Normal and pathological dynamics of platelets in humans

Abstract

We develop a mathematical model of platelet, megakaryocyte, and thrombopoietin dynamics in humans. We show that there is a single stationary solution that can undergo a Hopf bifurcation, and use this information to investigate both normal and pathological platelet production, specifically cyclic thrombocytopenia. Carefully estimating model parameters from laboratory and clinical data, we then argue that a subset of parameters are involved in the genesis of cyclic thrombocytopenia based on clinical information. We provide model fits to the existing data for both platelet counts and thrombopoietin levels by changing four parameters that have physiological correlates. Our results indicate that the primary change in cyclic thrombocytopenia is an interference with, or destruction of, the thrombopoietin receptor with secondary changes in other processes, including immune-mediated destruction of platelets and megakaryocyte deficiency and failure in platelet production. This study contributes to the understanding of the origin of cyclic thrombocytopenia as well as extending the modeling of thrombopoiesis.

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Acknowledgements

This research was supported by the NSERC (National Sciences and Engineering Research Council) of Canada through Discovery grants to JB, ARH, and MCM, and PGS-D program to MC. SRS thanks McGill University for a Science Undegraduate Research Award. GPL and MCM are especially grateful to Dr. Jayson Potts (UBC) for his initial contact that prompted the initiation of this research. We thank Prof. Jiguo Cao (SFU) for introducing us to LW.

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Correspondence to Gabriel P. Langlois.

Appendices

Appendix 1: Parameter estimation and constraints

This extensive appendix contains the details of the parameter estimation procedure for this model largely based on experimental data. First, in “Homeostasis relationships” section we consider the model at homeostasis. We then use TPO-knockout experimental observations in “TPO knock-out relationships” section to derive further parameter constraints. In “Parameters estimated from experimental data” section we provide estimates for other parameters directly from experimental data. Finally, in “Parameters calculated from experimental data and the model” section we calculate remaining parameters using experimental data and the relationships derived in “Homeostasis relationships” and “TPO knock-out relationships” sections.

Homeostasis relationships

Let \(Q^{*}\) denote the stem cell concentration, \(M_{e}^{*}\) the total megakaryocyte volume, \(P^{*}\) the platelet concentration, \(T^{*}\) the thrombopoietin concentration, \(\eta _{m}^{*}\) and \(\eta _{e}^{*}\), the rate of mitosis and endomitosis, respectively, and \(\tau _{m}\) and \(\tau _{e}\), the average time megakaryoblasts and megakaryocytes spend in the mitotic and endomitotic stages, respectively, at homeostasis. At this steady state, the equations for megakaryocyte production rate (3), platelet production rate (13), megakaryocyte volume (8), platelet balance (9), and thrombopoietin balance (10) become

$$\begin{aligned} m_{m}^{*}(\tau _{m})= & {} \kappa _{P}Q^{*}\mathrm {e}^{\eta _{m}^{*}\tau _{m}}, \end{aligned}$$
(26)
$$\begin{aligned} m_{e}^{*}(\tau _e)= & {} V_m\kappa _PQ^*\mathrm {e}^{\eta ^{*}_{m}\tau _m+\eta ^{*}_{e}\tau _e}, \end{aligned}$$
(27)
$$\begin{aligned} M_{e}^{*}= & {} V_{m}\kappa _{P}Q^{*}\mathrm {e}^{\eta _{m}^{*}\tau _{m}}\left( \frac{\mathrm {e}^{\eta _{e}^{*}\tau _{e}}-1}{\eta _{e}^{*}}\right) , \end{aligned}$$
(28)
$$\begin{aligned} \frac{D_{0}}{\beta _{P}}m_{e}^{*}(\tau _e)= & {} \gamma _{P}P^{*}+\alpha _{P}\frac{(P^{*})^{n_{P}}}{(b_{P})^{n_{P}}+(P^{*})^{n_{P}}}, \end{aligned}$$
(29)
$$\begin{aligned} T_{prod}= & {} \gamma _{T}T^{*}+\alpha _{T}(M_{e}^{*}+k_{S}\beta _{P}P^{*})\frac{(T^{*})^{n_{T}}}{(k_{T})^{n_{T}}+(T^{*})^{n_{T}}}. \end{aligned}$$
(30)

TPO knock-out relationships

The elimination of TPO gene or its receptor in mice reduces megakaryocyte and platelet levels to approximately 10% of normal (Sauvage et al. 1996), a finding also observed in humans (Kaushansky, private communication). Therefore, the model must have a steady state solution at 10% normal platelet and megakaryocyte levels when the thrombopoietin production rate and level are both zero, giving

$$\begin{aligned} \tau _{e}m_{m}^{*}(T=0,\tau _{m})=\frac{1}{10}\tau _{e}m_{m}^{*}(\tau _{m}), \end{aligned}$$

and

$$\begin{aligned} \frac{D_{0}}{\beta _{P}}m_{e}^{*}(T=0,\tau _{e})=\frac{1}{10}\gamma _{P}P^{*}+\alpha _{P}\frac{(P^{*})^{n_{P}}}{(10b_{P})^{n_{P}}+(P^{*})^{n_{P}}}. \end{aligned}$$

Using Eqs. (26) and (27), we rewrite these two relationships to obtain

$$\begin{aligned} \mathrm {e}^{\eta ^{min}_{m}\tau _m}= & {} \frac{1}{10}\mathrm {e}^{\eta ^{*}_{m}\tau _m}, \end{aligned}$$
(31)
$$\begin{aligned} \frac{D_{0}}{\beta _{P}}V_{m}\kappa _{P}Q^{*}\mathrm {e}^{\eta _{m}^{min}\tau _{m} + \eta _{e}^{min}\tau _{e}}= & {} \frac{1}{10}\gamma _{P}P^{*}+\alpha _{P}\frac{\left( P^{*}\right) ^{n_{P}}}{(10b_{P})^{n_{P}}+(P^{*})^{n_{P}}}. \end{aligned}$$
(32)

Parameters estimated from experimental data

Megakaryocyte compartment

Mackey (2001) estimated the homeostatic concentration of HSCs using data from cats and mice, giving an estimate of \(Q^{*}=1.1\times 10^{6}\hbox { cells/kg of body weight}\). We assume that humans have roughly this same number of stem cells per kg of body weight. We estimate the parameter \(\kappa _{P}\), the rate at which stem cells commit to the megakaryocyte lineage, from the model of stem cells dynamics Mackey (2001) and Bernard et al. (2003a) developed, and the assumption that stem cells differentiate at an equal rate into all blood lineages. This gives, to four significant digits, an estimate of \(\kappa _{P}\approx 0.0072419\hbox { cells/kg of body weight per day}\). See Craig et al. (2016) for more details.

Tomer and Harker (1996) measured the diameters of megakaryocytes in the bone marrow of 10 healthy individuals. They found that megakaryocytes of ploidy 2N (those megakaryocytes which have not yet undergone endomitosis) had a mean diameter of \(21 \pm 4\,\upmu \hbox {m}\), while the average megakaryocyte had a mean diameter of \(37 \pm 4\,\upmu \hbox {m}\). We set the average volume of a megakaryocyte of ploidy 2N to be approximately that of a sphere with mean diameter \(21\, \upmu \hbox {m}\), and hence, set \(V_{m}=4\pi (21)^{3}/24\,\hbox {fL}\). The parameter \(\tau _{e}\), the time a megakaryocyte spends in endomitosis, is estimated by various sources to range from 5 to 7 days (Finch et al. 1977; Kuter 2013). We take \(\tau _{e}=5\,\hbox {days}\).

