Hybrid Systems Modeling of Donor T‐Cell Responses and Graft‐Versus‐Host‐Disease After Posttransplantation Cyclophosphamide Administration

ABSTRACT

The optimal post‐transplant cyclophosphamide (PTCy) dose to prevent graft‐versus‐host disease (GVHD) after allogeneic hematopoietic cell transplant (HCT) is undefined. Data from a novel murine HLA‐haploidentical HCT model suggested that PTCy has a dose‐dependent effect associated with its efficacy in preventing GVHD, including reduced proliferation of conventional CD4⁺Foxp3⁻ T‐helper cells (Th) at Day 7, followed by the preferential expansion of regulatory CD4⁺CD25⁺Foxp3⁺ T cells (Tregs) at Day 21 after HCT. To better quantify the impact of the PTCy dose, we built a hybrid quantitative systems pharmacology (hybrid systems) model consisting of: (1) simulated pharmacokinetics of PTCy and its metabolites; (2) the donor immune system; and (3) a neural network mapping drug exposure and T‐cell profiles to clinical scores of acute GVHD. Data were collected from murine MHC‐haploidentical HCT studies of PTCy from 0 to 100 mg/kg/day on Days 3 and 4 after transplantation of donor cells. The final model successfully captured time‐varying changes in donor Tcytotoxic CD8⁺ T cells (Tc), Th, and Tregs up to Day 21. The clinical score for GVHD was associated with numerous descriptors, with the immediately preceding GVHD score, body weight, PTCy dose, and donor Treg concentration in the blood and in the liver having the greatest contribution. This hybrid systems model quantitatively captures the beneficial impact of PTCy on Tc, Th, Tregs, and acute GVHD after HCT. The mediation through PTCy and metabolite concentrations and Tc, Th, and Treg numbers is underlined, and the model may be further used to refine and improve PTCy regimens.

Study Highlights

• What is the current knowledge on the topic?

  • ○
    Post‐transplant cyclophosphamide (PTCy) is the predominant immunosuppressive used for hematopoietic stem cell transplant (HCT) to prevent graft‐versus‐host disease (GVHD). The optimal PTCy regimen is not known, and preclinical studies suggest better outcomes with intermediate dose levels. However, high‐dose PTCy is often used clinically.
• What question did this study address?

  • ○
    This study aimed to develop a hybrid systems model to identify the primary predictors of GVHD scores from preclinical studies testing a range of PTCy doses.
• What does this study add to our knowledge?

  • ○
    The validated hybrid systems model well captured the diverse temporal patterns of GVHD from preclinical studies. The model identified PTCy dose and regulatory T‐cell counts in blood and liver tissue (among others) as major predictors of GVHD scores.
• How might this change drug discovery, development, and/or therapeutics?

  • ○
    The final model provides a translational framework for potentially optimizing PTCy and other immunosuppression regimens to prevent GVHD and improve immune recovery following HCT.

1. Introduction

Allogeneic hematopoietic cell transplantation (HCT) is a curative procedure that replaces a patient's hematopoietic cells with those from a healthy donor, typically to exert an antitumor immune graft‐versus‐tumor (GVT) effect against cancer. However, the GVT effect is balanced with the immunologic reactions of the donor cells toward healthy patient tissues (graft‐versus‐host disease, GVHD), which remains a significant contributor to nonrelapse mortality [1, 2].

High‐dose post‐transplant cyclophosphamide (HD‐PTCy, 50 mg/kg/day on Days 3 and 4) is the predominant immunosuppression used for HCT, regardless of donor type or human leukocyte antigen (HLA) matching [3, 4, 5, 6, 7]. The optimal dosing and timing of PTCy administration remain uncertain. The timing of PTCy administration was mainly derived from murine major histocompatibility complex (MHC)‐matched skin allografting studies [8], in which a very high dose of 200 mg/kg was administered on Day 2 or 3 after cell infusion, yielding favorable outcomes. This approach was adapted to murine HCT, where it showed promising effects [9, 10, 11]. Initially, for the first clinical trial for HLA‐haploidentical HCT [12], a dose of 50 mg/kg was selected based on prior experience in treating aplastic anemia and because it was close to the maximum tolerated dose in humans [13]. Administering HD‐PTCy on Day 3 was chosen to reduce toxicity by spacing HD‐PTCy further from HCT conditioning. A subsequent phase II study demonstrated less extensive chronic GVHD when an additional dose of PTCy 50 mg/kg was administered on Day 4, thus establishing the current standard dosing schedule for HD‐PTCy of 50 mg/kg/day on Days 3 and 4 [5, 6, 7, 8, 14, 15, 16].

Preclinical studies in murine MHC‐haploidentical HCT models support a novel mechanism for how PTCy attenuates GVHD: (1) Direct impairment of alloreactive T‐cell function; and (2) Preferential recovery of regulatory T cells (Tregs) that control GVHD [8, 17]. These preclinical data also demonstrated that intermediate‐dose PTCy (ID‐PTCy, 25 mg/kg/day on Days 3 and 4) reduces clinical and histopathologic GVHD compared to high‐dose (50–100 mg/kg/day) or low‐dose (1–10 mg/kg/day) PTCy [17]. ID‐PTCy effectively reduced alloreactive conventional T‐helper (Th) proliferation and promoted preferential Treg recovery, which are crucial mechanisms in preventing GVHD [17, 18, 19, 20, 21, 22, 23]. ID‐PTCy was recently shown clinically in HLA‐haploidentical HCT to be very effective in preventing GVHD with faster engraftment and T‐cell recovery [24].

