Abstract
Human immunodeficiency virus (HIV) is a fast replicating ribonucleic acid virus, which can easily mutate in order to escape the effects of drug administration. Hence, understanding the basic mechanisms underlying HIV persistence in the body is essential in the development of new therapies that could eradicate HIV infection. Lymphoid tissues are the primary sites of HIV infection. Despite the recent progress in real-time monitoring technology, HIV infection dynamics in a whole body is unknown. Mathematical modeling and simulations provide speculations on global behavior of HIV infection in the lymphatic system. We propose a new mathematical model that describes the spread of HIV infection throughout the lymphoid tissue network. In order to represent the volume difference between lymphoid tissues, we propose the proportionality of several kinetic parameters to the lymphoid tissues’ volume distribution. Under this assumption, we perform extensive numerical computations in order to simulate the spread of HIV infection in the lymphoid tissue network. Numerical computations simulate single drug treatments of an HIV infection. One of the important biological speculations derived from this study is a drug saturation effect generated by lymphoid network connection. This implies that a portion of reservoir lymphoid tissues to which drug is not sufficiently delivered would inhibit HIV eradication despite of extensive drug injection.
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The authors are grateful to the referees for constructive and helpful comments and suggestions, which led to significant improvement of our original manuscript.
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Dedicated to Mats Gyllenberg on the occasion of his 60th birthday.
This research was partly supported by (i) the Japan Society for the Promotion of Science (JSPS) through the “Grant-in-Aid for Young Scientists B25871132 (to S.N.) and B25800092 (to S.I.)”, and received funding support from the Commissioned Research program of the Ministry of Health, Labour and Welfare, Japan (to S.N., H26-ShinkoJitsuyoka-General-016).
Appendix: Supplementary information
Appendix: Supplementary information
1.1 Quasi-steady state approximation
Let us consider a situation in which a virion’s turnover rate is significantly faster than that of a CD4 T cell. While we do not apply quasi-steady state approximation to the blood component. Then, quasi-steady state approximation can be applied to the third equation in (2.9). That is,
Equation (A.1) for \(j=1,2,\ldots ,N\) are then written:
If \(\det (\mathbf {I}-\frac{\rho _V{{\mathbf {M}}}_{\mathbf {L}}}{c}) \ne 0\), then
Assume that \(\rho _V{{\mathbf {M}}}_{\mathbf {L}}/c \approx {\mathbf {O}}\), where \({\mathbf {O}}\) is the zero matrix. Then we have
Substituting (A.4) into (2.9), system (2.9) is reduced to the following \(2\times N\)-dimensional ordinary differential equations:
Let \(E_{c}=({\bar{T}}_1,\ldots ,{\bar{T}}_N,{\bar{T}}_B,0,\ldots ,0)\) denote an infectious-free equilibrium for (A.5) with (2.8), and let T(t) and I(t) denote the total number of uninfected and infected CD4 T cells in the lymphoid tissue network at time t, respectively. That is,
Note that the total flux (input and output) of migrated cells between the lymphoid tissues is conserved. More precisely, it follows from (2.6) that
Adding the right hand side of system (A.5) and the first equation of (2.8) at this equilibrium, we find
Note that any solution of (A.8) with positive initial condition converges to \(\lambda /d\) as \(t \rightarrow \infty \). The explicit values of the components of \(E_c\) correspond to the stable equilibrium of the following system of ordinary differential equations
with the constraint:
1.2 Derivation of basic reproduction number
We derive a next generation matrix for (A.5), which will determine the possibility of HIV persistence. The next generation matrix is defined for the linearized system in (A.5) around \(E_{c}\). Define matrices \({\mathcal {F}}\) and \({\mathcal {V}}\) to be
and
The basic reproduction number is given by the spectral radius of next generation matrix \({\mathcal {F}}{\mathcal {V}}^{-1}\) (Diekmann et al. 1990; Driessche and Watmough 2002):
The basic reproduction number \(R_{0,\mathrm{{LN}}}\) is numerically calculated using the predefined function \(\mathtt {eigen()}\) in the statistical computation software \(\mathtt {R}\) (R Core Team 2014).
