Modelling 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.

Model comparison tests to determine data information content

“…We focus on modeling of the BK virus, a common pathogen (and major threat) found in kidney transplant patients-see [7] and the references therein. 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.…”

“…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 [7]. 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.…”

“…Hence, the concentration of susceptible cells reflects the health of the kidney.) 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 , the clearance rate δ C is defined as follows $${\delta }_{C}false(E_K,H_{S}false)=\frac{{\delta }_{C0}{\kappa }_{EK}}{E_K+{\kappa }_{EK}}·\frac{H_S}{H_S+{\kappa }_{CH}}.$$ Based on the above discussions and those in [7], the model is given as follows: $${Ḣ}_S={\lambda }_{HS}false(1-\frac{H_S}{{\kappa }_{HS}}false)H_S-\beta H_{S}V,$$ $${Ḣ}_I=\beta H_{S}V-{\delta }_{HI}{H}_I-{\delta }_{EH}{E}_V{H}_I,$$ $$\mathrm{V̇}={\rho }_V{\delta }_{HI}{H}_I-{\delta }_{V}V-\beta H_{S}V,$$ $${Ė}_V=false(1-{\epsilon }_{I}false)false[{\lambda }_{EV}+{\rho }_{EV}false(Vfalse)E_{V}false]-{\lambda }_{EV}{E}_V,$$ $${Ė}_K=false(1-{\epsilon }_{I}false)false[{\lambda }_{EK}+{\rho }_{EK}false(H_{S}false)E_{K}false]-{\lambda }_{EK}{E}_K,$$ $$Ċ={\lambda }_C-{\delta }_{C}false(E_K,H_{S}false)C,$$ with initial conditions ( H S (0), H I (0), V (0), E V (0), E K (0), C (0)) = ( H S 0 , H I 0 , V 0 , E V 0 , E K 0 , C 0 ).…”

“…The observed amount of free virus (DNA) in the blood is represented by ȳ i 1 , with corresponding measured time point $t_i^1$, i = 1, 2, …, n 1 , and ȳ i 2 is the observed amount of serum creatinine at time point $t_i^2$, i = 1, 2, …, n 2 . We define $y_i^1={\text{log}}_{10}false({{\mathrm{y̅}}_i}^{1}false)$, i = 1, 2, …, n 1 , and $y_i^2={{\mathrm{y̅}}_i}^2$, i = 1, 2, …, n 2 , with n = n 1 + n 2 = 8 + 16 data points in the data considered here and in [7]. Let $\mathrm{boldy}={false[y_1^1,\dots ,y_{n_1}^1,y_1^2,\dots ,y_{n_2}^{2}false]}^T$.…”

“…Based on our sensitivity analysis in [7], we felt we could reliably estimate 5 parameters including 𝒬 H = {β, ρ̄ EV , δ EV , δ EK , ρ EK }. We chose an additional sixth parameter to estimate to form 𝒬 and ran the corresponding inverse problems.…”