In Silico Study of the Influence of Intensity and Duration of Blood Flow Reduction on Cell Death Through Necrosis or Apoptosis During Acute Ischemic Stroke

Abstract

Ischemic stroke involves numerous and complex pathophysiological mechanisms including blood flow reduction, ionic exchanges, spreading depressions and cell death through necrosis or apoptosis. We used a mathematical model based on these phenomena to study the influences of intensity and duration of ischemia on the final size of the infarcted area. This model relies on a set of ordinary and partial differential equations. After a sensibility study, the model was used to carry out in silico experiments in various ischemic conditions. The simulation results show that the proportion of apoptotic cells increases when the intensity of ischemia decreases, which contributes to the model validation. The simulation results also show that the influence of ischemia duration on the infarct size is more complicated. They suggest that reperfusion is beneficial when performed in the early stroke but may be either inefficacious or even deleterious when performed later after the stroke onset. This aggravation could be explained by the depolarisation waves which might continue to spread ischemic damage and by the speeding up of the apoptotic process leading to cell death. The effect of reperfusion on cell death through these two phenomena needs to be further studied in order to develop new therapeutic strategies for stroke patients.

Appendix A: Equations of the Model

1.1 A.1 List of the Variables

In the following equations, t refers to the time and x to the place in the brain. Hence x is a vector of dimension 1, 2 or 3, depending on the dimension of the brain that we are considering. For example, in this paper we are working on a section of the brain, hence x = (x₁, x₂) is a two-coordinates vector.

η(t):

fraction of blood flow in the occluded vessel during the stroke in function of time t

BFlow (t, x):

proportion of blood flow that irrigates the cells at location x and time t

BFlow _vessel (x):

physiological proportion of blood flow carried to the cells at location x by the vessel that will be occluded

BFlow _others (x):

physiological proportion of blood flow carried to the cells at location x by the vessels that will remain sane

OEF (t, x):

oxygen extraction factor at time t in the cells at location x

OEF_∞ (BF):

stationary value of the oxygen extraction factor for cells that receive a blood flow BF

E_av (t, x):

available energy at time t in the cells at location x

E_{meta, used} (t, x):

energy used to maintain life at time t in the cells at location x

E_{homeo, used} (t, x):

energy used to maintain homeostasis at time t in the cells at location x

E_{SD, used} (t, x):

energy used to recover from spreading depressions at time t in the cells at location x

E_{act, used} (t, x):

energy used to have an electrical activity at time t in the cells at location x

E_rem (t, x):

remainder energy at time t in the cells at location x

Ca_a (t, x):

astrocytic calcium concentration difference between the cells at location x and time t and the resting state

Ca_n (t, x):

neuronal calcium concentration difference between the cells at location x and time t and the resting state

K_e (t, x):

extracellular potassium concentration difference between time t and location x and the resting state

Cap (t, x):

capacity of the cells at location x to recover from SD at time t

R (t, x):

strength of the recovery mechanism from spreading depression for the cells at location x and time t

$\Uptheta (t,{\text{x}})$ :

state of the cells at location x and time t

Repair (t, x):

ability to repair of the cells at location x and time t

Damage (t, x):

damage to the cells at location x and time t

A (t, x):

progress of the apoptosis process at time t in the cells at location x

1.2 A.2 List of the Equations

The significances of the parameters used in the following equations are given in section A.3.

1.2.1 A.2.1 Blood Flow

By definition of BFlow_vessel and BFlow_others, we have: BFlow_vessel (x) + BFlow_others (x) = 1.(see Fig. 1 for a representation of BFlow_others)

During the stroke, the blood flow at point x and time t is given by:

$$ {\text{BFlow}}(t, {\text{x}}) = \eta (t)\,{\text{BFlow}}_{\text{vessel}} ({\text{x}})\, + \,{\text{BFlow}}_{\text{others}} ( {\text{x)}} $$

(1)

with ${{\upeta}}(t) = {\kern 1pt} \,\left\{ {\begin{array}{*{20}c} {\eta_{0} \quad {\text{if}}\; t \le {\text{T}}_{\text{isch}} } \\ {1\quad {\text{if}}\; t > {\text{T}}_{\text{isch}} } \\ \end{array} } \right.$

1.2.2 A.2.2 Oxygen Absorption

The equations that govern the oxygen extraction factor (OEF) are:

$$ {\frac{{\partial \,{\text{OEF}}}}{\partial t}}(t ,\,{\text{x}}) = {\frac{{{\text{OEF}}_{\infty } ({\text{BFlow}}(t,{\text{x}})) - {\text{OEF}}(t,{\text{x}})}}{{{{\uptau}}_{\text{OEF}} }}} $$