With the goal of optimizing PTCy dosing to balance the bidirectional immunologic reactions in HCT patients, we built a quantitative systems pharmacology (hybrid systems) model of simulated pharmacokinetic data with observed T‐cell subset numbers and observed acute GVHD severity in a murine MHC‐haploidentical experimental HCT model. The prodrug cyclophosphamide and its metabolites have substantial pharmacokinetic variability; 4‐hydroxycyclophosphamide (4HCY), the principal precursor to the cytotoxic metabolite of CY, has a short half‐life in plasma that requires bedside processing [25]. However, collecting PK data for both CY and its metabolite with T‐cell subsets is challenging in preclinical models because of blood volume and serial sampling constraints. Here, the simulated concentrations of CY, 4HCY, and carboxyethylphosphoramide mustard (CEPM) were used in the development of a hybrid systems‐PTCy model based on relevant T‐cell subsets to provide a framework for simulating future designs of better preclinical studies. Furthermore, the hybrid systems‐PTCy model framework from murine data has the potential to be translated and used to optimize PTCy dosing in patients.

2. Methods

2.1. Study Data

The hybrid systems model was calibrated and later validated with published preclinical data from a T‐cell‐replete, MHC‐haploidentical, murine HCT model (B6C3F1 $\to$ B6D2F1) [17]. Briefly, recipient mice were irradiated 6–8 h before receiving $10\times {10}^6$ B6C3F1 T‐cell‐depleted bone marrow and $40\times {10}^6$ splenocytes from the donor mice. Recipient mice received phosphate‐buffered saline vehicle control (0 mg/kg) or PTCy at 5, 10, 25, 50, and 100 mg/kg/day administered intraperitoneally (IP) on Day 3 (~72 h after HCT) and Day 4 (~96 h after HCT). A summary of the experimental design for the data used to calibrate the hybrid systems model is in Figure 1.

FIGURE 1.

Experimental design of preclinical data used for calibration of the hybrid systems model developed to predict temporal GVHD scores. A full description is provided in Section 2.1. The color schema used for PTCy doses corresponds to that used in the publication of the preclinical data [17] and throughout this hybrid modeling paper. Dotted outlines imp simulated data. ^aT‐cell subsets were quantified in blood, peripheral LN, liver, and spleen. ^bIn mice whose tissues were evaluated at Day 200, blinded evaluations of GVHD scores and weights were assessed every 3 days. The neural network included data from PTCy, PK, and the T‐cell subsets along with the GVHD assessments to try to predict the GVHD scores. Day, hematopoietic cell transplant (HCT) day, with Day 0 being the day of donor graft infusion; GVHD, graft‐versus‐host disease; mg/kg/d, milligrams/kg/day; PK, pharmacokinetics of post‐transplantation cyclophosphamide (PTCy), 4‐hydroxycyclophosphamide, and carboxyethylphosphoramide mustard; TBI, total body irradiation; Tc, T cytotoxic cells; Th, T helper cells; Treg, T regulatory cells.

The pharmacokinetics of CY, 4HCY, and CEPM were not quantified. The concentration (blood) or absolute numbers (organs) of T‐cell subsets (i.e., CD8⁺ (cytotoxic, T_c), CD4⁺Foxp3⁻ (helper, T_h), and CD4⁺CD25⁺Foxp3⁺ regulatory [T_reg]) were quantified using a cell counter and flow cytometry on Days 7, 21, or 200 in blood, peripheral (axillary, brachial, cervical, and inguinal) lymph nodes (LNs), spleen, and liver. Each mouse provided only one data point. In mice whose tissues were evaluated at Day 200, blinded evaluations of GVHD scores and weights were assessed every 3 days, with GVHD scores being calculated based on a standardized scoring rubric including activity, posture, fur texture, skin integrity, and eye appearance [17].

2.2. Modeling Strategy

The overall stepwise hybrid modeling strategy is summarized in Figure 2. Briefly, a pharmacokinetic/pharmacodynamic (PK/PD) model was developed to simulate CY, 4HCY, and CEPM concentration‐time profiles, and then, after calibration to the observed data [17], simulate the concentration of T‐cell subsets in blood, spleen, LN, and liver, which were subsequently mapped to the observed GVHD scores. A previously published PK model for CY, 4HCY, and CEPM [26] was reproduced and slightly modified to simulate drug exposure in mice (Figure 2A). A time‐dependent signal transduction PD model [27] (Figure 2B) accounted for a time delay between drug exposure and stimulated removal of T cells. Donor T‐cell kinetics were modeled by adapting a prior T‐cell biodistribution model [28] (Figure 2C). The cell biodistribution model was modified to include thymic production of T cells following recovery, T‐cell proliferation, and apparent tissue capacities for T cells. Selected parameters in these submodels were calibrated to best capture the time‐course of sparse T‐cell observations from the preclinical study [17]. Finally, a neural network was developed to map the model inputs to the GVHD scores at the observed time points (Figure 2D). The overall primary model assumptions are listed in Table S1.

FIGURE 2.