Platelet compartment

Giles (1981) measured the mean platelet count and volume in 1011 healthy human adult blood specimens. He found the mean platelet count to be \(290\times 10^{9}\) platelets/L of blood and the mean platelet volume to be 8.6 fL, so we set \(\beta _{P}\) to be 8.6 fL. Since, on average, one third of the total platelet mass in the body is sequestered by the spleen (Aster 1966), we approximate the mean platelet count in the body is 1.5 times this amount. The Hill coefficient for the platelet-dependent removal of platelets is assumed to take the value \(n_P = 2\). Supposing that a healthy adult has roughly 5 L of blood per 70 kg of body weight gives \(P^{*}=1.5\times (5/70)\times (290\times 10^{9})\approx 3.1071\times 10^{10}\hbox { platelets/kg of body weight}\).

Thrombopoietin compartment

Normal TPO concentrations in humans range from 50 to 150 pg/ml of blood (Kuter 2013), and so we select the middle of the range as the homeostatic concentration, \(T^{*}=100\) pg/ml. Since only platelets in circulation, and not in the spleen, contribute to the binding of TPO, we set \(k_{S}=2/3\), which is the fraction of the platelet mass in circulation (Aster 1966). As there are two thrombopoietin binding sites on a TPO receptor (Hitchcock and Kaushansky 2014), we set \(n_{T}=2\) for the binding coefficient of thrombopoietin.

Parameters calculated from experimental data and the model

Megakaryocyte and platelet compartments

At equilibrium, the total megakaryocyte volume is

$$\begin{aligned} M_{e}^{*}=V_{m}\kappa _{P}Q^{*}\mathrm {e}^{\eta _{m}^{*}\tau _{m}} \left( \frac{\mathrm {e}^{\eta _{e}^{*}\tau _{e}}-1 }{\eta _{e}^{*}}\right) , \end{aligned}$$
(33)

wherein \(\eta _{e}^{*} = \eta _e(T^*)\). Tomer and Harker (1996) found that the average megakaryocyte in humans has a mean diameter of \(37 \pm 4 \,\upmu \hbox {m}\), giving an approximate mean volume of \(4\pi (37)^3/24\,\hbox {fL}\). Assuming the average megakaryocyte volume predicted by our model (total volume of megakaryocytes divided by the number of megakaryocytes) equals this value, we have from (33) that

$$\begin{aligned} \frac{M_{e}^{*}}{\tau _{e}m_{m}^{*}(\tau _{m})}=V_{m}\left( \frac{\mathrm {e}^{\eta _{e}^{*}\tau _{e}}-1}{\eta _{e}^{*}\tau _{e}}\right) \approx \frac{4\pi (37)^{3}}{24}, \end{aligned}$$
(34)

which can be rearranged as

$$\begin{aligned} \frac{\mathrm {e}^{\eta _{e}^{*}\tau _{e}}-1}{\eta _{e}^{*}\tau _{e}}=\left( \frac{37}{21}\right) ^{3}. \end{aligned}$$
(35)

Using the MATLAB (Mathworks 2015) function fsolve, which solves the equation \(F(x) = 0\) for x for some function F, we solved Eq. (35) for \(\tau _{e}\eta _{e}^{*}\), yielding \(\tau _{e}\eta _{e}^{*} \approx 2.788\). Since \(\tau _{e}\) is known, we have \(\eta _{e}^{*} \approx 0.5576\).

Using \(^{111}\) In-Oxine and \(^{111}\) In-tropolone (more reliable markers than the previously used \(^{51}\)Cr label), Tsan (1984) measured the mean platelet survival time \(\tau _P\) to be \(8.4 \pm 0.25\) days. We assume that the platelet production rate replenishes the full platelet population (those circulating and in the spleen) in about \(\tau _{P}\) days. Therefore,

$$\begin{aligned} \frac{1}{\tau _{P}}P^{*}\approx \frac{D_{0}}{\beta _{P}}m_{e}^{*}(t,\tau _{e})=\frac{D_{0}}{\beta _{P}}V_{m}\kappa _{P}Q^{*}\mathrm {e}^{\eta _{m}^{*}\tau _{m}+\eta _{e}^{*}\tau _{e}}, \end{aligned}$$
(36)

and, in particular,

$$\begin{aligned} \frac{1}{\tau _{P}}P^{*}=\gamma _{P}P^{*}+\alpha _{P}\frac{(P^{*})^{n_{P}}}{(b_{P})^{n_{P}}+(P^{*})^{n_{P}}}. \end{aligned}$$
(37)

Solving for \(\alpha _{P}\), we have

$$\begin{aligned} \alpha _{P}=P^{*}\left( \frac{1}{\tau _{P}}-\gamma _{P}\right) \left( 1+\left( \frac{b_{P}}{P^{*}}\right) ^{n_{P}}\right) . \end{aligned}$$
(38)

All parameters in (38) except for \(b_P\) and \(\gamma _P\) are known. The rates of removal of the platelets from the blood should be positive, implying that \(\gamma _P\) and \(\alpha _P > 0\). The latter requires that

$$\begin{aligned} 1/\tau _{P}-\gamma _{P}>0. \end{aligned}$$
(39)

One megakaryocyte sheds 1000–3000 platelets (Harker and Finch 1969). Assuming, on average, that one megakaryocyte sheds 2000 platelets, then the rate of production of megakaryocytes times the number of platelets shed per megakaryocyte equals roughly the rate of production of platelets:

$$\begin{aligned} m_{m}^{*}(\tau _{m})\times \text{2000 }=\kappa _{P}Q^{*}\mathrm {e}^{\eta _{m}^{*}\tau _{m}}\times 2000=\frac{1}{\tau _{P}}P^{*}, \end{aligned}$$
(40)

giving a rate of production of megakaryocytes of \(1.85\times 10^{6}\,\hbox {megakaryocytes/kg/day}\), which is close to the value of \(2 \times 10^{6}\,\hbox {megakaryocytes/kg/day}\) estimated to be the normal production rate of megakaryocytes (Finch et al. 1977).

The parameter \(D_0\), the fraction of megakaryocytes shedding platelets, can be solved for in (36) by equating (40) with (36) and using (27). This gives

$$\begin{aligned} D_{0}=2000\frac{\beta _{P}}{V_{m}}\mathrm {e}^{-\eta _{e}^{*}\tau _{e}}\approx 0.21829. \end{aligned}$$
(41)

Rearranging (40) and solving for \(\eta _{m}^{*}\tau _{m}\), we get

$$\begin{aligned} \eta _{m}^{*}\tau _{m}=\ln \left( \frac{1}{2000}\frac{1}{\kappa _{P}Q^{*}}\frac{1}{\tau _{P}}P^{*}\right) \approx 5.4394. \end{aligned}$$

Substituting in Eq. (31) and solving for \(\eta ^{min}_{*}\), we have

$$\begin{aligned} \eta _{m}^{min}=\frac{1}{\tau _{m}}\ln \left( \frac{1}{20{,}000}\frac{1}{\kappa _{P}Q^{*}}\frac{1}{\tau _{P}}P^{*}\right) . \end{aligned}$$
(42)

Using Eq. (1) at homeostasis to solve for \(\eta ^{max}_m\) in Eq. (31) gives

$$\begin{aligned} \eta _{m}^{max}=\eta _{m}^{min}+\frac{\ln (10)}{\tau _{m}}\left( 1+\frac{b_{m}}{T^*}\right) . \end{aligned}$$
(43)

We can now use the steady state equation for the platelet numbers in absence of thrombopoietin, Eq. (32), in combination with the expressions (40) and (41) to get

$$\begin{aligned} \frac{1}{10\tau _{P}}P^{*}\mathrm {e}^{(\eta _{e}^{min}-\eta _{e}^{*})\tau _{e}}=\frac{1}{10}\gamma _{P}P^{*}+\alpha _{P}\frac{(P^{*})^{n_{P}}}{(10b_{P})^{n_{P}}+(P^{*})^{n_{P}}}. \end{aligned}$$

Solving for \(\eta _{e}^{min}-\eta _{e}^{*}\), we get

$$\begin{aligned} \eta _{e}^{min}=\eta _{e}^{*}+\frac{1}{\tau _{e}}\ln \left[ \tau _{P}\gamma _{P}+10\tau _{P}\alpha _{P}\frac{(P^{*})^{n_{P}-1}}{(10b_{P})^{n_{P}}+(P^{*})^{n_{P}}}\right] . \end{aligned}$$
(44)

