (MMC). Additionally, the model considers the number of dead cells and the level of inflammation. The debris from the dead cells stimulates the immune response, which in turn affects the self-renewal rate of both HSCs and MSCs.

    The model consists of six ordinary differential equations one for each compartment; the number of HSC (x0), the number of HMC (x1), the number of MSC (y0), the number of MMC (y1), the number of dead cells (a), and the level of inflammation (s).

    The equations are of the general form,

    {Change in amount of acompartment per time}={rate of production timesthe producing source }{rate of elimination times theamount in the compartment}.

    and read specifically,

    ˙x0=rx(ϕxsαx)x0rmsx0, (2.1a)
    ˙x1=axAxx0dx1x1, (2.1b)
    ˙y0=ry(ϕysαy)y0+rmsx0, (2.1c)
    ˙y1=ayAyy0dy1y1, (2.1d)
    ˙a=dx0x0+dy0y0+dx1x1+dy1y1eaas, (2.1e)
    ˙s=rsaess+I, (2.1f)

    with

    αx=dx0+axrxandαy=dy0+ayry. (2.2)

    The expressions for the inhibitory niche feedback are chosen as Michaelis-Menten-like functions in contrast to [5],

    ϕxϕx(x0,y0)=11+cxxx0+cxyy0, (2.3a)
    ϕyϕy(x0,y0)=11+cyxx0+cyyy0. (2.3b)

    A stem cell can proliferate in three ways; symmetric self-renewal (resulting in two new stem cells), asymmetric self-renewal (resulting in one stem cell and one progenitor cell) and symmetric differentiation (resulting in two progenitor cells). The rate of self-renewal is denoted as rx and ry for HSC and MSC respectively. The self-renewal of stem cells is known to be inhibited by self-regulating niche feedback [19], resulting in decreased self-renewal when the level of stem cells in the bone marrow is high. Adopting the approach taken in [12], [20] and [21], this is implemented by Michaelis-Menten-like functions ϕx(x0,y0) and ϕy(x0,y0), shown in Eq (2.3b). Allowing the feedback to be different for HSC and MSC, the constants cxx and cxy capture the effects of HSC and MSC on the self-renewal of HSC, while cyx and cyy capture the corresponding effects on the self-renewal of MSC. Additionally, the inflammatory level also affects the self-renewal [22,23]. This leads a to self-renewal term per cell of rxϕxs and ryϕys for HSC and MSC respectively. The parameter cij describes the inhibitory strength of the signalling bone marrow niche feedback from cell type j onto cell type i. It is generally assumed that cyycyxcxycxx, since malignant cells are less sensitive to inhibitive niche feedback than hematopoietic cells [22,24].

    In [25], a multi compartmental model is proposed relying on a single external feedback mechanism. It is shown that the equilibrium level of mature cells depends only on the self-renewal parameters for the HSC and it is independent of the other compartments. Therefore, the progenitor cells are considered as intermediate steps between stem cells and mature cells, and are implicitly accounted for by multiplication factors Ax and Ay for HSC and MSC respectively. The rate at which the HSC reduces in transforming to HMC is denoted by ax while the similar rate for MSC transforming to MMC is denoted by ay. As such, the HMC and MMC accordingly increase with rates axAx and ayAy respectively. To account for the cell apoptosis, the four types of cells are removed with rates dx0, dx1, dy0 and dy1, for the corresponding cell types.

    Genetic mutations are by nature to be described as Poisson processes [26,27,28,29]. However, not all mutations are malignant; only mutation which happens on a particular location of the DNA, i.e. at specific amino acids causes a specific mutation, e.g. the JAK2V617F mutation. The probability for hitting a specific location is about 1/30000. In [30] the average mutation probability is estimated to 0.0035 per year, which corresponds to a specific mutation probability of 0.0035/30000=1.2107 per year. Thus, the probability for one specific malignant mutation is about 107 per cell per year. Moreover, the mutation is affected by the inflammation, s [31,32], which is explicitly stated, and resulting in the effective mutation rate rms. Assuming three sequential mutations are needed to generate a specific malignant stem cell the resulting probability becomes much higher (1025 per year per cell if the mutations are assumed independent). This could be implemented in the otherwise deterministic model but it would increase the computational cost, since it depends on both the probability of a single cell mutation and the number of potential mutating cells at a given time, which itself is determined by the preceding mutational history. To avoid such complications we initialize by having a single MSC and none MMC, and put the mutation rate to zero. This is justified by the fact that the probability of a single cell mutating is small compared to the self-renewal of the MSCs. Thus, the first mutation drives the development leaving a later identical mutation insignificant to the dynamics, which is confirmed by numerical simulations.

