# 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  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
##### References:
 [1] B. M. Adams, H. T. Banks, H. Kwon and H. T. Tran, Dynamic multidrug therapies for HIV: Optimal and STI control approaches, Mathematical Biosciences and Engineering, 1 (2004), 223-241. doi: 10.3934/mbe.2004.1.223. [2] H. T. Banks, Modeling and Control in the Biomedical Sciences, Springer-Verlag, NY, 1975. [3] H. T. Banks, R. A. Everett, S. Hu, N. Murad and H. T. Tran, Mathematical and statistical model misspecifications in modeling immune response in renal transplant recipients, Inverse Problems in Science and Engineering, 26 (2018), 1-18. doi: 10.1080/17415977.2017.1312363. [4] H. T. Banks, S. Hu, T. Jang and H. D. Kwon, Modeling and optimal control of immune response of renal transplant recipients, Journal of Biological Dynamics, 6 (2012), 539-567. doi: 10.1080/17513758.2012.655328. [5] H. T. Banks, S. Hu, K. Link, E. S. Rosenberg, S. Mitsuma and L. Rosario, Modeling immune response to BK virus infection and donor kidney in renal transplant recipients, Inverse Problems in Science and Engineering, 24 (2016), 127-152. doi: 10.1080/17415977.2015.1017484. [6] H. T. Banks, S. Hu and W. C. Thompson, Modeling and Inverse Problems in the Presence of Uncertainty Taylor/Francis-Chapman/Hall-CRC Press, Boca Raton, FL, 2014. [7] H. T. Banks, H. Kwon, J. A. Toivanen and H. T. Tran, A state-dependent Riccati equation-based estimator approach for HIV feedback control, Optimal Control Application and Methods, 27 (2006), 93-121. doi: 10.1002/oca.773. [8] H. T. Banks and H. T. Tran, Mathematical and Experimental Modeling of Physical and Biological Processes, Taylor/Francis-Chapman/Hall-CRC Press, Boca Raton, FL, 2009. [9] B. E. Chambers and R. A. Wingert, Renal progenitors: Roles in kidney disease and regeneration, World Journal of Stem Cells, 8 (2016), 367-375. doi: 10.4252/wjsc.v8.i11.367. [10] L. Chatenoud, Precision medicine for autoimmune disease, Nature Biotechnology, 34 (2016), 930-932. doi: 10.1038/nbt.3670. [11] J. Colaneri, An overview of transplant immunosuppression-history, principles, and current practices in kidney transplantation, Nephrology Nursing Journal, 41 (2014), 549-560. [12] N. Elfadawy, S. M. Flechner, X. Liu, J. Schold, D. Tian, T. R. Srinivas, E. Poggio, R. Fatica, R. Avery and S. B. Mossad, The impact of surveillance and rapid reduction in immunosuppression to control BK virus-related graft injury in kidney transplantation, Transplant International, 26 (2013), 822-832. doi: 10.1111/tri.12134. [13] S. Lenhart and J. T. Workman, Optimal Control Applied to Biological Models, Chapman and Hall/CRC/Taylor and Francis Group, Boca Raton, FL, 2007. [14] Y. Fujigaki, Different modes of renal proximal tubule regeneration in health and disease, World Journal of Nephrology, 1 (2012), 92-99. doi: 10.5527/wjn.v1.i4.92. [15] G. A. Funk, R. Gosert, P. Comoli, F. Ginevri and H. H. Hirsch, Polyomavirus BK replication dynamics in vivo and in silico to predict cytopathology and viral clearance in kidney transplants, American Journal of Transplantation, 8 (2008), 2368-2377. [16] G. A. Funk and H. H. Hirsch, From plasma BK viral load to allograft damage: Rule of thumb for estimating the intra-renal cytopathic wear, Clinical Infectious Diseases, 49 (2009), 989-990. doi: 10.1086/605538. [17] G. A. Funk, J. Steiger and H. H. Hirsch, Rapid dynamics of polyomavirus type BK in renal transplant recipients, The Journal of Infectious Diseases, 193 (2006), 80-87. doi: 10.1086/498530. [18] R. S. Gaston, Immunosuppressive Therapy: The Scientific Basis and Clinical Practice of Immunosuppressive Therapy in the Management of Transplant Recipients, Extending Medicare Coverage for Preventive and Other Services (Appendix D, Part I), Editors: M. J. Field, R. L. Lawrence and L. Zwanziger, National Academies Press US, Washington (DC), 2000. [19] H. H. Hirsch, C. B. Drachenberg, G. Steiger