Overall modeling strategy and detailed sub‐model schematics. (A) Simulated pharmacokinetics of cyclophosphamide (CY) and its metabolites, 4‐hydroxycyclophosphamide (4HCY) and carboxyethylphosphoramide mustard (CEPM), in mouse peripheral (p), central (c), and brain extracellular fluid (ecf) compartments using a previously published model [2]. (B) Predicted $\mathrm{HC}{\mathrm{Y}}_{\mathrm{c}}$ concentrations were used as a driving force for drug‐mediated effects on donor T‐cell kinetics. A time delay between drug exposure and effects was recapitulated using four transit compartments ($M_n$). The red dashed arrows indicate drug‐induced stimulation of T‐cell removal. (C) A donor T‐cell kinetic model was constructed by modifying and calibrating a prior murine T‐cell biodistribution model [6]. Migration of T‐cell subsets ($i$), which were helper (h, CD4⁺Foxp3⁻), cytotoxic (c, CD8⁺), and regulatory (reg, CD4⁺CD25⁺Foxp3⁺) T cells ($i=h,c,\mathit{reg}$), between blood and other tissues was defined. The blue‐framed compartments denote two of the three organs affected by acute GVHD. (D) A neural network model was developed to describe the GVHD score time‐series data and to identify key features that predicted the GVHD scores. BWT, simulated body weight; Dose, PTCy dose; Pre‐S, average of the observed GVHD score at the time point preceding the GVHD score (S); S, GVHD score; T‐cell, simulated amount of T‐cell subsets in blood, liver, LN, and spleen, at the time of GVHD assessment.

2.3. Hybrid Model Development

2.3.1. Pharmacokinetic Model

The plasma concentrations of CY, 4HCY, and CEPM at PTCy doses of 5, 10, 25, 50, and 100 mg/kg/day, administered IP on Days 3 and 4 after transplant, were simulated using a published population PK model [26]. Methods S1 and Table S2 summarize the model equations and fixed parameter values. Simulations with the prior model and parameter values regenerated the published PK profiles (data not shown). PTCy was introduced into the model using a 9 min ($t_{\inf }=$ 9 min) zero‐order infusion, K ₀, defined in the following equation:

$$K_0=\frac{\text{Dose}}{M_{\mathrm{CY}}{t}_{\inf }{V}_{c,\mathrm{CY}}}$$ (1)

where $3\le t<3+t_{\inf }$ and $4\le t<4+t_{\inf }$; otherwise K ₀ = 0. The unit of time, $t$, is days; $V_{c,\mathrm{CY}}$ is the central volume of CY (L/kg); and the molecular weight of CY, $M_{\mathrm{CY}}$, is 0.261 ($\mathrm{mg}/\text{μmol}$).

2.3.2. Pharmacodynamic Model

CY is a prodrug; thus, 4HCY is the principal precursor to the cytotoxic metabolite [25] and was used as the driver of the drug effect on donor T cells, $E_{\text{drug}}$, and the time delay of the drug effect on T cells was described by a series of transit compartments $M_n$ where $n=1,2,\dots ,4$ as shown in Figure 2B. The operable equations are defined as:

$$E_{\text{drug}}\left(\mathrm{HC}{\mathrm{Y}}_c\right)=\frac{E_{\max }·\mathrm{HC}{\mathrm{Y}}_c^{\gamma }}{\mathrm{E}{\mathrm{C}}_{50}^{\gamma }+\mathrm{HC}{\mathrm{Y}}_c^{\gamma }}$$ (2)

$$\frac{d{M}_1}{\mathrm{dt}}=\left(E_{\text{drug}}-M_1\right)/\tau$$ (3)

$$\frac{d{M}_n}{\mathrm{dt}}=\left(M_{n-1}-M_n\right)/\tau \left(n=2,3,4\right)$$ (4)

where the pharmacokinetics of $\mathit{HC}{Y}_c$ is defined by Equation (S4), $\gamma$ is the Hill coefficient, $E_{\max }$ represents the maximal drug effect, $\mathrm{E}{\mathrm{C}}_{50}$ is the concentration of $\mathit{HC}{Y}_c$ producing half‐maximal drug effect, and $\tau$ is the mean transit time for each compartment [29]. The signal in the last transit compartment (M ₄) was used to stimulate the removal of T cells, as shown in Equations ((5), (6), (7)). The parameters for drug effect, specifically $E_{\max },\mathrm{E}{\mathrm{C}}_{50},\tau ,$ and γ were calibrated to the observed T‐cell counts on Day 7.

2.3.3. Tissue Migration of Donor T‐Cell Subsets

The migration of T‐cell subsets ($i$): helper, cytotoxic, and Tregs ($i=h,c,\mathit{reg}$), between blood (compartment number or $k=1$) and other tissues, such as spleen ($k=2$), LN ($k=3$), liver ($k=4$), intestine ($k=5$), and lung ($k=6$), was defined by modifying a prior base model that was developed for the recirculation of T cells in mice (Figure 2C) [28]. Migration kinetics were assumed to be organ‐specific but independent of the T‐cell subset. Three model modifications included the addition of T‐cell proliferation using a density‐dependent proliferation rate constant ($g_i$) [30], thymic production of T cells following recovery (${\lambda }_i$) [31], and apparent tissue capacities for T cells ($I_{\mathit{ki}}$). The following differential equations give the final donor T‐cell kinetic model:

$$\begin{array}{c}\frac{d{x}_{1i}}{\mathrm{dt}}={\lambda }_i\left(t\right)-x_{1i}\left(d_i+{\sum }_{k=2}^6{m}_{1k}\times I_{\mathit{ki}}\left(t\right)\right)+{\sum }_{k=4,6}{m}_{k1}{x}_{\mathit{ki}}\hfill \\ +{\sum }_{k=2,3,5}{m}_{k1}{x}_{\mathit{kli}}+g_i{x}_{1i}-M_4\times x_{1i}\hfill \end{array}$$ (5)

$$\frac{d{x}_{\mathit{ki}}}{\mathrm{dt}}=m_{1k}\times I_{\mathit{ki}}\left(t\right)\times x_{1i}-m_{k1}{x}_{\mathit{ki}}+g_i{x}_{\mathit{ki}}-M_4\times x_{\mathit{ki}}\left(k=4,6\right)$$ (6)