    The number of dead cells has an up-regulatory effect on the immune response denoted rs. External environmental factors also influence the inflammatory level. This is captured in the model by the term I. Throughout we take I>0, as a perfect sterile environment is an utopic idealization. This term may vary over time due to environmental changes, but in our mathematical analysis we will consider I as piecewise constant. The inflammation, s, is down-regulated naturally by the eliminating rate es.

    Additionally, the change in the amount of dead cells per time is given by the death rate times the number of cells minus the clearance by the immune system. As given in [33] clearance is described by a second order equation eaas since the engulfed immune cells have to meet the dead cells debris to mediate endocytosis. Thus, clearance is bilinear in both a and s representing the activity of the immune system, eliminating the dead cells with an elimination rate ea.

    Initial conditions for the Cancitis model in equations are needed for the given system of differential equations (2.1–2.3b). Here, we mainly focus on the model after the first mutation, i.e. with y0(0)=1, y1(0)=0, rm=0, and the remaining variables as those in the healthy steady state (see below). All other parameter values are assumed to be positive.

    The stable steady states are attractors in the six dimensional phase space. This motivates systematic study of the existence and location of steady states and how this is affected by perturbing parameter values.

    Motivated by the biology where the number of cells and concentrations are required to be non-negative numbers, we will use the terminology that a steady state is admissible if and only if all the components are non-negative, i.e. if and only if a steady state is in the non-negative octahedron.

    Consider the system of Eqs (2.1–2.3b). For steady state solutions, ˙x0=˙x1=˙y0=˙y1=˙a=˙s=0,

    (ˉsαx(1+cxxˉx0+cxyˉy0))ˉx0=0, (2.4a)
    (ˉsαy(1+cyxˉx0+cyyˉy0))ˉy0=0, (2.4b)
    ˉx1=axAxˉx0dx1, (2.4c)
    ˉy1=ayAyˉy0dy1, (2.4d)
    dx0ˉx1+dx1ˉx1+dy0ˉy0+dy1ˉy1eaˉaˉs=0, (2.4e)
    ˉa=esrs(ˉsIes). (2.4f)

    The values of x0, x1, y0, y1, a and s at steady state are denoted as ˉx0, ˉx1, ˉy0, ˉy1, ˉa and ˉs respectively. The admissible steady states can be classified as,

    ● a trivial steady state if and only if ˉx0=ˉy0=0,

    ● a (purely) hematopoietic steady state if and only if ˉy0=0 and ˉx0>0,

    ● a (purely) malignant steady state if and only if ˉx0=0 and ˉy0>0, or

    ● a co-existing steady state if and only if ˉx0>0 and ˉy0>0.

    The admissibility of steady states necessitates certain inequalities to be fulfilled, which leads to restriction on the parameters, e.g. from Eq (2.4f), non-negativity of ˉa requires ˉsIes. I is assumed to be positive thus Ies>0. Hence

    ˉsIes>0, (2.5)

    for any admissible steady state. Note that, ˉx1 and ˉy1 are non-negative if and only if ˉx0 and ˉy0 are non-negative, respectively. Substituting expressions of ˉx1 and ˉy1 in Eq (2.4e) gives,

    eaˉaˉs=βxˉx0+βyˉy0, (2.6)

    where βx=axAx+dx0 and βy=ayAy+dy0. Thus, Eqs (2.4e–2.4f) and (2.6) result in a second order equation in ˉs having the general solution,

    ˉs±=I2es(1±1+ζ(βxˉx0+βyˉy0)), (2.7)

    where ζ=4rseseaI2>0. ˉs is negative for positive ˉx0 or ˉy0. In case, (ˉx0,ˉy0)=(0,0), Eq (2.7) leads to ˉs=0 contradicting ˉs>0. Thus, s=ˉs is not biologically realizable and we therefore put ˉs=ˉs+ in the further analysis. Note that, for non-trivial steady states, ˉs>Ies. In addition, non-negativity of ˉx0 and ˉy0 implies non-negativity of ˉx1, ˉy1, ˉs and ˉa.

    Hence, the existence of admissible steady states ˉx0, ˉy0, ˉx1, ˉy1, ˉs and ˉa follows from Eqs (2.4a), (2.4b) and (2.7).

    Below we make a complete analysis of the existence of various steady states depending on how I relates to the remaining parameters. This choice is due to the fact that the external inflammatory stimuli I is of great interest in health care and to elucidate consequences of using inflammation inhibitors as part of treatment.