We can solve for \(\eta ^{max}_{e}\) via Eq. (5) at steady state:

$$\begin{aligned} \eta _{e}^{max}=\eta _{e}^{min}+(\eta _{e}^{*}-\eta _{e}^{min})\left( 1+\frac{b_{e}}{T^{*}}\right) . \end{aligned}$$
(45)

Thrombopoietin compartment

Using Eq. (30), the homeostasis relationship for the thrombopoietin concentration, we can solve for \(\alpha _T\), yielding

$$\begin{aligned} \alpha _{T}=\frac{T_{prod}-\gamma _{T}T^{*}}{M_{e}^{*}+\beta _{P}P^{*}}\left( 1+\left( \frac{k_{T}}{T^{*}}\right) ^{n_{T}}\right) . \end{aligned}$$
(46)

Parameters fit from experimental data

From the above calculations, it remains to estimate eight more parameters: \(\tau _m\), \(b_m\), \(b_{e}\), \(\gamma _P\), \(b_P\), \(T_{prod}\), \(\gamma _T\), and \(k_T\). The first five parameters pertain to megakaryocyte and platelet dynamics, while the last three pertain to thrombopoietin dynamics.

We digitized data from Wang et al. (2010) for the circulating platelet and TPO levels in healthy patients following a \(1\,\upmu \text{ g/kg } \text{ of } \text{ bodyweight }\) intravenous infusion of Romiplostim, a TPO mimetic with similar physiological activity to TPO. We then fitted the parameters by simulating the response of our model (11)–(14) to an infusion of \(1\, \upmu \hbox {g/kg}\) of TPO and minimizing the squared error between data and simulation.

Specifically, the platelet and TPO data points (and error bars, when available) were interpolated and evaluated at 1000 points in each time interval (ranging from 0 to 42 days for the platelet data and 0 to 1 day for the TPO data), yielding the vectors of points \({\mathbf {P}}_{data}\) and \({\mathbf {T}}_{data}\). To simulate the response of our model to a \(1\,\upmu \text{ g/kg }\) intravenous infusion of thrombopoietin, we ran the numerical algorithm described in “Appendix 7” with initial history functions \((P_h, T_h) = (P^*, T^*)\) and initial conditions \(P_0 = P^*\) and \(T_0 = T_{data}(t=0)\). We then evaluated the solution of our model at the interpolated points \({\mathbf {P}}_{data}\) and \({\mathbf {T}}_{data}\) to obtain the model points \({\mathbf {P}}_{model}\) and \({\mathbf {T}}_{model}\), respectively.

For parameter estimation, we minimized the fitting error

$$\begin{aligned} Err = \frac{\Vert \frac{2}{3}{\mathbf {P}}_{model}(t) - \mathbf P_{data}\Vert _{2}}{\max ({\mathbf {P}}_{data})} + \frac{\Vert \mathbf T_{model}-{\mathbf {T}}_{data}\Vert _{2}}{\max ({\mathbf {T}}_{data})}, \end{aligned}$$
(47)

by using the fmincon function in MATLAB (Mathworks 2015) to find the set of parameters that minimizes (47). (The factor of 2 / 3 accounts for the fraction of platelets that circulate in blood in our model.) The fit obtained from this procedure is shown in Fig. 9.

Fig. 9
figure9

Fit of our model to the platelet and thrombopoietin data from Wang et al. (2004) following a \(1\, \upmu \hbox {g/kg}\) intravenous infusion of TPO

Appendix 2: Proof of existence and uniqueness of a positive steady state solution

Here, we prove that the model has a unique, positive steady state solution by showing that Eqs. (9) and (10) have a unique positive fixed point.

As discussed in “Appendix 1”, Eqs. (9) and (10) at steady state (homeostasis) are given by

$$\begin{aligned} \frac{D_{0}}{\beta _{P}}m_{e}^{*}(T^*,\tau _e)=\gamma _{P}P^{*}+\alpha _{P}\frac{(P^{*})^{n_{P}}}{(b_{P})^{n_{P}}+(P^{*})^{n_{P}}}, \end{aligned}$$
(48)

and

$$\begin{aligned} T_{prod}=\gamma _{T}T^{*}+\alpha _{T}(M_{e}^{*}(T^*)+k_{S}\beta _{P}P^{*})\frac{(T^{*})^{n_{T}}}{(k_{T})^{n_{T}}+(T^{*})^{n_{T}}}, \end{aligned}$$
(49)

where

$$\begin{aligned} m_{e}^{*}(T^*,\tau _e)= & {} V_m\kappa _PQ^*\mathrm {e}^{\eta ^{*}_{m}(T^*)\tau _m+\eta ^{*}_{e}(T^*)\tau _e},\\ M_{e}^{*}(T^*)= & {} V_{m}\kappa _{P}Q^{*}\mathrm {e}^{\tau _{m}\eta _{m}^{*}(T^*)}\left( \frac{\mathrm {e}^{\eta _{e}^{*}(T^*)\tau _{e}}-1}{\eta _{e}^{*}(T^*)}\right) ,\\ \eta _{m}(T(t))= & {} \eta _{m}^{min}+\left( \eta _{m}^{max}-\eta _{m}^{min}\right) \frac{T}{b_{m}+T}, \end{aligned}$$

and

$$\begin{aligned} \eta _{e}(T(t))=\eta _{e}^{min}+\left( \eta _{e}^{max}-\eta _{e}^{min}\right) \frac{T}{b_{e}+T}. \end{aligned}$$

Unlike most population and blood cell regulation models, notice that \((P^*= 0, T^*=0)\) is not an equilibrium of Eqs. (48) and (49). This is because of the nonzero constant production rate of thrombopoietin reflected in the first term of (49), and is analogous to infectious disease models for which there is a constant influx of susceptibles (see, for example, Sect. 2.1.2 of Keeling and Rohani 2008).

Although finding \((P^*,T^*)\) involves solving two highly nonlinear equations in two unknowns, we can use Eq. (49) to solve for \(P^*\) explicitly in terms of \(T^*\), yielding

$$\begin{aligned} \begin{aligned} P^*&= \frac{1}{k_S\beta _P }\left[ \frac{T_{prod} - \gamma _T T^*}{\alpha _T}\left( 1+\left( \frac{k_T}{T^*}\right) ^{n_T}\right) \right. \\&\left. \quad - V_{m}\kappa _{P}Q^{*}\mathrm {e}^{\eta _{m}^{*}(T^*)\tau _{m}}\left( \frac{\mathrm {e}^{\eta _{e}^{*}(T^*)\tau _{e}}-1}{\eta _{e}^{*}(T^*)}\right) \right] . \end{aligned} \end{aligned}$$
(50)

Denote the right-hand-side of Eq. (50) by \(\mathscr {L}(T^*)\). Writing

$$\begin{aligned} \mathscr {L}_1(T^*) = \frac{T_{prod} - \gamma _T T^*}{\alpha _T}\left( 1+\left( \frac{k_T}{T^*}\right) ^{n_T}\right) \end{aligned}$$

and

$$\begin{aligned} \mathscr {L}_2(T^*)=V_{m}\kappa _{P}Q^{*}\mathrm {e}^{\eta _{m}^{*}(T^*)\tau _{m}}\left( \frac{\mathrm {e}^{\eta _{e}^{*}(T^*)\tau _{e}}-1}{\eta _{e}^{*}(T^*)}\right) , \end{aligned}$$

we can rewrite Eq. (50) as

$$\begin{aligned} \displaystyle P^* = \mathscr {L}(T^*)=\frac{1}{k_S\beta _P }\left[ \mathscr {L}_1(T^*) - \mathscr {L}_2(T^*)\right] . \end{aligned}$$