$$\frac{d{x}_{k1i}}{\mathrm{dt}}=m_{1k}\times I_{\mathit{ki}}\left(t\right)\times x_{1i}-m_{k1}{x}_{k1i}+g_i{x}_{k1i}-M_4\times x_{k1i}\left(k=2,3,5\right)$$ (7)

$$\frac{d{x}_{\mathit{kji}}}{\mathrm{dt}}=m_{k1}{x}_{k\left(j-1\right)i}-m_{k1}{x}_{\mathit{kji}}\left(k=2,3,5,j=2,\cdots ,l\right)$$ (8)

with $m_{\mathit{ij}}$ representing the first‐order migration rate constant for the migration of T cells from compartment $i$ to compartment $j$, M ₄ is the cell removal term driven by drug exposure (Equations (2), (3), (4)), $d_i$ is the T‐cell death rate constant, and the density‐dependent proliferation rate constant, $g_i$, is defined as:

$$g_i=\frac{r_i}{1+x_{\mathit{ki}}/h_i}\left(k=1,\dots ,6\right)$$ (9)

where $r_i$ is the growth rate constant, $h_i$ is the amount of T cells reaching half‐maximal proliferation, and $r_i$ and $h_i$ were assumed to be independent of tissue type. Moreover, the residence of T cells in the spleen, LN, and intestine is gamma‐distributed [28], and transit compartments ($l=2$) were used according to the original model derivation as shown in Equation (8). Parameters ($m_{21}$, $m_{31}$, $m_{41}$, and $h_{\mathrm{reg}}$) were estimated by using observed T‐cell subset observations on Days 7 and 21 in the control group (PTCy = 0 mg/kg).

Thymic‐dependent T‐cell recovery occurred within the first month after transplant [23], including the MHC‐haploidentical HCT model. Therefore, a thymic output function [31], ${\lambda }_i\left(t\right)$, was built into the model, as shown in the first term of Equation (5), was defined as:

$${\lambda }_i\left(t\right)={\lambda }_{0,i}\frac{1-\exp \left[\frac{-2t}{{\lambda }_b}\right]}{1+\exp \left[{\lambda }_r\left(1-\frac{t}{{\lambda }_b}\right)\right]}$$ (10)

where ${\lambda }_{0,i}$ is the zero‐order thymic output of T‐cell subset $i$, ${\lambda }_b$ is the time after transplant that the thymic output reaches half of ${\lambda }_{0,i}$, and ${\lambda }_r$ is the increased rate of T cells. ${\lambda }_b$ was estimated by using literature data for % donor T cells that are bone marrow derived [23]; ${\lambda }_r$ was calibrated such that thymic output occurs around Day 21 [23]; then ${\lambda }_{0,i}$ terms were calibrated to observations for T‐cell subsets on Days 21 and 200 in all of the PTCy groups (i.e., with or without PTCy administration). The capacity of T cells in the spleen, LN, and liver was introduced into the base model to better capture the T‐cell amount on Day 200 after transplant. The physiologic rationale for this capacity is that T lymphocyte migration into LN is regulated by the density of cell surface L‐selectin, and there is a physical capacity within the capsule, both of which may be saturable [32]. Increased T‐cell concentrations above saturating values do not promote a further increase in T‐cell recruitment [32]. The function, $I_{\mathit{ki}}\left(t\right)$, was introduced to regulate the migration of T‐cell subsets ($i$) from blood to tissue ($k$) as shown in Equations ((5), (6), (7)):

$$I_{\mathit{ki}}\left(t\right)=\left\{\begin{array}{cc}1-\frac{x_{\mathit{ki}}^6}{x_{\mathit{ki}0}^6+x_{\mathit{ki}}^6}& \left(k=2,3,4\right)\\ 1& \left(k=5,6\right)\end{array}\right.$$ (11)

where $x_{\mathit{ki}}$ is the amount of T cells in tissue, and $x_{\mathit{ki}0}$ represents the amount of T cells resulting in 50% inhibition, the values of which were determined by the maximum median value observed over Days 7, 21, and 200 for each tissue. The exponent in Equation (11) was set to a relatively high default value (i.e., 6) to represent a step‐function owing to the sparseness of the data and lack of identifiability.

To compare the model‐based predictions with the observed T‐cell subset concentrations in blood, the number of T cells in circulation was divided by the blood volume, $V_b\left(t\right)$, which was calculated from body weight $W\left(t\right)$ using to the following equation:

$$V_b\left(t\right)=W\left(t\right)\times 77$$ (12)

where the conversion factor of 77 is in units of $\mathrm{\mu L}/\mathrm{g}$[33]. $W\left(t\right)$ is a spline function as defined in Methods S2. The parameters for $W\left(t\right)$ were estimated by fitting the observed body weights. A comparison of fitted and observed body weight profiles is shown in Figure S2, and the parameter estimates are listed in Data S2.

2.3.4. Neural Network Model of GVHD Scores

All neural networks were fully connected and three‐layered, including input, hidden, and output layers, as shown in Figure 2D. Two neural networks were built: first, a full model using all the input features (Table 2) and second, a final model with a subset of these features. The input features are summarized in Table 2, and include the simulated PK in the central compartments (i.e., HCYc and CEPMc), PTCy dose cohort, simulated body weights, time of the observed GVHD score measurement, average observed GVHD score at the previous time (i.e., PreScore), and T‐cell subsets (i.e., Treg in blood, liver, LN, spleen; Th in blood, liver, LN, spleen; and Tc in blood, liver, LN, spleen) simulated at the time of the GVHD score. The output was set as the average GVHD scores, denoted as $S\left(x\right)$, and can be mathematically represented as:

$$S\left(x\right)=f\left(f\left(x\times W_1+b_1\right)\times W_2+b_2\right)$$ (13)

where $x$ is the input vector, $W_1$ and $W_2$ are matrices denoting the weights between the first and second layer, $b_1$ and $b_2$ are vectors representing the biases of the first and second layer, and $f$ represents an activation function. The rectified linear unit (ReLU) function was used as the activation function. The Python library MLPRegressor in scikit‐learn was used for model training. The model was then evaluated in the test datasets.

