Modeling Immune Response to BK Virus Infection and Donor Kidney in Renal Transplant Recipients

Abstract

In this paper we develop and validate with bootstrapping techniques a mechanistic mathematical model of immune response to both BK virus infection and a donor kidney based on known and hypothesized mechanisms in the literature. The model presented does not capture all the details of the immune response but possesses key features that describe a very complex immunological process. We then estimate model parameters using a least squares approach with a typical set of available clinical data. Sensitivity analysis combined with asymptotic theory is used to determine the number of parameters that can be reliably estimated given the limited number of observations.

1 Introduction

According to the OPTN/SRTR 2011 Annual Report [15], 17,604 kidney transplants were performed in the United States between 2010 and 2011. Overall, there were 54,599 active candidates on the waiting list for kidney transplants, roughly 3-fold more than those that underwent transplant. These numbers reflect trends consistent with previous years, in which the number of patients waiting for transplants greatly exceeds the availability of organs. Given these facts, and the fact that as of June 30, 2011, 164,200 adults in the U.S. were surviving with a functioning kidney graft, about twice as many as a decade earlier, optimal care of renal transplant patients is of great importance.

To reduce risk of allograft rejection, the standard of care for renal transplant recipients involves life-long pharmacological immunosuppression, making patients susceptible to opportunistic infections. Specifically, this therapy can render the recipients susceptible to an array of viral pathogens and may also reactivate latent viruses. For some time, human polyomavirus type 1, named “BK virus” (BKV), has been a common pathogen found in kidney transplant patients. BKV is one of the two human polyomaviruses and was first discovered in 1970 in the urine of a kidney transplant patient with the initials B.K. [22]. This double stranded non-enveloped DNA virus with icosahedral capsids asymptomatically infects more than 90% of the adult population worldwide and establishes a state of non-replicative infection [25, 26, 29], or latent state. The infection is established in the kidneys and peripheral blood, specifically the renal tubular epithelial and urothelial cells, where replication-permissive cells express the viral capsid proteins followed by virion assembly in the nucleus [16]. This process eventually causes host cell lysis and the release of infectious progeny, deeming BKV replication as cytopathic, leading to a new round of active infection and latency. A high-level of BKV replication in the kidney results in a complication known as PVAN (polyomavirus associated nephropathy) [18, 20, 21, 23]. It is characterized by viral cytopathic changes of renal tubular epithelial cells, with enlarged nuclei, cell rounding, detachment and denudation of basal membranes [32]. Increasing prevalence rates of PVAN (1–10%) have been reported, with allograft dysfunction and loss in greater than 50% of cases [27, 35]. Therefore, PVAN is viewed as one of the leading causes of renal allograft loss in the first two years of transplantation. Unfortunately, there are no available licensed anti-polyoma viral drugs and treatment relies on improving immune function to control BKV replication [18]. Given BKV infection is a significant health threat to immunosuppressed renal transplant patients, patient outcomes might be improved with the use of mathematical modeling to predict the course of the disease in individuals and recommend optimized treatment strategies.

Mathematical modeling is widely used and historically accepted in the physical science and engineering communities as an aid in the understanding of complex phenomena. Specifically, we highlight the use of mathematical modeling with experimental investigations to enhance the understanding of biological processes. The process involves a sequences of steps: (i) empirical observations, experiments and data collection; (ii) formalization of the biological model; (iii) abstraction of mathematical model; (iv) formalization of uncertainty and use of a statistical model; (v) model analysis; (vi) changes in understanding; and (vii) design of new experiments [12]. Given the highly iterative process of mathematical modeling and the recently developed quantitative techniques such as real-time PCR measurements of viral load, flow cytometry of T cell subsets and ELISPOT for virus-specifical T cell function, it is feasible to couple mathematical modeling with biological experimentation to investigate the dynamics of viral infection and the cellular immune response.

Here we give a brief review of recent works that use mathematical modeling to investigate viral infection in relation to organ transplant dynamics. Funk et al., in [21], summarized investigations of a retrospective analysis of BKV plasma load in renal transplant recipients undergoing allograft nephrectomy or changes in immunosuppressive regimens. PCR measurements of viral DNA are applied to a standard mathematical model for viral load decay kinetics to estimate the half-life and doubling time of BKV as well as clearance and growth rates [21]. This model addresses purely BKV replication. The next iteration in the modeling process requires the development of dynamic models that account for interactions between viral infection and host cell populations. Funk, Gosert, et al., in [19] extended a 1-compartment model to a 2-compartment model with six state variables describing BKV replication dynamics in renal tubular epithelial cells and in urothelial cells. Estimation of parameters was based on population level clearance, proliferation, etc., rates. The study presented a basic model integrating two replication sites, the kidney and the urinary tract, and derived four variants which incorporated coupled and decoupled dynamics of the two sites. It remains unknown whether the two replicate sites are in fact coupled, however, results in [19] suggest that viral expansion was best explained by models where BKV replication started in the kidney followed by urothelial amplification and then reached an equilibrium amongst both replication sites. The model does not address the response of the immune system to viral infection and donor kidney and little to no statistically-based model validation or calibration as proposed in our efforts here was carried out.

Other viral infections have been studied in relation to organ transplant dynamics. In particular, two models have been developed involving human cytomegalovirus (HCMV) infection. Kepler et al., in [28] developed a dynamic model describing the pathogenesis of primary HCMV infection in immunocompetent and immunosuppressed patients at the cellular/viral mechanistic scale for application to individual clinical data and patient health. The model incorporates dynamics of the viral load, immune response as well as actively-infected, susceptible and latently-infected cells. Results highlight the necessity of longitudinal data for multiple state variables for robust parameter estimation. In addition, Banks et al., in [8] developed a model to describe the immune response to both HCMV viral infection and introduction of a donor kidney in a renal transplant recipient. Dynamics of the viral load, susceptible and infected host cells, the immune response specific to viral infection and the transplant, and creatinine are incorporated into the model. Delineation between the cellular immune response to HCMV infection and the alloreactive immune response to the transplanted kidney as well as the incorporation of creatinine dynamics are vital additions to the dynamic model.

In the present effort, we develop a mathematical model of BKV infection and renal transplant dynamics at the cellular/viral mechanistic scale for application to renal transplant immunosuppressed individual clinical data. Specifically, we adapt dynamics of the HCMV model in [8] to allot for more specific BKV infection features. We eliminate the use of an antiviral treatment term, incorporate the effect of the alloreactive immune response and the presence of susceptible host cells on the clearance of creatinine, and add the effect of susceptible host cells on the concentration of allospecific CD8+ T cells. In contrast to the model in [19], we assume that the two BKV replication sites, the kidney and the urinary tract, are decoupled, focussing on the kidney as the main replication site. This choice was not only made given the inconclusive literature but the data types available. Examining BKV replication within the urinary tract would require BKV urine data, which is not included in our available data sets. We use a typically available data set to pursue statistically-based model validation or calibration in attempts to discern the specific information content one might expect in such a data set.

The remainder of this paper is organized as follows. In Section 2, we present the biological model for which we base our mathematical model describing the immune response to both BKV infection and the introduction of a donor kidney in a renal transplant patient. An overview of clinical data acquired from collaborators is also given in Section 2. Model calibration and analysis is detailed in Section 3, where we give details regarding a log-transformed system, provide a detailed procedure for sensitivity analysis as well as outline our iterative inverse problem methodology. Details regarding computation of standard errors and confidence intervals are also discussed in Section 3. Results are given and discussed as well. In Section 4 we summarize efforts and conclusions drawn as well as suggest future research efforts.

2 Mathematical model description and data

In this section we describe the dynamics of the viral load V, susceptible H_S and infected H_I host cells, BKV-specific E_V and allospecific E_K effector CD8+ T cells and serum creatinine C with a brief description of the underlying biological model for which we base our mathematical model. Table 1 lists the state variables and Figure 1 diagrams the intracellular dynamics embodied in the model.

Table 1.

Description of state variables.

State	Description	Units
HS	concentration of susceptible host cells	cells/mL
HI	concentration of infected host cells	cells/mL
V	concentration of free BKV	copies/mL
EV	concentration of BKV-specific CD8+ T cells	cells/mL
EK	concentration of allospecific CD8+ T cells that target kidney	cells/mL
C	concentration of serum creatinine	mg/dL

Figure 1.