We now show that \(\mathscr {L}\) is a monotone decreasing function of \(T^*\), and thus defines an injective function of \(P^*\). First, notice that \(\mathscr {L}_1\) is a monotone decreasing function of \(T^*\), approaching \(+\infty \) as \(T^* \rightarrow 0\), becoming negative-valued when \(T^*~>~ T_{prod}/\gamma _T\), and approaching \(-\infty \) as \(T^* \rightarrow \infty \). Second, as both \(\eta _m(T^*)\) and \(\eta _e(T^*)\) are monotone increasing in \(T^*\) (taking values in the intervals \((\eta _m^{min},\eta _m^{max})\) and \((\eta _e^{min},\eta _e^{max})\), respectively), the terms

$$\begin{aligned} V_m\kappa _PQ^*\mathrm {e}^{\eta ^{*}_{m}(T^*)\tau _m}\quad \text{ and } \quad \frac{\mathrm {e}^{\eta ^{*}_{e}(T^*)\tau _e}-1}{\eta ^{*}_{e}(T^*)} \end{aligned}$$

in \(\mathscr {L}_2(T^*)\) are both monotone increasing whenever \(\tau _e\eta ^{*}_{e}(T^*) ~>~ 0\), which holds by definition in our model. As \(\mathscr {L}_2\) is the product of two monotone increasing functions, it is also a monotone increasing function. Taken together, these results show that \(\mathscr {L}\) defines an injective function of \(P^*\).

Now consider Eq. (48) and denote its left-hand-side and right-hand-side by \(g(T^*)\) and \(h(P^*)\), respectively. The function h is clearly monotone increasing in \(P^*\), starting from 0 and approaching \(+\infty \) as \(P^* \rightarrow \infty \). Using the previous argument on \(\eta ^{*}_m(T^*)\) and \(\eta ^{*}_e(T^*)\), we have that g is a monotone increasing function of \(T^*\) and uniformly bounded above and below by

$$\begin{aligned} \frac{D_0V_m\kappa _PQ^*}{\beta _P} \mathrm {e}^{\eta _m^{min}\tau _m + \eta _e^{min}\tau _e} \le g(T^*) \le \frac{D_0V_m\kappa _PQ^*}{\beta _P} \mathrm {e}^{\eta _m^{max}\tau _m + \eta _e^{max}\tau _e}. \end{aligned}$$

Finding a positive stationary solution in \((P^*, T^*)\) therefore amounts to finding a value of \(T^*\) satisfying

$$\begin{aligned} h(\mathscr {L}(T^*))=g(T^*) \end{aligned}$$
(51)

and then defining the corresponding value of \(P^*\) being given by Eq. (50). As \(h(\mathscr {L}(T^*))\) is monotone decreasing from \(+\infty \) to negative values and \(g(T^*)\) is monotone increasing between two positive values, there is a unique positive solution to Eq. (51). That is, Eqs. (17) and (18) have a unique positive steady state solution \((P^*,T^*)\).

Appendix 3: Linearization of the thrombopoiesis equations and bifurcation analysis

We take the model in Sect. 4, which has a differential equation for the platelets, (17), and one for the thrombopoietin, (18), depending only on P and T. From “Appendix 2”, the model has a unique positive equilibrium, \((P^*, T^*)\). Linearizing about the equilibrium, we let \(x(t) = P(t)-P^*\) and \(y(t) = T(t)-T^*\). We use Taylor expansions for both the exponential function and \(\eta _m(T(s))\) to obtain the following linear approximation:

$$\begin{aligned} \begin{aligned} \exp \left[ \int _{t - \tau _e - \tau _m}^{t - \tau _e} \eta _m(T(s)) \, \mathrm {d}s\right]&\approx \exp \left[ \int _{t - \tau _e - \tau _m}^{t - \tau _e} (\eta _m(T^*) + \partial _{T}\eta _m(T^*)y(s)) \, \mathrm {d}s\right] \\&= \mathrm {e}^{\eta _m(T^*)\tau _m}\exp \left[ \partial _{T}\eta _m(T^*)\int _{t - \tau _e - \tau _m}^{t - \tau _e} y(s) \, \mathrm {d}s\right] \\&\approx \mathrm {e}^{\eta _m(T^*)\tau _m}\left( 1 + \partial _{T}\eta _m(T^*)\int _{t - \tau _e - \tau _m}^{t - \tau _e} y(s) \, \mathrm {d}s\right) . \end{aligned} \end{aligned}$$

Similarly,

$$\begin{aligned} \exp \left[ \int _{t - \tau _e}^{t} \eta _e(T(s)) \, \mathrm {d}s\right]\approx & {} \mathrm {e}^{\eta _e(T^*)\tau _e}\left( 1 + \partial _{T}\eta _e(T^*)\int _{t - \tau _e}^{t} y(s) \, \mathrm {d}s\right) ,\\ \exp \left[ \int _{t - a - \tau _m}^{t - a} \eta _m(T(s)) \, \mathrm {d}s\right]\approx & {} \mathrm {e}^{\eta _m(T^*)\tau _m}\left( 1 + \partial _{T}\eta _m(T^*)\int _{t - a - \tau _m}^{t - a} y(s) \, \mathrm {d}s\right) , \end{aligned}$$

and

$$\begin{aligned} \exp \left[ \int _{t - a}^{t} \eta _e(T(s)) \, \mathrm {d}s\right] \approx \mathrm {e}^{\eta _e(T^*)a}\left( 1 + \partial _{T}\eta _e(T^*)\int _{t - a}^{t} y(s) \, \mathrm {d}s\right) . \end{aligned}$$

Linearizing the integral product in Eq. (18), we obtain the following approximation:

$$\begin{aligned}&\int _0^{\tau _e}\exp \left[ \int _{t - a - \tau _m}^{t - a} \eta _m(T(s)) \, \mathrm {d}s\right] \exp \left[ \int _{t - a}^{t} \eta _e(T(s)) \, \mathrm {d}s\right] \, \mathrm {d}a \\&\quad \approx \int _0^{\tau _e} \mathrm {e}^{\eta _m(T^*)\tau _m + \eta _e(T^*)a} \left( 1 + \partial _{T}\eta _m(T^*)\int _{t - a - \tau _m}^{t - a} y(s) \, \mathrm {d}s\right) \left( 1 + \partial _{T}\eta _e(T^*)\int _{t - a}^{t} y(s) \, \mathrm {d}s\right) \, \mathrm {d}a \\&\quad \approx \mathrm {e}^{\eta _m(T^*)\tau _m}\left[ \frac{\mathrm {e}^{\eta _e(T^*)\tau _e} - 1}{\eta _e(T^*)} + \partial _{T}\eta _m(T^*)\int _0^{\tau _e}\mathrm {e}^{\eta _e(T^*)a}\left( \int _{t - a - \tau _m}^{t - a} y(s) \, \mathrm {d}s\right) \, \mathrm {d}a \right. \\&\quad \quad +\, \partial _{T}\eta _e(T^*)\int _0^{\tau _e} \mathrm {e}^{\eta _e(T^*)a}\left( \int _{t - a}^{t} y(s) \, \mathrm {d}s\right) \, \mathrm {d}a\Biggr ]. \end{aligned}$$

These results can be used to find the linearization of the platelet and thrombopoietin equations. The platelet equation with only the constant and linear terms (higher order terms dropped) is given by

$$\begin{aligned} \begin{aligned} \frac{\mathrm {d}x}{\mathrm {d}t}&= A_2\left[ 1 + \partial _{T}\eta _m(T^*)\int _{t - \tau _e - \tau _m}^{t - \tau _e} y(s) \, \mathrm {d}s + \partial _{T}\eta _e(T^*)\int _{t - \tau _e}^{t} y(s) \, \mathrm {d}s\right] \\&\quad -\, \gamma _P(x + P^*) - \bigl (F(P^*) + \partial _{P}F(P^*)x\bigr ), \end{aligned} \end{aligned}$$
(52)

where

$$\begin{aligned} A_2 = \frac{D_0V_m\kappa _PQ^*}{\beta _P}\mathrm {e}^{\eta _m(T^*)\tau _m+\eta _e(T^*)\tau _e}. \end{aligned}$$