    Proposition 1. A trivial steady state E0 always exists,

    E0=(0,0,0,0,0,Ies). (2.8)

    Proof. Follow directly from Eqs (2.4e) and (2.5).

    Hematopoietic steady states may exist depending on the rest of the parameter values. As above we chose the inflammatory stimuli I as the leading parameter and make a complete analysis of possible hematopoietic steady states. The analysis of the existence of the hematopoietic steady states depends crucially on the following lumped parameters,

    IH=2esrsβxeacxxrsβxeacxxαx, (2.9)
    ζH1=2esαxrsβxeacxxαx, (2.10)
    ζH2=esαx, (2.11)
    ζH3=rsβxeseacxx, (2.12)

    the last two always being positive.

    Proposition 2. Two hematopoietic steady states EH± may exist in the following cases,

    If α2xζH34 then EH+ exists if and only if I>0.

    If ζH34<α2xζH3 then EH+ exists if and only if IIH.

    If ζH3<α2x then EH+ exists if and only if I>ζH2.

    If α2xζH34 then EH exists if and only if IζH2.

    If ζH34<α2xζH3 then EH exists if and only if IHIζH2.

    If ζH3<α2x then EH does not exist.

    In case of existence, EH+=(ˉx0H+,ˉx1H+,0,0,ˉaH+,ˉsH+) is given by ˉx0H+=12escxxαx(IζH1+(ζH1I)24esαx(ζH2I)), ˉsH+=αx(1+cxxˉx0H+), ˉaH+=βxˉx0H+eaˉsH+, and ˉx1H+=axAxˉx0H+dx1 whereas EH=(ˉx0H,ˉx0H,0,0,ˉaH,ˉsH) is given by ˉx0H=12escxxαx(IζH1(ζH1I)24esαx(ζH2I)), ˉsH=αx(1+cxxˉx0H), ˉaH=βxˉx0HeaˉsH, and ˉx1H=axAxˉx0Hdx1.

    Proof. A hematopoietic steady state EH follows from Eqs (2.4a) and (2.7) with y0=y1=0 as possible positive solutions to

    ˉx20H+1escxxαx(ζH1I)ˉx0H+1esc2xxαx(ζH2I)=0. (2.13)

    For the solutions to (2.13) to be real,

    (ζH1I)24ζH2(ζH2I). (2.14)

    In case IζH2, (2.14) is always fulfilled. In case I<ζH2, (2.14) is equivalent to

    I2+2rsβxeacxxαxI+rsβxeacxxαx(rsβxeacxxαx4esαx)0. (2.15)

    Solving for I we get,

    IIH=γx(α2xζH34), (2.16)

    with

    γx=4esrsβxeacxxα2xrsβxeacxxαx+2esrsβxeacxx>0. (2.17)

    From Eqs (2.14) and (2.16) it follows that the solutions to Eq (2.13) are real for IζH2 or IHI<ζH2 in case IH<ζH2.

    Whenever the solutions to (2.13) are real, they are given by

    ˉx0H±:=12escxxαx(IζH1±(ζH1I)24esαx(ζH2I)), (2.18)

    which depends on the sign of the following five quantities,

    ζH1=2esαx(α2xζH32), (2.19)
    IH=γx(α2xζH34), (2.20)
    Δ12=ζH1ζH2=esαx(α2xζH3), (2.21)
    ΔH1=ζH1IH=esαx+ζH3(α2xζH3), and  (2.22)
    ΔH2=IHζH2=esαx(αx+ζH3)2(α2xζH3)20, (2.23)

    where the last one immediately implies that the criteria for real solutions of Eq (2.13) is IIH. If the solutions, ˉx0H±, are positive and real, then the formulas for the remaining variables easily follow from Eqs (2.4a–2.4f).

    To continue we first consider ˉx0H+ and afterwards ˉx0H.

    For α2xζH34, it follows from Eqs (2.19)–(2.22) that ζH1<0, ζH1<ζH2, IH0, and ζH1<IH. Thus, ˉx0H+>0 if and only if I>0.

    For ζH34<α2xζH32, it follows from Eqs (2.19)–(2.22) that ζH10, ζH1<ζH2, IH>0, and ζH1<IH. Thus, ˉx0H+>0 if and only if I>IH.

    For ζH32<α2xζH3, it follows from Eqs (2.19)–(2.22) that ζH1>0, ζH1ζH2, IH>0, ζH1<IH, and IHζH2. Thus, ˉx0H+>0 if and only if I>IH.