TABLE 2.

Input features to the neural networks.

Input	Description	Full model	Final model
4HCYc	Simulated 4HCY in central compartment	Yes	No
CEPMc	Simulated CEPM in the central compartment	Yes	No
Dose	PTCy dose administered	Yes	Yes
BWT	Simulated bodyweight	Yes	Yes
Time	Day after transplant	Yes	Yes
PreScore	GVHD score observed at the previous time point	Yes	Yes
Treg, blood	Simulated regulatory T‐cell amount in the blood	Yes	Yes
Treg, liver	Simulated regulatory T‐cell amount in the liver	Yes	Yes
Treg, spleen	Simulated regulatory T‐cell amount in the spleen	Yes	No
Treg, LN	Simulated regulatory T‐cell amount in the lymph nodes	Yes	No
Tc, blood	Simulated cytotoxic T‐cell amount in the blood	Yes	No
Tc, liver	Simulated cytotoxic T‐cell amount in the liver	Yes	No
Tc, spleen	Simulated cytotoxic T‐cell amount in the spleen	Yes	No
Tc, LN	Simulated cytotoxic T‐cell amount in the lymph node	Yes	No
Th, blood	Simulated helper T‐cell amount in the blood	Yes	No
Th, liver	Simulated helper T‐cell amount in the liver	Yes	No
Th, spleen	Simulated helper T‐cell amount in the spleen	Yes	No
Th, LN	Simulated helper T‐cell amount in the lymph node	Yes	No

During the training process, each set of inputs and a single output measurement constituted a training pair. Data for PTCy doses 0, 5, 25, and 100 mg/kg were used for training or calibration; PTCy 10 and 50 mg/kg were used for validation. Each input feature was scaled by MinMaxScalar in scikit‐learn to a given range between 0 and 1. The stochastic gradient‐based optimizer, adam, was implemented for weight optimization. Max iteration was set to 6000, but convergence was reached earlier than the max iteration. The random state was set to 100 for reproducibility purposes. Mean squared error was used to measure model performance. SHAP values were used to determine the relative contribution of each input feature to the GVHD model prediction [34].

2.4. Data Processing and Parameter Estimation

The robust Z‐score method, also known as the median absolute deviation method, was adopted to detect outliers (robust Z‐score > 3), which were excluded from model development [35]. The model was solved numerically by using odeint, a function in the scipy package (https://scipy.org/) in Python 3, which solves a system of ordinary differential equations by implementing lsoda from the FORTRAN library odepack. The model parameters were estimated by adopting a naïve pooled modeling approach and minimizing the objective function, $\psi \left(\theta \mathit{,}\xi \right)$, with respect to the unknowns $\theta$ and $\xi$: [36]

$$\psi \left(\theta ,\xi \right)=\sum _{l=1}^N\ln \left[{\sigma }_l\left(\theta ,\xi \right)\right]+\frac{1}{2}\sum _{l=1}^N{\left[\frac{y_{l,\mathit{obs}}-y_{l,\mathit{\text{pred}}}\left(\theta \right)}{{\sigma }_l\left(\theta ,\xi \right)}\right]}^2$$ (14)

The Nelder–Mead method was used for minimization in the scipy package, and ${\sigma }_l\left(\theta ,\xi \right)$ is the error model of consideration. In this study, a proportional error model {${\sigma }_l\left(\theta ,\xi \right)=\xi \times y_{l,\mathit{\text{pred}}}\left(\theta \right)$} was evaluated to describe the unexplained residual variability in the data. The model was qualified using visual inspection of fitted profiles. All model programming codes can be accessed in Data S1, and parameter values for PK and PK/PD models are listed in Table 1 and Table S2.

TABLE 1.

Final PK/PD model parameter values for donor T‐cell subset distribution and PTCy pharmacodynamics.