The thrombopoietin equation can be written as

$$\begin{aligned} \frac{\mathrm {d}y}{\mathrm {d}t}= & {} T_{prod} - \gamma _T(y + T^*) - \alpha _T\left( A_1\left[ E_1 + \partial _{T}\eta _m(T^*)\int _0^{\tau _e} \mathrm {e}^{\eta _e(T^*)a}\left( \int _{t - a - \tau _m}^{t - a} y(s) \, \mathrm {d}s\right) \, \mathrm {d}a \right. \right. \\&+\, \partial _{T}\eta _e(T^*)\int _0^{\tau _e} \mathrm {e}^{\eta _e(T^*)a}\left( \int _{t - a}^{t} y(s) \, \mathrm {d}s\right) \, \mathrm {d}a\biggr ] + k_S\beta _P(x + P^*)\Biggr )(G(T^*) + \partial _{T}G(T^*)y), \end{aligned}$$

where

$$\begin{aligned} A_1 = V_m\kappa _PQ^*\mathrm {e}^{\eta _m(T^*)\tau _m} \quad \mathrm{and} \quad E_1 = \frac{\mathrm {e}^{\eta _e(T^*)\tau _e} - 1}{\eta _e(T^*)}. \end{aligned}$$

The thrombopoietin equation with only the constant and linear terms (higher order terms dropped) is given by:

$$\begin{aligned} \begin{aligned} \frac{\mathrm {d}y}{\mathrm {d}t}&= T_{prod} - \gamma _T(y + T^*) - \alpha _T\Biggl [(A_1E_1 + k_S\beta _PP^*)G(T^*) + (A_1E_1 + k_S\beta _PP^*)\partial _{T}G(T^*)y \\&\quad + k_S\beta _PG(T^*)x + A_1G(T^*)\Biggl (\partial _{T}\eta _m(T^*)\int _0^{\tau _e} \mathrm {e}^{\eta _e(T^*)a}\left( \int _{t - a - \tau _m}^{t - a} y(s) \, \mathrm {d}s\right) \, \mathrm {d}a \\&\quad + \partial _{T}\eta _e(T^*)\int _0^{\tau _e} \mathrm {e}^{\eta _e(T^*)a}\left( \int _{t - a}^{t} y(s) \, \mathrm {d}s\right) \, \mathrm {d}a\Biggr )\Biggr ], \end{aligned} \end{aligned}$$
(53)

By the definition of an equilibrium, the constant terms in Eqs. (52) and (53) sum to zero, yielding the linear equations for platelets, (19), and thrombopoietin, (21), given in Sect. 4.1.

Details for the characteristic equation

We examine the integral terms in Eqs. (19) and (21), using the exponential form for \(y(t) = \mathrm {e}^{\lambda t}\). There are four integrals, which we evaluate below:

$$\begin{aligned}&\int _{t - \tau _e - \tau _m}^{t - \tau _e} \mathrm {e}^{\lambda s} \, \mathrm {d}s = \frac{\mathrm {e}^{\lambda (t-\tau _e)}}{\lambda }\left( 1 - \mathrm {e}^{-\lambda \tau _m}\right) ,\\&\int _{t - \tau _e}^{t} \mathrm {e}^{\lambda s} \, \mathrm {d}s = \frac{\mathrm {e}^{\lambda t}}{\lambda }\left( 1 - \mathrm {e}^{-\lambda \tau _e}\right) ,\\&\int _0^{\tau _e} \mathrm {e}^{\eta _e(T^*)a}\left( \int _{t - a - \tau _m}^{t - a} \mathrm {e}^{\lambda s} \, \mathrm {d}s\right) \, \mathrm {d}a = \frac{\mathrm {e}^{\lambda t}\left( 1 - \mathrm {e}^{-\lambda \tau _m}\right) }{\lambda (\lambda - \eta _e(T^*))} \left( 1 - \mathrm {e}^{\eta _e(T^*)\tau _e-\lambda \tau _e}\right) , \end{aligned}$$

and

$$\begin{aligned} \int _0^{\tau _e} \mathrm {e}^{\eta _e(T^*)a}\left( \int _{t - a}^{t}\mathrm {e}^{\lambda s} \, \mathrm {d}s\right) \, \mathrm {d}a = \frac{\mathrm {e}^{\lambda t}}{\lambda }\left( \frac{\mathrm {e}^{\eta _e(T^*)\tau _e}-1}{\eta _e(T^*)} + \frac{\mathrm {e}^{-(\lambda -\eta _e(T^*))\tau _e}-1}{\lambda - \eta _e(T^*)}\right) . \end{aligned}$$

These expressions are used in the terms \(L_2(\lambda )\) and \(L_4(\lambda )\) in the characteristic equation for the linear functional Eq. (23).

If Eq. (24) is multiplied by \(\lambda (\lambda - \eta _e)\), then the terms in the denominator can be eliminated (at the expense of introducing the roots \(\lambda = 0\) and \(\eta _e\)). The first polynomial piece becomes

$$\begin{aligned} (\lambda + L_1)(\lambda + C_1)\lambda (\lambda - \eta _e)= & {} \lambda ^4 + (L_1 + C_1 - \eta _e)\lambda ^3\\&+ \left( L_1C_1 - (L_1+C_1)\eta _e\right) \lambda ^2 - L_1C_1\eta _e\lambda . \end{aligned}$$

Next we consider the portion \(L_4(\lambda ) - C_1\) in (24)

$$\begin{aligned} (L_4(\lambda ) - C_1)\lambda (\lambda - \eta _e)= & {} C_2\partial _{T}\partial _{T}\eta _m(T^*)\left( 1 - \mathrm {e}^{-\lambda \tau _m}\right) \left( 1 - \mathrm {e}^{-(\lambda -\eta _e) \tau _e}\right) \\&+\, C_2\partial _{T}\eta _e(T^*)\left( \frac{\mathrm {e}^{\eta _e \tau _e}-1}{\eta _e}(\lambda -\eta _e) + (\mathrm {e}^{-(\lambda -\eta _e)\tau _e}-1)\right) \\= & {} C_2\partial _{T}\eta _m(T^*)\left( 1 - \mathrm {e}^{-\lambda \tau _m} - \mathrm {e}^{\eta _e \tau _e}\mathrm {e}^{-\lambda \tau _e} + \mathrm {e}^{\eta _e \tau _e}\mathrm {e}^{-\lambda (\tau _e+\tau _m)}\right) \\&+\, C_2\partial _{T}\eta _e(T^*)\left( \frac{\mathrm {e}^{\eta _e \tau _e}-1}{\eta _e}\right) (\lambda -\eta _e) \\&+\, C_2\partial _{T}\eta _e(T^*)(\mathrm {e}^{\eta _e \tau _e}\mathrm {e}^{-\lambda \tau _e} - 1). \end{aligned}$$

We take the previous expression and multiply by \(\lambda + L_1\) and define

$$\begin{aligned} D_1= & {} \frac{C_2\partial _{T}\eta _e(T^*)\left( \mathrm {e}^{\eta _e \tau _e} - 1\right) }{\eta _e}, \quad D_2 = C_2\mathrm {e}^{\eta _e \tau _e}(\partial _{T}\eta _e(T^*)-\partial _{T}\eta _m(T^*)), \quad \mathrm{and} \\ D_3= & {} C_2\mathrm {e}^{\eta _e \tau _e}\partial _{T}\eta _m(T^*). \end{aligned}$$

