A mathematical model for dynamics of soluble form of DNAM-1 as a biomarker for graft-versus-host disease

DNAM-1 (CD226) is an activating immunoreceptor expressed on T cells and NK cells and involved in the pathogenesis of acute graft-versus-host disease (aGVHD) after allogeneic hematopoietic stem cell transplantation (allo-HSCT). We previously reported that a soluble form of DNAM-1 (sDNAM-1) is generated by shedding from activated T cells. Moreover, higher serum levels of sDNAM-1 in patients before allo-HSCT is a predictive biomarker for the development of aGVHD based on the retrospective univariate and multivariate analyses in allo-HSCT patients. However, it remains unclear how the serum levels of sDNAM-1 are regulated after allo-HSCT and whether they are associated with the development of aGVHD. Here, we constructed a mathematical model to assess the dynamics of sDNAM-1 after allo-HSCT by assuming that there are three types of sDNAM-1 (the first and the second were from alloreactive and non-alloreactive donor lymphocytes, respectively, and the third from recipient lymphocytes). Our mathematical model fitted well to the data set of sDNAM-1 in patients (n = 67) who had undergone allo-HSCT and suggest that the high proportion of the first type of sDNAM-1 to the total of the first and second types is associated with high risk of the development of severe aGVHD. Thus, sDNAM-1 after allo-HSCT can be a biomarker for the development of aGVHD.


Introduction
Acute graft-versus-host disease (aGVHD) is a major complication of allogeneic hematopoietic stem cell transplantation (allo-HSCT). Although the mechanism underlying the development of aGVHD has been extensively studied in vitro and in vivo [1][2][3], the diagnosis of and treatment for aGVHD are still problematic.
DNAM-1, also known as CD226, is an activating immunoreceptor expressed on CD4 + T cells, CD8 + T cells, natural killer (NK) cells, and monocytes [4]. We and others demonstrated a1111111111 a1111111111 a1111111111 a1111111111 a1111111111 that DNAM-1 plays an important role in the development of aGVHD in mouse models [5,6]. Moreover, we have recently identified a soluble form of DNAM-1 (sDNAM-1), which is shed from the membrane type of DNAM-1 expressed on the cell surface of activated T lymphocytes, in human sera [7]. We performed retrospective univariate and multivariate analyses of serum levels of sDNAM-1 in patients before and after allo-HSCT at a single center (n = 71) [7]. We demonstrated that cumulative incidences of all grade (grade I-IV) and grade II-IV aGVHD in patients with high maximal serum levels of sDNAM-1 (�30 pM) in the 7 days before allo-HSCT were significantly higher than those in patients with low maximal serum levels of sDNAM-1 (<30 pM) in the same period, and concluded that the serum levels of sDNAM-1 can be a predictive biomarker for the development of aGVHD [7]. However, it remains unclear how the dynamics of serum levels of sDNAM-1 after allo-HSCT is regulated and whether it is associated with the development of aGVHD.
In this study, we constructed a mathematical model to assess the dynamics of serum levels of sDNAM-1, and revisited the data set of sDNAM-1, which had been analyzed previously [7], particularly after, rather than before, allo-HSCT to be applied by the mathematical model. We show that sDNAM-1 after allo-HSCT can be a biomarker for the development of aGVHD.

Patients, samples, and inclusion criteria
Serum samples were obtained from 156 patients at the Tokyo Metropolitan Cancer and Infectious Diseases Center, Komagome Hospital, Japan, between March 2009 and November 2011. Data from some of the patients had already been analyzed [7]; we basically followed their methods in terms of such features as informed consent and sample handling but here we included other patients according to our new criteria which is shown in Fig 1. Written informed consent was obtained from each patient in accordance with the Declaration of Helsinki. This study was approved by the ethics committee of the University of Tsukuba (approval No., 505) and Tokyo Metropolitan Cancer and Infectious Diseases Center, Komagome Hospital (approval No., 571). Soluble DNAM-1 in sera was measured by sandwich enzyme-linked immunosorbent assay (ELISA), as described [7]. The data were collected from about 7 days before allo-HSCT up to a maximum of 249 days after allo-HSCT. Since our mathematical model required eight parameters and at least one parameter within 100 days and 7 days after allo-HSCT, respectively, we excluded patients who did not meet these criteria. As a result, 67 patients (GVHD (+) = 48 patients, GVHD (-) = 19 patients) satisfied our criteria (Fig 1); their characteristics are listed in Table 1.

