Mathematical Immunology of Virus Infections pp 97152  Cite as
Modelling of Experimental Infections
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Abstract
This chapter aims to give a clear idea of how mathematical analysis for experimental systems could help in the process of data assimilation, parameter estimation and hypothesis testing.

estimation of model parameters for the ‘virus–host’ system,

understanding kinetic regulation of virus infection dynamics,

prediction of various phenotypes of virus infections and antigenspecific immune responses,

testing specific hypothesis about the feedback regulation of Tcell responses.
The material of this chapter is based on our previous work published in [1, 2, 3, 4, 9, 51, 84].
4.1 Why Experimental Infections?
Experimental systems of various types are used in fundamental immunology to unravel the complex cellular interactions of the immune responses. In vivo systems, which involve the whole animal provides the most natural experimental conditions. However, the in vivo systems have many unknown and uncontrollable interactions that add ambiguity to the interpretation of empirical data. The study of the immune system in vertebrates requires a suitable animal model. For most basic research in immunology, mice have been the experimental animal of choice. To control experimental variation caused by differences in the genetic background of experimental animals, immunologists work with inbred or knockout or knockin strains that are genetically identical animals produced by inbreeding. Hundreds of different strains of mice are available these days, e.g. CBA, BALB, C57BL/6, etc.
 Descriptive

qualitative and quantitative characterization of process dynamics;

 Explanatory

interpretation of the experimental observations,

understanding the numbers game;

 Predictive

testable predictions; suggestion of new experiments,

sensitivity performance quantification,

hidden effects.

4.2 The LCMV System: Gold Standard for Infection Biology^{1}
One of the beststudied model systems of viral infections is the lymphocytic choriomeningitis virus (LCMV) infection in mice.
4.2.1 Immunobiology of LCMV
LCMV is an RNA virus of the Arenaviridae family that is noncytopathic in vivo, i.e. the virus itself does not cause direct damage to cells and tissues. This feature enables relating any damage that appears in the course of an infection to host immune responses against the virus. Another important feature of the LCMV model system is the existence of several wellcharacterized viral strains that differ in their replicative capacity, host range (cell tropism and mouse strain) and experimental routes of infection (intracranial versus intraperitoneal (i.p.) or intravenous (i.v.)) and thus show different infection outcomes. This enables directly linking easily measurable viral dynamic properties to pathogenic consequences and studying the kinetic mechanisms of chronic infections.

back in 1974/75, Zinkernagel and Doherty demonstrated that cytotoxic Tlymphocytes (CTLs) recognize foreign antigens only in the context of proteins of the major histocompatibility complex (MHC) [5, 6]. For this finding of MHC restriction, they were awarded the Nobel Prize in 1996.

with the help of knockout mice, the mechanism of CTLmediated destruction of LCMVinfected target cells in vivo was directly linked to perforin, a poreforming protein contained in granules of this cell type [7, 8].

fundamental properties of ‘memory’ of the adaptive immune response have been understood, in particular, the requirements for CTL memory to prevent the establishment of a persistent LCMV infection [9].

NK cells of the innate immune response have been recognized as an important regulator of the helper Tcell support for antiviral CTL [10].

a critical role of organized secondary lymphoid organs in the induction of naive T and B cells and subsequent virus control was established [11].

the concept of immunopathology, that is the damage of tissues and organs due to the antiviral immune response rather than the infecting virus itself, was established. Mediators of immunopathology include CTL, macrophages, neutrophils and interferons [12, 13, 14].

based on the amino acid similarities between viral antigens and host proteins, the socalled molecular mimicry, viral infections can trigger autoimmunity and influence the course of subsequent infections with other viral pathogens [15, 16, 17].

important observations towards an acute versus a persistent infection outcomes were made as shown in Fig. 4.1 [18, 19, 20, 21].
Which infection fate is followed depends on the infecting viral dose and the viral strain and thus can be easily directed experimentally. LCMV persistence is associated with CTL exhaustion, a reversible, nonfunctional state of CTL. CTL exhaustion is a physiological consequence of persistent antigen exposure and has been observed both in persistent human viral infections and in cancers, the LCMV system was instrumental to understand infection fate regulation in general terms. As CTL exhaustion can be reversed by antibodies against PD1 or PDL1 that block the negative signalling pathway, novel immunotherapeutic modalities arose which show exciting promises as antiviral and anticancer therapies [22, 23, 24].
The LCMV infection model system offers sufficient experimental data to develop mathematical models in a problemoriented manner. The mathematical modeldriven studies of LCMV resulted in experimentally testable predictions concerning the mechanisms of the infection control, for example (i) threshold numbers of initial specific CTL precursors to protect from a chronic LCMV infection outcome, (ii) minimal number of antigenpresenting DCs in spleen for robust induction of CTL responses, and (iii) the effect of virus growth rate on the magnitude of the clonal expansion of CTLs, to name just the major of them.

Family: Arenaviridae;

Strains: Docile, Traub, WE, Aggressive, Armstrong, Clone 13;

Host: mice, hamsters; humans: acute hemorrhagic fever;

Target cells: macrophages and lymphocytes;

Cytopathicity in vivo: noncytopathic;

CTL responses play a dominant role in virus clearance: appear early and are high;

Neutralizing antibody responses: appear only late after infection;

Immunopathology is a recovery fee: is observed in spleen, liver, central nervous system.
4.2.2 Basic Mathematical Model of LCMV Infection

V(t) virus titer in spleen at time t (pfu/ml);

\(E_p(t)\) number of virusspecific precursor CTLs in spleen at time t (cell/ml);

E(t) number of virusspecific effector CTLs in spleen at time t (cell/ml);

W(t) cumulative virus antigen load in spleen at time t (pfu/ml).

\(V(t)=0,~t \in [\tau ^*,0),~V(0)={10^2, \, 10^4, \, 10^7}\) pfu/ml, \(\tau ^*=\max [\tau , \tau _A]\),

\(E_p(t)=265\) cell/ml, \(t \in [\tau ^*,0]\),

\(E_e(0)=0\) cell/ml,

\(W(0) = 0\) pfu/ml.
In the equation for V(t), the first term on the righthand side describes the virus growth with an upper limit K due to the limited amount of sensitive tissue cells supporting virus replication, and the second term takes into account the clearance of viruses due to lysis of virusinfected cells by effector CTLs.
In the equation for \(E_p(t)\), the first term describes the maintenance of virusspecific precursor CTL at a certain level through their export from thymus and death in the periphery. The second term accounts for an increase in the number of CTL precursors resulting from virusinduced proliferation with the inhibitory effect of cumulative virus load on clonal expansion. The last term describes activationinduced cell death by apoptosis.
The dynamics of \(E_e(t)\) is determined by the appearance of mature effector CTLs due to the division and differentiation of antigenstimulated precursor CTLs with the downregulation of the differentiation process of CTLp due to high virus antigen load (the first term); the decrease in the number of effector CTLs as a consequence of lytic interactions with virusinfected target cells (the second term); the activationinduced cell death of effector CTLs and natural death of effector CTL due to their finite lifespan (two last terms).
In the equation for W(t), the first term describes the increase in the total viral antigen load due to virus spread in the host and the second one accounts for the decrease of the inhibitory effect of high virus loads on the virusspecific CTLs as the virus is eliminated.
The model is based upon a fundamental assumption which reflects results of empirical analysis [25] that continuous exposure of virusspecific CTLs to LCMV induces a sequence of proliferation, anergy and activationinduced cell death by apoptosis. The balance between the above processes depends on the cumulative viral load and shifts towards the anergy and death phenotype in a high viral load infection.
This lowdimensional model is based on (i) a Verhulst logistic form for virus growth; (ii) secondorder virus elimination kinetics by CTLs; (iii) the Holling type II response curve for CTLs expansion with a time lag representing cell division time and antigenindependent production/death of CTLs in the immune system (homeostasis).
The LCMV infection model parameters and their bestfit estimates. We considered one pfu is one infectious particle
Parameter  Biological meaning  Units  Bestfit estimate 