The results are

$$\begin{aligned} \begin{aligned} (\lambda + L_1)(L_4(\lambda ) - C_1)\lambda (\lambda - \eta _e)&= (\lambda + L_1) \left( C_2\partial _{T}\eta _m(T^*)\right. \\&\quad \left. + D_1(\lambda - \eta _e) - C_2\partial _{T}\eta _e(T^*)\right) \\&\quad - (\lambda + L_1)C_2\partial _{T}\eta _m(T^*)\mathrm {e}^{-\lambda \tau _m} \\&\quad + (\lambda + L_1)D_2\mathrm {e}^{-\lambda \tau _e} + (\lambda + L_1)D_3\mathrm {e}^{-\lambda (\tau _e + \tau _m)} \end{aligned} \end{aligned}$$

and

$$\begin{aligned} L_2(\lambda )L_3\lambda (\lambda -\eta _e)= & {} L_3(\lambda -\eta _e)A_2 \left( \partial _{T}\eta _m(T^*)\mathrm {e}^{-\lambda \tau _e}\right. \\&\left. - \partial _{T}\eta _m(T^*)\mathrm {e}^{-\lambda (\tau _e + \tau _m)} + \partial _{T}\eta _e(T^*) - \partial _{T}\eta _e(T^*)\mathrm {e}^{-\lambda \tau _e}\right) . \end{aligned}$$

Multiplying by \(\lambda (\lambda -\eta _e)\) produces a quartic exponential polynomial in the eigenvalues, which can be analyzed using techniques we have developed earlier (Mahaffy 1982). The characteristic equation can be written as

$$\begin{aligned}&\lambda ^4 + K_3\lambda ^3 + K_2\lambda ^2 + K_1\lambda + K_0 + (\alpha _1\lambda + \alpha _0)\mathrm {e}^{-\lambda \tau _m}\\&\quad + (\beta _1\lambda + \beta _0)\mathrm {e}^{-\lambda \tau _e} + (\gamma _1\lambda + \gamma _0)\mathrm {e}^{-\lambda (\tau _e + \tau _m)} = 0. \end{aligned}$$

We examine the terms above and obtain the following coefficients:

$$\begin{aligned} \begin{aligned} K_0&= L_1C_2(\partial _{T}\eta _m(T^*) - \partial _{T}\eta _e(T^*)) - D_1L_1\eta _e + L_3A_2\partial _{T}\eta _e(T^*)\eta _e, \\ K_1&= -L_1C_1\eta _e + C_2(\partial _{T}\eta _m(T^*) + \partial _{T}\eta _e(T^*) + D_1(L_1 - \eta _e) - L_3A_2\partial _{T}\eta _e(T^*), \\ K_2&= L_1C_1 - (L_1+C_1)\eta _e + D_1, \\ K_3&= L_1 + C_1 - \eta _e, \\ \alpha _0&= -C_2\partial _{T}\eta _m(T^*)L_1, \\ \alpha _1&= -C_2\partial _{T}\eta _m(T^*), \\ \beta _0&= D_2L_1 - L_3A_2\eta _e(\partial _{T}\eta _e(T^*)-\partial _{T}\eta _m(T^*)), \\ \beta _1&= D_2 + L_3A_2(\partial _{T}\eta _e(T^*)-\partial _{T}\eta _m(T^*)), \\ \gamma _0&= D_3L_1 + L_3A_2\eta _e\partial _{T}\eta _m(T^*) \\ \gamma _1&= D_3 - L_3A_2\partial _{T}\eta _m(T^*). \end{aligned} \end{aligned}$$

Appendix 4: Parameter sensitivity of the model for healthy subjects

The characteristic equation (24) from the linear analysis is used to study the sensitivity of each parameter near its normal value. We developed MATLAB programs from the linear analysis to compute the leading eigenvalues of the model. With the normal parameters of Table 1 we obtained the equilibrium \((P^*, T^*) = (31.071, 100)\), which has the leading pair of eigenvalues \(\lambda _1 = -0.058953 \pm 0.053015i\). It follows that the model with these eigenvalues is asymptotically stable. It should be noted that the frequency is equivalent to a period of 118.5 days, which is far from the observed oscillation periods in either the healthy subjects or patients with cyclic thrombocytopenia.

To analyze the sensitivity of the model to the various parameters, we used our MATLAB code to find the new equilibrium and leading eigenvalues as we varied each parameter by ±10%. Table 6 gives a complete listing of how the equilibrium changes and leading eigenvalues shifts with all of the individual parameter changes. We are most interested in stability changes, so the lowest ratio of the real part of the leading eigenvalues gives the largest change in the direction of a Hopf bifurcation.

Table 6 Parameter sensitivity to changes in the normal parameters of the leading pair of eigenvalues \(\lambda _1\)

For the sensitivity analysis, if we only consider the movement of the leading pair of eigenvalues toward the imaginary axis, then the smallest values in the 7th column (ratio Re) give the greatest shift toward instability. The parameter changes that destabilize the model most are (in descending order) increasing \(b_P\), decreasing \(\alpha _P\), decreasing \(k_T\), increasing \(\beta _P\), increasing \(k_S\), increasing \(b_m\), and decreasing \(T_{prod}\).

As an experiment to extend this analysis, we chose to increase or decrease all seven of these parameters by 20% to see what happened to the equilibrium and the leading eigenvalues. The result of all seven changes resulted in the equilibrium \((P^*, T^*) = (20.772, 85.117)\) and leading eigenvalues \(\lambda _1 = -0.02555 \pm 0.06563i\). We note that this more than halves the distance of the real part toward the imaginary axis, and also the frequency shifts the period to 95.74 days.

The analysis shows that near normal, there are several parameters to which the model is very insensitive. Surprisingly, this includes all of the delay parameters, \(\tau _e\) and \(\tau _m\). It is also quite insensitive to changes in \(\gamma _P\), \(\gamma _T\), and \(b_e\). However, our numerical study shows that \(\tau _e\) and \(\tau _m\) can have effects on the imaginary part.

This analysis shows that the leading eigenvalues have the wrong frequency for the observed oscillations in cyclic thrombocytopenia patients. This suggests the need to examine the next, second leading pair of eigenvalues for this model. Its frequency is closer to the range of interest and provides a starting point for a Hopf bifurcation study of our cyclic thrombocytopenia patients. Again, with the normal parameters the second eigenvalues are \(\lambda _2 = -0.11375 \pm 0.35888i\), which gives a quasiperiod near 17.5 days. Table 7 shows the effects on this pair of eigenvalues as the parameters are changed by ±10%.

Table 7 Parameter sensitivity to changes in the normal parameters of the second leading pair of eigenvalues \(\lambda _2\)

For the sensitivity analysis, we examine the movement of the second leading pair of eigenvalues, \(\lambda _2\), toward the imaginary axis. Again, the smallest values in the 7th column (ratio Re) give the greatest shift toward instability. The most destabilizing changes for this pair of eigenvalues occur by (in descending order) increasing \(\tau _e\), decreasing \(b_m\), increasing \(k_S\), increasing \(\beta _P\), increasing \(\tau _m\), decreasing \(b_P\), and decreasing \(\gamma _P\). We note that the delays \(\tau _e\) and \(\tau _m\) affect the movement of the real part of these eigenvalues.

To extend this analysis, we chose to increase or decrease all seven of these parameters by 20% to see what happened to the equilibrium and eigenvalues. The result of all seven changes resulted in the equilibrium \((P^*, T^*) = (26.505, 87.236)\) and eigenvalues \(\lambda _2 = -0.09281 \pm 0.3091i\). Note that the distance of the real part toward the imaginary axis is slightly more than for the leading pair with its most significant parameters, and also the frequency shifts the period to 20.33 days.

Appendix 5: Hopf bifurcation for CT patients

This appendix continues the studies of the CT patients from Sect. 5.4. The four parameters are varied linearly between the normal state and the best fitting parameters for several CT patients. Again, our numerical methods tracked the changes in the equilibria and the pairs of eigenvalues, which result in Hopf bifurcations leading to the cyclic behavior observed in the CT patients.