    For ζH3<α2x are ζH1>0, it follows from Eqs (2.19)–(2.23) that ζH1>ζH2, IH>0, ζH1>IH, and IHζH2. Thus, ˉx0H+>0 if and only if I>ζH2.

    Similar, ˉx0H is real if and only if IIH and ζH1<I<ζH2.

    For α2xζH34, it follows from Eqs (2.19) and (2.20) that ζH1<0<ζH2, and IH0. Thus, ˉx0H>0 if and only if I<ζH2.

    For ζH34<α2xζH32, it follows from Eqs (2.19), (2.20) and (2.23) that ζH10<ζH2, IH>0, and IH<ζH2. Thus, ˉx0H>0 if and only if IHI<ζH2.

    For ζH32<α2xζH3, it follows from Eqs (2.19)- (2.21) and (2.23) that 0<ζH1<ζH2, IH>0, and IH<ζH2. Thus, ˉx0H>0 if and only if IHI<ζH2.

    For ζH3<α2x, it follows from Eqs (2.20) and (2.21) that ζH1>ζH2 and IH>0. Thus, ˉx0H>0 if and only if ζH1<I<ζH2, which is a contradiction.

    The conditions for the existence of the hematopoietic steady states are summarized in Table 1.

    Table 1.  Summarizing necessary and sufficient criteria for admissibility of EH. The first column conditions how αx is related to ζH3, the middle column shows the existence conditions for EH+ and the last column shows the existence conditions for EH± explicitly in terms of I.
    For Only EH+ if Both EH+ and EH if
    α2x<ζH34 I>ζH2 IζH2
    ζH34<α2x<ζH3 I>ζH2 IHIζH2
    ζH3<α2x I>ζH2

     | Show Table
    DownLoad: CSV

    Malignant steady states may exist depending on therange of the parameters. As above we chose the inflammatory stimuli I as our leading parameter and make a complete analysis of possible malignant steady states. The analysis of the existence of the malignant steady states depends crucially on the following lumped parameters,

    IH=2esrsβyeacyyrsβyeacyyαy, (2.24)
    ζL1=2esαyrsβyeacyyαy, (2.25)
    ζL2=esαy, (2.26)
    ζL3=rsβyeseacyy, (2.27)

    the last two being positive.

    Proposition 3. Two malignant steady states EL± may exist in the following cases,

    If α2yζL34 then EL+ exists if and only if I>0.

    If ζL34<α2yζL3 then EL+ exists if and only if IIL.

    If ζL3<α2y then EL+ exists if and only if I>ζL2.

    If α2yζL34 then EL exists if and only if IζL2.

    If ζL34<α2yζL3 then EL exists if and only if ILIζL2.

    If ζL3<α2y then EL does not exist.

    In case of existence, EL+=(0,0,ˉy0L+,ˉy1L+,ˉaL+,ˉsL+) is given by ˉy0L+=12escyyαy(IζL1+(ζL1I)24esαy(ζL2I)), ˉsL+=αy(1+cyyˉy0L+), ˉaL+=βyˉy0L+eaˉsL+, and ˉy1L+=axAxˉy0L+dy1 whereas EL=(0,0,ˉy0L,ˉy0L,ˉaL,ˉsL) is given by ˉy0L=12escyyαy(IζL1(ζL1I)24esαy(ζL2I)), ˉsL=αy(1+cyyˉy0L), ˉaL=βyˉy0LeaˉsL, and ˉy1L=axAxˉy0Ldy1.

    Proof. Due to symmetry in indices x and y, the proof for the malignant case is equivalent to that for the hematopoietic case except index H has to be substituted by L.

    The result is summarized in Table 2.

    Table 2.  Summarizing necessary and sufficient criteria for admissibility of EL. The first column conditions how αy is related to ζL3, the middle column shows the existence conditions for EL+ and the last column shows the existence conditions for EL± explicitly formulated in terms of I.
    For Only EL+ if Both EL+ and EL if
    α2y<ζL34 I>ζL2 IζL2
    ζL34<α2y<ζL3 I>ζL2 ILIζL2
    ζL3<α2y I>ζL2

     | Show Table
    DownLoad: CSV

    The existence of a co-existing steady state

    EC=(ˉx0C,ˉx1C,ˉy0C,ˉy1C,ˉaC,ˉsC),

    is far more cumbersome to deal with, since a wealth of sub-cases may arrive depending on various inequality-relations between the parameters. To avoid many tedious but straight forward calculations we limit ourself to the non-degenerate cases where ζC1=αycyxαxcxx0 and ζC2=αycyyαxcxy0.