\(\beta \)  Replication rate constant of viruses  1/day  3.35 
\(\gamma _{VE}\)  Rate constant of virus clearance due to effector CTLs  ml/(cell day)  \(1.34 \times 10^{6}\) 
K  Virus carrying capacity of spleen  particles/ml  \(4.82 \times 10^7\) 
\(\tau \)  Duration of CTL division cycle  day  0.6 
\(b_p\)  Rate constant of precursor CTL stimulation  ml/(particle day)  \(7.73 \times 10^{5}\) 
\(b_e\)  Rate constant of precursor CTL stimulation  ml/(particle day)  \(7.73 \times 10^{4}\) 
\(\theta _{p}\)  Cumulative viral load threshold for anergy induction in precursor CTL  particle/ml  \(3.25 \times 10^{6}\) 
\(\theta _{e}\)  Cumulative viral load threshold for anergy induction in differentiation of CTL  particle/ml  \(3 \times 10^{5}\) 
\(\alpha _{E_p}\)  Rate constant of precursor CTL natural death  1/day  0.542 
\(\alpha _{E_e}\)  Rate constant of effector CTL natural death  1/day  0.01 
\(E_p^0\)  Homeostatic concentration of LCMVspecific precursor CTL in spleen of unprimed mice  cell/day  265 
\(\tau _{AP}\)  Duration of commitment of CTLs for apoptosis  day  5.6 
\(\alpha _{AP}\)  Rate constant of precursor CTL apoptosis  (ml/particle)\(^2\)/day  \(7.5 \times 10^{16}\) 
\(\alpha _{EP}\)  Rate constant of effector CTL apoptosis  (ml/particle)\(^2\)/day  \(4.36 \times 10^{14}\) 
\(b_{W}\)  Rate constant of the cumulative viral load growth  1/day  1 
\(\alpha _{W}\)  Rate constant of the restoration from the inhibitory effect of cumulative viral load  1/day  0.11 
The phenomenon of exhaustion in the model is defined as disappearance of CTL activity and the functional impairment of virusspecific CTLs. The exhaustion of antiviral CTL responses is modelled as a stepwise process observed in an overwhelming infection with LCMVDocile. Following the initial activation, LCMVspecific T cells become anergic for 3–5 days and then disappear because of activationinduced cell death (apoptosis). (Of note, the observed lack of Tcell functionality was in time of the described experiments termed anergy; however, this functional state of T cells has been studied in more detail and shown to be a nonresponsive state after continuous antigen exposure that is now termed exhausted; for a detailed discussion, see Wherry and Kurachi [26]).
The single characteristic that appeared to be sufficient to control conventional versus exhaustive responses of CTLs was the cumulative viral load (cvl) since the beginning of the infection. The increase of cvl above a certain threshold value in conjunction with the high viral load in the host for about 5 days results in the shift of the infection phenotype from an acute with recovery to a chronic infection.
4.2.3 Viral Parameters: Impact on the Infection Phenotype
4.2.3.1 Why Does LCMVWE Strain Fail to Cause Exhaustion of CTLs After i.v. Infection of C57BL/6 Mice?
It is known that some LCMV isolates (WE or Armstrong) do not induce viral persistence after highdose i.v. infection. However, under certain conditions LCMVWE can also establish persistent infection like in congenital LCMVWE carrier C57BL/6 mice. The calibrated model can be used to examine the shape and position of the dose of infectionclearance curve for the LCMVWE isolate. To this end, using additional data on the growth kinetics of LCMVWE, one needs to quantify the exponential growth rate of the virus in spleen and the carrying capacity value. These appear to be smaller than in the cases of LCMVD, i.e. \(\beta =2.57\) pfu ml\(^{1}\) day\(^{1}\) and \(K = 0.18\times 10^8\) pfu ml\(^{1}\), respectively. Neglecting the differences in CTL stimulation rate due to the variation in amino acids of the LCMVGP epitope, the infection dosedependent clearance curve for LCMVWE can be computed as shown in Fig. 4.5. It suggests that the threshold dose of infection separating clearance and persistence phenotypes is around \(7.0 \times 10^6\) pfu. This is an order of magnitude larger than the virus population reaching the spleen after i.v. infection (see 4.5). The difference provides an explanation of why LCMVWE fails to cause chronic infection after i.v. injection of \(10^7\) pfu. Thus, minor variations between the distinct LCMV strains in the values of virus multiplication parameters might underline the about \(10^3\) increase in the virus dose threshold separating the two phenotypes of the virus–mouse interaction, i.e. virus clearance and persistent infection.
4.2.3.2 Can Underwhelming Infection Lead to Chronic Persistence?
The experimental analysis of the clonal expansion of CTLs in C57BL/6 mice to LCMV strains (Armstrong, WEArmstrong, WE, Traub and Docile) differing in their replication rate confirmed that there is a bellshaped relationship between the LCMV growth rate and the peak CTL response (see Fig. 4.6). Both slow and fast replicating LCMV strains produce weaker CTL responses. Thus, a mechanism of virus persistence by sneaking surveillance due to slow replication kinetics can be hypothesized. The ‘underwhelming’ infection mechanism (supplementing the ‘overwhelming’ infection) fits the abovementioned concept of the sensitivity of immune responses to perturbations.
4.2.3.3 LowLevel LCMV Persistence^{2}
How can the virus population persist in the face of CTL memory? There is a diverse array of biological mechanisms that are used by viruses to escape complete elimination by the immune system, ranging from those based on a limited growth, celltocell passage without maturation, localization in an immunologically privileged site, integration into the host cell chromosome to those based on decreasing immune detection and destruction, e.g. via downregulation of MHCrestricted antigen presentation [29, 30, 31]. In terms of kinetics, the implication is that the replication rate and CTLmediated elimination rate of LCMV (represented by \(\beta \) and \(\gamma _{VE}\), respectively) might well be reduced during transition from the acute to the lowlevel persistence phase. Indeed, available data on the growth kinetics of LCMV after immune therapy of a persistent viral infection [32] or in CD4\(^{+}\) T cell or Bcelldeficient mice [33] show a much lower rate of viral growth compared to the acute infection.
For the model analysis, we use the software package DDEBIFTOOL [34, 35]. DDEBIFTOOL is a MATLAB package (The MathWorks, Inc.) for bifurcation analysis of systems of DDEs with several discrete delays. The package can be used to compute and analyse the stability of steadystate and periodic solutions of a given system as well as to study the dependence of these solutions on system parameters via continuation.
4.2.3.4 SteadyState Solutions
Dependence of the steadystate solution \(S^*\) on a physical parameter (a component of p) can be studied by computing a branch of steadystate solutions as a function of the parameter using a continuation procedure [34]. The stability of the steady state can change during continuation whenever characteristic roots cross the imaginary axis. Generically a fold bifurcation (or turning point) occurs when a real characteristic root passes through zero and a Hopf bifurcation occurs when a pair of complex conjugate characteristic roots crosses the imaginary axis. Once a Hopf point is detected it can be followed in a twoparameter space using an appropriate determining system [34]. In this away, for instance one computes the stability region of the steadystate solution in the twoparameter space (if no other bifurcations occur in this region).
Some information about the numerical values of virus and CTL population densities for the steady states in the stability region shown in Fig. 4.9 (left) is given in Fig. 4.10. Figure 4.10 (left) presents the regions in the \((\beta ,\alpha _{E_p})\)plane where virus persists below the detection limit (\(V<1000\) pfu/ml) and below 100 pfu/ml. One can see that the value of V almost does not depend on \(\beta \) unless \(\beta \) gets close to 0 (see also Fig. 4.8) and virus can persist at a very low level if the death rate of CTLp (\(\alpha _{E_p}\)) is small enough.
4.2.3.5 Periodic Solutions
A periodic solution \(S^*(t)\) is a solution which repeats itself after a finite period T, i.e. \(S^*(t+T)=S^*(t)\) for all \(t>0\). A discrete approximation to a periodic solution on a mesh in [0, T] and its period are computed as solutions of the corresponding periodic boundary value problem using piecewise polynomial collocation [37, 38]. Adaptive mesh selection (the lengths of the mesh subintervals are adapted to the solution gradient) allows the computation of solutions with steep gradients.