Parameter (unit)	Definition	Value	RSE (%)	Source
$m_{12}$ (1/h)	Rate of entrance from blood to spleen	5.4	—	Ganusov and Tomura [28]
$m_{21}$ (1/h)	Rate of exit from spleen to blood	0.531	14.4	Estimateda
$m_{13}$ (1/h)	Rate of entrance from blood to LN	0.7	—	Ganusov and Tomura [28]
$m_{31}$ (1/h)	Rate of exit from LN to blood	0.883	14.3	Estimated a
$m_{14}$ (1/h)	Rate of entrance from blood to liver	8.61	—	Ganusov and Tomura [28]
$m_{41}$ (1/h)	Rate of exit from liver to blood	0.237	15.6	Estimated a
$m_{15}$ (1/h)	Rate of entrance from blood to intestine	5.5	—	Ganusov and Tomura [28]
$m_{51}$ (1/h)	Rate of exit from intestine to blood	0.1	—	Ganusov and Tomura [28]
$m_{16}$ (1/h)	Rate of entrance from blood to lung	23.64	—	Ganusov and Tomura [28]
$m_{61}$ (1/h)	Rate of exit from lung to blood	1.7	—	Ganusov and Tomura [28]
$d_c$ (1/day)	Death rate of cytotoxic T cells	0.018	—	Westera 2013 [37], Borghans 2018 [38]
$d_h$ (1/day)	Death rate of helper T cells	0.023	—	Westera 2013 [37], Borghans 2018 [38]
$d_{\mathit{reg}}$ (1/day)	Death rate of regulatory T cells	0.014	—	Milanez‐Almeida 2015 [39]
$r_c$ (1/day)	Proliferation rate of cytotoxic T cells	1	—	den Braber et al. [30]
$r_h$ (1/day)	Proliferation rate of helper T cells	1	—	den Braber et al. [30]
$r_{\mathit{reg}}$ (1/day)	Proliferation rate of regulatory T cells	1	—	den Braber et al. [30]
$h_c$ (cells)	Half‐maximal repression, cytotoxic T cells	$3.0\times {10}^4$	—	den Braber et al. [30]
$h_h$ (cells)	Half‐maximal repression, helper T cells	$2.4\times {10}^4$	—	den Braber et al. [30]
$h_{\mathit{reg}}$ (cells)	Half‐maximal repression, regulatory T cells	$4.3\times {10}^3$	24.6	Estimated a
$E_{\mathit{max}}$ (1/day)	Maximum of drug effect	350	—	Calibrated b
$E{C}_{50}\left(\mathit{\mu M}\right)$	Half‐maximal of drug effect	85	—	Calibrated b
$\tau$ (day)	Transit time in time‐delay drug effect model	0.4	—	Calibrated b
$\gamma$	Hill coefficient in $E_{\text{drug}}$	4	—	Calibrated b
${\lambda }_b$(day)	Time after transplant that thymic output reaches half of ${\lambda }_{0,i}$	28.7	4.64	Estimated c
${\lambda }_r$(−)	The increase rate of T cells	20	—	Calibrated c
${\lambda }_{0,c}$ (cells/day)	The zero‐order thymic output of cytotoxic T cells	$4.65\times {10}^9$	6.84	Estimated c
${\lambda }_{0,h}$ (cells/day)	The zero‐order thymic output of helper T cells	$4.65\times {10}^9$	—	Assumed ${\lambda }_{0,h}\mathit{=}{\lambda }_{0,c}$
${\lambda }_{0,\mathit{reg}}$ (cells/day)	The zero‐order thymic output of regulatory T cells	$6.87\times {10}^7$	9.68	Estimated c
$x_{2c0}$ (cells)	The capacity of cytotoxic T cells in the spleen	$3.31\times {10}^6$	—	Maximum of observations
$x_{2h0}$ (cells)	The capacity of helper T cells in the spleen	$7.37\times {10}^6$	—	Maximum of observations
$x_{2\mathit{reg}0}$ (cells)	The capacity of regulatory T cells in the spleen	$6.76\times {10}^4$	—	Maximum of observations
$x_{3c0}$ (cells)	The capacity of cytotoxic T cells in the LN	$8.11\times {10}^5$	—	Maximum of observations
$x_{3h0}$ (cells)	The capacity of helper T cells in the LN	$1.41\times {10}^6$	—	Maximum of observations
$x_{3\mathit{reg}0}$ (cells)	The capacity of regulatory T cells in the LN	$1.05\times {10}^5$	—	Maximum of observations
$x_{4c0}$ (cells)	The capacity of cytotoxic T cells in the liver	$3.97\times {10}^6$	—	Maximum of observations
$x_{4h0}$ (cells)	The capacity of helper T cells in the liver	$2.73\times {10}^6$	—	Maximum of observations
$x_{4\mathit{reg}0}$ (cells)	The capacity of regulatory T cells in the liver	$7.30\times {10}^4$	—	Maximum of observations
$\xi$	Parameter in proportional error model	0.821	10.4	Estimated a

Abbreviation: LN, lymph node.

^a Parameters were re‐estimated by implementing observations for T‐cell subsets on Days 7 and 21 without administering PTCy.

^b Calibrated to Day 7 observations (see Section 2, PD section).

^c ${\lambda }_b$ was estimated by using literature data for % donor T cells that are bone marrow derived in Patterson et al. [23]; ${\lambda }_r$ was calibrated such that thymic output occurs around Day 21. Then ${\lambda }_{0,i}$ were estimated by implementing observations for T‐cell subsets on Days 21 and 200 with or without PTCy administration.

3. Results

3.1. Plasma Pharmacokinetics

Because the simulated CY, 4HCY, and CEPM concentrations decreased to the lower limit of quantitation after 24 h (Figure S1), no further direct drug effects on T cells by Days 21 through 200 post‐transplant were assumed based on drug washout kinetics and the decay of the pharmacological signal (M ₄ → 0).

3.2. Effect of PTCy on T‐Cell Subsets in Various Tissues

A continuous function of body weight was required to account for blood volume, and simple spline functions accurately captured body weight profiles for all treatment cohorts (Figure S2). The observed and model‐fitted kinetic profiles of the donor T‐cell subsets in blood, LN, spleen, and liver are shown in Figure 3; notably, T‐cell subset data were available for PTCy 0, 5, 25, and 100 mg/kg/day only at Days 7 and 21. Table 1 lists the model parameter values, and all estimated terms had acceptable relative standard errors (RSE < 25%). The model captured the concentration of these donor T‐cell subsets in the vehicle control group. Owing to the limited data sampling, parameters for the effect of PTCy were calibrated to capture observations on Day 7 following PTCy administration. After testing various possible models for drug effect, the time‐dependent signal transduction model shown in Figure 2B was the best at describing observations on Day 7. The model described the T‐cell subset concentrations in various tissues reasonably well, except after PTCy 100 mg/kg in the LN and spleen. Various models were also tested to improve the fit of the donor T‐cell subset concentrations in various tissues up to Day 200, and the capacity limitation function (Equation 11) stabilized the T‐cell concentrations at later times (Figure S3).