    From Eqs (2.4a and 2.4b), a linear relation between ˉx0 and ˉy0 directly follows,

    ζC1ˉx0C+ζC2ˉy0CζC3=0, (2.28)

    where ζC3=αxαy. Thus, for the non-degenerate cases,

    ˉy0C=ζC1ζC2(ζC3ζC1ˉx0C), (2.29)

    which geometrically corresponds to a straight line through (0,ζC3ζC2) and (ζC3ζC1,0). Hence, two generic cases arrive, for (0,ζC1ζC2) corresponding to positive slope, ζC1ζC2<0 corresponding to negative slope, ζC1ζC2>0. The first case defines a half line in the positive octahedron and in this case ˉx0C(max{0,ζC3ζC1};) and ˉy0C(max{0,ζC3ζC2};). The second case corresponds to either no admissible solution (if and only if ζC3ζC2<0 and ζC3ζC1<0) or a line segment in the positive octahedron which requires that ζC3ζC2>0 and ζC3ζC1>0 and in that case are ˉx0C(0,ζC3ζC1) and ˉy0C(0,ζC3ζC2). From Eq (2.4a) and (2.29),

    ˉsC=m1ˉx0C+m0, (2.30)

    with m0=αx(cxyζC3ζC2+1) and m1=αx(cxxcxyζC1ζC2). Before continuing, it is emphasized that ζ1, ζ2, ζ3, m0, and m1 all are independent of I but may be positive, negative or in case of m0 and m1 zero. From Eq (2.7) it follows that a real and positive ˉs exist for (ˉx0C,ˉy0C)R+×R+,

    ˉsC=I2es(1+1+ζ0I2(βxˉx0C+βyˉy0C)) (2.31)

    where ζ0=4rsesea>0. Similarly, a negative real root exists. Substituting (2.29) into (2.31) give,

    ˉsC=I2es+(I2es)2+ζ0βyζC34e2sζC2+ζ04e2s(βxβyζC3ζC2)ˉx0C. (2.32)

    Combining Eq (2.30) and (2.32) results in,

    n0I+n1ˉx0C=I2+n2+n3ˉx0C, (2.33)

    where n0=2esm0, n1=2esm1, n2=ζ0βyζC3ζC2 and n3=ζ0(βxβyζC1ζC2).

    Note that Eq (2.33) has no real solution if either of f(x)=n0I+n1x and g(x)=I2+n2+n3x are negative. Thus, if both f(x) and g(x) are positive, Eq (2.33) is equivalent to,

    n21x2+(2n1(n0I)n3)x+(n20n22n0I)=0, (2.34)

    which may have up to two real positive solutions. Hence, there can be at most two coexistence steady states. More specifically,

    x0C+=2n1(n0I)n32n21+(2n1(n0I)n32n21)2n20n22n0In1, (2.35)

    is positive if and only if f(x0C+)>0, g(x0C+)>0, and

    n20n22n0In1<0 or 2n1(n0I)n3n21<0. (2.36)

    Similar,

    x0C=2n1(n0I)n32n21(2n1(n0I)n32n21)2n20n22n0In1, (2.37)

    is positive if and only if f(x0C)>0, g(x0C)>0, and

    2n1(n0I)n32n21>n20n22n0In1>0 and 2n1(n0I)n32n21<0. (2.38)

    Note, some possibilities of equality signs in the inequalities are left out for simplification reasons. Equality may occur on a set of measure zero which is unlikely for a noisy biological system and including these possibilities makes the analysis much more messy. For practical purposes one may first calculate the two (possibly complex) roots x of Eq (2.34) and afterwards examine whether these are real and positive, whether f(x)>0 and g(x)>0, and whether the corresponding ˉy0C calculated from Eq (2.29) is positive, thus the remaining component of EC will be positive too and the steady state admissible.

    Continuing analytically is possible but becomes somehow cumbersome and instead we point out that for any choice of parameter values, there can be at most two coexistence steady states, their existence and value depending on the admissibility of x0C+ (Eq (2.35)) and x0C (Eq (2.37)).

    In this section we analytically and numerically examine the stability properties of the various admissible steady states of Eq (2.1) in terms of selected parameters.

    The Jacobian of the trivial steady states E0 is a triangular matrix and four of the six eigenvalues, dx1,dy1,es and Ieaes, are negative but the two, rxes(Iesαx) and ryes(Iesαy), may be positive, negative, zero. Thus, by the Hartman-Grobman Theorem [34]

    Lemma 1. E0 is asymptotically stable if I<esmin{αx,αy}, whereas it is unstable if I>esmin{αx,αy}.