Table 2 in Sect. 5.2 shows the best parameter fit to \(\tau _{e}\), \(\alpha _{P}\), \(\alpha _{T}\), and \(k_{T}\) for the CT patients of Connor and Joseph (2011), Kimura et al. (1996), and Zent et al. (1999). As was done with the CT patient of Bruin et al. (2005) in Sect. 5.4, a hyperline in the 4D-parameter space from the normal parameter values to each of the parameter sets for these three CT patients was followed, and the numerical values of the equilibrium \((P^{*}\), \(T^{*})\) and eigenvalues \(\lambda \) were tracked at each set of parameter values. As before, the eigenvalues tracked were the ones from the second leading pair of the healthy subject, which is the pair that undergoes a Hopf bifurcation as the hyperline extends to any of the CT patients. The results are displayed in Fig. 10.

Fig. 10
figure10

The curves on the left show the evolution of the equilibrium from healthy subject to CT patient as parameters vary. The curves on the right follow the eigenvalues. The rows represent the evolution from healthy subject to the CT patients of Connor and Joseph (2011), Kimura et al. (1996), and Zent et al. (1999), respectively

The equilibrium for the normal parameter set is \((P^{*}, T^{*}) = (31.071, 100.00)\). As the parameter sets move along each of the hyperlines, the equilibria first shift slowly in an arc toward the origin. For all of our examples, this first arc of equilibria takes over 98% of the hyperline, that is, in Eq. (25) if the arc is created by \(t \in [0, t_1)\), then \(t_1 > 0.98\). For an unknown reason (likely a transition in one of the Hill functions), the equilibrium rapidly shifts away from this slow path toward the origin along a different trajectory. This new direction roughly doubles back, but heads to the different states of equilibria for each of the different patients. It is along this rapidly evolving path that the Hopf bifurcation occurs. Figure 10 (and Fig. 8) shows that the cusp-like behavior is similar in all cases, but the evolving paths are distinct for each patient. This complicates the interpretation of how the cyclic thrombocytopenia is explained through the parameters. Table 8 gives the values for the equilibria of the Hopf bifurcation along with the equilibria for the best fitting parameters of the different CT patients.

Table 8 The second and third columns give the equilibria and eigenvalues at the Hopf bifurcation

We numerically solve Eq. (24), starting at the eigenvalues for the normal case with \(\lambda = -0.11375 \pm 0.3588i\). In all cases, the eigenvalues create an arc with the imaginary part decreasing, while the real part first increases then decreases. This arc is created quite slowly and follows the slowly evolving equilibria above. When the equilibria start evolving rapidly, the values of the eigenvalues rapidly shift with increasing real part. Specifically, the real part turns around and increases very rapidly to the Hopf bifurcation. The different CT patients have slightly different changes in their eigenvalues, particularly in what happens to the imaginary part. All cases of the CT patients have their eigenvalues with positive real part, which is to be expected. The frequency of the eigenvalues varies from 0.1641 to 0.2698, yielding periods in the range of 23.3 to 38.3 days, which are consistent with the simulations in Sect. 5.2.

Appendix 6: Fitting of cyclic thrombocytopenia patient data

In this appendix, we describe the statistical procedure used in Sect. 5 to fit the parameters \(\tau _{e}\), \(\alpha _P\), \(\alpha _T\), and \(k_T\) of our model to 15 published platelet and TPO data sets of patients with CT (Bruin et al. 2005; Cohen and Cooney 1974; Connor and Joseph 2011; Engström et al. 1966; Helleberg et al. 1995; Kimura et al. 1996; Kosugi et al. 1994; Morley 1969; Rocha et al. 1991; von Schulthess and Gessner 1986; Skoog et al. 1957; Wilkinson and Firkin 1966; Yanabu et al. 1993; Zent et al. 1999). Of these 15 data sets, only four contained both platelet and TPO data (Bruin et al. 2005; Connor and Joseph 2011; Kimura et al. 1996; Zent et al. 1999). We chose to analyze data sets of untreated CT patients only, as treatments may have altered platelet or TPO dynamics or both and is thus outside the scope of this model.

We first introduce some notation before describing the ABC-MCMC algorithm and the fitting procedure (Marjoram et al. 2003). Let \({\theta }\) be the vector of all the parameters, that is, \({\theta }=(\tau _{e}, \alpha _P, \alpha _T, k_T)^\top .\) Let \(\mathscr {D}\) denote the observed data, including \({\mathbf {P}}_{data}\) and \({\mathbf {T}}_{data}\). Let \(\mathscr {D}^{'}\) denote the model response, including \({\mathbf {P}}_{model}\) and \({\mathbf {T}}_{model}\). Let \(\rho (\mathscr {D}^{'}, \mathscr {D})\) denote the distance between \(\mathscr {D}^{'}\) and \(\mathscr {D}\). Let \(\epsilon \) be a prefixed threshold. The gist of the fitting procedure for a data set is as follows.

  1. 1.

    Choose an initial set of values of parameters for \({\theta }\).

  2. 2.

    Propose a move from the current value of \(\theta \) to \({\theta }^{'}\) according to a transition kernel \(q(\cdot |\theta )\).

  3. 3.

    Simulate \(\mathscr {D}^{'}\) using the model with parameters \({\theta }^{'}\).

  4. 4.

    If \(\rho (\mathscr {D}^{'}, \mathscr {D})\le \epsilon \), go to step 5, and otherwise stay at \({\theta }\) and return to step 2.

  5. 5.

    Calculate

    $$\begin{aligned} \alpha ({\theta }, \theta ^{'}) = \min \left( 1, \frac{\pi ({\theta }^{'}) q(\theta |{\theta }^{'})}{\pi (\theta ) q({\theta }^{'}|\theta )}\right) , \end{aligned}$$

    update \({\theta }\) to \(\theta ^{'}\) with this probability, and store the value of \({\theta }^{'}\).

  6. 6.

    Repeat steps 2–5 using \({\theta }'\) as the new initial set of values of parameters for a sufficient number of times, and finally pick \({\theta }'\) that minimizes \(\rho (\mathscr {D}^{'}, \mathscr {D})\) among all stored values of \({\theta }'\).

To measure the distance between the simulated data \(\mathscr {D}^{'}\) and the observed data \(\mathscr {D}\), we use the following sum of squared errors (SSE):

$$\begin{aligned} \rho (\mathscr {D}^{'}, \mathscr {D}) = \frac{\Vert \frac{2}{3}{\mathbf {P}}_{model} - {\mathbf {P}}_{data}\Vert _{2}}{\Vert {\mathbf {P}}_{data}\Vert _{2}} + \frac{\Vert {\mathbf {T}}_{model} -{\mathbf {T}}_{data}\Vert _{2}}{\Vert {\mathbf {T}}_{data}\Vert _{2}}, \end{aligned}$$
(54)

where the factor of 2/3 accounts for the fraction of platelets that circulate in blood in our model.

For the fits in this paper, in step 1 we chose the initial parameters so that the model generated oscillations, as discussed in Sect. 5.1, and gave a rough approximation to the data set. These parameters defined the initial data \(\mathscr {D}\), and the initial SSE was computed using Eq. (54). We fixed the threshold \(\epsilon \) to 1.15 times the initial SSE and computed steps 2–5 for 250 successful iterations before choosing the vector of parameters \(\theta \) minimizing \(\rho (\mathscr {D}^{'}, \mathscr {D})\).

In step 5 of the ABC-MCMC implementation, we use a uniform distribution as the prior distribution for the parameters, which leads to \(\pi ({\theta }^{'})/\pi (\theta )=1\). In addition, we choose a Gaussian distribution to be the transition kernel, which implies \(q(\theta |{\theta }^{'})/q(\theta ^{'}|{\theta })=1\). As a result, \(\alpha (\theta , {\theta }^{'})\) is simplified to 1.

The stationary distribution of the MCMC chain is the posterior distribution of the parameters given the simulated data are close enough to the observed data (\(\rho (\mathscr {D}^{'}, \mathscr {D})\le \epsilon \)). In other words, with a good initial choice of parameter values, this ABC-MCMC algorithm guarantees convergence to a good fit.