FIGURE 3.

Donor T‐cell subset kinetics in blood, liver, LN, and spleen, for 21 days after transplantation with or without PTCy administered on Days 3 and 4. $T_c$: Cytotoxic (CD8⁺) T cells; $T_h$: Helper (CD4 ⁺ Foxp3⁻) T cells; $T_{\mathit{reg}}$: Regulatory (CD4⁺CD25⁺Foxp3⁺) T cells. Open circles denote the median value of observations; upper (lower) error bar denotes the 75th (25th) percentiles of the observed data.

3.3. Neural Network Modeling of Clinical GVHD Scores

Neural network models were developed to learn and predict GVHD scores from input features (Table 2), including drug exposure metrics and T‐cell subset profiles in various tissues and plasma as a function of time. In addition to building a predictive model, the neural network was also used to identify key model input features that predict the GVHD score patterns. The outputs from the PK/PD model were used as input features into the neural network and mapped to GVHD scores. The GVHD score from the previous time step, current observation time, and the numbers of T‐cell subsets in blood, spleen, LN, and liver were also evaluated. The neural network was trained using the PTCy 0, 5, 25, and 100 mg/kg dose cohorts and then subsequently tested on the PTCy 10 and 50 mg/kg dose cohorts for validation purposes.

To determine the optimal number of hidden layers in the neural network, the lowest mean squared error for both the training and test datasets was used (Figure 4A). The nine nodes in the hidden layer resulted in the best performance, and the fitted GVHD scores for the training dataset are shown in Figure 4B. The importance of predictors for the GVHD scores was calculated using SHAP values, and the results are shown in Figure 4C. The six most important features to predict the GVHD score were: GVHD score at last measurement (PreScore), body weight (BWT), PTCy dose, Tregs in blood and liver (T_{reg, blood} and T_{reg, liver}), and observation time. Predictions for the test dataset from the full model agreed well with observations (Figure 4D).

FIGURE 4.

Diagnostics and predictive performance of the full neural network model for predicting GVHD scores. (A) The mean squared error for training and test datasets is considered for various numbers of nodes in the hidden layer. (B) Comparison of observed and fitted profiles for the training dataset with nine nodes in the hidden layer. (C) Order of importance of inputs for predicting GVHD scores according to the mean of the absolute value of SHAP values for the full model. (D) Comparison of observed and predicted profiles for the test dataset using the full model.

For the final model, the six most important features from the full model (i.e., PreScore, BWT, PTCy dose, T_{reg, blood}, T_{reg, liver}, and time) were retained. Mean squared errors for training and test datasets were evaluated, and the performance of various numbers of nodes in the hidden layer is shown in Figure 5A. A total of 15 nodes in the hidden layer led to the best performance in the final model, which had the same predictive performance as the full model. The fitted and predicted GVHD scores for the training and test datasets from the final model are shown in Figure 5B,C, and the GVHD scores were predicted well by the final model.

FIGURE 5.

Predictive performance of the final neural network model to predict GVHD scores. (A) Mean squared error for training and test datasets (solid line) following various numbers of nodes in the hidden layer. Dashed lines represent the mean squared error for training and test datasets from the best full model (Figure 4). (B) Comparison of observed and fitted profiles for the training dataset with 15 nodes in the hidden layer. (C) Comparison of observed and predicted profiles for the test dataset using the final model.

4. Discussion

The main findings from this hybrid systems model are: (1) the simulated pharmacokinetics of PTCy and its metabolites served as reasonable driving functions for constructing a PK/PD model of T‐cell immunomodulation over the first 21 days after HCT (Figure 3); (2) the engraftment of donor T cells could be characterized using signal transduction and mechanistic lymphocyte modeling including donor T‐cell and thymic outputs; and (3) a neural network identified that the GVHD score was associated with numerous descriptors, with the GVHD score immediately preceding it, the concentration of Tregs in the liver and blood, PTCy dose, and body weight having the greatest contribution to variability in the GVHD scores. Removing the preceding GVHD score from the neural network inputs substantially reduced the predictive performance of the initial and final models, and the only change to the SHAP analysis was that Tregs in the LN became the second most important descriptor (data not shown). The PreScore was retained in the final model, and further research into the contribution of Tregs in the LN is warranted.

Patients with malignancy receive HCT in the hope of benefiting from the GVT effect of the donor immune cells, treating the recipient's cancer. At present, GVT and GVHD continue to be linked, although preclinical and clinical studies indicate that Tregs can protect from GVHD without interfering with the GVT effect of HCT [40, 41, 42]. The mechanistic insights gained from this hybrid systems model, which characterizes the association of Tregs in different tissues with GVHD, provide a framework for further study of GVHD and GVT in HCT and optimizing outcomes. The hybrid systems model was built using preclinical data with one HCT conditioning regimen and donor graft, with the only variation being the PTCy dose. The preclinical data of Wachsmuth et al. [17] suggest that PTCy has dose‐dependent effects that are associated with its efficacy in preventing GVHD, including reduced proliferation of Th at Day 7, followed by the preferential expansion of Treg at Day 21 after HCT in murine HLA‐haploidentical HCT models [17, 18]. These preclinical data were used to calibrate the hybrid systems model successfully. This model can be used to test hypotheses regarding the impact of changing the PTCy dose, the timing of PTCy administration relative to donor T‐cell infusion, adding specific adjunct immunosuppressants to PTCy, and/or different allograft sources [43] to further reduce proliferation of Th on Day 7 or expand Tregs by Day 21. Such hypotheses can be further tested by virtually mimicking patient data (or so‐called digital twins) as exemplified by the recent use of systems modeling for the human immune system [44]. Whether delayed thymic output in humans compares with that of mice will require additional testing and potential modifications to the model.