    At EH± the Jacobian for the hematopoietic states can be calculated (see Supplementary) and the resulting sixth order characteristic equation shows that EH± are stable for

    ˉsH<αy(1+cyxˉx0H). (3.1)

    However, this is not the generic case, since αy<αx (and cyxcxx), which contradicts ˉsH=αx(1+cxxˉx0H). Intensive numerical investigations shows that EH± are unstable.

    The stability of EL is similar to that for the hematopoietic steady state except that it is stable if

    ˉsL<αx(1+cxyˉy0L), (3.2)

    which is fulfilled in the generic case, since αx<αy (and cxycyy). This follows from ˉsL=αy(1+cyyˉy0L). The Jacobian may be found in supplementary.

    Lastly, consider the co-existing steady state. The Jacobian at EC may be found in supplementary. However, it is hard to prove any result analytically and we therefore do the stability investigation numerically the in next section.

    In this section, we focus on numerical results. The default values of parameters used in Figure 2 are given in Table 3. The values are the same as given in [18].

    Figure 2.  Admissibility and stability of the steady states depending on the parameters I and C for different values of R. Crossing a solid curve implies a change in which type of stable steady state exists i.e. trivial, malignant, hematopoietic or coexistence. Crossing a dotted curves implies the same steady state is stable in both regions but the number of steady states is changed. The stable steady states are written as subscript of E and unstable steady states are written as superscript of E.
    Table 3.  Default parameter values and their lumped counterpart.
    Parameter Value Unit Parameter Value Unit
    rx 8.7104 day1 ry 1.3103 day1
    ax 1.1105 day1 ay 1.1105 day1
    Ax 4.71013 - Ay 4.71013 -
    dx0 2103 day1 dy0 2103 day1
    dx1 129 day1 dy1 129 day1
    cxx 5.6105 - cyx 5.2105 -
    cxy 5.4105 - cyy 5.0105 -
    es 2 day1 rs 3104 day1
    ea 2109 day1 I 7 day
    αx 2.31 - αy 1.54 -
    βx 5.17108 - βy 5.17108 -
    ζ0 1.21012 - ζC1 4.9105 -
    ζC2 7.4146104 - ζC3 0.7646 -
    n0 9.76 - n1 0.51 -
    n2 0.64 - n3 6.61104 -
    ζH1 8.65 - ζH2 4.62 -
    ζL1 5.18 - ζL2 3.09 -
    ζH3 0.69 - ζL3 0.7646 -

     | Show Table
    DownLoad: CSV

    The model has been investigated for various choices of parameter values. In Figure 2, clusters of five important parameters, C=cxxcyy, R=ζH2ζL2=αxαy and I are considered to investigate the number of steady states and their stability. In the default case R>1 (Figure 2a), a trivial steady state always exists, and for low inflammation, i.e., I<ζL2 it is stable otherwise it is unstable. For I>ζL2, a purely malignant steady state becomes admissible. For values of I where the trivial and the malignant steady states are admissible, the malignant steady state is stable whereas the trivial steady state is unstable. An unstable hematopoietic steady state becomes admissible as I becomes larger than the threshold value ζH2, and increasing I further causes emergence of a stable co-existing steady state while the malignant steady state becomes unstable. Thus, for I>ζH2 and C sufficiently small, four steady states appear namely the trivial, the hematopoietic, the malignant and the co-existing steady states where the co-existing steady state is stable and the rest are unstable. This illustrates that the co-existing steady state depends on I, C and R. Increasing C from a small, initial value makes the co-existing steady state vanish and the malignant steady state becomes stable whereas the trivial and the hematopoietic steady states remain unstable. Secondly, consider the second case where R=1 implying that ζH2=ζL2 (Figure 2b). Increasing I across this value generates an unstable hematopoietic steady state and a malignant steady state simultaneously. For C<1 the malignant steady state is unstable, and a stable coexistence steady state is created as I increase past ζH2. For C>1 no coexistence steady state is created, instead the malignant steady state is stable. Hence, for R=1, decreasing C may change the topology from a stable malignant steady state to a stable coexistence steady state i.e. improving the prognosis from disease escape to disease equilibrium. The stable co-existing steady state bifurcates from the trivial steady state and remains stable until C=1. As C exceeds 1, the co-existing steady state disappears, the malignant steady state becomes stable and the trivial and the hematopoietic steady state become unstable.