Definition of grading system of acute GVHD and transplantation procedures
The GVHD grade of each patient was determined as previously described [8,9]. The transplantation procedures are described previously [7].

Mathematical model and statistical analysis
sDNAM-1 was detected in the sera of healthy people as well as patients before and after allo-HSCT. sDNAM-1 might be produced by either donor cells, recipient cells or both. Therefore, we modeled the dynamics of those cells using ordinary differential equations. Differential equations have been used for describing the virus and cell growth [10,11]. In our research, we firstly formulate the mathematical model to explain the dynamics of sDNAM-1. Secondly, we fitted this model to the patients' data and got each patient's parameter using Package FME [12]  Assessing goodness of fit to data and contribution of first type of sDNAM-1 to aGVHD. We compared 2 mathematical models to assess their goodness of fit to the data. The model 1 is consisted of two secretors of sDNAM-1 (i.e., donor lymphocytes that persistently produce sDNAM-1 and recipient lymphocytes that had produced it before all-HSCT). On the other hand, the model 2 is consisted of three secretors of sDNAM-1 (i.e., donor lymphocytes that transiently release sDNAM-1 in addition to those in the model 1). Here, we used AIC, which can be used to compare two mathematical models [13].

Mathematical model of sDNAM-1 dynamics
sDNAM-1 in sera of patients that received allo-HSCT can be derived from lymphocytes of either a donor, a recipient or both. Therefore, we assumed that sDNAM-1 in sera of patients consists of three types of sDNAM-1. The first and second types are transiently and persistently released from donor lymphocytes, respectively. The third one is a residual sDNAM-1 derived from recipient's lymphocytes before allo-HSCT. Since it is difficult to discriminate the origins of serum sDNAM-1 detected by ELISA, we performed a computational simulation to estimate the concentration of each type of sDNAM-1. In our mathematical model, we denoted x 1 (t) as the concentration of the first type of sDNAM-1. We adopted a gamma distribution for x 1 (t), because the gamma function is a unimodal distribution and has a flexible shape, unlike the normal distribution. It can be used to express the feature of first type of sDNAM-1, namely transient growth. The second and third types of sDNAM-1 follows a logistic equation and an exponential decay pattern, which we denoted them as x 2 (t) and x 3 (t), respectively. The model is shown as follows: Definition of each parameter are given in S1 Table in the Supplementary Methods. x 1 (t) +x 2 (t)+x 3 (t) expresses the total amount of sDNAM-1. We fitted this model to the observed data and obtained the estimated parameters.

Definition of R day_n and comparison of R day_n between patients' group with or without aGVHD
We defined R day_n using a mathematical expression as follows: Integration of x 1 (t) gives an area under the curve (AUC). The AUC of x 1 (t) from 0 to day_n is the amount of sDNAM-1 that is secreted by the donor lymphocytes during n days after allo-HSCT. In order to assess the effect of sDNAM-1 from donor lymphocytes on acute GVHD, we excluded the third type of sDNAM-1, which is derived from recipient lymphocytes. Since donor-derived allo-reactive T cells attack the recipient's tissues during acute GVHD [14], it is reasonable to focus on the first and second type of sDNAM-1 derived from donor lymphocytes. Therefore, R day_n is the proportion of sDNAM-1, which is released transiently from donor lymphocytes. We plotted each distribution of R day_n . The R day_n distribution of patients with aGVHD (GVHD (+)) tended to shift to the right, whereas the distribution of patients without aGVHD (GVHD (-)) tended to shift to the left (Fig 2). We also compiled boxplots of R day_n for GVHD (+) and GVHD (-) (n = 20, 30, 40, and 50). R day_n of GVHD (+) was higher than that of GVHD (-) for each day (n = 20, 30, 40, and 50) (Fig 3). We performed t-test and Wilcoxon test and calculated the 95% CIs of mean & median of differences between GVHD (+) and GVHD (-) groups. The GVHD (+) group showed higher R day_20 , R day_30 , R day_40 , and R day_50 than did the GVHD (-) group ( Table 2). The differences in each estimated mean between the two groups exceeded 30% ( Table 2). In addition, we analyzed the relationship between R day_n (n = 20, 30, 40, and 50) with the grades of aGVHD. We found that the higher the grade, the more the value of R day_n (n = 20, 30, 40, and 50) increased (Fig 4), suggesting that R day_n reflected the grade of aGVHD. In addition, we also found the association of R day_n with aGVHD in the skin and gastrointestinal tract, but not in the liver (S2-S4 Figs, S2-S4  Tables). In contrast, we did not observe the association of R day_n with the factors of patient background (e.g., with or without CMV infection, total body irradiation, and HLA matching, and treatment for GVHD prophylaxis) (S5-S8 Tables).