(2)

$$ {\text{OEF}}_{\infty } ({\text{BF}}) = {\text{OEF}}_{\text{average}} - {\text{OEF}}_{\text{var}} \,\tanh \left( {{\frac{{{\text{BF}} - {\text{B}}_{0} }}{{{{\updelta}}_{\text{OEF}} }}}} \right) $$

(3)

1.2.3 A.2.3 Energy

The available energy for a cell is assumed to be proportional to the oxygen extracted from blood:

$$ {\text{E}}_{\text{av}} (t,{\text{x}}) = {\text{Bflow}}(t,{\text{x}})\,{\text{OEF}}(t,{\text{x}}) $$

(4)

This available energy is used for the four main tasks: maintaining metabolism, maintaining homeostasis, recovering from spreading depressions and creating an electrical activity:

$$ {\text{E}}_{\text{meta,used}} (t,{\text{x}}) = {\text{E}}_{\text{meta}} \,\tanh \left( {{\frac{{{\text{E}}_{\text{av}} (t,{\text{x}})}}{{{\text{E}}_{\text{meta}} }}}} \right) $$

(5)

$$ {\text{E}}_{\text{homeo,used}} (t,{\text{x}}) = {\tfrac{1}{2}}{\text{E}}_{\text{homeo}} \left( {\,1 + \tanh \left( {{\frac{{{\text{E}}_{\text{av}} (t,{\text{x}}) - {{{\text{E}}_{\text{meta}} - {\text{E}}_{\text{homeo}} } \mathord{\left/ {\vphantom {{{\text{E}}_{\text{meta}} - {\text{E}}_{\text{homeo}} } 2}} \right. \kern-\nulldelimiterspace} 2}}}{{{{{\text{E}}_{\text{homeo}} } \mathord{\left/ {\vphantom {{{\text{E}}_{\text{homeo}} } 2}} \right. \kern-\nulldelimiterspace} 2}}}}} \right)} \right) $$

(6)

$$ {\text{E}}_{\text{SD,used}} (t,{\text{x}}) = {\tfrac{1}{2}}{\text{E}}_{\text{SD}} \left( {1 + \tanh \left( {{\frac{{{\text{E}}_{\text{av}} (t,{\text{x}}) - {{{\text{E}}_{\text{meta}} - {\text{E}}_{\text{homeo}} - {\text{E}}_{\text{SD}} } \mathord{\left/ {\vphantom {{{\text{E}}_{\text{meta}} - {\text{E}}_{\text{homeo}} - {\text{E}}_{\text{SD}} } 2}} \right. \kern-\nulldelimiterspace} 2}}}{{{{{\text{E}}_{\text{SD}} } \mathord{\left/ {\vphantom {{{\text{E}}_{\text{SD}} } 2}} \right. \kern-\nulldelimiterspace} 2}}}}} \right)} \right) $$

(7)

$$ {\text{E}}_{\text{act,used}} (t,{\text{x}}) = {\tfrac{1}{2}}{\text{E}}_{\text{act}} \left( {1 + \tanh \left( {{\frac{{{\text{E}}_{\text{av}} (t,{\text{x}}) - {{{\text{E}}_{\text{meta}} - {\text{E}}_{\text{homeo}} - {\text{E}}_{\text{SD}} - {\text{E}}_{\text{act}} } \mathord{\left/ {\vphantom {{{\text{E}}_{\text{meta}} - {\text{E}}_{\text{homeo}} - {\text{E}}_{\text{SD}} - {\text{E}}_{\text{act}} } 2}} \right. \kern-\nulldelimiterspace} 2}}}{{{{{\text{E}}_{\text{act}} } \mathord{\left/ {\vphantom {{{\text{E}}_{\text{act}} } 2}} \right. \kern-\nulldelimiterspace} 2}}}}} \right)} \right) $$

(8)

with ${\text{E}}_{\text{need}} = {\text{E}}_{\text{meta}} + {\text{E}}_{\text{homeo}} + {\text{E}}_{\text{SD}} + {\text{E}}_{\text{act}}$