Stability of a periodic solution is determined by the spectrum of the linear socalled Monodromy operator, which integrates the variational equation (4.8) around \(S^*(t)\) from time \(t=0\) over the period T. Any nonzero eigenvalue \(\mu \) of this operator is called a characteristic (Floquet) multiplier. Furthermore, \(\mu =1\) is always an eigenvalue and it is referred to as the trivial Floquet multiplier. A discrete approximation of this operator, a matrix M, is obtained using the collocation equations. The eigenvalues of M form approximations to the Floquet multipliers.
A branch of periodic solutions can be traced as a function of a system parameter using a continuation procedure [34]. The branch can be started from a Hopf point or from an initial guess (e.g. resulting from time integration). Bifurcations of periodic solutions occur when Floquet multipliers move into or out of the unit circle. Generically this is a turning point when a real multiplier crosses through 1, a period doubling point when a real multiplier crosses through \(1\) and a torus bifurcation when a complex pair of multipliers crosses the unit circle.
In the neighbourhood of a Hopf bifurcation point, solutions which belong to a branch of periodic solutions emanating from this point oscillate around the steadystate value corresponding to the Hopf point. Hence, Hopf points with low values of V can be sources of periodic solutions with oscillatory lowlevel viral persistence. We use the Hopf point shown in Fig. 4.8 as our ‘basic Hopf point’ and study the existence of oscillatory patterns in viral persistence by computing branches of periodic solutions emanating from this point as a function of the parameters listed in Table 4.1. Note that we depict periodic solutions on the time interval [0, 1], i.e. after time is scaled by the factor \(T^{1}\) with T the period of the solution.
Influence of \(b_p\). Larger values of \(b_p\) increase the region in the \((\beta ,\alpha _{E_p})\)plane where oscillatory solutions with \(V_{\max }<10^3\) pfu/ml exist, see region A in Fig. 4.13. However, due to a high sensitivity of the amplitude of oscillations to changes in values of \(\beta \) and \(\alpha _{E_p}\) the region where \(V_{\min }>10\) pfu/ml (B) remains quite small. Although two stable periodic solutions coexist in a part of region A, one of them is not biologically realistic because of very large values of the amplitude and the period of oscillations. Note that the upper left part of region A is bounded by the curve of turning points (which ends at a Hopf point), i.e. for the corresponding values of \(\beta \) and \(\alpha _{E_p}\) periodic solutions lose stability before \(V_{\max }\) reaches \(10^3\) pfu/ml.
The effect of other parameters can be briefly summarized as follows.
Influence of \(\alpha _{E_e}\). As \(\alpha _{E_e}\) decreases from 0.3 (Hopf point) to 0.1, the period of oscillations increases to 22 days and \(V_{\max }\) increases to 950 pfu/ml. For \(\alpha _{E_e}=0.1\), Hopf bifurcation occurs at \(\beta \approx 1.54\), which implies that for \(\alpha _{E_e}=0.1\) the value of \(V_{\max }\) grows from 129 to 950 pfu/ml as \(\beta \) changes from 1.54 to 1.675. Hence, for \(\alpha _{E_e} \in [0.1, 0.4]\) the size of the region in \((\beta , \alpha _{E_p})\)plane where \(V_{\max }<10^3\) pfu/ml is also quite small, and the location of this region with respect to the corresponding Hopf curve is similar to the one shown in Fig. 4.12.
Influence of \(\tau \). As \(\tau \) increases from 0.6 (Hopf point), the amplitude of oscillations grows rapidly and \(V_{\max }\) reaches \(10^3\) pfu/ml at \(\tau \approx 0.8\). At this point the period of oscillations is about 15 days. For \(\tau =0.8\) Hopf bifurcation occurs at \(\beta \approx 1.3\), which implies that for \(\tau =0.8\) the value of \(V_{\max }\) grows from 129 to \(10^3\) pfu/ml as \(\beta \) changes from 1.3 to 1.675. Hence for \(\tau =0.8\) the size of the region in \((\beta ,\alpha _{E_p})\)plane where \(V_{\max }<10^3\) pfu/ml is also quite small and the location of this region with respect to the corresponding Hopf curve (see Fig. 4.9) is similar to the one shown in Fig. 4.12.
Variations of parameters \(b_d\) and \(\gamma _{VE}\) within some admissible ranges (see Table 2 in [2] for details) have much lesser impact on the amplitude of oscillations compared to variations of \(\beta ,\alpha _{E_p}\) and \(\tau \) and do not change it significantly.
Overall, we found that the periodic solutions with V varying in between 10 and \(10^3\) pfu/ml exist in quite narrow intervals of \(\beta \) and \(\alpha _{E_p}\) values and the amplitude of oscillations grows rapidly as parameters \(\beta ,~\alpha _{E_p}\) and \(\tau \) increase. So the model predicts that oscillatory patterns in low level viral persistence (with virus population varying in between 1 and 100 pfu/spleen) are possible for quite ‘special’ combinations of the rates of virus growth and precursor CTLs death because of a high sensitivity of the amplitude of oscillations to changes in the above parameters.
The main result of our analysis is that unless LCMV replication rate does not reduce to smaller values, as compared to that during the acute phase of primary infection, a lowlevel persistence in the face of CTL memory as an equilibrium state is not possible: the virus will either be cleared or establishes a high viral load chronic infection, both outcomes depend on the initial dose of infection and the relative kinetics of viral growth. The extent of reduction needed depends on the responder status of the host, in particular, the lifespan of CTL memory subsets, duration of CTL division cycle, activation thresholds of CTL for proliferation, and differentiation. Since the virus remains the same during acute and persistence phase (it should not acquire attenuating mutations) we propose that the reduction in LCMV replication rate resulting in the lowlevel persistence could be either due to changes in the host cells, e.g. mediated by type I interferons, or intrinsic features of the virus replicatory cycle [39], which slow down the virus growth. This mechanism seems to be in agreement with virus reappearance after in CD4\(^+\) T cell help deficient mice, since the deficiency primarily impairs the LCMVspecific \(IFN{\gamma }\) production by CTLs and CD4\(^+\) T cells.
4.2.4 Role of CD8\(^+\) T Cells: Protection, Exhaustion, Immunopathology
LCMV infection of mice is a highly dynamic process with high sensitivity to variation in both host and virus parameters: virus control and functional CTL memory versus virus persistence and complete exhaustion of virusspecific CTL precursors reflect the two extreme ends of this spectrum. While both of these outcomes are of limited pathological consequences for the host, extensive Tcellmediated immunopathology represents an unfortunate intermediate in the balance of virus–host interactions. Important host and virus parameters that determine the outcome of infection include those controlled by MHC and nonMHC genes, presumably affecting Tcell precursor frequencies and Tcell responsiveness, and virus strain, the route and dose of infection affecting the kinetics of initial virus multiplication and virus distribution. Thus, the susceptibility to the establishment of a virus carrier state is increased with lower CTL responses (low responder status) and slower CTL expansion on the one hand and the ability of the virus to spread rapidly and widely on the other hand.
4.2.4.1 How Many Precursor T Cell are Needed to Protect Against Chronic Infection?^{3}
The calibrated mathematical model allows the examination of the effects of variations in virus dose and initial CTLp number on the phenotype of the LCMV infection. Two basic outcomes of the infection can be assessed: virus clearance, i.e. virus titer on day 20 (\(V_{min}\)) less than the detection limit of 30 pfu per gram of spleen associated with an elevated number of CTLp versus virus persistence (\(V_{min} \ge 30\) pfu/g of spleen) and exhaustion of virusspecific CTL.