BKV model.

Active BKV infection targets both renal tubular epithelial cells and urothelial cells. For this model, however, we focus on one BKV target, the renal tubular epithelial cells. Susceptible host cells, the uninfected kidney tubular epithelial cells, H_S, in the absence of infection, are assumed to proliferate through the term ${\lambda }_{HS}(1-\frac{H_S}{{\kappa }_{HS}})H_S$, indicating that new epithelial cells are derived from proliferation of existing H_S. Proliferation is modeled by logistic dynamics with λ_HS being the maximum proliferation rate and κ_HS is the cell density at which proliferation shuts off. A loss of susceptible cells, H_S, due to viral infection which occurs by cell-to-cell transmission, is represented by the term βH_SV. Here we assume that one copy of virion infects one cell. Infected host cells or BKV replicating cells, H_I, lyse due to the cytopathic effect of BK virus, represented by the term δ_HIH_I and produce ρ_V virions before death. In addition, infected host cells are eliminated by the BK-specific effector CD8+ T cells with rate term δ_EH E_V H_I. Free virus is naturally cleared at the rate δ_V by the body and a loss of viral concentration is experienced through the infection of susceptible host cells.

The cellular immune response is the key defense against the BK-viral infection. The terms λ_EV and δ_EV represent the source and death rates of BK-specific effector CD8+ T cells. The concentration of BK-specific CD8+ T cells increases in response to the presence of free virus through the term ρ_EV E_V, where ρ_EV is a bounded positive increasing function of free virus concentration. Specifically, ρ_EV (V) = (ρ̄_EV V)/(V + κ_V) is a saturating nonlinearity with positive constants ρ̄_EV and κ_V. The alloreactive immune response to the transplanted kidney is incorporated into the model via a state variable, E_K, which denotes the effector CD8+ T cells that specifically target the transplant. The source rate for E_K, λ_EK, is dependent upon the HLA donor/recipient matching conducted prior to transplantation. Similar to the BK-specific CD8+ T cells, the concentration of allospecific CD8+ T cells increases in response to the presence of susceptible host cells H_S, since BK-virus targets kidney cells, represented by the term ρ_EK E_K, where ρ_EK (H_S) = (ρ̄_EK H_S)/(H_S + κ_KH) with positive constants ρ̄_EV and κ_KH. The death rate of E_K is represented by δ_EK.

Finally, we discuss the role of creatinine in the model. Creatinine is a waste product in the blood resulting from muscle activity and is removed by the healthy kidney. Therefore, serum creatinine concentration C is used as a surrogate for glomerular filtration rate (GFR), a commonly used index of kidney function [30]. The production rate of C is represented by λ_C and when the kidney is impaired, creatinine is not effectively filtered and its concentration increases. To account for the negative effect of the alloreactive immune response E_K on the kidney and the positive effect of susceptible cells H_S (Recall that the renal allograft is a site of replication. Hence, the concentration of susceptible cells reflects the health of the kidney.), the clearance rate δ_C is defined as follows

$${\delta }_C(E_K,H_S)=\frac{{\delta }_{C0}{\kappa }_{EK}}{E_K+{\kappa }_{EK}}·\frac{H_S}{H_S+{\kappa }_{CH}}.$$

Table 2 lists the parameters used in the model.

Table 2.

Model parameters and their corresponding descriptions.

Parameter	Description	Unit
λHS	proliferation rate for HS	1/day
κV	saturation constant	copies/mL
κHS	saturation constant	cells/mL
λEK	source rate of EK	cells/(mL·day)
β	infection rate of HS by V	mL/(copies·day)
δEK	death rate of EK	1/day
δHI	death rate of HI by V	1/day
λC	production rate for C	mg/(dL·day)
ρV	# virions produced by HI before death	copies/cell
δC0	maximum clearance rate for C	1/day
δEH	elimination rate of HI by EV	mL/(cells·day)
κEK	saturation constant	cells/mL
δV	natural clearance rate of V	1/day
κCH	saturation constant	cells/mL
λEV	source rate of EV	cells/(mL·day)
ρ̄EK	maximum proliferation rate for EK	1/day
δEV	death rate of EV	1/day
κKH	saturation constant	cells/mL
ρ̄EV	maximum proliferation rate for EV	1/day
εI	efficacy of immunosuppressive drugs

Based on the above discussion, the model is given as follows.

$${Ḣ}_S={\lambda }_{HS}(1-\frac{H_S}{{\kappa }_{HS}})H_S-\beta H_{S}V,$$ (1)

$${Ḣ}_I=\beta H_{S}V-{\delta }_{HI}{H}_I-{\delta }_{EH}{E}_V{H}_I,$$ (2)

$$\dot{V}={\rho }_V{\delta }_{HI}{H}_I-{\delta }_{V}V-\beta H_{S}V,$$ (3)

$${Ė}_V=(1-{\epsilon }_I)[{\lambda }_{EV}+{\rho }_{EV}(V)E_V]-{\delta }_{EV}{E}_V,$$ (4)

$${Ė}_K=(1-{\epsilon }_I)[{\lambda }_{EK}+{\rho }_{EK}(H_S)E_K]-{\delta }_{EK}{E}_K,$$ (5)

$$Ċ={\lambda }_C-{\delta }_C(E_K,H_S)C.$$ (6)

with initial conditions,

$$(H_S(0),H_I(0),V(0),E_V(0),E_K(0),C(0))=(H_{S0},H_{I0},V_0,E_{V0},E_{K0},C_0).$$ (7)

We note that (1)–(4) describe the immune response to the viral infection coupled with (5) and (6) describing the immune response to the transplanted kidney. Here ε_I represents the efficacy of immunosuppressive drugs and is assumed to be scaled to less than or equal to 1. This variable serves as the controller of the system to achieve balance between under-suppression and over-suppression of the patient’s immune system.

2.1 Data

The data for our investigation is from Massachusetts General Hospital (MGH) and our discussions here consists of one patient record, TOS003. The data collection is performed as part of the patient’s routine medical care. Visits include pre-transplantation evaluation, day of transplant, and post-transplantation visits. Day of transplant and post-transplantation visits are pertinent for model validation. Record TOS003 describes immunosuppression regimen, renal function recorded in plasma creatine (mg/dL) levels, and infectious complications given in BKV viral plasma loads (in DNA copies/mL). TOS0003 associated data consists of eight viral load measurements and sixteen serum creatinine measurements. It should be noted that the recipient was diagnosed with “BK virus reactivation” over the course of the first three months post-transplantation. It was documented that TOS003 was given immunosuppressive treatment upon organ transplantation and monitored accordingly. We note that the efficacy of the immunosuppression is a function of time, given the evidence in the data, however, it is difficult to quantify the efficacy of the immunosupression from the drug regimen records. The dosage, type, and combination of drugs are changed after each visit. Hence, for simplicity, we assume that ε_I can be approximated by the following piecewise constant function.

$${\epsilon }_I(t)=\{\begin{array}{cc}{\epsilon }_1,\hfill & t\in [0,21],\hfill \\ {\epsilon }_2,\hfill & t\in (21,60],\hfill \\ {\epsilon }_3,\hfill & t\in (60,120],\hfill \\ {\epsilon }_4,\hfill & t\in (120,450],\hfill \end{array}$$ (8)

where the time frames are chosen based on the drug regimen record. It is also important to note that we are assuming test results displaying “None detected” equate to the detection limit 20 copies/mL of free virus in the system [21]. In addition, no test quantifying the BK viral load was conducted pre-transplant or the day of transplant, however, due to the seroprevalence of BKV in the population, it is assumed that the patient had a latent infection.