The remaining energy is used for other secondary tasks:

$$ {\text{E}}_{\text{rem}} (t,{\text{x}}) = {\text{E}}_{\text{av}} (t,{\text{x}}) - {\text{E}}_{\text{meta,used}} (t,{\text{x}}) - {\text{E}}_{\text{homeo,used}} (t,{\text{x}}) - {\text{E}}_{\text{SD,used}} (t,{\text{x}}) - {\text{E}}_{\text{act,used}} (t,{\text{x}}) $$

(9)

So we have:

$$ {\text{E}}_{\text{av}} (t,{\text{x}}) = {\text{E}}_{\text{meta,used}} (t,{\text{x}}) + {\text{E}}_{\text{homeo,used}} (t,{\text{x}}) + {\text{E}}_{\text{SD,used}} (t,{\text{x}}) + {\text{E}}_{\text{act,used}} (t,{\text{x}}) + {\text{E}}_{\text{rem}} (t,{\text{x}}) $$

1.2.4 A.2.4 Ionic Exchanges

For simplicity reasons, the variables t and x have not been written in the following equations.We recall the definition of the Laplacian $\Updelta = \sum\nolimits_{k = 1}^{n} {{\frac{{\partial^{2} }}{{\partial x_{k}^{2} }}}}$ where n is the dimension of the space. We also define 1_{A>a} = 1 if A > a and 1_{A>a} = 0 if A ≤ a.

The ionic exchanges between intra- and extracellular spaces are governed by the following equations:

$$ \begin{aligned} {\frac{{\partial \,{\text{Ca}}_{\text{a}}}}{{\partial {\text{t}}}}} - {{\upnu}}_{\text{Ca}} \Updelta {\text{Ca}}_{\text{a}} = & - {\text{k}}_{{{\text{Ca}}_{\text{a}}}} {\text{E}}_{\text{homeo,used}} \tanh \left({{\frac{{{\text{Ca}}_{\text{a}} }}{{{\text{Ca}}_{\text{a}}^{ 0}}}}} \right) + {\text{h}}_{\text{K,Ca}}{\text{E}}_{\text{homeo,used}} {\text{K}}_{\text{e}} - {\text{k}}_{\text{gap}} ({\text{Ca}}_{\text{a}} - {\text{Ca}}_{\text{n}} ) \\ & + \,{\text{c}}_{\text{grad,Ca}}({\text{E}}_{\text{homeo,used}} - {\text{E}}_{\text{homeo}})({\text{Ca}}_{\text{a}} - \left[ {\text{Ca}} \right]_{\text{ext}}) - {{\upalpha}}_{\text{Ca,a}} {\text{Ca}}_{\text{a}}^{2} -{{\upalpha}}_{\text{rec,a}} {\text{E}}_{\text{SD,used}}{\text{R}}\,{\text{Ca}}_{\text{a}} \\\end{aligned} $$

(10)

$$ {\frac{{\partial \,{\text{Ca}}_{\text{n}} }}{{\partial {\text{t}}}}} = - {\text{k}}_{{{\text{Ca}}_{\text{n}} }} {\text{E}}_{\text{homeo,used}} \tanh \left( {{\frac{{{\text{Ca}}_{\text{n}} }}{{{\text{Ca}}_{\text{n}}^{ 0} }}}} \right) + {\text{k}}_{\text{gap}} ({\text{Ca}}_{\text{a}} - {\text{Ca}}_{\text{n}} ) - {{\upalpha}}_{\text{Ca,n}} {\text{Ca}}_{\text{n}}^{ 2} - {{\upalpha}}_{\text{rec,n}} {\text{E}}_{\text{SD,used}} {\text{R}}\,{\text{Ca}}_{\text{n}} $$

(11)

$$ \begin{aligned} {\frac{{\partial \,{\text{K}}_{\text{e}} }}{{\partial {\text{t}}}}} - {{\upnu}}_{\text{K}} \Updelta {\text{K}}_{\text{e}} = & - {\text{k}}_{{{\text{K}}_{\text{e}} }} {\text{E}}_{\text{homeo,used}} \tanh \left( {{\frac{{{\text{K}}_{\text{e}} }}{{{\text{K}}_{\text{e}}^{ 0} }}}} \right) + {\text{h}}_{\text{Ca,K}} {\text{E}}_{\text{homeo,used}} {\text{Ca}}_{\text{n}} \\ & + \,{\text{c}}_{\text{grad,K}} ({\text{E}}_{\text{homeo,used}} - {\text{E}}_{\text{homeo}} )({\text{K}}_{\text{e}} - \left[ {\text{K}} \right]_{\text{int}} ) - {{\upalpha}}_{\text{Ke}} {\text{K}}_{\text{e}}^{ 2} - {{\upalpha}}_{\text{rec,e}} {\text{E}}_{\text{SD,used}} {\text{R}}\,{\text{K}}_{\text{e}} \\ \end{aligned} $$