a minimal threshold number of about 2550 naive LCMVspecific CTL precursors are necessary for control of infections in the range of \(110^{4}\) pfu;

with a tenfold higher dose, a 100fold increase is required to restore virus control;

in highdose infection (above \(10^6\) pfu), elevations in CTLp were found to be detrimental as they changed the outcome of infection from harmless virus persistence to lethal immunopathology.
The opportunity to compare the model predictions and experimental allows one to define the limitations of the model as a predictive tool related to the fact that it neglected virus spread outside the spleen. While this assumption is presumably justified for lowdose infection, it is responsible for the fact that the model does not account for the significant immunopathology observed after infections with higher doses.
Since the model neglects spread of virus to extralymphatic organs, it is not suited to predict the extent of immunopathology associated with virus clearance from these tissues. The model requires organoriented extention to be relevant for examination of the balance between protection and immunopathology by effector memory versus naive precursor CTLs against intravenous or peripheral infections.
4.2.4.2 Modelling LCMVAssociated Liver Disease^{4}
Infection of mice LCMV represents an example of a systemic infectious process, where the localization, dose and time of availability of virus antigens are important parameters determining the outcome of infection by affecting the antiviral immune response and pathological consequences of the cytotoxic T lymphocyte (CTL) mediated destruction of virusinfected cells [40, 41]. It provides an experimental model system for studying diseases mediated by cytotoxic activity of effector CTL against cells expressing virus antigen such as diabetes [42, 43], aplastic anemia [44], choriomeningitis [45], liver disease [46], to name just few of them. A classical example is the LCMVWEinduced liver hepatitis in mice [47].

the route of infection (intravenous, intracerebral, intrahepatic, etc.),

tissue tropism of the virus,

the dose of infection and

immune status of the host.
 1.
systemic virus spread,
 2.
lymphocyte migration during immune responses to tissue sites outside the spleen and
 3.
the pathological consequences of virus elimination via perforindependent CTLmediated destruction of infected cells.
In this section, we formulate a mathematical model to investigate the demands to CTL memory for protection against LCMV infection with minimal immunopathology. To address the immunopathology question, the basic model of LCMV infection in spleen was extended to consider additional organs, i.e. blood and liver. Such extension should allow to examine the severity of LCMVassociated CTLinduced hepatitis.
Formulation of a multicompartmental mathematical model integrating the kinetics of LCMV spread in various tissues of mice with effector CTL activation and trafficking allow one to specify the parameters which have to be achieved for CTL vaccination/immunization to ensure virus elimination with minimal immunopathology versus vaccination for disease. To keep the mathematical model in accord with what is experimentally controlled [47], one can consider the dynamics of two enzymes signalling liver cells destruction, AST(t) and ALT(t) as disease characteristics.

virus titer in spleen, blood and liver: \(V_{Spleen}(t), \ V_{Blood}(t), \ V_{Liver}(t)\);

precursor CTLs in spleen: \(E_{p}(t)\);

recirculating effector CTLs in spleen, blood and liver: \(E_{e,Spleen}(t), \ E_{e,Blood}(t), \ E_{e,Liver}(t)\);

cumulative viral load in spleen: W(t);

liver enzymes levels in blood: \(AST(t), \ ALT(t)\).
List of the model parameters estimated for systemic LCMVWE infection in CB57BL/6 mice
Parameter  Biological meaning  Units  Bestfit estimate 