3 Model calibration and analysis

The model contains 6 variables and 29 constant parameters (model parameters and initial conditions). Equations (1)–(6) were first written as a vector system

$$\dot{\overline{\mathbf{x}}}=\mathbf{ḡ}(\overline{\mathbf{x}};\overline{\mathbf{p}},{\overline{\mathbf{x}}}_0),$$

$$\overline{\mathbf{x}}(0)={\overline{\mathbf{x}}}_0,$$

where x̄ = [H_S, H_I, V, E_V, E_K, C]^T is the vector of model states, p̄ = [λ_HS, β, δ_EH, δ_V, ρ̄_EV, δ_EV, δ_EK, δ_C0, κ_HS, δ_HI, ρ_V, λ_EV, κ_V, λ_EK, λ_C, κ_EK, κ_CH, ρ̄_EK, κ_KH, ε₁, ε₂, ε₃, ε₄]^T ∈ ℝ²³ is the vector of model parameters, and x̄₀ = [H_S0, H_I0, V₀, E_V0, E_K0, C₀]^T ∈ ℝ⁶ is the set of initial conditions. Due to the scale difference between model states, we adopted the approach used in [1, 3]: solutions were determined for a log-transformed system. We remark that this is common in the biological literature to use log scale to deal with very large numbers; for example, in HIV studies and other viral infection studies both the viral load level (which can be as high as millions of copies/mL) and the changes in the viral load level can be large and hence they are often reported and analyzed in log scale (e.g., see [24, 31, 33]). Since model parameter values and initial conditions are in different scales, a subset of the model parameters and initial conditions are also log-transformed to the log scale. This log-transformation approach is used to convert all the analyzed quantities roughly to the same scale. It is worth noting that whenever we carry out a log-transformation for a quantity, we only log-transform its corresponding numerical value. In other words, log₁₀(x̄) is always understood as a shorthand notation for log₁₀((x̄ [units])/(1 [units])) so the argument of log function is really dimensionless. An alternative approach, a common practice in engineering, to deal with the scale difference among analyzed quantities is to normalize them so that the numerical values of the resulting quantities vary between 0 and 1, e.g., x = (x̄ − x̄_min)/(x̄_max − x̄_min), where x stands for quantity x̄ after normalization and subscripts min and max respectively denote the minimal and maximal values of x̄. However, for most biological applications the minimal and maximal values of the analyzed quantities are usually unknown and may vary among individuals, which is true for the problem we presented here. Thus, this normalization approach would result in a large number of extra unknown parameters that need to be identified (recall that the model contains 6 model states and 29 constant parameters). Considering the limited number of observations available to us, this approach can provide significant, if not impossible, challenges for the inverse problem investigated below and thus will not be considered in this paper.

Let,

$$x_i={\text{log}}_{10}({\overline{x}}_i),i=1,2,3,4,5,$$

$$x_6={\overline{x}}_6,$$

$$x_{0i}={\text{log}}_{10}({\overline{x}}_{0i}),i=1,2,3,4,5,$$

$$x_{06}={\overline{x}}_{06},$$

$$p_j={\text{log}}_{10}({\overline{p}}_j),j=1,2,\dots ,19,$$

$$p_j={\overline{p}}_j,j=20,21,22,23.$$

Then we have the system

$$\mathbf{ẋ}=\mathbf{g}(\mathbf{x};\mathbf{p},{\mathbf{x}}_{\mathbf{0}}),\mathbf{x}(\mathbf{0})={\mathbf{x}}_{\mathbf{0},}$$ (9)

where g = (g₁, g₂, ⋯ g₆)^T is given by

$$g_i(\mathbf{x};\mathbf{p},{\mathbf{x}}_{\mathbf{0}})=\frac{{10}^{-x_i}}{\text{ln}(10)}{ḡ}_i(\overline{\mathbf{x}};\overline{\mathbf{p}},{\overline{\mathbf{x}}}_0),i=1,2,\dots 5,$$

$$g_6(\mathbf{x};\mathbf{p},{\mathbf{x}}_{\mathbf{0}})={ḡ}_6(\overline{\mathbf{x}};\overline{\mathbf{p}},{\overline{\mathbf{x}}}_0).$$

We remark that by using the above log-transformed system (9) one can resolve a problem of states becoming unrealistically negative in solving model (1)–(6) due to round-off error: nonnegative solutions of this model should stay so throughout numerical simulation. This log-transformation approach also enables the changes in model states due to the changes in parameters to be more measurable. In addition, this enables more easily formulation of a reasonable stopping criterion for the inverse problem and can also make the associated optimization algorithm converge faster (vastly different magnitudes of analyzed quantities increase condition numbers and hence the associated optimization algorithm may require more iterations to converge). From a statistical point of view, log-transformation is also a standard technique to render the observations more nearly normally distributed, and it is also commonly used to render heteroscedastic measurement errors more homoscedastic (e.g., see [13, 14, 37]) so that the resulting inverse problem is easier to implement (in ordinary least squares approaches as opposed to generalized least squares approaches, e.g., see [11, 37] for details).

3.1 Forward simulations

Forward simulations were carried out by numerically solving the log transformed version of model equations (9) in Matlab using the ODE variable order variable step-size solver ode15s over a time course of 450 days. Parameter values used are listed in Table 3. The values of these parameters were either derived from published experimental studies or through simulation to acquire a benchmark value. For those parameters whose values can be found or derived from the literature, specifics are detailed below.

1. The parameter ρ_V, which quantifies the number of virions released by BKV-infected cells, was taken to be ρ_V ≈ 6000 copies/cell [19, 20, 21, 36].

2. The measured BK viral half-life was found to be ${\mathbf{t}}_{\frac{\mathbf{1}}{\mathbf{2}}}\approx \mathbf{1.1}\mathbf{h}\mathbf{r}\mathbf{s}-\mathbf{9}\mathbf{\text{days}}$. This implies that the total clearance rate of BK is from 0.077 to 15.1232 per day. In the simulations, we set δ_V = 0.4 per day [19, 20].

Table 3.

Initial guess for parameter values θ = (p̄, x̄₀) used in the parameter estimation simulations.

Parameter	Value	Parameter	Value
λHS	0.03 per day	κHS	1 × 103 cells/mL
β	8 × 10−8 mL/(copies·day)	δHI	0.08 per day
δEH	1.5 × 10−3 mL/(cells·day)	ρV	6 × 103 copies/cell
δV	0.4 per day	λEV	1 × 10−3 cells/(mL·day)
ρ̄EV	0.3 per day	κV	100 copies/mL
δEV	0.1 per day	λEK	2 × 10−3 cells/(mL·day)
δEK	0.1 per day	λC	6 × 10−3 mg/(dL·day)
δC0	0.01 per day	κEK	0.2 cells/mL
κCH	10 cells/mL	ρ̄EK	0.2 per day
κKH	85 cells/mL	ε1	0.1
ε2	0.38	ε3	0.6
ε4	0.3	HS0	5 × 103 cells/mL
HI0	60 cells/mL	V0	5 × 104 copies/mL
EV0	0.04 cells/mL	EK0	0.4 cells/mL
C0	1.07 mg/dL

3.2 Sensitivity analysis

In practice, one may be in a situation of estimating a large number of unknown parameters with a limited data set. Such a situation is true in our case. To alleviate some of this difficulty, sensitivity analysis has been widely used in inverse problem investigations [2, 3, 4, 5, 6, 12, 34]. This process identifies the model parameters and initial conditions to which the model outputs are most sensitive. Specifically, sensitivity analysis provides insight into how changes in the parameters can affect the model output. In addition, this framework can be used to construct confidence intervals for parameter estimates using asymptotic properties of estimators.

To compute the sensitivities of model outputs to the model parameters and initial conditions, one proceeds to determine the sensitivity of each model state x_i to each parameter p_j and initial condition x_0j. These are defined as the derivatives of the model states with respect to the parameters, ∂x_i/∂p_j and ∂x_i/∂x_0j, which satisfy in our case

$$\frac{d}{dt}\frac{\partial x_i}{\partial p_j}=\sum _{l=1}^6\frac{\partial g_i}{\partial x_l}\frac{\partial x_l}{\partial p_j}+\frac{\partial g_i}{\partial p_j},$$

$$\frac{\partial x_i}{\partial p_j}(0)=0,i=1,2,\dots 6,j=1,2,\dots ,23,$$ (10)

$$\frac{d}{dt}\frac{\partial x_i}{\partial x_{0j}}=\sum _{l=1}^6\frac{\partial g_i}{\partial x_l}\frac{\partial x_l}{\partial x_{0j}},$$

$$\frac{\partial x_i}{\partial x_{0j}}(0)={\delta }_{ij},i=1,2,\dots 6,j=1,2,\dots ,6,$$ (11)

where

$${\delta }_{ij}=\{\begin{array}{cc}1,\hfill & i=j,\hfill \\ 0,\hfill & i\ne j.\hfill \end{array}$$

The sensitivities $\frac{\partial \mathbf{x}}{\partial \mathbf{p}}$ and $\frac{\partial \mathbf{x}}{\partial {\mathbf{x}}_{\mathbf{0}}}$ can be calculated by solving (9), (10), (11) simultaneously in Matlab using ode15s where the derivatives $\frac{\partial \mathbf{g}}{\partial \mathbf{x}},\frac{\partial \mathbf{g}}{\partial \mathbf{p}}$ are obtained through automatic differentiation (AD) using myAD and tssolve in Matlab. In our case, we have data for both the free virus, V, and serum creatinine, C, states. Therefore, we only need to compute the sensitivities corresponding to states x_i, i = 3, 6. Initial conditions and model parameter values used are listed in Table 3. Results of the sensitivity analysis informed us as to which parameters were to be estimated. These findings along with the corresponding confidence intervals for these estimated parameters are described below.