The hybrid systems model also facilitates optimizing T‐cell subsets and/or postgrafting immunosuppression, which aligns with the objectives of Project Optimus, a key initiative by the US Food and Drug Administration (FDA) aimed at optimizing drug dosages in oncology. Project Optimus seeks to replace the maximum tolerated dose method by proactively evaluating dose–response relationships to enhance efficacy and safety, ultimately maximizing the benefit‐to‐risk ratio for patients [45]. With the preclinical data indicating that an intermediate dose of PTCy is associated with lower GVHD rates than HD‐PTCy [17], PTCy provides an ideal framework for demonstrating the benefits of hybrid systems modeling to HCT.

Over the past 20 years, PTCy has enabled safe HCT of HLA‐partially‐mismatched allografts [4]. Recently, it became a standard of care for GVHD prevention in HCT recipients of either related or unrelated HLA‐matched donors [5, 6]. Despite its widespread acceptance, the optimal dosing and timing for administering PTCy in patients remain uncertain. The current body weight‐based dosing method for CY results in significant variability in the area under the plasma concentration‐time curve (AUC) for CY and its metabolites, including 4HCY, which is the precursor to the main cytotoxic metabolite, phosphoramide mustard [25]. Notably, PTCy dose and local T‐cell subset numbers were of more importance in predicting the GVHD score compared to the simulated concentrations of 4HCY and CEPM in the central compartment (Figure 4C). However, the pharmacokinetic variability in inbred mice is expected to be much smaller than in patients. A key step in translating this hybrid systems model from mice to humans will be to measure the concentrations of PTCy and its metabolites in patients to characterize the effect of varying PTCy doses upon Th and Treg counts in patients.

In addition, when translating this model from mice to humans, careful consideration is needed for the scaling of the donor cell dose used in the murine preclinical model to the donor cell dose administered to patients. At present, the optimal method to scale the donor cell dose from preclinical models to patients is not apparent. Insights can be gained from analyses of transgene product expression in humans, which indicated that allometric scaling using body weight^−0.25 had the greatest prediction accuracy for human transgene product prediction from intravenous viral vectors to locally delivered viral vectors [46]. Whether the actual PTCy dose needs to be scaled from murine to human HCT studies is also unknown and will need to be tested.

An additional potential application of this hybrid systems model is to optimize the immunosuppressants, typically mycophenolate mofetil, tacrolimus, cyclosporine, and/or sirolimus, used to prevent GVHD in combination with PTCy in patients. To date, all immunosuppressants used in HCT are characterized by large intra‐ and interindividual pharmacokinetic variability and by narrow therapeutic indices [2, 47]. The association between clinical outcomes and the systemic exposure of these immunosuppressants should be evaluated [48]. In a retrospective single‐center study of 349 patients who received PTCy and mycophenolate mofetil with either tacrolimus (n = 185) or sirolimus (n = 164), early immunosuppression (i.e., sirolimus or tacrolimus) concentrations were not associated with acute GVHD incidence, but higher immunosuppression concentrations were associated with worse outcomes [49]. The translation of this hybrid systems model to patients may provide a mechanistic rationale to lower acute or chronic GVHD after PTCy‐based regimens in patients [46].

Our modeling was limited by three factors: (1) the number of data points for T‐cell subsets, which was constrained by the available data, (2) the lack of serial sampling from the same mice, which was necessary owing to the low blood volume available from the mouse model and the interest in examining T‐cell kinetics within hematopoietic and GVHD‐target tissues, and (3) serial sampling of organ data is not feasible and thus resulted in one time point per mouse, which led to the use of a naïve pooled approach for estimation purposes. Another limitation is that only live mice were scored for GVHD, which might result in potential bias toward surviving mice at the late timepoints. However, survival was relatively high in the 10, 25, and 50 mg/kg groups (70%, 100%, and 90%), and deaths in the 50, 100, and 200 mg/kg groups were owing to toxicity rather than severe GVHD [17]. Notwithstanding these limitations, our modeling does address contemporary clinical questions regarding PTCy. Also, the important role of the donor T‐cell subsets in various mouse tissues confirms previously published modeling [28]. Ultimately, we anticipate that this hybrid systems modeling will serve as a springboard to optimize PTCy and other adjunct immunosuppression required to effectively prevent GVHD, while optimizing immune recovery necessary for pathogen‐specific immunity and GVT.

Author Contributions

H.‐H.F., J.S.M., C.G.K., and D.E.M. wrote the manuscript, performed the research, analyzed the data, and contributed a new analytical tool. J.S.M., C.G.K., and D.E.M. designed the research.

Funding

This publication was supported by the National Institutes of Health under the Award Number U01CA239373 and by the Intramural Research Program of the National Cancer Institute of the National Institutes of Health. The content is solely the authors' responsibility and does not necessarily represent the official views of the National Institutes of Health.

Conflicts of Interest

The authors declare no conflicts of interest.

Supporting information

Table S1: Primary model assumptions.

Table S2: Pharmacokinetic model parameters.

Methods S1. Pharmacokinetic (PK) model.

Methods S2. Time course of body weight function.

Figure S1: Simulated plasma pharmacokinetics of cyclophosphamide (CY) and its metabolites.

Figure S2: Spline functions describing the time course of measured body weight, (𝑡).

Figure S3: Donor T cell subtype kinetics in blood, spleen, lymph node, and liver after posttransplant.

CY (PTCy) administered on Days 3 and 4.

Data S1: A folder containing the Python code and its accompanying notebook for the hybrid systems model.

Data S2: Supporting Information.

Data S3: Supporting Information.