\(\beta \)  Replication rate constant of viruses in spleen  1/day  4.7 
\(V_{mvc}\)  Maximal virus concentration in spleen  pfu/ml  \(6.5 \times 10^8\) 
\(\beta \)  Replication rate constant of viruses in liver  1/day  2.1 
\(V_{mvc}\)  Maximal virus concentration in spleen  pfu/ml  \(3.0 \times 10^8\) 
\(\gamma _{VE}\)  Rate constant of virus clearance due to effector CTLs  ml/(cell day)  \(2.5 \times 10^{5}\) 
\(\theta _{VE}\)  CTL number of halfmaximal virus clearance rate  cell/ml  \(2.6 \times 10^{5}\) 
\(E^0_p\)  Homeostatic concentration of LCMVspecific CTLs in spleen of unprimed mouse  cell/ml  1100 
\(\alpha _{E_p}\)  Rate constant of natural death for precursor CTLs  1/day  0.068 
\(b_p\)  Rate constant of CTL stimulation  ml/(pfu day)  \(2 \times 10^{3}\) 
\(\tau \)  Duration of CTL division cycles  day  1.0 
\(b_d\)  Rate constant of CTL differentiation  ml/(pfu day)  \(2 \times 10^{2}\) 
\(\theta _p\)  Cumulative viral load threshold for anergy induction in precursor CTLs (proliferation process)  pfu/ml  \(1 \times 10^6\) 
\(\theta _E\)  Cumulative viral load threshold for anergy induction in effector CTLs (differentiation process)  pfu/ml  \(5.5 \times 10^5\) 
\(\alpha _{PCD}\)  Rate constant of effector CTL death after virus clearance below a threshold  l/day  0.3 
\(\theta _{PCD}\)  Extent of virus elimination at which the passive cell death is in effect  l/day  1.0 
\(\alpha _{E_e}\)  Rate constant of natural death for effector CTLs  pfu/ml  0.068 
\(\tau _A\)  Duration of commitment of CTLs for apoptosis  day  9.1 
\(\alpha _{AP}\)  Rate constant of apoptosis for precursor CTLs  (ml/pfu)\(^2\)/day  \(1.0 \times 10^{13}\) 
\(\alpha _{AE}\)  Rate constant of apoptosis for effector CTLs  (ml/pfu)\(^2\)/day  \(3 \times 10^{14}\) 
\(b_W\)  Rate constant of viral load increase  1/day  1.7 
\(\alpha _W\)  Rate constant of restoration from the inhibitory effect of virus load  1/day  0.4 
\(E^{Sat}_{EP}\)  Saturation rate constant for CTL expansion  cell/ml  \(1.0 \times 10^7\) 
\(\rho _{AST}\)  Rate constant of AST release into blood from CTL destroyed infected liver cell  U/l/(pfu cell day) ml\(^2\)  \(1.0\times 10^{9}\) 
\(\rho _{ALT}\)  Rate constant of ALT release into blood from CTL destroyed infected liver cell  U/l/(pfu cell day) ml\(^2\)  \(0.7\times 10^{9}\) 
\(\alpha _{AST}\)  Decay rate of AST in blood  1/day  0.5 
\(\alpha _{ALT}\)  Decay rate of ALT in blood  1/day  0.5 
Transfer rates (\(hr^{1}\)) of LCMV between Blood, Spleen and Liver compartments
Organ  Blood  Spleen  Liver 

Blood  −0.74  \(0.33 \times 10^{3}\)  \(0.27 \times 10^{4}\) 
Spleen  \(0.5 \times 10^{3}\)  \(0.33 \times 10^{3}\)  0 
Liver  \(0.74 \times 10^{2}\)  0  \(0.27 \times 10^{4}\) 
Trafficking rates (\(day^{1}\)) of effector CTLs between Blood, Spleen and Liver compartments
Organ  Blood  Spleen  Liver 

Blood  \(20\)  0.25  0.2 
Spleen  10  \(0.25\)  0 
Liver  10  0  \(0.2\) 
The example of compartmental dynamics of LCMV infection predicted by the model is shown in Fig. 4.16a. It presents the simulation of the CTLmediated liver disease after i.v. infection of C57BL/6 mice with \(2\times 10^5\) pfu of LCMVWE. The model predicts that: (i) virus growth in the liver proceeds at a slower rate than in spleen and the viremia lasts for about 1 week; (ii) CTL response in spleen eliminates the splenic virus in about 10 days starting from day 4; (iii) it takes 3 days more to overcome virus replication in the liver and this time lag is needed for effector CTL to accumulate in liver above the threshold number, estimated to be \(1.24 \times 10^5\) cells per ml of liver, for which the basic reproductive ratio of the virus in liver becomes less than one; (iv) the serum enzyme levels start to rise at high rate by day 5 after infection.
The dynamics and outcome of LCMV infection after peripheral route of infection is quite different. In Fig. 4.16b, the simulation of a direct injection of \(2\times 10^5\) pfu of LCMVWE into the liver of C57BL/6 mouse is shown. The model predicts that virus extensively replicates in liver reaching the maximum possible titer of \(3\times 10^8\) pfu/ml by day 5, which implies that all target cells get infected. Virus growth in the spleen is decreased and delayed by about 1 day as compared to the i.v. infection, and therefore, the splenic CTL response starts later. By day 5, when effector CTLs accumulate in the liver in large number, the destruction of all the infected hepatocytes results in a fulminant immunopathology as is manifested in the model by the enormous elevation of AST and ALT levels. Therefore, this particular combination of viral and host parameters leads to a lethal outcome.
Adoptive transfer experiments demonstrated that virusspecific CTLs are crucial in production of LCMVassociated hepatitis. We examined the ‘doseeffect’ relationship between the number of effector CTLs injected into blood from one side and the peak serum AST levels and the time until virus in spleen declines below detection limit of 100 pfu/ml on the other side. The scenario of experimental i.v. infection of a naive C57BL/6 mouse with \(2\times 10^5\) pfu of LCMVWE accompanied with adoptive transfer of effector CD8\(^+\) T cells at day 0 was mathematically modelled to determine the maximum serum AST level. The predicted effect of the number of transferred effector CTL and peak AST is shown in Fig. 4.17 Left, (b). It suggests that a higher number of injected effectors decreases the severity of clinical disease, and injection of about \(10^3\) cells is enough to reduce the AST level below 500 U/l. The time required to eliminate virus below detection limit displays a nonmonotone pattern, it declines from 14 to 7 days as the number of transferred effector CTL increases from 0 to \(10^5\). Further increase of transferred CTL above \(4\times 10^5\) cells leads to a rapid virus elimination within 1 day with no signs of disease. Note, that the narrow suboptimal range of transferred CTL represents the situation when the basic reproductive ration of virus infection is close to 1.
The effect of increase in number of virusspecific precursor CTL in spleen (the responder immune status) on the severity of LCMVWE induced liver disease and the time of virus elimination is summarized in Fig. 4.17 Left, (c). A lifelong virus persistence and CTL exhaustion are predicted by the model as an outcome of systemic infection with \(2\times 10^5\) pfu of mice with less than 20 precursor CTLs in spleen. The minimal number of precursor CTL to clear an infection is about 100 per spleen. The time needed for virus elimination decreases with the increase in the number of precursor CTLs but does not go below 4 days, in contrast to the case of effector CTLs. For initial numbers of precursor CTLs in spleen ranging from 30 to 100 cells the outcome of the highdose infection would be a severe or fatal hepatitis, reflecting an unfavourable combination of viral and host parameters.
Figure 4.17 right, (c) predicts the effect of the LCMVspecific precursor CTLs present in the spleen at the moment of infection on the peak AST level and virus elimination time. Low numbers (less than 20 cells per spleen) of CTLp are associated with virus persistence, CTL exhaustion and no symptoms of hepatitis. For the initial number of CTLp in between 20 and \(10^5\) cells a severe or fatal hepatitis would be an outcome of the intrahepatic infection with \(2\times 10^5\) pfu. Only the population of splenic precursor CTLs larger than \(10^5\) cells would provide protection against virus persistence, but at the expense of a marked damage of the infected liver. Even with \(10^7\) LCMVspecific precursor CTLs in the spleen, the mouse would need at least 4 days to eliminate the virus and the associated immunopathology would still be above 1000 U/l. This time lag is required for them to get activated and generate sufficient number of effector CTLs.
Overall, a ‘Complete’ characterization of the outcome of virus–host organism interaction with a mathematical model requires consideration of not only the immune response and viral dynamics, but also some characteristics of tissue damage. A new ‘spatial’ dimension can be introduced into the model via compartmental analysis.
The extended model quantitatively predicts that there is a range for the initial number of precursor CTLs in spleen for which an elevation in the clonal size is accompanied by an increase of disease immunopathology. Thus, it overcomes the predictive limitations of a singlecompartmental model as discussed in Sect. 4.2 and reflects what was described experimentally as ‘vaccination for disease’ [16].
4.3 Parameters Defining a Robust DCInduced CTL Expansion^{5}
Successful vaccination depends on the availability of specific antigens, efficient delivery of these antigens, and their optimal presentation to T cells within secondary lymphoid organs. The growing knowledge of the molecular identity of tumourspecific antigens has opened new avenues for effective cancer vaccines [52]. Immunotherapeutic approaches based on adoptive transfer of dendritic cells (DC) expressing relevant antigens may be used for active mobilization of cellular immune responses (CTLs, Thelper cells and NK cells) against tumours. DCbased immunotherapeutic approaches appear particularly promising because DC migrate to the Tcell zones of secondary lymphoid organs where they efficiently initiate both Th and CTL responses [53, 54]. The extraordinary efficacy of DC to prime immune responses is shown by the fact that only \(10^2  10^3\) antigenpresenting DCs in the spleen are sufficient to achieve protective levels of CTL activation in mice [55]. A series of preclinical experimental studies in mice demonstrated that antitumour immunity can be induced using DC [56, 57, 58, 59]. This preclinical experience has been translated into the performance of a variety of clinical trials, which have shown that application of DC is safe and that clinical efficacy of this treatment strategy can be obtained [60, 61, 62].
The efficacy of this active immunization depends on the complex biology of the DC life cycle and their interaction with T cells. The kinetics of this interaction and its sensitivity to relevant parameters are still incompletely understood. These parameters include antigen loading, DC maturation stage, frequency and route of DC injection, frequency and activation status of T cells, and the homing rate of DC to and their persistence within lymphoid tissues. However, the major quantitative parameters of the DC–CTL interaction (e.g. the elimination kinetics of DC by CTL, the threshold for T cell activation, and the impact of DC on T cell homing and recirculation) require further analysis.
 1.
initial data collection and model establishment by data assimilation;
 2.
evaluation of effects of varied parameters in a range that is easily accessible to the model prediction but not experimental measurement;
 3.
model predictions on DCbased immunization and experimental validation.
4.3.1 The Experimental Model of LCMV gp33Specific CTL Induction