It is worth emphasizing that sensitivity coefficients obtained above are for the log-transformed system (9). We note that in the case where x_i = x̄_i, p_j = log₁₀(p̄_j) and x_0j = log₁₀(x_0j) we have

$$\frac{\partial x_i}{\partial p_j}=\frac{\partial x_i}{\partial {\overline{x}}_i}\frac{\partial {\overline{x}}_i}{\partial {\overline{p}}_j}\frac{\partial {\overline{p}}_j}{\partial p_j}=\text{log}(10){\overline{p}}_j\frac{\partial {\overline{x}}_i}{\partial {\overline{p}}_j},\frac{\partial x_i}{\partial x_{0j}}=\frac{\partial x_i}{\partial {\overline{x}}_i}\frac{\partial {\overline{x}}_i}{\partial {\overline{x}}_{0j}}\frac{\partial {\overline{x}}_{0j}}{\partial x_{0j}}=\text{log}(10){\overline{x}}_{0j}\frac{\partial {\overline{x}}_i}{\partial {\overline{x}}_{0j}}.$$

Hence, the sensitivity coefficients $\frac{\partial x_i}{\partial p_j}$ and $\frac{\partial x_i}{\partial x_{0j}}$ obtained here are really the so-called relative sensitivity coefficients in [34]. We also note that in the case where x_i = log₁₀(x̄_i), p_j = log₁₀(p̄_j) and x_0j = log₁₀(x̄_0j) we have

$$\frac{\partial x_i}{\partial p_j}=\frac{\partial x_i}{\partial {\overline{x}}_i}\frac{\partial {\overline{x}}_i}{\partial {\overline{p}}_j}\frac{\partial {\overline{p}}_j}{\partial p_j}=\frac{{\overline{p}}_j}{{\overline{x}}_i}\frac{\partial {\overline{x}}_i}{\partial {\overline{p}}_j},\frac{\partial x_i}{\partial x_{0j}}=\frac{\partial x_i}{\partial {\overline{x}}_i}\frac{\partial {\overline{x}}_i}{\partial {\overline{x}}_{0j}}\frac{\partial {\overline{x}}_{0j}}{\partial x_{0j}}=\frac{{\overline{x}}_{0j}}{{\overline{x}}_i}\frac{\partial {\overline{x}}_i}{\partial {\overline{x}}_{0j}}.$$

Hence, the sensitivity coefficients $\frac{\partial x_i}{\partial p_j}$ and $\frac{\partial x_i}{\partial x_{0j}}$ obtained here are really the so-called dimensionless sensitivity coefficients in [34]. For more information on the relative sensitivity coefficients and dimensionless sensitivity coefficients, we refer the interested reader to [34].

3.3 Parameter estimation

The observed amount of free virus (DNA) in the blood is represented by ${\overline{y_i}}^1$, with corresponding measured time point $t_i^1$, i = 1, 2, …, n₁ and ${\overline{y_i}}^2$ is the observed amount of serum creatinine at time point $t_i^2$, i = 1, 2, …, n₂. We define $y_i^1={\text{log}}_{10}({\overline{y_i}}^1)$, i = 1, 2, …, n₁ and $y_i^2={\overline{y_i}}^2$, i = 1, 2, …, n₂. Following standard inverse problem procedures [11, 12, 13, 14, 34, 37], we define random variables Y_i¹ and Y_i² and formulate the statistical model as follows:

$${Y_i}^1=f_1(t_i^1;{\mathbf{q}}_{\mathbf{0}})+{\epsilon }_i^1,i=1,2,\dots ,n_1,$$ (12)

$${Y_i}^2=f_2(t_i^2;{\mathbf{q}}_{\mathbf{0}})+{\epsilon }_i^2,i=1,2,\dots ,n_2,$$

where n₁ = 8 and n₂ = 16. Here $f_1(t_i^1;{\mathbf{q}}_{\mathbf{0}})=x_3(t_i^1;{\mathbf{q}}_{\mathbf{0}})$ and $f_2(t_i^2;{\mathbf{q}}_{\mathbf{0}})=x_6(t_i^2;{\mathbf{q}}_{\mathbf{0}})$, where q₀ ∈ ℝ^κ denotes the hypothesized “true values” of the model parameters and initial conditions that need to be estimated (κ is a positive integer and denotes the number of model parameters and initial conditions to be estimated). The observation errors ${\epsilon }_i^1$, i = 1, 2, …, n₁ and ${\epsilon }_i^2$, i = 1, 2, …, n₂ are assumed to be independent and identically distributed (i.i.d) with zero mean and constant variance ${\sigma }_0^2$. Therefore, q₀ can be properly estimated by using an ordinary least squares (OLS) technique

$$\hat{\mathbf{q}}=\text{arg}\underset{\mathbf{q}\in 𝒬}{\text{min}}[\sum _{i=1}^{n_1}{|f_1(t_i^1;\mathbf{q})-y_i^1|}^2+\sum _{i=1}^{n_2}{|f_2(t_i^2;\mathbf{q})-y_i^2|}^2],$$ (13)

where 𝒬 is some compact set of admissible values in ℝ^κ.

Note that we have a large number of parameters (29 parameters) with little experimental data (n₁+n₂ = 24 time observations). Hence, in attempts to produce reliable estimates, the parameter estimation was implemented in an iterative process similar to that used in Adams et al., [1], Banks et al., [3]. This process entailed identifying the most influential or sensitive parameter subset at each step. Quantification of influence is determined by sensitivity rankings based on the quantities

$$S_V(q_j)=\sqrt{\sum _{k=1}^{n_1}{\left(\frac{\partial x_3(t_k^1)}{\partial q_j}\right)}^2},j=1,2,\dots \kappa ,$$

$$S_C(q_j)=\sqrt{\sum _{k=1}^{n_2}{\left(\frac{\partial x_6(t_k^2)}{\partial q_j}\right)}^2},j=1,2,\dots \kappa .$$ (14)

Again, κ denotes the number of model parameters and initial conditions to be estimated.

Initially, we computed sensitivity rankings for all parameters with respect to the viral load and creatinine state variables using (14), where parameters values are chosen as those given in Table 3. Our findings are given in Table 6 Appendix A. Given these results in Table 6, we identified the parameters that are the most influential in the viral load state variables and the most influential in the creatinine state variables. We subsequently designed an iterative inverse problem algorithm that utilizes the sensitivity rank information to propose parameters sets that we wish to estimate. All sensitivity ranking results for the following algorithm are found in the tables in Appendix A.

Table 6.

Sensitivity rankings for 29 estimated parameters.

SV	Parameter	SC	Parameter
0	δEK	1.8375 × 10−8	κV
0	δC0	2.5563 × 10−6	λEV
0	λEK	6.1953 × 10−5	κCH
0	λC	1.4577 × 10−4	EV0
0	κEK	1.8694 × 10−4	δEH
0	κCH	1.9313 × 10−4	λHS
0	ρ̄EK	5.6584 × 10−4	V0
0	κKH	7.3820 × 10−4	HS0
0	EK0	1.9130 × 10−3	HI0
0	C0	4.1888 × 10−3	δEV
4.2652 × 10−3	V0	4.4437 × 10−3	λEK
0.016519	HI0	0.011330	δHI
0.025134	HS0	0.019960	ρ̄EV
0.025148	λEV	0.063921	δV
0.087472	λHS	0.071157	ρV
1.4476	ε4	0.084435	β
5.3296	κV	0.14688	κHS
14.003	EV0	0.34333	κEK
15.359	δHI	0.42737	EK0
18.007	δEH	0.51850	κEK
131.04	ε1	0.68038	δC0
280.99	δV	1.2403	ε4
282.13	ρV	1.3561	ε1
292.00	ε3	3.1602	C0
296.00	κHS	3.5369	ε2
300.73	β	5.1086	ε3
446.59	ε2	49.962	λC
1433.4	δEV	59.907	δEK
4680.9	ρ̄EV	75.318	ρ̄EK

Parameters were estimated using the Matlab routine lsqnonlin. It is important to note that the use of overbar notation when referencing model parameters and initial conditions refers to the original scale. This is done to simplify notation when reporting estimated and fixed parameters within the algorithm. We remind the reader that all simulations were conducted using the log-transformed system with log-scaled parameter values as previously discussed.