(12)

The recovery mechanisms (from spreading depressions) are governed by the following equations:

$$ {\frac{{\partial \,{\text{Cap}}}}{{\partial {\text{t}}}}} = 1_{{\left\{ {{\text{Ca}}_{\text{a}} > {\text{c}}_{\text{a}} } \right\} \cup \left\{ {{\text{Ca}}_{\text{n}} > {\text{c}}_{\text{n}} } \right\} \cup \left\{ {{\text{K}}_{\text{e}} > {\text{c}}_{\text{e}} } \right\}}} \,{\frac{{1 - {\text{Cap}}}}{{{\text{T}}_{\text{o}} }}} - 1_{{\left\{ {{\text{Ca}}_{\text{a}} < {\text{c}}_{\text{a}} } \right\} \cup \left\{ {{\text{Ca}}_{\text{n}} < {\text{c}}_{\text{n}} } \right\} \cup \left\{ {{\text{K}}_{\text{e}} < {\text{c}}_{\text{e}} } \right\}}} \,{\frac{\text{Cap}}{{{\text{T}}_{\text{f}} }}} $$

(13)

$$ {\frac{{\partial \,{\text{R}}}}{{\partial {\text{t}}}}} = 1_{{\left\{ {{\text{Cap}} > {\text{c}}_{\text{cap}} } \right\}}} \,{\frac{{1 - {\text{R}}}}{{{{\uptau}}_{\text{o}} }}} - 1_{{\left\{ {{\text{Cap}} < {\text{c}}_{\text{cap}} } \right\}}} \,{\frac{\text{R}}{{{{\uptau}}_{\text{f}} }}} $$

(14)

1.2.5 A.2.5 Cell Death

In order to determine if a cell survives or not, we quantify the state of a cell as follows:

$$ {\frac{\partial \,\Uptheta }{{\partial {\text{t}}}}}(t,{\text{x}}) = \,\text{Re} {\text{pair}}(t,{\text{x}})(1 - \Uptheta (t,{\text{x}})) - {\text{Damage}}\,(t,{\text{x}})\Uptheta (t,{\text{x}}) $$

(15)

Repair quantifies the cell ability to “repair”:

$$ {\text{Repair}}(t,{\text{x}}) = {\text{R}}_{ 0} ({\text{E}}_{\text{meta,used}} (t,{\text{x}}) + {\text{E}}_{\text{homeo,used}} (t,{\text{x}}))\,\exp \left( {{\text{E}}_{ 0} ({\text{E}}_{\text{meta,used}} (t,{\text{x}}) + {\text{E}}_{\text{homeo,used}} (t,{\text{x}}))} \right) $$

(16)

Damage quantifies the cell damage due to the excess of intracellular calcium:

$$ {\text{Damage}}(t,{\text{x}}) = {\text{D}}_{ 0} {\frac{{{\text{Ca}}_{\text{n}} (t,{\text{x}})}}{{{\text{E}}_{ 1} + {\text{Ca}}_{\text{n}} (t,{\text{x}})}}} $$

(17)

If $\Uptheta (t,{\text{x}}) > \Uptheta_{\text{apop}}$, the apoptosis has not begun: A = 0. If $\Uptheta (t,{\text{x}}) \le \Uptheta_{\text{apop}}$, the apoptosis process progresses:

$$ {\frac{{\partial \,{\text{A}}}}{{\partial {\text{t}}}}}(t,{\text{x}}) = {\text{A}}_{ 0} {\frac{{{\text{E}}_{\text{av}} (t,{\text{x}})}}{{{\text{E}}_{ 2} + {\text{E}}_{\text{av}} (t,{\text{x}})}}} $$

(18)

Conclusion:

• If $\Uptheta$ stays above $\Uptheta_{\text{apop}}$, the cells are alive.

• If A passes over A_apop before $\Uptheta$ passes above $\Uptheta_{\text{nec}}$, the cells are dead by apoptosis.

• If $\Uptheta$ passes above $\Uptheta_{\text{nec}}$ before A passes over A_apop, the cells are dead by necrosis.

1.3 A.3 List of the Parameters

See Tables 2, 3, 4, 5.

Table 2 Blood flow and oxygen absorption

Table 3 Energy

Table 4 Ionic exchanges

Table 5 Cell death