Activated CD8\(^+\) 62L\(^\) Tcells staining with the gp33tetramer (tet+) in spleen

Quiescent CD8\(^+\)CD62L\(^+\)tet\(^+\) cells in spleen \(E_m(t)\);

The availability of adoptively transferred DC for productive interaction with Tcells within secondary lymphoid organs was quantified. To this end 51Crlabelled H8DC were injected i.v. into naive recipient mice, and the accumulated radioactivity was determined in spleen at different time points using established protocols [65].
The data set for homing of adoptively transferred DC from blood to spleen has been published elsewhere [84].
4.3.2 Mathematical Model for DCInduced Systemic Dynamics of CTL Responses
Mathematical models for the interaction of antigenpresenting cells (APC) and T cells developed so far, consider mainly the stimulatory aspects of the interaction of APC and T cells [67, 68]. However, CTLmediated killing of the antigenpresenting DC is probably a key process in the downregulation of adaptive immune responses [63, 69, 70]. The positive amplification effect of antigenpresenting DC on the CTL population and the negative feedback from CTL on DC numbers implies that the cell population dynamics of the CTL–DC system in vivo most likely reflects a predator–prey type of interaction (Fig. 4.18). Mathematical modelling facilitates the analysis of the following issues: (i) suitability of the predator–preytype framework for the dynamics of the DC–CTL system in vivo; (ii) estimation of thresholds for DCmediated CTL induction and trafficking; (iii) analysis of sensitivity of CTL dynamics to various parameters (e.g. halflife of DC and the initial number of precursor T cells); and (iv) role of TCR avidity in the robustness of CTL priming.
 1.
DC do not recirculate from lymphoid organs into the blood after intravenous injection.
 2.
Adoptively transferred DC are in mature state.
 3.
DCmediated induction of antigenspecific CTL is due to their interaction in the spleen.
 4.
DC do not divide in secondary lymphoid organs.
 5.
DC decay due to a short lifespan and their killing by activated CTL.
 6.
The population of antigenspecific CTLs in spleen is split into quiescent (nave or central memorylike) and activated CTL (effector or effector memorylike).
 7.
CTL recirculate among spleen, blood and peripheral organs (e.g. liver).
To formulate the systemic model, we follow a building block approach and calibrate submodels (1) for initial DC distribution, (2) DC–CTL population dynamics in spleen, and (3) the compartmental dynamics of CTL responses.
4.3.2.1 Initial DC Migration
Transfer rates (\(hr^{1}\)) of DC between Blood, Spleen, Liver and Lung compartments
Organ  Blood  Spleen  Liver  Lung 

Blood  −1.124  0  0  0 
Spleen  0.12  0  0  0 
Liver  0.38  0  0  0 
Lung  0.16  0  0  \(0.0911\) 
4.3.2.2 DC–CTL Interaction in Spleen
The estimated parameters of the mathematical model of H8DCinduced CTL population dynamics
Parameter  Biological meaning  Units  Bestfit estimate 

\(\mu _{BS}^{H8DC}\)  Transfer rate of H8DCs from blood to spleen  1/day  2.832 
\(\alpha _{D}\)  Decay rate of gp33 expressing DCs  1/day  0.23 
\(b_{DE}\)  Per capita elimination rate of H8DCs by activated CTLs  ml/(cell day)  \(0.487 \times 10^{5}\) 
\(E^{naive}\)  The number of naive gp33specific CTLs contributing to primary clonal expansion  cell  370 
\(\tau _d\)  Duration of preprogrammed CTL division cycle  day  1 
\(\alpha _{E_a}\)  Rate constant of activated CTLs death  1/day  0.12 
\(\alpha _{E_m}\)  Rate constant of resting memory CTLs death  1/day  0.01 
\(b_p\)  Maximal expansion factor of activated CTLs per day  1/day  12 
\(\theta _{D}\)  Threshold in DC density in the spleen for halfmaximal proliferation rate of CTL  cell/ml  \(2.12 \times 10^{3}\) 
\(r_{am}\)  Rate constant of reversion of activated CTLs  1/day  0.01 
\(b_{a}\)  Activation rate constant of quiescent CTLs by DCs  ml/(cell day)  \(1.05 \times 10^{3}\) 
\(\theta _{shut}\)  Threshold in DC density in the spleen for halfmaximal transfer rate of CTL from spleen to blood  cell/ml  13.0 
The model predicts that the threshold of DC density for halfmaximal CTL expansion rate in the spleen is about 200 cells per spleen which explains the rather small effects in the chosen dose range on the magnitude of the CTL response. The amplification factor of the CTL expansion is about 12 cells per day implying that the preprogramming effect probably lasts for three to four divisions. The estimate of per capita CTLmediated elimination rate of the DC (\(b_{DE}\)) suggests that the threshold number of activated CTLs eliminating about 50% of antigenpresenting DCs per day is about \(1.4 \times 10^4\) per spleen. Furthermore, the model predicts that about \(7\%\) of the activated CTLs enter the memory pool.
4.3.2.3 Compartmental Dynamics of CTL Responses
Trafficking rates (\(day^{1}\)) of \(tet^+\) CTLs between Blood, Spleen and Liver compartments
Organ  Blood  Spleen  Liver 