• Step 1: We fixed model parameters and initial conditions q̄_U1 = [λ_EV, λ_EK, κ_EK, κ_CH, κ_KH, H_S0, H_I0, V₀, E_K0]^T ∈ ℝ⁹ using corresponding values found in Table 3 based on sensitivity rankings in Table 6. Next, we estimated the remaining parameters q̄_E1 = [λ_HS, β, δ_EH, δ_V, ρ̄_EV, δ_EV, δ_EK, δ_C0, κ_HS, δ_HI, ρ_V, κ_V, λ_C, ρ̄_EK, ε₁, ε₂, ε₃, ε₄, E_V0, C₀]^T ∈ ℝ²⁰ with initial values found in Table 3 and obtained q̂_E1 using the OLS procedure in (13). Results ${\theta }_{\mathit{\text{opt}}}^1={[q_{U1},{\hat{q}}_{E1}]}^T\in {ℝ}^{29}$ are found in Table 10. We will refer to this experiment as “parameters estimation # 1”.

• Step 2: We performed a sensitivity analysis for the 20 estimated parameters in Step 1 using corresponding values in ${\theta }_{\mathit{\text{opt}}}^1$. The resulting sensitivity rankings are presented in Table 8. We then fixed an additional 5 parameters chosen from these 20 parameters based on the sensitivity rankings. That is, we fixed 14 model parameters and initial conditions q̄_U2 = [q̄_U1, δ_EH, κ_V, ε₁, E_{V 0}, C₀]^T ∈ ℝ¹⁴ using corresponding values found in ${\theta }_{\mathit{\text{opt}}}^1$. We estimated the remaining parameters q̄_E2 = [λ_HS, β, δ_V, ρ̄_EV, δ_EV, δ_EK, δ_C0, κ_HS, δ_HI, ρ_V, λ_C, ρ̄_EK, ε₂, ε₃, ε₄]^T ∈ ℝ¹⁵ with initial values found in ${\theta }_{\mathit{\text{opt}}}^1$ and obtained q̂_E2 using OLS procedure. Results ${\theta }_{\mathit{\text{opt}}}^2={[q_{U2},{\hat{q}}_{E2}]}^T\in {ℝ}^{29}$ are found in Table 12. We will refer to this experiment as “parameters estimation # 2”.

• Step 3: We performed a sensitivity analysis for the 15 estimated parameters in Step 2 using corresponding values in ${\theta }_{\mathit{\text{opt}}}^2$. The resulting sensitivity rankings are presented in Table 9. We then fixed an additional 5 parameters chosen from these 15 parameters based on the sensitivity rankings. That is, we fixed 19 model parameters and initial conditions q̄_U3 = [q̄_U2, λ_HS, δ_C0, δ_HI, λ_C, ε₂]^T ∈ ℝ¹⁹ using corresponding values found in ${\theta }_{\mathit{\text{opt}}}^2$. We then estimated the remaining parameters q̄_E3 = [β, δ_V, ρ̄_EV, δ_EV, δ_EK, κ_HS, ρ_V, ρ̄_EK, ε₃, ε₄]^T ∈ ℝ¹⁰ with initial values found in ${\theta }_{\mathit{\text{opt}}}^2$ and obtained q̂_E3 using the OLS procedure. Results ${\theta }_{\mathit{\text{opt}}}^3={[q_{U3},{\hat{q}}_{E3}]}^T\in {ℝ}^{29}$ are found in Table 14. We will refer to this experiment as “parameters estimation # 3”.

• Step 4: We performed a sensitivity analysis for the 10 estimated parameters in Step 3 using corresponding values in ${\theta }_{\mathit{\text{opt}}}^3$. The resulting sensitivity rankings are presented in Table 7. We then fixed an additional 5 parameters chosen from these 10 parameters based on the sensitivity rankings. That is, we fixed 24 model parameters and initial conditions q̄_U4 = [q̄_U3, δ_V, κ_HS, ρ_V, ε₃, ε⁴]^T ∈ ℝ²⁴ using corresponding values found in ${\theta }_{\mathit{\text{opt}}}^3$. Next, we estimated the remaining parameters q̄_E4 = [β, ρ̄_EV, δ_EV, δ_EK, ρ̄_EK]^T ∈ ℝ⁵ with initial values found in ${\theta }_{\mathit{\text{opt}}}^3$ and obtained q̂_E4 using the OLS procedure. Results ${\theta }_{\mathit{\text{opt}}}^4={[q_{U4},{\hat{q}}_{E4}]}^T\in {ℝ}^{29}$ are found in Table 4. We will refer to this experiment as “parameters estimation # 4”.

Table 10.

Parameter estimation # 1 results for TOS003 viral load and serum creatinine data on [0, 450].Top half of the table gives fixed parameters whereas the bottom half of the table displays the estimated parameters.

Parameter	${\theta }_{\mathit{\text{opt}}}^1$	Parameter	${\theta }_{\mathit{\text{opt}}}^1$
log10 λEV	−3.0000	log10 λEK	−2.6990
log10 κEK	−0.69897	log10 κCH	1.0000
log10 κKH	1.9294	log10 HS0	3.6990
log10 HS0	1.7782	log10 V0	4.6990
log10 EK0	−0.39794
log10 λHS	−1.5291	log10 β	−7.0633
log10 δEH	−2.7500	log10 δV	−0.43203
log10 ρ̄EV	−0.60674	log10 δEV	−0.96806
log10 δEK	−0.99357	log10 δC0	−1.8907
log10 κHS	3.0032	log10 δHI	−1.0681
log10 ρV	3.6349	log10 κV	2.2569
log10 λC	−2.1994	log10 ρ̄EK	−0.78374
ε1	0.10085	ε2	0.36754
ε3	0.60763	ε4	0.35925
log10 EV0	−1.4234	C0	0.67629

Table 8.

Sensitivity rankings for 20 estimated parameters.

SV	Parameter	SC	Parameter
0	δEK	1.4997 × 10−7	κV
0	δC0	1.2963 × 10−4	EV0
0	λC	1.9144 × 10−4	δEH
0	ρ̄EK	0.012164	δEV
0	C0	0.026468	ρ̄EV
0.66447	EV0	0.029674	δHI
1.0155	δEH	0.040367	λHS
4.6235	ε1	0.32979	δV
5.7545	κV	0.37520	ρV
12.076	λHS	0.42114	β
15.216	δHI	0.48602	κHS
17.445	ε2	0.64429	ε1
40.518	ε3	1.5687	ε2
173.58	ε4	1.9628	ε3
175.34	κHS	2.2956	C0
183.78	δV	7.8627	δC0
205.82	ρV	9.0800	λC
235.35	β	53.932	ε4
639.54	δEV	145.34	ρ̄EK
1022.7	ρ̄EV	168.46	δEK

Table 12.

Parameter estimation # 2 results for TOS003 viral load and serum creatinine data on [0, 450]. Top half of the table gives fixed parameters whereas the bottom half of the table displays the estimated parameters.