and E. Ramos, Polyomavirus-associated nephropathy in renal transplantation: Critical issues of screening and management, Polyomaviruses and human diseases. Georgetown: Landes Bioscience, 2006,117-147. [20] Harvard Stem Cell Institute, The Kidney Repair Shop, Stem Cell Lines, Spring 2008. [21] A. James and R. B. Mannon, The cost of transplant immunosuppressant therapy: Is this sustainable?, Current Transplant Reports, 2 (2015), 113-121. [22] T. S. Jang, J. Kim, H. Kwon and J. Lee, Hybrid on-off controls for an HIV model based on a linear control problem, Journal of the Korean Mathematical Society, 52 (2015), 469-487. doi: 10.4134/JKMS.2015.52.3.469. [23] L. E. Keshet, Mathematical Models in Biology, Siam Classics in Applied Mathematics Series, 2005. doi: 10.1137/1.9780898719147. [24] G. M. Kepler, H. T. Banks, M. Davidian and E. S. Rosenberg, A model for HCMV infection in immunosuppressed patients, Mathematical and Computational Modeling, 49 (2009), 1653-1663. doi: 10.1016/j.mcm.2008.06.003. [25] M. Prague, D. Commenges and R. Thiébaut, Dynamical models of biomarkers and clinical progression for personalized medicine: the HIV context, Advanced Drug Delivery Reviews, 65 (2013), 954-965. [26] H. Schättler and U. Ledzewicz, Optimal Control for Mathematical Models of Cancer Therapies: An Application of Geometric Methods, Springer, New York, 2015. doi: 10.1007/978-1-4939-2972-6. [27] W. D. Shlomchik, Graft-versus-host disease, Nature Reviews Immunology, 7 (2007), 340-352. [28] Department of Laboratory Medicine, Yale New Haven Hospital Medical Centre, Quantitative BK virus DNA PCR for Diagnosis in Compromised Hosts, Clinical Virology Laboratory Newsletter, 14, June 2005. [29] Department of Laboratory Medicine, Yale New Haven Hospital Medical Centre, BK virus Nephropathy, Genome Variability and the Pitfalls of PCR Surveillance, Clinical Virology Laboratory Newsletter, 20, Jan 2011.

show all references

##### References:
 [1] B. M. Adams, H. T. Banks, H. Kwon and H. T. Tran, Dynamic multidrug therapies for HIV: Optimal and STI control approaches, Mathematical Biosciences and Engineering, 1 (2004), 223-241. doi: 10.3934/mbe.2004.1.223. [2] H. T. Banks, Modeling and Control in the Biomedical Sciences, Springer-Verlag, NY, 1975. [3] H. T. Banks, R. A. Everett, S. Hu, N. Murad and H. T. Tran, Mathematical and statistical model misspecifications in modeling immune response in renal transplant recipients, Inverse Problems in Science and Engineering, 26 (2018), 1-18. doi: 10.1080/17415977.2017.1312363. [4] H. T. Banks, S. Hu, T. Jang and H. D. Kwon, Modeling and optimal control of immune response of renal transplant recipients, Journal of Biological Dynamics, 6 (2012), 539-567. doi: 10.1080/17513758.2012.655328. [5] H. T. Banks, S. Hu, K. Link, E. S. Rosenberg, S. Mitsuma and L. Rosario, Modeling immune response to BK virus infection and donor kidney in renal transplant recipients, Inverse Problems in Science and Engineering, 24 (2016), 127-152. doi: 10.1080/17415977.2015.1017484. [6] H. T. Banks, S. Hu and W. C. Thompson, Modeling and Inverse Problems in the Presence of Uncertainty Taylor/Francis-Chapman/Hall-CRC Press, Boca Raton, FL, 2014. [7] H. T. Banks, H. Kwon, J. A. Toivanen and H. T. Tran, A state-dependent Riccati equation-based estimator approach for HIV feedback control, Optimal Control Application and Methods, 27 (2006), 93-121. doi: 10.1002/oca.773. [8] H. T. Banks and H. T. Tran, Mathematical and Experimental Modeling of Physical and Biological Processes, Taylor/Francis-Chapman/Hall-CRC Press, Boca Raton, FL, 2009. [9] B. E. Chambers and R. A. Wingert, Renal progenitors: Roles in kidney disease and regeneration, World Journal of Stem Cells, 8 (2016), 367-375. doi: 10.4252/wjsc.v8.i11.367. [10] L. Chatenoud, Precision medicine for autoimmune disease, Nature Biotechnology, 34 (2016), 930-932. doi: 10.1038/nbt.3670. [11] J. Colaneri, An