1.3 Two lymphoid tissues model
Numerical computations in Sect. 4 implies existence of the lower limit of \(R_{0,LN}\). To analytically prove existence of the lower limit, we consider a simplified system in which only two lymphoid tissues exist and are connected each other. The corresponding system of differential equations is given by
Let \(E_0=({\bar{T}}_1,\bar{T_2},0,0)\) denote an equilibrium in which no infection occurs. In the similar way in “Derivation of basic reproduction number”, matrices \({\mathcal {F}}_2\) and \({\mathcal {V}}_2\) for system (A.14) are explicitly given by
and
A decline curve of basic reproduction number \(R_{0,2}\) (left) and numerical simulation result of \(I_1\) for \(\alpha =0.1\) (red solid) and \(\alpha =0.5\) (green dashed) (right). Left Existence of plateau at low \(\alpha \). Right \(R_{0,2}\) plays a role as the threshold to determine infected cell persistence or eradication. Numerical values of parameters used are \(\alpha \in (0.016,0.5)\) (left), \(\lambda _1=0.2\), \(\lambda _2=0.1\), \(\delta =2.0\), \(m_1=10.0\), \(m_2=5.0\), \(d=0.01\) (color figure online)
The basic reproduction number defined for \(E_0\) of system (A.14) is given by the spectral radius of next generation matrix \({\mathcal {F}}_2{\mathcal {V}}_{2}^{-1}\):
Note that \(R_{0,2}\) is explicitly given as
Note that \(R_{0,2}\) is monotonically increasing with respect to \(\alpha \):
Consider a situation that drug treatment reduces \(\alpha \). By taking the limit \(\alpha \rightarrow 0\), \(R_{0,2}\) converges to
If migration rates \(m_1\) and \(m_2\) are sufficiently larger than \(\delta \), then
(A.21) indicates that the basic reproduction number is less than 1 but it is close to 1 if cellular/viral migration are sufficiently fast. For instance, we use the following numerical values to calculate the limit of \(R_{0,2}\) with respect to the limit \(\alpha \rightarrow 0\):
Then \(\bar{R}_{0,2} \simeq 0.8\). The threshold value of \(\alpha \) which determines whether \(R_{0,2}>1\) or not is \(\alpha ^* \simeq 0.12\). To simulate drug therapy, numerical values of basic reproduction number \(R_{0,2}\) are calculated with respect to (A.22) and different \(\alpha \). The left panel of Fig. 8 shows a decline curve of \(R_{0,2}\) with respect to the decrease of \(\alpha \). Saturation effect described as the plateau of the curve at low values of \(\alpha \) exists. Moreover, numerical simulation results with \(\alpha =0.5>\alpha ^*\) and \(\alpha =0.1<\alpha ^*\) shown in the right panel of Fig. 8 indicate that \(R_{0,2}\) plays a role as the threshold to determine whether infected cells persist or not. Importantly, although parameter values used in numerical computations of \(R_{0,2}\) have less biological meaning, qualitatively similar characteristics to the network model under single drug treatment therapy in Sect. 4 are reproduced with two lymphoid tissues model (A.14).
1.4 Scaling
The units of variables T, I and V are (cells/ml), (cells/ml) and (copies/ml), respectively (see Table 1 in Appendix). Because the total amount of blood accounts for about 8 % of total body mass, the average man (60 kg) contains approximately 5 l (5 kg) of blood (Alberts et al. 2008, Table 22-1 Blood Cells). We assume that about 2 % of white blood cells are found in the peripheral blood (Westermann and Pabst 1990; Vrisekoop et al. 2008). In addition, because the average number of lymphocytes in the peripheral blood is \(1.0\times 10^{6}\) cells/ml, the total number of lymphocytes in a body is estimated to be
Let x, y and z denote the number of uninfected CD4 T cells, infected CD4 T cells and virions in a body, respectively. Define the conversion ratio \(\gamma \) to be
Then the relationship between any two variables is
Parameters \(\beta \) and \(\omega \) in the (T, I, V)-system are the product of \(\gamma \) and the corresponding parameters in the un-scaled system. The other parameters remain unchanged.