    In the remaining panels, R<1, which implies that a stable hematopoietic steady state is created as the first transition to appear when increasing I from low values past the threshold value ζH2. Simultaneously, the trivial steady state becomes unstable. In Figure 2c where R=0.97 the hematopoietic steady state remains stable for low values of C until I passes a threshold value where a stable coexistence steady state is created leaving the hematopoietic steady state unstable.

    For larger values of C there is no coexistence steady state. Instead, as I is increased, a region of bistability appears with a stable hematopoietic steady state and a stable malignant steady state. Increasing I further the hematopoietic steady state becomes unstable. Hence, to reduce disease load, in the case of R<1, and large values of C and I, it may be optimal treatment to reduce the C value prior to reducing the inflammatory level to avoid being stuck in the bassin of attraction of the malignant steady state.

    In Figure 2d where R=0.93, the coexistence steady state no longer appears, the region of bistability has shrunk and a hematopoietic stable steady state is more dominant.

    In Figure 2e and f, R is decreased to 0.77 and 0.5 respectively, and the bistability region is no longer visible. For I>ζH2 a hematopoietic steady state is the only stable steady state. Figure 2 indicates that reducing C and R should be targets of intervention. A reduction of I may improve prognosis as well, for example for parameter values as in 2c.

    A mechanism-based model published in [5] - the Cancitis model - describing the interaction of the hematopoietic cells, malignant cells and inflammation is analysed here. A thorough mathematical investigation of the model is presented in this paper which did not appear previously. We conducted an analytical analysis of the steady states and showed that four kinds of steady states may exist i.e. trivial, hematopoietic, malignant and co-existing steady states. We characterized the stability of each of these steady states and identified transitions conditions in the number of steady states and in their stability. Trivial, hematopoietic, malignant and coexistence steady states all appear for some parameter values. The steady states are highly relevant as all trajectories appear to approach a steady state after some time - see Figure 3. The case of bistability is visualized in the bottom right panel of Figure 3, with the basin of attraction shown in the (x0,y0)-plane using initial condition (x1,y1,a,s)=(4×1011,4×1011,600,2). The initial conditions for x0 and y0 are varied in a range 1105. The malignant steady state has a large bassin of attraction (region (i)), while region (ii) marks the bassin of attraction for the hematopoietic steady state.

    Figure 3.  The first three panels illustrate that when a unique, stable steady state exists, it is globally attracting (based on a numerical argument). U(t) denotes the solution of the six-dimensional model 2.1. In the top left panel, ||U(t)EH||||U(0)EH|| is plotted against time for three different initial conditions. It corresponds to the region where EH is stable with C=0.8, R=0.97 and I=6. ||U(t)EH||||U(0)EH|| tending to zero for large time implies that U(t) is close to EH for large time. The top right panel shows the stability of EL with C=0.2, R=1.5 and I=4. The bottom left panel shows the stability of EC with C=0.1, R=1.5 and I=7. The bottom right panel corresponds to bi-stability of EH and EL with C=1.4, R=0.97. The solution to the 6D model is projected onto the x0 and y0 plane. Region (i) denotes the set of initial conditions with trajectories converging to EL whereas region (ii) denotes that trajectories converge to EH. Black circles show four steady states, E0, EH, EL and EC, where filled circle shows stable steady states and empty circle shows unstable steady states.

    The intuitive interpretation in most bio-medical literature attributes the main cause for cancer development to the frequency of stem cell division. Another main cause is the regulatory feedback that allows stem cells residing in niche to further divide into blood cell required in blood stream. Our investigation is in agreement with this perception and quantifies this intuitive concept. Furthermore, it shows that stem cell population is important to target in treatment to prevent disease progression.

    In [14] and [15] a model without immune interaction is presented. The authors discuss a fraction similar to R and show that it is important for the dynamics of the system. It has been shown [15] that the leukemic cell load can be temporarily reduced if the growth of HSC is larger than that of leukemic cells for cytokine-dependent AML.

    It is generally assumed that cyycxx since malignant cells might be less sensitive to environmental crowding [22] and [24]. The ratio C of inhibition of the hematopoietic relative to malignant cells is one of several important prognostic markers. For large values of I, bi-stable and mono-stable regions depend upon C. It can be observed in Figure 2 that for small values of C, i.e., cyycxx, either the hematopoietic steady state is stable or the co-existing steady state is stable which can be interpreted as a good prognosis. However, large values of C may lead to a worse situation, e.g. in one case, the malignant steady state is stable or there exists bi-stability of the hematopoietic and the malignant steady states (see Figure 2c). In addition to the ratio of inhibitive niche feedback, the ratio R is also important to consider, since it determines how robust the hematopoietic condition may be and how disastrously a potential blood cancer disease will develop. Thus for R>1 we have a more serious situation than for R<1 showing that if this reproduction ratio exceeds the threshold R0=1, it is more disastrous than if it is below R0.