Parameter	${\theta }_{\mathit{\text{opt}}}^2$	Parameter	${\theta }_{\mathit{\text{opt}}}^2$
log10 δEH	−2.7500	log10 λEV	−3.0000
log10 κV	2.2569	log10 λEK	−2.6990
log10 κEK	−0.69897	log10 κCH	1.0000
log10 κKH	1.9294	ε1	0.10085
log10 HS0	3.6990	log10 HS0	1.7782
log10 V0	4.6990	log10 EV0	−1.4234
log10 EK0	−0.39794	C0	0.67629
log10 λHS	−1.5205	log10 β	−7.0671
log10 δV	−0.43233	log10 ρ̄EV	−0.60429
log10 δEV	−0.96680	log10 δEK	−0.99302
log10 δC0	−1.8520	log10 κHS	3.0060
log10 δHI	−1.0704	log10 ρV	3.6344
log10 λC	−2.1805	log10 ρ̄EK	−0.78540
ε2	0.36575	ε3	0.61164
ε4	0.36374

Table 9.

Sensitivity rankings for 15 estimated parameters.

SV	Parameter	SC	Parameter
0	δEK	0.0097914	δEV
0	δC0	0.021305	ρ̄EV
0	λC	0.026802	δHI
0	ρ̄EK	0.027472	λHS
10.683	λHS	0.26025	δV
13.489	δHI	0.29872	ρV
18.399	ε2	0.33683	β
42.884	ε3	0.37132	κHS
168.53	κHS	1.5572	ε2
174.92	δV	1.7727	ε3
186.84	ε4	7.6471	δC0
195.44	ρV	8.2445	λC
223.38	β	42.258	ε4
688.21	δEV	110.23	ρ̄EK
1104.2	ρ̄EV	130.27	δEK

Table 14.

Parameter estimation # 3 results for TOS003 viral load and serum creatinine data on [0, 450]. Top half of the table gives fixed parameters whereas the bottom half of the table displays the estimated parameters.

Parameter	${\theta }_{\mathit{\text{opt}}}^3$	Parameter	${\theta }_{\mathit{\text{opt}}}^3$
log10 λHS	−1.5205	log10 δEH	−2.7500
log10 δC0	−1.8520	log10 δHI	−1.0704
log10 λEV	−3.0000	log10 κV	2.2569
log10 λEK	−2.6990	log10 λC	−2.1805
log10 κEK	−0.6990	log10 κCH	1.0000
log10 κKH	1.9294	ε1	0.1009
ε2	0.3658	log10 HS0	3.6990
log10 HS0	1.7782	log10 V0	4.6990
log10 EV0	−1.4234	log10 EK0	−0.3979
C0	0.6763
log10 β	−7.0675	log10 δV	−0.4295
log10 ρ̄EV	−0.6005	log10 δEV	−0.9635
log10 δEK	−0.9945	log10 κHS	3.0111
log10 ρV	3.6327	log10 ρ̄EK	−0.7850
ε3	0.5999	ε4	0.3649

Table 7.

Sensitivity rankings for 10 estimated parameters.

SV	Parameter	SC	Parameter
0	δEK	0.011400	δEK
0	ρ̄EK	0.024951	ρ̄EK
1.0137	δV	0.30254	δV
40.538	ε3	0.34492	ρV
171.61	ε4	0.38744	β
175.47	κHS	0.45276	κHS
205.94	ρV	1.8521	ε3
235.60	β	50.789	ε4
624.14	δEV	135.23	ρ̄EK
996.73	ρ̄EV	157.37	δEK

Table 4.

Parameter estimation # 4 results for TOS003 viral load and serum creatinine data on [0, 450]. Top half of the table gives fixed parameters whereas the bottom half of the table displays the estimated parameters.

Parameter	${\theta }_{\mathit{\text{opt}}}^4$	Parameter	${\theta }_{\mathit{\text{opt}}}^4$
log10 λHS	−1.5205	log10 δEH	−2.7500
log10 δV	−0.4295	log10 δC0	−1.8520
log10 κHS	3.0111	log10 δHI	−1.0704
log10 ρV	3.6327	log10 λEV	−3.0000
log10 κV	2.2569	log10 λEK	−2.6990
log10 λC	−2.1805	log10 κEK	−0.6990
log10 κCH	1.0000	log10 κKH	1.9294
ε1	0.1009	ε2	0.3658
ε3	0.5999	ε4	0.3649
log10 HS0	3.6990	log10 HS0	1.7782
log10 V0	4.6990	log10 EV0	−1.4234
log10 EK0	−0.3979	C0	0.6763
log10 β	−7.0674	log10 ρ̄EV	−0.6006
log10 δEV	−0.9636	log10 δEK	−0.9945
log10 ρ̄EK	−0.7853

Next, we outline a method to quantify the uncertainty in our parameter estimations. Two methods that have been widely used in the literature to quantify uncertainty in parameter estimates are asymptotic theory and bootstrapping. Both have been investigated and compared in Banks et al. [7] for problems with different form and level of noise. It was found that asymptotic theory is always faster computationally than bootstrapping and there is no clear advantage in using bootstrapping versus asymptotic theory when the constant variance using OLS is assumed. For these reasons, we will use asymptotic theory [4, 37] to quantify the uncertainty in our parameter estimations.

We calculated standard errors and confidence intervals [11, 12] in order to quantify the uncertainty in parameter estimates. To compute these values, we must define some terms. Recall the statistical model defined in (12). Let

$$ℱ(\mathbf{q})={(f_1(t_1^1;\mathbf{q}),f_1(t_2^1;\mathbf{q}),\dots ,f_1(t_{n_1}^1;\mathbf{q}),f_2(t_1^2;\mathbf{q}),f_2(t_2^2;\mathbf{q}),\dots ,f_2(t_{n_2}^2;\mathbf{q}))}^T.$$

Then the sensitivity matrix χ(q) is an (n₁ + n₂) × κ matrix, where N = n₁ + n₂ is the total number of viral and creatinine data points and κ is the number of estimated parameters, with its (i, j)th element being defined as

$${\chi }_{ij}(\mathbf{q})=\frac{\partial {ℱ}_i(\mathbf{q})}{\partial q_j},i=1,2,\dots n_1+n_2,j=1,2,\dots ,\kappa ,$$ (15)

where ℱ_i is the ith element of ℱ, and q_j is the jth element of q. Given the data ${\{y_i\}}_{i=1}^N=\{y_1^1,y_2^1,\dots ,y_{n_1}^1,y_1^2,y_2^2,\dots y_{n_2}^2\}$ and the resulting parameter estimate q̂, the variance ${\sigma }_0^2$ can be approximated by

$${\sigma }_0^2\approx {\hat{\sigma }}^2=\frac{1}{N-\kappa }\sum _{j=1}^N{[y_j-{ℱ}_j(\hat{\mathbf{q}})]}^2.$$ (16)

With these values, we can calculate

$$\hat{\mathrm{\Sigma }}(\hat{\mathbf{q}})={\hat{\sigma }}^2{[\chi {(\hat{\mathbf{q}})}^T\chi (\hat{\mathbf{q}})]}^{-1}.$$ (17)

This matrix is known as the covariance matrix and is used to compute the standard errors for each element of q̂ given by

$$S{E}_k(\hat{\mathbf{q}})=\sqrt{{\hat{\mathrm{\Sigma }}}_{kk}(\hat{\mathbf{q}})},k=1,2,\dots \kappa .$$ (18)

Hence, the 100(1 − α)% confidence intervals are given by

$$[{\hat{q}}_k-t_{1-\alpha /2}S{E}_k(\hat{\mathbf{q}}),{\hat{q}}_k+t_{1-\alpha /2}S{E}_k(\hat{\mathbf{q}})].$$ (19)

We determine t_1−α/2 by Prob{T ≥ t_1−α/2 } = α/2, where T has a student’s t distribution t^N−κ with N − κ degrees of freedom.

For the following results, we chose α = 0.05. Results obtained for parameter estimation steps #1, #2, #3 are presented in Appendix B. We observe that parameter estimations # 1–3 produced good fits to the data (as seen in Figures 4–6). However, there is substantial uncertainty in at least some components of the estimates for all three cases (see Tables 11, 13, 15). The reliability of parameter estimates depends on the number of parameters estimated. Specifically, $\underset{i}{\text{max}}|{\gamma }_i|$ increases as κ increases, where γ_i denotes the absolute ratio of the standard error to the corresponding estimate for the ith estimated parameter. This is in agreement with common understanding that for a fixed number of observations, the estimation accuracy decreases as the number of parameters increases [11]. Further discussions of these results are found in Appendix B.

Figure 4.

