Immunosuppressant treatment dynamics in renal transplant recipients: an iterative modeling approach

Neha Murad; H. T. Tran; H. T. Banks; R. A. Everett; Eric S. Rosenberg

doi:10.3934/dcdsb.2018274

Article Contents

2019, Volume 24, Issue 6: 2781-2797. Doi: 10.3934/dcdsb.2018274

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Immunosuppressant treatment dynamics in renal transplant recipients: an iterative modeling approach

1.
Center for Research in Scientific Computation, North Carolina State University, Raleigh, NC 27695-8212, USA
2.
Department of Mathematics and Statistics, Haverford College, Haverford, PA 19041, USA
3.
Massachusetts General Hospital and Harvard Medical School, Departments of Pathology and Medicine, Boston, MA 02114, USA

Received: March 2018

Revised: May 2018

Early access: October 2018

Published: June 2019

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Abstract

Finding the optimal balance between over-suppression and under-suppression of the immune response is difficult to achieve in renal transplant patients, all of whom require lifelong immunosuppression. Our ultimate goal is to apply control theory to adaptively predict the optimal amount of immunosuppression; however, we first need to formulate a biologically realistic model. The process of quantitively modeling biological processes is iterative and often leads to new insights with every iteration. We illustrate this iterative process of modeling for renal transplant recipients infected by BK virus. We analyze and improve on the current mathematical model by modifying it to be more biologically realistic and amenable for designing an adaptive treatment strategy.

Keywords:

Mathematics Subject Classification: 92C50, 92B05, 37N25, 34A34.

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Figure 1. Schematic of the iterative modeling process [8]

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Figure 2. Model diagram of the BKV virus affecting renal cells [5]

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Figure 3. Plot illustrating the balance between under and over suppression of the immune response

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Figure 4. BK viral load in blood, creatinine, susceptible and infected cell dynamics for highest and lowest immunosuppressant dosages

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Figure 5. Model simulations for both iterations of modeling for different $\epsilon_I$ values

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Table 1. Description of state variables

State	Description	Unit
${H_S}$	Concentration of susceptible graft cells	cells/mL
${H_I}$	Concentration of infected graft cells	cells/mL
$V$	Concentration of free BKV	copies/mL
$E_V$	Concentration of BKV-specific CD8+ T-cells	cells/mL
${E_K}$	Concentration of allospecific CD8+ T-cells that target kidney	cells/mL
$C$	Concentration of serum creatinine	mg/dL

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Table 3. Original (Iteration Ⅰ) and new model (Iteration Ⅱ) parameters.

Parameter	Description	Unit	Iteration Ⅰ	Iteration Ⅱ	Justification
$\lambda_{HS}$	Proliferation rate for $H_S$	/day	$0.03$	-
$\kappa_V$	Half saturation constant	copies/mL	$180$	$10^6$	See A
$\kappa_{HS}$	Saturation constant	cells/mL	$1025$	-
$\tilde{\beta}$	Attack rate on $H_S$ by $E_K$	mL/(cells$\cdot$day)	-	$0.0001$
$\lambda_{EK}$	Source rate of $E_K$	cells/(mL$\cdot$day)	$0.002$	$285$	[4] See B
$\beta$	Infection rate of $H_S$ by $V$	mL/(copies$\cdot$day)	$8.22 \times 10^{-8} $	$8.22 \times 10^{-8}$
$\delta_{EK}$	Death rate of $E_K$	/day	$0.103 $	$0.09$
$\delta_{HI}$	Death rate of $H_I$ by $V$	/day	$0.085 $	$0.085$
$\lambda_{C}$	Production rate for $C$	mg/(dL$\cdot$day)	$0.01$	$0.01 $
$\rho_V$	# Virions produced by $H_I$ before death	copies/cells	$4292.4$	$15000$	$3-44, 000$ [16]
$\delta_{C0}$	Maximum clearance rate for $C$	/day	$0.014 $	$0.2 $	[4]
$\delta_{EH}$	Elimination rate of $H_I$ by $E_V$	mL/(cells$\cdot$day)	$0.0018$	$0.0018$
$\kappa_{EK}$	Half saturation constant	cells/mL	$0.2$	-	.
$\delta_{V}$	Natural clearance rate of $V$	/day	$0.37$	$0.05$	$0.04-15.12$ [5,15,16]
$\kappa_{CH}$	Half saturation constant	cells/mL	$10$	$10^4$	See A
$\lambda_{EV}$	Source rate of $E_V$	cells/(mL$\cdot$day)	$0.001$	$285$	[4] See B
$\bar \rho_{EK} $	Maximum proliferation rate for $E_K$	/day	$0.164 $	$0.137$
$\delta_{EV}$	Death rate of $E_V$	/day	$0.11$	$0.17 $
$\kappa_{KH}$	Half saturation constant	cells/mL	$84.996 $	$10^3 $	See A
$\bar \rho_{EV}$	Maximum proliferation rate for $E_V$	/day	$0.25$	$0.36$
$V^*$	Threshold concentration of BKV	copies/mL	-	$1000$
${E_K}^*$	Threshold concentration of Allospecific CD8+ T-cells	cells/mL	-	$2500$
${E_V}^*$	Threshold concentration of BKV specific CD8+ T-cells	cells/mL	-	$500$

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Table 2. Summary of cell dynamics under influence of immunosuppression

$\epsilon$	CD8+ T-cells	BKV	Infected cells	Susceptible cells	Creatinine
Low	$\uparrow$	$\downarrow$	$\downarrow$	$\downarrow$	$\uparrow$
High	$\downarrow$	$\uparrow$	$\uparrow$	$\downarrow$	$\uparrow$

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Table 4. Initial conditions both original (Iteration Ⅰ) and new (Iteration Ⅱ)

State	Iteration Ⅰ IC	Iteration Ⅱ IC	Justification
${H_S}_0$	$5\times10^3$ cells/mL	$1025$ cells/mL	Assume ${H_S}_0$ = $\kappa_{HS}$ from [5]
${H_I}_0$	$60$ cells/mL	$2\times10^{-16}$ cells/mL	Trace infection before transplant
${V}_0$	$5\times10^4$ copies/mL	$1200$ copies/mL	Minimal V of $10, 000$ copies/mL for low BK viremia detection [12,28,29]
${E_K}_0$	$0.04$ cells/mL	$2\times10^{-16}$ cells/mL	Negligible amounts of Allospecific CD8+ T-cells
${E_V}_0$	$0.4$ cells/mL	$100$ cells/mL	Low level of BKV specific CD8+ T-cells
$C_0$	$1.07$ mg/dL	$ 0.7$ mg/dL	Range 0.6 -1.1 [4]

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