    The JAK2V617F allele burden is expected to increase due to the expansion of malignant cells. The JAK2V617F allele burden is interpreted as the ratio of malignant cells to the total number of mature cells. The model predicted JAK2V617F allele burden is shown in Figure 4 for the region where EC is stable. Perturbation of a parameter may improve or impair prognosis when the coexistence point is the stable attractor. The top panel of Figure 4, shows that decreasing C and R improve prognosis by lowering the allele burden. Contrarily, increasing I, causes a decay in allele burden. This suggests that inflammatory inhibitors could counteract treatments in this case. In other cases, increasing I typically leads to a worse prognosis, considering Figure 2.

    Figure 4.  Allele burden (the ratio of ˉy1c to the total number of ˉx1c and ˉy1c) at the steady state for the region where the co-existing steady state is stable. On the top, left and right panels show that by increasing C and R, allele burden also increases. On the contrary, increasing I reduces the allele burden. In the top panel R=1.5 and I=7, in the top right, C=0.1 and I=7 and in the bottom panel, R=1.5 and C=0.1.

    The model presented here may inform clinical practice to make group specific treatment protocols with particular focus on the inflammatory components which may accelerate or dampen the disease progression. Interventions should address decreasing C and R and potentially I but the latter depends on the remaining parameter values as adverse effects may be observed.

    No conflict of interest.

    Stability analysis of Steady states:

    At EH± the Jacobian of the purely hematopoietic steady state becomes,

    JEH=[a110a1300a16a21a22000000a3300000a43a4400a51a52a53a54a55a560000a65a66] (6.1)

    where

    a11=rx(ˉsH(1+cxxˉx0H)2αx),

    a13=rxcxyˉsHˉx0H(1+cxxˉx0H)2,

    a16=rxˉx0H(1+cxxˉx0H),

    a21=axAx,

    a22=dx1,

    a33=ry(ˉsH1+cyxˉx0Hαy),

    a43=ayAy,

    a44=dy1,

    a51=dx0,

    a52=dx1,

    a53=dy0,

    a54=dy1,

    a55=eaˉsH,

    a56=eaˉaH,

    a65=rs,

    a66=es,

    and rest of the elements of JEH are zero.

    At EL± the Jacobian of the purely malignant steady state.

    JEL=[a1100000a21a220000a310a3300a3600a43a4400a51a52a53a54a55a560000a65a66] (6.2)

    where

    a11=rx(ˉsL1+cxyˉy0Lαx),

    a21=axAx,

    a22=dx1,

    a31=rycyxˉsLˉy0L(1+cyyˉy0L)2,

    a33=ry(ˉsL(1+cyyˉy0L)2αy),

    a36=ryˉy0L1+cyyˉy0L.

    a43=ayAy.

    a44=dy1.

    a51=dx0.

    a52=dx1.

    a53=dy0.

    a54=dy1.

    a55=eaˉsL.

    a56=eaˉaL.

    a65=rs.

    a66=es

    and rest of the elements of array are zero.

    At EC± the Jacobian of the co-existing steady state becomes,

    JEC=[a110a1300a16a21a220000a310a3300a3600a43a4400a51a52a53a54a55a560000a65a66] (6.3)

    where

    a11=rx(ˉsC(1+cxxˉx0C+cxyˉy0C)2αx),

    a13=rxcxyˉsCˉx0C(1+cxxˉx0C+cxyˉy0C)2,

    a16=rxˉx0C(1+cxxˉx0C+cxyˉy0C),

    a21=axAx,

    a22=dx1,

    a31=rycyxˉsCˉy0C(1+cyxˉx0C+cyyˉy0C)2,

    a33=ry(ˉsC(1+cyxˉx0C+cyyˉy0C)2αy),

    a36=ryˉy0C(1+cyxˉx0C+cyyˉy0C),

    a43=ayAy,

    a44=dy1,

    a51=dx0,

    a52=dx1,

    a53=dy0,

    a54=dy1,

    a55=eaˉsC,

    a56=eaˉaC,

    a65=rs,

    a66=es

    and rest of the elements of the JEC are zero.



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