Appendix 7: Numerical analysis

Numerical simulations of the system of equations (12) and (13) are necessary both to illustrate results for any parameter set, and also as part of the parameter fitting described in “Appendix 6”. Accurate numerical solution of these equations is complicated by their structure as distributed delay differential equations (DDEs) with the terms \(m_{e}(t,\tau _{e})\) and \(M_{e}(t)\) both defined by integrals of the solution functions from time \(t-\tau _{e}-\tau _{m}\) to t, with \(M_{e}(t)\) requiring the computation of the integral of the product of the exponential of two integrals.

Traditional Runge–Kutta methods for ordinary differential equations (ODEs) only define a numerical approximation to the solution on a discrete set of time points. This is problematical when the solution is required at off mesh time values, which arises for example for the accurate evaluation of integrals, as is the case in our problem. Continuous Runge–Kutta (CRK) methods were developed (Bellen and Zennaro 2003; Hairer et al. 1993) to produce continuous output suitable for the numerical solution of both ODEs and DDEs. These are the methods currently most often used to solve discrete DDEs, with the Matlab (Mathworks 2015) software package containing built in functions (dde23, ddesd) for the solution of discrete constant delay and state-dependent variable delay DDEs. However, these methods are not appropriate for problems with vanishing or distributed delays, because they become fully implicit. To see how this arises suppose the system (12)–(13) is already solved up to time \(t_n\) and consider the computation of the next step. To compute the jth stage of the CRK method for the next step requires the computation of the right-hand side of the system of equations at time \(t_n+c_jh\), where \(h=t_{n+1}-t_n\) is the step-size of the method and the \(c_j\) are the abscissa of the CRK method. But this requires us to evaluate \(m_{e}(t_n+c_jh,\tau _{e})\) and \(M_{e}(t_n+c_jh)\), for which we need integrals of T(t) up to time \(t_n+c_jh\), but until all the stages of the current step are computed we only have the solution of T(t) available up to time \(t_n\). This problem does not arise in the first order forward Euler method (which has just one stage with \(c_1=0\)), but all higher order CRK methods become fully implicit. Ad-hoc methods for approximating the missing integral result in a reduction in the order of the method, and so the only CRK methods that are appropriate for our problem are the first order forward Euler method or higher order implicit methods. To obtain accurate solutions efficiently we do not use such methods.

Methods that remain explicit for distributed DDEs and DDEs with vanishing delays were first proposed by Tavernini (1971), and have more recently been developed into a class of methods called Functional Continuous Runge–Kutta (FCRK) methods (Bellen et al. 2009; Maset et al. 2005). In Bellen et al. (2006) it was proposed to apply FCRK methods to distributed DDEs in biomathematics, but we are not aware of any implementation of FCRK methods for distributed DDEs before the current work.

To solve the system (12)–(13) we implemented the explicit two-stage second order Heun FCRK method proposed in Cryer and Tavernini (1972). This has Butcher tableau

which defines the parameters \(c_{i}\), \(b_{i}(\alpha )\) and \(a_{i j}(\alpha )\) in the explicit FCRK method

$$\begin{aligned} K_{i}= & {} hG(t_{n} + c_{i}h, Y^i_{t_{n} + c_{i}h}) \\ Y^{i}(s)= & {} u_{n}(s) \text { for } s \le t_{n} \quad Y^{i}(t_{n} + \alpha h) = u_{n}(t_{n}) + \sum \limits _{j=1}^{i-1} a_{i j}(\alpha ) K_{j} \text { for } \alpha \in (0, 1] \\ u_{n+1}(s)= & {} u_{n}(s) \text { for } s \le t_{n} \quad u_{n+1}(t_{n} + \alpha h) = u_{n}(t_{n}) + \sum \limits _{i=1}^s b_{i}(\alpha )K_{i} \text { for } \alpha \in (0, 1] \end{aligned}$$

where the \(K_{i}\) are said to be the stage variables. In the method, \(u_{n}\) is the numerical solution with step-size h generated by the nth step of the method defined up to time \(t_{n}\) for the delayed functional differential equation

$$\begin{aligned} u'(t) = G(t, u_{t}) \text { for } t > t_{0}, \quad u(t) = \phi (t) \text { for } t \le t_{0}, \end{aligned}$$

where \(\phi \) is said to be the starting data, \(t_{0}\) is the initial time, and \(u_{t}\) is the continuous function \(u_{t}(\sigma ) = u(t + \sigma )\) for \(\sigma \in [-\tau , 0]\), where \(\tau \) is the largest delay in the system (which is \(\tau =\tau _e+\tau _m\) for our problem).

The fundamental difference between CRKs and FCRKs which enables the FCRK methods to remain explicit is that the FCRKs are endowed with a continuous approximation \(Y^i(t)\) associated with each stage, whereas the CRK methods only have the single continuous approximation \(u_{n+1}(t)\) defined once the step is computed.

To implement this FCRK method for the system (12)–(13), the integrals need to be evaluated numerically to sufficient accuracy to maintain the convergence order of the method. Although the composite trapezoidal rule would be sufficient for second order accuracy we used fourth order composite methods, to allow for the possible later implementation of a fourth order FCRK method. It is necessary to evaluate the integrals on the same computational mesh as the underlying FCRK method. Since the numerical approximation \(T_{n+1}(t)\) to T(t) is smooth on each interval \([t_n,t_{n+1}]\), but not differentiable at the mesh points \(t_n\), the functions \(\eta _m(T_{n+1}(t))\) and \(\eta _m(T_{n+1}(t))\) will also not be differentiable at the mesh points, and the convergence theory of the composite quadrature methods will break down unless the mesh points \(t_n\) of the FCRK method are included as quadrature points for the integration. The exact solutions of delayed functional differential equations also have discontinuous derivatives at breaking points as outlined in Bellen and Zennaro (2003), which requires certain time points to be included in the computational mesh. But because of the additional smoothing afforded by the integrals in (12)–(13) the only breaking point that needs to be included in the mesh to obtain second order convergence is the initial point \(t_0\).

Taking account of these considerations we used Simpson’s method to evaluate \(M_e(t_n+c_jh)\) using values of \(m_e(t_n+c_jh,kh)\) and \(m_e(t_n+c_jh,(k+1/2)h)\), where \(c_j=0\) or 1 for Heun’s method, and \(k\in \{0,1,...,N\}\) is an integer with the step-size h of the FCRK method chosen so that \(\tau _e=Nh\). The necessary values of \(m_e(t,a)\) were obtained from (13), evaluating the integrals in this formula using Milne’s method. To evaluate the nested integrals efficiently we store values of \(\eta _m(T(t))\) and \(\eta _e(T(t))\) at relevant points (\(t_n+kh/4\) for \(k=0,1,2,3\)) and also store integrals which would otherwise be recomputed at multiple steps, including

$$\begin{aligned} \int _{t_j-\tau _m}^{t_j} \eta _m(T(s)) \, \mathrm {d}a, \end{aligned}$$

which appears in \(m_e(t_j+kh,kh)\) and hence is required to evaluate \(M_e(t_{j+k})\) for each of \(k=0,1,\ldots ,N\).

The method is written to return a function handle that is created from the continuous approximation to the solution generated by the method. This is returned to the user, which allows for the evaluation of P(t) and T(t) at all points in the computational interval (not just at mesh points).

The explicit fourth order FCRK method proposed in Cryer and Tavernini (1972) could be similarly implemented, but requires 6 stages to evaluate one step to fourth order, and so is not as advantageous as the 4-stage fourth order RK method for ODEs, and would require the computation of some complicated integrals over 6 stages.

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Langlois, G.P., Craig, M., Humphries, A.R. et al. Normal and pathological dynamics of platelets in humans. J. Math. Biol. 75, 1411–1462 (2017). https://doi.org/10.1007/s00285-017-1125-6

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Keywords

  • Platelet regulation dynamics
  • Thrombopoiesis
  • Megakaryopoiesis
  • Cyclic thrombocytopenia
  • Dynamic diseases
  • Delay differential equations

Mathematics Subject Classification

  • 37N25
  • 92B99
  • 92C30
  • 37G15