Using ${\theta }_{\mathit{\text{opt}}}^1$ in Table 10 for parameter estimation # 1, (Left) Plot of model solution and viral load data on [0, 450]. (Right) Plot of model solution and serum creatinine data on [0, 450].

Figure 6.

Using ${\theta }_{\mathit{\text{opt}}}^3$ in Table 14 for parameter estimation # 3, (Left) Plot of model solution and viral load data on [0, 450]. (Right) Plot of model solution and serum creatinine data on [0, 450].

Table 11.

Parameter estimates, asymptotic standard errors, confidence intervals, and the ratios γ of standard error to the corresponding estimate for parameter estimation # 1.

Parameter	Estimate	SE	95% CI	γ
log10 λHS	−1.5291	321.79	[−745.52, 742.46]	210.45
log10 β	−7.0633	530.68	[−1.2340 × 103, 1.2199 × 103]	75.131
log10 δEH	−2.7500	7.7607 × 105	[−1.7943 × 106, 1.7943 × 106]	2.8221 × 105
log10 δV	−0.4320	6.7490 × 103	[−1.5604 × 104, 1.5603 × 104]	1.5622 × 104
log10 ρ̄EV	−0.6067	997.13	[−2.3060 × 103, 2.3047 × 103]	1.6434 × 103
log10 δEV	−0.9681	2.0163 × 103	[−4.6627 × 103, 4.6608 × 103]	2.0828 × 103
log10 δEK	−0.9936	2.5211 × 103	[−5.8298 × 103, 5.8278 × 103]	2.5374 × 103
log10 δC0	−1.8907	254.42	[−590.10, 586.32]	134.56
log10 κHS	3.0032	1.7893 × 103	[−4.1339 × 103, 4.1399 × 103]	595.81
log10 δHI	−1.0681	2.2293 × 103	[−5.1553 × 103, 5.1531 × 103]	2.0871 × 103
log10 ρV	3.6349	8.7049 × 103	[−2.0122 × 104, 2.0129 × 104]	2.3948 × 103
log10 κV	2.2569	5.6586 × 103	[−1.3080 × 104, 1.3085 × 104]	2.5073 × 103
log10 λC	−2.1994	121.88	[−284.00, 279.60]	55.416
log10 ρ̄EK	−0.7837	2.4214 × 103	[−5.5990 × 103, 5.5974 × 103]	3.0895 × 103
ε1	0.1009	1.5278 × 103	[−3.5321 × 103, 3.5323 × 103]	1.5148 × 104
ε2	0.3675	128.05	[−295.69, 296.43]	348.40
ε3	0.6076	1.6913 × 103	[−3.9097 × 103, 3.9109 × 103]	2.7835 × 103
ε4	0.3593	142.45	[−328.99, 329.71]	396.53
log10 EV0	−1.4234	9.3880 × 105	[−2.1705 × 106, 2.1705 × 106]	6.5953 × 105
C0	0.6763	0.16971	[0.28393, 1.0687]	0.25094

Table 13.

Parameter estimates, asymptotic standard errors, confidence intervals, and the ratios γ of standard error to the corresponding estimate for parameter estimation # 2.

Parameter	Estimate	SE	95% CI	γ
log10 λHS	−1.5205	2.8635	[−6.7692, 3.7282]	1.8832
log10 β	−7.0671	64.938	[−126.10, 111.96]	9.1888
log10 δV	−0.4323	464.46	[−851.79, 850.92]	1074.3
log10 ρ̄EV	−0.6043	53.623	[−98.895, 97.686]	88.737
log10 δEV	−0.9668	79.892	[−147.41, 145.48]	82.636
log10 δEK	−0.9930	36.222	[−67.388, 65.402]	36.477
log10 δC0	−1.8520	3.7012	[−8.6362, 4.9323]	1.9985
log10 κHS	3.0060	32.931	[−57.357, 63.369]	10.955
log10 δHI	−1.0704	74.871	[−138.31, 13.617]	69.944
log10 ρV	3.6344	550.96	[−1006.3, 1013.5]	151.60
log10 λC	−2.1805	2.0517	[−5.9412, 1.5802]	0.9409
log10 ρ̄EK	−0.7854	26.666	[−49.664, 48.093]	33.952
ε2	0.3657	10.447	[−18.784, 19.516]	28.564
ε3	0.6116	30.913	[−56.052, 57.275]	50.541
ε4	0.3637	14.389	[−26.011, 26.739]	39.559

Table 15.

Parameter estimates, asymptotic standard errors, confidence intervals, and the ratios γ of standard error to the corresponding estimate for parameter estimation # 3.

Parameter	Estimate	SE	95% CI	γ
log10 β	−7.0675	3.1628	[−12.637, −1.4978]	0.44751
log10 δV	−0.4295	11.135	[−20.038, 19.179]	25.924
log10 ρ̄EV	−0.6005	1.3981	[−3.0626, 1.8617]	2.3285
log10 δEV	−0.9635	1.9509	[−4.3990, 2.4721]	2.0249
log10 δEK	−0.9945	0.41297	[−1.7218, −0.26729]	0.41524
log10 κHS	3.0111	1.7573	[−0.083412, 6.1057]	0.58359
log10 ρV	3.6327	12.5245	[−18.423, 25.688]	3.4477
log10 ρ̄EK	−0.7850	0.40087	[−1.4910, −0.079107]	0.51064
ε3	0.5999	0.59392	[−0.44597, 1.6458]	0.9900
ε4	0.3649	0.22157	[−0.025269, 0.75510]	0.60718

Results corresponding to parameter estimation # 4 associated with estimating 5 parameters are given in Figure 2, Table 4, and Table 5. We observe from Figure 2 that we obtain good fits to the data, where the model outputs are obtained using parameter values given in Table 4. Table 5 illustrates the estimate, the standard error, the 95% confidence intervals and the absolute ratio of the standard error to the corresponding estimate (denoted by γ). From Table 5, we can see that γ_i < 0.2, i = 1, 2 …, 5. These parameter estimates for β, ρ̄_EV, δ_EV, δ_EK, ρ̄_EK are reliable. Therefore by Figure 2 and Table 5 we may infer that the goodness-to-fit is reasonable.

Figure 2.

Using ${\theta }_{\mathit{\text{opt}}}^4$ in Table 4 for parameter estimation # 4, (Left) Plot of model solution and viral load data on [0, 450]. (Right) Plot of model solution and serum creatinine data on [0, 450].

Table 5.

Parameter estimates, asymptotic standard errors, confidence intervals, and the ratios γ of standard error to the corresponding estimate for parameter estimation # 4.

Parameter	Estimate	SE	95% CI	γ
log10 β	−7.0674	8.0555 × 10−3	[−7.0813, −7.0534]	1.1398 × 10−3
log10 ρ̄EV	−0.6006	0.034668	[−0.6605, −0.5406]	0.057728
log10 δEV	−0.9636	0.044640	[−1.0407, −0.8864]	0.046328
log10 δEK	−0.9945	0.128860	[−1.2173, −0.7717]	0.129580
log10 ρ̄EK	−0.7853	0.140580	[−1.0283, −0.5422]	0.179020

The next step is then to validate our model with these 5 free parameters. There are a number of methods in the literature that can be used for model validation (e.g., see [17, 38] and the reference therein). These include data-splitting methods, cross-validation methods and bootstrapping methods. Specifically, both data-splitting methods and cross-validation methods use part of the data (training data) for parameter estimation and the rest of the data (test data) for validation. However bootstrapping involves repeatedly fitting the model using the bootstrapping samples and evaluating the performance of the resulting fitted model with the original data. In other words, bootstrapping samples are used as training data and the original data is used as test data. We thus see that bootstrapping method uses the entire data set for parameter estimation and hence it is very efficient and is especially useful for the case where one has a limited number of observations or even with a single (one patient only) clinical data set. Recall that the sample size of our single patient data set is small. Hence, bootstrapping is chosen to validate our model.

Let $J(\mathbf{q};{\{y_j\}}_{j=1}^N)=\frac{1}{N}({\sum }_{j=1}^N|y_j-{ℱ}_j(\mathbf{q}){|}^2)$, where ${\{y_j\}}_{j=1}^N=\{y_1^1,y_2^1,\dots ,y_{n_1}^1,y_1^2,y_2^2,\dots y_{n_2}^2\},ℱ(\mathbf{q})={(f_1(t_1^1;\mathbf{q}),f_1(t_2^1;\mathbf{q}),\dots ,f_1(t_{n_1}^1;\mathbf{q}),f_2(t_1^2;\mathbf{q}),f_2(t_2^2;\mathbf{q}),\dots ,f_2(t_{n_2}^2;\mathbf{q}))}^T$ and N = n₁ + n₂. The procedures for using bootstrapping to calculate the predication error are given as follows.