1.5 Parameter values
The default values for the parameters are summarized in Table 1 in Appendix. We use the reference values \(d=0.01\) and \(\delta =0.7\) for the per capita death rates of the uninfected and infected CD4 T cells, respectively. The normal count of CD4 T cells contained in 1 ml of peripheral blood is approximately \(1.0\times 10^6\) in a healthy individual. Since about 2 % of white blood cells are found in blood, we assume that the total number of CD4 T cells is in a steady state \(\lambda /d\) under the absence of infection. Hence, we have \(\lambda =5.0\times 10^5\) and \(T(0)+T_B(0)=\lambda /d\). We need to point out the limitation of the current model formulation. [R2-1] The current model does not distinguish CD4 T cell subsets. In quantitative estimates of human immune cell turnover rates using in vivo kinetic deuterium labeling (Vrisekoop et al. 2008), an estimate for naive T cell turnover rate is in the order of 0.001 [/day]. Hence the choice of the numerical value \(d=0.01\) in Table 1 in Appendix might be inappropriate. Regardless of the choice of \(\lambda \) and d, however, our numerical simulation results under the absence of infection shows consistent results with clinical observations that a T cell level in the peripheral blood is \(1.0 \times 10^6\) per ml. Values \(\beta \) and \(\omega \) vary. Because the HIV infection’s basic reproduction number is estimated to be between 8 and 10, the ranges of \(\beta \) and \(\omega \) are determined so that they cover the estimated value for \(R_{0,\mathrm{{LN}}}\). Numerical values of free and cell-to-cell infection rate \(\beta \) and \(\omega \) are kept in the same order for the following reasons. In Komarova et al. (2013), in vitro cell culture system was used to infect target cells (Jurkat cells expressing CXCR4) by HIV. Experimental data of a single round HIV infection with/without shaking were fitted with their mathematical model which incorporates two sources of infection: free and cell-to-cell transmission. The ratio of the free to cell-to-cell infection rate (\(\omega /\beta \) in our notation) for several experiments indicated that estimated values take 1.0 on average, suggesting that the free and cell-to-cell infection rates are in the same order (Table 1 in Komarova et al. 2013). The number of HIV virions produced from one infected cell during its life expectancy is known as its burst size. Because the life expectancy of infected cells is \(1/\delta \simeq 1\), the burst size roughly corresponds to the per capita production rate of the virions p. According to the estimate in Chen et al. (2007), burst size in vivo is approximately \(1.0 \times 10^4\) per ml. In addition, given \(1.0 \times 10^4\) infected cells per ml, it follows from the burst size estimate that the estimated number of virions is \(1.0 \times 10^8\). This contradicts clinical observations, in which the estimated number of mRNA copies of HIV is \(1.0 \times 10^{5}\) on average. One possible interpretation of this is given in Boer and Perelson (2013), where the clearance of HIV is suggested to be faster than the usual estimates obtained using lymphoid tissue. In this paper, we adjust the value of p in order to produce clinically relevant observations. The decay rate of virions \(c = 30.0\) is also taken from past literature. Because the volume distribution of lymphoid tissues is currently unknown, the exponential distribution was assumed in our study. Additionally, because the total number of infected CD4 T cells and particles of HIV at infection initiation are also unknown, we use the numerical values in Bajaria et al. (2004) as our reference: \(I^0=I(0)=1.0\) (assumed) and \(V^0=V(0)=10\) (Bajaria et al. 2004). To determine the lymphoid tissue structure and migration rate of cells/virions, numerical values of the population migration rate from the blood to lymphoid tissues \(m_{BL}\) recurrence interval \(\tau \), mean neighborhoods \({\bar{n}}\), mean connections \({\bar{e}}\), and mean lymphoid tissue volume size \({\bar{C}}\) must be specified. We set \(m_{BL}=1000\), \(\tau =1.0\), \({\bar{n}}=5\), \({\bar{e}}=10\), \({\bar{C}}=10\) (mm: axial diameter), respectively. These numerical values are written in Table 1 in Appendix and captions of Figs. 3, 4, 5, and 7. In silico lymphoid tissue network is then generated under the rules described in Sects. 2.3 and 2.4.
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Nakaoka, S., Iwami, S. & Sato, K. Dynamics of HIV infection in lymphoid tissue network. J. Math. Biol. 72, 909–938 (2016). https://doi.org/10.1007/s00285-015-0940-x
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DOI: https://doi.org/10.1007/s00285-015-0940-x
Keywords
- Mathematical Modeling
- Numerical computation
- HIV infection
- Lymphoid tissue network
- The basic reproduction number
- Combinational drug treatment