Blood  \(1 \)  [0.012, 0.112]  0.51 
Spleen  0.022  \([0.012, 0.112]\)  0 
Liver  0.1  0  \(0.51\) 
\(\mathbf{M}^{E_i} = \left( \begin{array}{ccc} \mu _{BB} &{} \mu _{SB}(D_{Spleen}(t)) &{} \mu _{LB} \\ \mu _{BS} &{} \mu _{SB}(D_{Spleen}(t)) &{} 0 \\ \mu _{BL} &{} 0 &{} \mu _{LB} \\ \end{array} \right) \), with \(\mu _{SB}(D_{Spleen}(t)) = \mu ^*_{SB} + \frac{\varDelta \mu }{1 + D_{Spleen}(t)/\theta _{shut}}\). Here, the DCdependent migration rate from the spleen to the blood takes into account the trapping effect. The input/output vectorfunction
The estimated trafficking rate parameters for CTLs are listed in Table 4.7. The computed curves of CTL dynamics versus the experimental data are shown in Fig. 4.21. A critical feature for the systemic response is that CTL transfer rates from spleen to blood appear to be DCdensity dependent. To describe the observed CTL kinetics, one needs to consider the possibility of the DCdependent retention of T cells. Thus, the model predicts a trapping effect, which reduces the export rate of CTLs to blood by about tenfold above a threshold of about 10 H8DCs present in the spleen and equally applies to quiescent and activated CTL. The model predicts that 89% of peptidespecific CTL leave the blood compartment daily to organs other than the spleen and liver.
The sensitivity analysis of the model solutions suggests that Tcell receptor avidity, the halflife of DC, and the rate of CTLmediated DC elimination are the major control parameters for optimal DCinduced CTL responses. For induction of high avidity CTLs, the number of adoptively transferred DC was of minor importance once a threshold of approximately 200 cells per spleen had been reached. As discussed before, the major objective of DCbased immunization is the maximal expansion and longterm maintenance of high numbers of antigenspecific T cells. Thus, the model can be applied to study the patterns of CTL population dynamics following repeated injection of H8DC. Two sequential applications of \(2 \times 10^4\) DCs at days 0, and 40 induce a robust CTL response with only a weak boosting effect. The model predicts that as long as significant numbers of activated (or memory cells with a faster activation kinetics than that of naive) CTLs persist which ensure rapid elimination of antigenexpressing DCs, any further application of DCs has only a limited enhancement effect. Nevertheless, such repeated DC application is apparently necessary to maintain high levels of activated CTLs.
4.4 MHV Infection: How Robust Is the IFN Type IMediated Protection?^{7}
Human infections with highly virulent viruses, such as 1918 influenza or SARScoronavirus, represent major threats to public health. The initial innate immune responses to such viruses have to restrict virus spread before the adaptive immune responses fully develop. Therefore, it is of fundamental practical importance to understand the robustness and fragility of the early protection against such virus infections mediated by the type I interferon response. The inherent complexity of the virus–host system suggests the application of mathematical modelling tools to predict the sensitivity of the kinetics and severity of infection to variations in virus and host parameters.
4.4.1 Immunobiology of MHV Infection
It has been demonstrated experimentally that pDCs are the major cell population generating IFN\(\alpha \) during the initial phase of mouse coronavirus infection [76]. Importantly, mainly macrophages (\(M\phi \)) and, to a lesser extent conventional DCs, respond most efficiently to the pDCderived type I IFN and thereby secure containment of MHV within secondary lymphoid organs (SLOs) [78]. Thus, the type I IFNmediated crosstalk between pDCs and \(M\phi \) represents an essential cellular pathway for the protection against MHVinduced liver disease. In systems biology terms, MHV infection triggers a complex array of processes at different biological scales such as protein expression, cellular migration or pathological organ damage. To focus on the front edge of the virus–host interaction, the modellingbased analysis specifically addresses the early dynamics (i.e. the first 48 h) of the type I IFN response to MHV since this is decisive for the outcome of the infection. The reductionists view of the most essential processes underlying the early systemic dynamics of MHV infection, liver pathology and the first wave of type I IFN production is summarized in Figs. 4.22 and 4.23.
4.4.2 Setting up a Mathematical Model
To describe quantitatively the structure, dynamics and the operating principles that permit pDCs to initially shield the host against an overwhelming spread of the cytopathic MHV infection, one can follow a systems biology approach. First, the system dynamics is decomposed into a set of elementary, welldocumented processes such as virus replication, target cell turnover and IFNI decay, as well as the production of virus and IFNI by infected cells (Fig. 4.22). This allows one to quantify the individual decay rates, the virus–target cell interaction parameters and the protective effect of IFNI. Once these elementary modules of virus–target cell interactions were calibrated, one can use them as building blocks to set up an integrated mathematical model of pDCmediated type I IFN responses against MHV infection in mice.

Kinetics of virus, IFNI and cells in vitro,

Basic IFNI response to infection of target cells,

Compartmental dynamics of virus growth,

Systemic model of MHV infection and IFNI response.

IFNI I(t) and unifected/infected pDCs and macrophages \(C^{pDC}(t), C^{M\phi }(t), C_{V}^{pDC}(t), C_{V}^{M\phi }(t)\) in spleen,

systemic dynamics of the virus in spleen, blood and liver \(V_{S}(t), V_{B}(t), V_{L}(t)\).

dynamics of liver enzyme AST in blood.
Estimated parameters of the mathematical model MHV infection and type I IFN response
Parameter  Biological meaning  Units  Bestfit estimate 