overview of transplant immunosuppression-history, principles, and current practices in kidney transplantation, Nephrology Nursing Journal, 41 (2014), 549-560. [12] N. Elfadawy, S. M. Flechner, X. Liu, J. Schold, D. Tian, T. R. Srinivas, E. Poggio, R. Fatica, R. Avery and S. B. Mossad, The impact of surveillance and rapid reduction in immunosuppression to control BK virus-related graft injury in kidney transplantation, Transplant International, 26 (2013), 822-832. doi: 10.1111/tri.12134. [13] S. Lenhart and J. T. Workman, Optimal Control Applied to Biological Models, Chapman and Hall/CRC/Taylor and Francis Group, Boca Raton, FL, 2007. [14] Y. Fujigaki, Different modes of renal proximal tubule regeneration in health and disease, World Journal of Nephrology, 1 (2012), 92-99. doi: 10.5527/wjn.v1.i4.92. [15] G. A. Funk, R. Gosert, P. Comoli, F. Ginevri and H. H. Hirsch, Polyomavirus BK replication dynamics in vivo and in silico to predict cytopathology and viral clearance in kidney transplants, American Journal of Transplantation, 8 (2008), 2368-2377. [16] G. A. Funk and H. H. Hirsch, From plasma BK viral load to allograft damage: Rule of thumb for estimating the intra-renal cytopathic wear, Clinical Infectious Diseases, 49 (2009), 989-990. doi: 10.1086/605538. [17] G. A. Funk, J. Steiger and H. H. Hirsch, Rapid dynamics of polyomavirus type BK in renal transplant recipients, The Journal of Infectious Diseases, 193 (2006), 80-87. doi: 10.1086/498530. [18] R. S. Gaston, Immunosuppressive Therapy: The Scientific Basis and Clinical Practice of Immunosuppressive Therapy in the Management of Transplant Recipients, Extending Medicare Coverage for Preventive and Other Services (Appendix D, Part I), Editors: M. J. Field, R. L. Lawrence and L. Zwanziger, National Academies Press US, Washington (DC), 2000. [19] H. H. Hirsch, C. B. Drachenberg, G. Steiger and E. Ramos, Polyomavirus-associated nephropathy in renal transplantation: Critical issues of screening and management, Polyomaviruses and human diseases. Georgetown: Landes Bioscience, 2006,117-147. [20] Harvard Stem Cell Institute, The Kidney Repair Shop, Stem Cell Lines, Spring 2008. [21] A. James and R. B. Mannon, The cost of transplant immunosuppressant therapy: Is this sustainable?, Current Transplant Reports, 2 (2015), 113-121. [22] T. S. Jang, J. Kim, H. Kwon and J. Lee, Hybrid on-off controls for an HIV model based on a linear control problem, Journal of the Korean Mathematical Society, 52 (2015), 469-487. doi: 10.4134/JKMS.2015.52.3.469. [23] L. E. Keshet, Mathematical Models in Biology, Siam Classics in Applied Mathematics Series, 2005. doi: 10.1137/1.9780898719147. [24] G. M. Kepler, H. T. Banks, M. Davidian and E. S. Rosenberg, A model for HCMV infection in immunosuppressed patients, Mathematical and Computational Modeling, 49 (2009), 1653-1663. doi: 10.1016/j.mcm.2008.06.003. [25] M. Prague, D. Commenges and R. Thiébaut, Dynamical models of biomarkers and clinical progression for personalized medicine: the HIV context, Advanced Drug Delivery Reviews, 65 (2013), 954-965. [26] H. Schättler and U. Ledzewicz, Optimal Control for Mathematical Models of Cancer Therapies: An Application of Geometric Methods, Springer, New York, 2015. doi: 10.1007/978-1-4939-2972-6. [27] W. D. Shlomchik, Graft-versus-host disease, Nature Reviews Immunology, 7 (2007), 340-352. [28] Department of Laboratory Medicine, Yale New Haven Hospital Medical Centre, Quantitative BK virus DNA PCR for Diagnosis in Compromised Hosts, Clinical Virology Laboratory Newsletter, 14, June 2005. [29] Department of Laboratory Medicine, Yale New Haven Hospital Medical Centre, BK virus Nephropathy, Genome Variability and the Pitfalls of PCR Surveillance, Clinical Virology Laboratory Newsletter, 20, Jan 2011.
Schematic of the iterative modeling process [8]
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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