1. Using the estimate q̂ obtained in the parameter estimation #4 to calculate $J(\hat{\mathbf{q}};{\{y_j\}}_{j=1}^N)$.

2. 1. Define the standardized residuals

      $${\stackrel{-}{r}}_j=\sqrt{\frac{N}{N-{\kappa }_{\theta }}}(y_j-{ℱ}_j(\hat{\mathbf{q}})),j=1,2,3,\dots ,N.$$

      Set m = 1.

   2. Create a bootstrapping sample of size N using random sampling with replacement from {r̄₁,…,r̄_N} to form a bootstrapping sample $\{r_1^{(m)},\dots ,r_N^{(m)}\}$.

   3. Create bootstrap sample points

      $$y_j^{(m)}={ℱ}_j(\hat{\mathbf{q}})+r_j^{(m)},j=1,2,3,\dots ,N.$$

   4. Obtain a new estimate q̂^(m) from the bootstrapping sample ${\{y_j^{(m)}\}}_{j=1}^N$ using OLS. Calculate $J({\hat{\mathbf{q}}}^{(m)};{\{y_j^{(m)}\}}_{j=1}^N)$ and $J({\hat{\mathbf{q}}}^{(m)};{\{y_j\}}_{j=1}^N)$.

   5. Set m = m+1 and repeat steps (b)–(d) until m > M (e.g., typically M = 200 as in our calculations below).

3. Calculate the estimate of optimism (a measure of how much worse our model fits to the new data compared to the training data)

   $$O=\frac{1}{M}(\sum _{m=1}^{M}J({\hat{\mathbf{q}}}^{(m)};{\{y_j\}}_{j=1}^N)-\sum _{m=1}^{M}J({\hat{\mathbf{q}}}^{(m)};{\{y_j^{(m)}\}}_{j=1}^N)).$$

4. Calculate the optimism adjusted predication error as $J(\hat{\mathbf{q}};{\{y_j\}}_{j=1}^N)+O$.

Using the above procedures, we found that the optimism is 3.2667 × 10⁻⁴ and the optimism adjusted predication error is 3.4248×10⁻³. This implies that our model with 5 free parameters is reasonably validated.

To gain some information on the dynamics of other model states (that is, H_S, H_I, E_V and E_K), we plotted these states versus time in Figure 3, where the results were obtained with parameter values given in Table 4. From this figure, we observe that the number of infected cells (indicated in the left panel) increases at the very beginning and then decreases until it settles down to a small value. This agrees with the viral load data shown in Figure 2 as the only source for production of new virus is from the infected cells. We also observe that the number of BK-specific CD8+ T cells E_V (shown in the right panel of Figure 3) increases at the beginning in response to the presence of large amount of free BK virus and then decreases sharply when the viral load decreases to a small level. The trend of increasing-decreasing-increasing of allospecific CD8+ T cells E_K (indicated in the right panel of Figure 3) perfectly explains the pattern shown in the serum creatinine data (see the right panel of Figure 2). This is because E_K has a negative effect on the health of the kidney and the damage to the kidney results in increases in the serum creatinine level (as explained in Section 2).

Figure 3.

Model solution obtained with parameter values given in Table 4: (left) results for H_S and H_I, (right) results for E_V and E_K.

4 Concluding remarks and future research effort

Our efforts here to develop a mathematical model with statistically based model validation can be properly viewed as an algorithmic approach to determining the information content in a specific experimental data set. That is, we illustrate one approach to determining how many and which model parameters can be estimated with confidence from a given data set. In this context we have developed a mechanistic mathematical model to describe the immune response to both BKV infection and a donor kidney, and have estimated model parameters and initial conditions using the clinical data. Due to the large number of unknown parameters and limited number of observations, one is unable to reliably estimate all the parameters. To alleviate this difficulty, we employed sensitivity analysis combined with asymptotic theory of estimators to determine the number of parameters that can be reliably estimated. Numerical results show that we were able to reliably estimate five parameters with the given eight viral load data points and sixteen creatinine data points. This clearly demonstrates that one requires more informative longitudinal data, particularly observations on other model variables, in order to fully validate the model. In this context an immediate future effort would be to use optimal design methods (e.g., [9, 10]) to determine when an experimenter should take measurements and what model variables to measure. The obtained results should then be used to guide the future data collection efforts.

Once we obtain sufficient amount of informative longitudinal data to reliably estimate all the unknown parameters, another immediate future effort is to quantify uncertainties of model response, which is due to uncertainty in parameter estimators. Based on the asymptotic theory, we know that parameter estimators are multivariate normally distributed. Thus, this would involve uncertainty quantification of the system of ordinary differential equations (9) driven by a multivariate normally distributed random vector (referred to as a random differential equation). To do this, one could use the fact that the probability density function of solutions to random differential equations is described by the integral of the solution to a hyperbolic partial differential equation with integrator being the cumulative distribution function of a multivariate normally distributed random vector (see [11, Section 7.3.2.2] for details). An alternative approach to do this is to use Monte Carlo methods or other numerical methods such as stochastic Galerkin methods and stochastic collocation methods (e.g., see [11, Section 7.3] and the references therein for details).

Appendix A

Results corresponding to the sensitivity analysis conducted during Step 1- Step 4 of the iterative inverse problem algorithm are given in Table 6, Table 7, Table 8, and Table 9.

Appendix B

Results corresponding to parameter estimation # 1 associated with estimating 20 parameters are found in Figure 4, Table 10, and Table 11. We can see from Figure 4 that the estimate ${\theta }_{\mathit{\text{opt}}}^1$ produced relatively good fits for both the viral and serum creatinine data. However, we observe from Table 11 that the ratio of standard errors to the corresponding estimate, γ, is huge for all parameters except C₀. This means that the estimates for all the parameters except C₀ are not reliable. One possible explanation for this result is the size of the condition number of the Fisher information, χ(q̂)^T χ(q̂). In this particular case, it is 2.0604×10¹⁷, obtained using Matlab function cond. This is considerably large and suggests that the matrix is very close to being singular. This presents an issue when calculating the standard errors, since the inverse of the Fisher matrix is required.

Results corresponding to parameter estimation # 2 associated with estimating 15 parameters are found in Figure 5, Table 10, and Table 13. Figure 5 also shows relatively good fits for both the viral and serum creatinine data. We observe, however, from Table 13 that γ is large for parameters λ_HS, β, ρ̄_EV, δ_EV, δ_EK, δ_C0, κ_HS, δ_HI, ρ_V, ρ̄_EK, ε₂, ε₃, ε₄. Yet, 1 < γ < 2 for λ_HS, δ_C0 and 0.9 < γ < 1 for λ_C. We note that max_i |γ_i| = 1074.3, which is much less than that corresponding to parameter estimation # 1. This suggests that there is less uncertainty in the estimates obtained compared to the previous case. It is important to note that the condition number of the Fisher matrix is 1.6620 × 10¹¹ and has decreased.

Figure 5.

Using ${\theta }_{\mathit{\text{opt}}}^2$ in Table 12 for parameter estimation # 2, (Left) Plot of model solution and viral load data on [0, 450]. (Right) Plot of model solution and serum creatinine data on [0, 450].

Results corresponding to parameter estimation # 3 associated with estimating 10 parameters are given in Figure 6, Table 14 and Table 15. Figure 6 shows relatively good fits for both the viral and serum creatinine data. However, we observe from Table 15 that γ is large for parameter δ_V, 1 < γ < 4 for parameters ρ̄_EV, δ_EV, ρ_V and 0.4 < γ < 1 for parameters β, δ_EK, κ_HS, ρ̄_EK, ε₃, ε₄. Note that max_i |γ_i| = 25.924, which is much less than that corresponding to parameter estimation # 1, 2. This means that we have less uncertainty in parameter estimates compared to that of the other two cases. Similar to the previous two cases, the condition number of the Fisher matrix is 1.9733 × 10⁸ and has decreased.