A Mathematical Model of Cytomegalovirus (CMV) Infection

in Transplant Patients

Grace M. Kepler

Center for Research in Scientific ComputationNorth Carolina State University

Outline

• Significance

• Modeling goals

• Mathematical/Biological model

• Parameter approximation

• Numerical results

• Conclusions

Transplantation Numbers (UNOS, 2005)

• More than 27,000 organs transplanted

• Approximately 90,000 waiting for organs

• 153,000 living with functioning organ transplant

The number of individuals waiting for, receiving, or living with a transplanted organ(s) is significant.

Life-long immunosuppression is the standard of care for transplant

patients.

• Common pathogens (eg., influenza)

• Opportunistic infections (eg., Listeria)

• Latent infections (eg., HCV, VZV, CMV)

Immunocompromised individuals are susceptible to infections from

CMV infection

• Most significant threat to patient and graft health

• Directly or indirectly causes:– allograft rejection– decreased graft and patient survival– predisposition to opportunistic infections

and malignancies

Facts about CMV

• A herpes virus• 50-90% of adults are infected (geographic

variation)• Primary infection in immunocompetent individuals

is generally asymptomatic (some get mononucleosis-like illness )

• Establishes lifelong latent infection• Latent infection is well control by healthy immune

systems• Reactivation rare in healthy individuals

CMV Infection Risk In Transplantation

Donor Recipient Type

D+ R- primary

D- R+ reactivation

D+ R+ superinfection

D- R- risk with exposure

Optimal care of individuals with transplanted organs is important

• No universal agreement among transplant centers about

– Prophylactic vs. preemptive antiviral treatment

– Optimal duration of antiviral treatment

• Optimal treament may vary among

subpopulations (eg., D+ R- vs D- R+)

Modeling goals

• Create a within-patient dynamic model of CMV infection

• Describe dynamics of cell and viral populations with ODEs

( ) ( , ; )x t f x t

Model parameters

Modeling goals

• Individualized medicine– model equations are the same for each individual– model parameter values may vary among individuals

– individuals are characterized by their particular set of parameter values

– parameter values for each individual determine their particular infection dynamics

– allowing prediction for each individual

Longitudinal data

Viral load data for one individual.

Censored data

Parameter estimationEstimate model parameters from the data.

PredictionUse the model and characterstic parameters to predict infection dynamics.

Modeling goal – Population predictions

• Estimate characteristic parameters for many individuals using longitudinal data

Modeling goal – Population predictions

• Create a probabilistic model to describe parameter distributions

Modeling goal – Population predictions

• Use the probabilistic model to create virtual patients

• Predict population behavior (eg., treatment regimens)

Antiviral treatment

Modeling goal – Population predictions

• Use a stochastic model to sample the parameter distributions (virtual patients)

• Predict population behavior (eg., treatment regimens) Antiviral treatment

Modeling goal – Population predictions

• Use a stochastic model to sample the parameter distributions (virtual patients)

• Predict population behavior (eg., treatment regimens) Antiviral treatment

Modeling considerations

• Start simply, capture most salient biological features– a model that can describe primary, latent,

and reactivated infections in healthy or immunocompromised individuals

• Use clinical measurements to inform the model

• Model cell and viral populations in the blood

Math/Bio model - virions

VIRIONS (free virus)

Math/Bio model – susceptible cells

SUSCEPTIBLE CELLS (monocytes)

cell replication and death

ACTIVELY -INFECTED CELLS

Math/Bio model - actively-infected cells

Math/Bio model – viral-induce cell lysis

Math/Bio model – immune response

CMV-SPECIFIC IMMUNE EFFECTOR CELLS

Math/Bio model – immune suppression

Math/Bio model – lysing of infected cells

LATENTLY- INFECTED CELLS

Math/Bio model – latently-infected cells

Math/Bio model – reactivation

reactivation of monocytes upon differentiation

Math/Bio model – cell replication/death

State Variables

Variable Description Units

V virions virions/L-blood

E virus-specific immune effector cells

cells/L-blood

RI actively-infected cells cells/L-blood

RS susceptible cells cells/L-blood

RL latently-infected cells cells/L-blood

Mathematical equations

0

0

(1 ) 1

(1 )

1

1

I S

S E

I S I S I L I

SS S S

S

LL L L I

L

V n R cV fkR V

EE E V

e

R kR V R mER R R

RR R kR V

r

RR R R R

r

ò

ò

Clinical data

• Real-time quantitave PCR measurements of viral DNA in plasma ( )

• Antigenemia assay ( )

• PBMC depleted ELISPOT assay ( )

Longitudinal measurements

VIR

E

Statistical framework

• Intra-subject variation of observations – assay errors– physiological fluctuations– assay limits (cesored data)

Parameter approximations

• Physiological information • Experimental measurements

• Auxilliary parameters

• Using reduced models for specific time regimes

Unknown parameters

Viral load decay0, , ,S Lr r

, , ,H Dt t E V

, , , Ek c m

, , , ,n e

Emery1999

Parameter approximations

• Provide initial values for parameter estimation when data is available

• Allow exploratory simulations of model behavior

Immunocompetent

Primary infection

( 0)s

Initial conditions:4 2( , , , , ) (1 10 ,0,0,4 10 ,0)I S LV E R R R

Immunocompetent

E

• The latent infection state is characterized by the equilibrium levels of the state variables following primary infection.

( 0)s

V

IR( , , , , )I S LV E R R R

Latent infection

Immunosuppression

0s Primary infection

Immunosuppression

0.4s Primary infection

Immunosuppression

0.7s Primary infection

Immunosuppression - Latency

D-R+ Transplant Scenario

• The donor tissue has no CMV virions or latently-infected cells (D-)

• Prior to transplantation, recipient has a latent CMV infection, characterized by low levels of V, RI, and RL that is controlled by the immune effector cells E

• After transplantation, pharmacolgical immunosuppression can result in a secondary (reactivated) CMV infection

Reactivation

E

V

IR

0.7s

Immune suppression of an individual with a latent CMV infection

Conclusion

• Created a mathematical model for CMV infection in both immunocompetent and immunocompromised individuals

• Identified data that can be collected to inform the model• Approximated values for most of the model parameters• Model exhibits primary, latent, and secondary (reactivated)

infections• Latent infection is characterized by low-level viral load and

actively-infected cells• Simulation of reactivated infection approximates CMV

infection in D-R+ transplant patients

CMV infection in other immunocompromised individuals

• Most common congenital infection – can result in developmental and sensory

disabilites

• Retinitis infection in AIDS patients.

• CMV CTL-inflation may be a cause of immunosuppression in elderly individuals

Challenges

• Get data– parameter estimation– predictive capability

• Further model development– other transplant situations (eg., D+R-)– HLA type, antiviral treatment,...

Collaborators

• Tom Banks, CRSC, NCSU

• Marie Davidian, CQSB, NCSU

• Eric Rosenberg, MGH, Harvard