# American Institute of Mathematical Sciences

June  2019, 24(6): 2781-2797. doi: 10.3934/dcdsb.2018274

## 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 Published  June 2019 Early access  October 2018

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.

Citation: Neha Murad, H. T. Tran, H. T. Banks, R. A. Everett, Eric S. Rosenberg. Immunosuppressant treatment dynamics in renal transplant recipients: an iterative modeling approach. Discrete & Continuous Dynamical Systems - B, 2019, 24 (6) : 2781-2797. doi: 10.3934/dcdsb.2018274
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##### References:
]">Figure 1.  Schematic of the iterative modeling process [8]
]">Figure 2.  Model diagram of the BKV virus affecting renal cells [5]
Plot illustrating the balance between under and over suppression of the immune response
BK viral load in blood, creatinine, susceptible and infected cell dynamics for highest and lowest immunosuppressant dosages
Model simulations for both iterations of modeling for different $\epsilon_I$ values
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
 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
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 - $500$
 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 - $500$
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 $\downarrow$ $\uparrow$ $\uparrow$ $\downarrow$ $\uparrow$
 $\epsilon$ CD8+ T-cells BKV Infected cells Susceptible cells Creatinine Low $\uparrow$ $\downarrow$ $\downarrow$ $\downarrow$ $\uparrow$ High $\downarrow$ $\uparrow$ $\uparrow$ $\downarrow$ $\uparrow$
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]
 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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