\(\rho _{V}^{DC}\)  Virus production rate by pDC  pfu/cell/h  1.7 
\(\rho _{V}^{M\phi }\)  Virus production rate by \(M\phi \)  pfu/cell/h  36.7 
\(\rho _{I}^{DC}\)  Type I IFN production rate by pDC  pg/cell/h  \(4.4 \times 10^{4}\) 
\(\rho _{I}^{M\phi }\)  Type I IFN production rate by \(M\phi \)  pg/cell/h  \(1.0 \times 10^{6}\) 
\(\theta _{pDC}\)  The threshold for 50% reduction of virus production rate by type I IFN  pg/ml  45.8 
\(\theta _{M\phi }\)  The threshold for 50% reduction of virus production rate by type I IFN  pg/ml  0.97 
\(\sigma _{V}^{DC}\)  Infection rate of pDC  cell/pfu/h  \(1.3 \times 10^{6}\) 
\(\sigma _{V}^{M\phi }\)  Infection rate of \(M\phi \)  cell/pfu/h  \(0.9 \times 10^{7}\) 
\(\tau ^{pDC}_{V}\)  Virus production delay by pDC  h  5.96 
\(\tau ^{M\phi }_{V}\)  Virus production delay by pDC  h  5.99 
\(\tau ^{pDC}_{I}\)  Type I IFN production delay by pDC  h  5.77 
\(\tau ^{M\phi }_{I}\)  Type I IFN production delay by \(M\phi \)  h  5.8 
\(d_{0CV}^{pDC}, \ k_{CV}^{pDC}\)  Gompertz death rate parameters for infected pDC  1/h  0.2, 0.087 
\(d_{0CV}^{M\phi }, \ k_{CV}^{M\phi }\)  Gompertz death rate parameters for infected \(M\phi \)  1/h  0.049, 0.057 
\(\mu _{BS}\)  Virus transfer rate from blood to spleen  1/h  3.46 
\(\mu _{BL}\)  Virus transfer rate from blood to liver  1/h  0.018 
\(\mu _{SB}\)  Virus transfer rate from spleen to blood  1/h  0.91 
\(\mu _{LB}\)  Virus transfer rate from liver to blood  1/h  0.61 
\(\mu _{BO}\)  Virus elimination rate from blood  1/h  1.22 
\(\beta _{L}\)  Virus growth rate in liver  pfu/ml/h  0.78 
\(K_{L}\)  Carrying capacity of the liver  pfu/ml  \(10^7\) 
\(\rho _{A}\)  Rate constant of ALT release into blood  IU/l  \(0.68 \times 10^{3}\) 
\(d_{A}\)  Decay rate of ALT release in blood  1/h  0.16 
\(A_{*}\)  Physiological level of ALT in blood  IU/l  25 
4.4.3 Parameter Estimates and Sensitivity Analysis
The bestfit parameter estimates of the model characterize the concentration of IFNI which is required to inhibit by twofold the production of virus by the infected cells. It appears that the pDC and M\(\phi \) differ with respect to their sensitivity to the protective effect of interferon, so that the 50% reduction threshold concentrations are about 46 pg/ml and 1 pg/ml, respectively. The per capita type I IFN secretion rate also differs substantially between pDC and M\(\phi \), being 15586 molec/h and 106 molec/h, respectively. The sensitivity analysis suggests a high protective capacity of single pDCs which protect \(10^3\)–\(10^4\) \(M\phi \) from cytopathic viral infection localized to spleen. The model allows one to determine the minimal protective unit of preactivated pDCs in spleen to be around 200 cells which can rescue the host from severe disease. The modelling results suggest that the spleens capability to function as a sink for the virus produced in peripheral target organs remains operational as long as viral mutations do not permit accelerated growth in peripheral tissues.
Overall, the modelling results suggest that the pDC population in spleen ensures a robust protection against virus variants which substantially downmodulate type I IFN secretion. However, the ability of pDCs to protect against severe disease caused by virus variants exhibiting an enhanced liver tropism and higher replication rates appears to be rather limited. Taken together, this system immunology analysis suggests that antiviral therapy against cytopathic viruses should primarily limit viral replication within peripheral target organs.
4.5 Identifying a Feedback Regulating Proliferation and Differentiation of CD4\(^+\) T Cells^{8}
In response to antigens, specific Tcell clones rapidly increase in size and then steeply decline, approaching relatively stable frequencies higher than those of the naive cell population. It was discovered by W.E. Paul’s team (see data presented in [79]) that there is a loglinear relation between the CD4\(^+\) Tcell precursor number (PN) and the factor of expansion (FE), with a slope of \({\sim } 0.5\) over a range of 3–30,000 antigenspecific precursors per mouse. The experimental results suggested an inhibition mechanism of precursor expansion either by competition for specific antigenpresenting cells or by the action of other antigenspecific cells in the same microenvironment. Mathematical modelling can be used to identify the specific functions underlying the feedback regulation of the observed clonal dynamics.
One can conclude that the feedbackregulated balance of growth and differentiation hypothesis, although requiring definite experimental characterization of the hypothetical cell phenotypes and molecules involved in the identified regulation, can explain the kinetics of CD4\(^+\) Tcell responses to antigenic stimulation. We note that a mathematical model based on a different hypothesis (e.g. ‘grazing of peptideMHC complexes’) was proposed to explain the same phenomenon although in a semiquantitative manner [83]. However, no evidence of its consistency with all available data sets that were described and analysed in [4, 79] was presented.
In conclusion, while a multitude of mathematical models can be generated to describe any given immunological phenomenon, it is crucial to always link it to available experimental data. If model and data are in good agreement, then the model may help to generate new hypothesis of underlying mechanisms and provide further testable predictions. In addition, as outlined in Sect. 4.5, a model may also strongly support a novel hypothesis that was brought up ad hoc from immunological considerations.
Footnotes
 1.
Material of subsections (4.2.2–4.2.4) uses the results of our studies from Bocharov, Modelling the dynamics of LCMV infection in mice: conventional and exhaustive CTL responses. J. Theor. Biol. 192, 283–308, Copyright \(\copyright \) 1998; Ehl et al., The impact of variation in the number of CD8+Tcell precursors on the outcome of virus infection. Cell. Immunol. 189, 67–73, Copyright \(\copyright \) 1998; Bocharov et al., Modelling the dynamics of LCMV infection in mice: II. Compartmental structure and immunopathology. J. Theor. Biol. 221, 349–78, Copyright \(\copyright \) 2003; Luzyanina et al., Low level viral persistence after infection with LCMV: a quantitative insight through numerical bifurcation analysis. Math. Biosci. 173, 1–23, Copyright \(\copyright \) 2001, with permission from Elsevier and the results of the studies from Proc. Natl. Acad. Sci. USA. (PNAS USA), Bocharov et al., Feedback regulation of proliferation vs. differentiation rates explains the dependence of CD4 Tcell expansion on precursor number, 108, 3318–23, Copyright \(\copyright \) 2011 with permission from PNAS USA.
 2.
The material of this subsection uses the results from Luzyanina et al., Low level viral persistence after infection with LCMV: a quantitative insight through numerical bifurcation analysis. Math. Biosci. 173, 1–23, Copyright \(\copyright \) 2001, with permission from Elsevier.
 3.
Material of this subsection uses the results of our studies from Ehl et al., The impact of variation in the number of CD8+Tcell precursors on the outcome of virus infection. Cell. Immunol. 189, 67–73, Copyright \(\copyright \) 1998, with permission from Elsevier.
 4.
Material of this subsection uses the results of our studies from Bocharov et al., Modelling the dynamics of LCMV infection in mice: II. Compartmental structure and immunopathology. J. Theor. Biol. 221, 349–78, Copyright \(\copyright \) 2003, with permission from Elsevier.
 5.
Material of this section uses the results from Ludewig et al., Eur J Immunol. 34 (2004), 2407–18.
 6.
(see for details Bocharov et al., (2005): A Mathematical Approach for Optimizing Dendritic CellBased Immunotherapy. In: Adoptive Immunotherapy. Methods and Protocols, Eds. Ludewig B. and Hoffmann M.W. (Humana Press) 109: 19–34).
 7.
Material of this section uses the results Bocharov et al., PLoS Pathog. 6 (2010), e1001017.
 8.
Material of this subsection uses the results of our studies from Proceedings of the National Academy of Sciences of the United States of America (PNAS USA), Vol. 108, Bocharov et al., Feedback regulation of proliferation vs. differentiation rates explains the dependence of CD4 Tcell expansion on precursor number, Pages 3318–3323, Copyright \(\copyright \) 2011 with permission from PNAS USA.
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