Half-life of sDNAM-1
Based on our mathematical model, the half-life of sDNAM-1 was calculated as log2 m . The mean of half-life of sDNAM-1 was 12.5 +/-9.2 days (n = 67). This calculation formula did not take the continuous production of sDNAM-1, which is variable in each patient, into account, resulting in the relatively high deviation of the half-life.

Days elapsed from HSCT to the development of aGVHD
We plotted a histogram of the days elapsed from allo-HSCT to the onset of aGVHD. The mean number of days was 23 (S1

Comparison between the models 1 and 2 using AIC
We compared 2 mathematical models to assess their goodness of fit to the data. The model 1 is consisted of two secretors of DNAM-1 (i.e., donor and recipient lymphocytes). On the other hand, the model 2 is consisted of three secretors of DNAM-1 (i.e., two types of donor lymphocytes that transiently and persistently release sDNAM-1 and recipient lymphocytes). The percentages of patients suitable for the models 1 and 2 is 29.9% and 70.1%, respectively (pvalue = 0.0013) ( Table 3). These results suggest that the model 2 consisting of three types of sDNAM-1 explained the dynamics of the observed sDNAM-1 better than did the model 1 consisting of two types of sDNAM-1.
In the current study, we constructed a mathematical model to assess the dynamics of sDNAM-1 in the sera of patients after allo-HSCT and applied it to the data set of sDNAM-1, which had been previously analyzed [7]. For a mathematical model, we assumed that there were three types of sDNAM-1. The first and second types of sDNAM-1 are released from alloreactive and non-alloreactive donor lymphocytes, respectively. The third type is residual one released from recipient lymphocytes before allo-HSCT. Based on these three types of sDNAM-1, we constructed a mathematical model for the dynamics of sDNAM-1 using either these three or two (the second and the third) types of sDNAM-1 and found that the model using the three types efficiently fitted into the data better than the model using two types of sDNAM-1, demonstrating that the first type of sDNAM-1 is required for the construction of a better   n = 20, 30, 40, and 50 days).
https://doi.org/10.1371/journal.pone.0228508.t002 mathematical model. By using this model, we showed that the greater the proportion of the first type of sDNAM-1 to the total of the first and second types of sDNAM-1 during n days (R day_n ) (n = 20, 30, 40, and 50) after allo-HSCT was, the more likely the patient developed aGVHD. R day_n was a reliable index for high sensitivity of and specificity to the development of aGVHD. We also showed that R day_n correlated well with the GVHD severity. Thus, R day_n is a key biomarker for the development of aGVHD when sDNAM-1 was analyzed repeatedly over time after allo-HSCT. Considering that allogeneic effector T cells derived from donors proliferate rapidly after transplantation and play a central role in the pathogenesis of aGVHD [24,25][26], donor lymphocytes producing the first type of sDNAM-1 are likely to be allogeneic effector T cells.
On the other hand, donor lymphocytes that persistently release sDNAM-1 might be non-effector T cells. This scenario is in agreement with the previous report that certain T cells showed clonal expansions after allo-HSCT particularly in patients with aGVHD as demonstrated by the T cell antigen receptor repertory analyses [27]. Our study showed that our mathematical model for sDNAM-1 dynamics provides a useful biomarker for the development of aGVHD. However, a larger prospective study is required to Representative patients Data of sDNAM-1 dynamics after transplantation are shown. The purple dots indicate the observed data. The blue line models the total amount of sDNAM-1; it represents the full model (= x 1 (t)+x 2 (t) +x 3 (t)). The green line models the first type of sDNAM-1, namely the transiently type (= x 1 (t)). The yellow-green line models the second type of sDNAM-1, namely the persistent type (= x 2 (t)). The red line models the third type of sDNAM-1, namely the residual type produced by recipient (= x 3 (t)). Thus, the blue line (total concentration of sDNAM-1) consists of the green, yellow-green, and red lines combined. Patients ID1 and ID2 did not develop aGVHD, whereas patients ID3 and ID4 did develop it.